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Gottfried Wilhelm Leibniz,
the humanist agenda and
the scientific method
Kundan Misra
This book is based on a thesis by Kundan Misra which satisfied the requirements for the
degree of Master of Science (Research), University of New South Wales.
The project was conducted under the supervision of Professor James Franklin. The research
started in 2008, the thesis was submitted for examination in August 2011, changes requested
by examiners were completed in September 2012, and in the same month the university
decided to award the degree.
© Copyright 2012
Licenced under the Creative Commons Attribution Licence
For licence details, see www.creativecommons.org.au
The author may be contacted at kundan at austi dot org or kundanmisra at yahoo dot com
Abstract
Modernity began in Leibniz’s lifetime, arguably, and due to the efforts of a group of
philosopher-scientists of which Leibniz was one of the most significant active contributors.
Leibniz invented machines and developed the calculus. He was a force for peace, and
industrial and cultural development through his work as a diplomat and correspondence with
leaders across Europe, and in Russia and China. With Leibniz, science became a means for
improving human living conditions. For Leibniz, science must begin with the “God’s eye
view” and begin with an understanding of how the Creator would have designed the universe.
Accordingly, Leibniz advocated the a priori method of scientific discovery, including the use
of intellectual constructions or artifices. He defended the usefulness and success of these
methods against detractors. While cognizant of Baconian empiricism, Leibniz found that an
unbalanced emphasis on experiment left the investigator short of conclusions on efficient
causes. Leibniz worked outside, but complemented, the current of formal reasoning and
empiricism which was developing in scientific circles during his lifetime. He supported the
development of methods for calculation and demanded precise reasoning, while arguing that it
was folly to omit the Neoplatonic orientation from science. Indeed, without Neoplatonism
there would be no modernity. Leibniz’s Neoplatonic course complemented his work with
machines. Leibniz crystallised the Neoplatonic orientation as a pragmatic humanist agenda,
and merged it with national imperatives for developing science. Leibniz’s policy orientation is
aligned with the Hermetic conception of Man as magus, who ultimately can control even the
stars. The industrial-scientific age which followed Leibniz is a testament to the success of his
life’s work.
Acknowledgements
I first met Professor James Franklin in 2007 to discuss the possibility of a project that
investigates the metaphysical and philosophical assumptions of Leibniz in his mathematical
work, most notably the calculus. I did not realise how arduous the journey was going to be, or
where in history it was going to take me. Professor Franklin provided intellectual support,
guidance and critical suggestions to keep the project going and to keep me focussed on the
original aim. The project would have reeled off the rails without Professor Franklin’s vast
knowledge of mathematics, the history of science and philosophy. The research touched many
areas and Professor Franklin always ensured that I avoided spending too much time on
backstreets and in blind alleys while allowing new ideas to be picked up in those segues.
There were many segues!
Thank you Professor Franklin for supporting me in this odyssey. Some segues became major
new paths. Thank you for your sensitivity in striking the delicate balance between allowing
new ideas to germinate while ensuring existing ideas were developed to maturity. Thank you
for your encouragement and wisdom in confronting and overcoming the many intellectual and
research obstacles.
I appreciate the recommendations of Postgraduate Review Committee in each annual review
which helped greatly in making the following year fruitful.
The thesis examiners clearly spent many hours poring over the thesis and in writing frank and
constructive comments. The examiners’ feedback exposed and proposed many ways to
improve the thesis, which I have acted upon. For example, greater depth of research into the
origins of Neoplatonism and the relevance of Hermetism was encouraged. The result was a
new chapter which is now Chapter 2 along with improvements throughout the thesis.
Page 1
Contents
Preface ....................................................................................................................................... 9
Abbreviations .......................................................................................................................... 11
Terminology ............................................................................................................................ 14
Chapter 1: Introduction......................................................................................................... 23
Chapter 2: Neoplatonism and the awakening of science .................................................... 26
Chapter 3: Ushering in modernity ........................................................................................ 42
Chapter 4: Idea of truth......................................................................................................... 72
Chapter 5: The Neoplatonist and Empiricist schools ......................................................... 82
Chapter 6: Discovery and deduction .................................................................................. 109
Chapter 7: Science by thought, and reality and substance .............................................. 124
Chapter 8: The ongoing role of thought and the ongoing creation of the Best .............. 148
Chapter 9: A priorism in science ......................................................................................... 178
Chapter 10: Conclusion – Leibniz’s humanist and Neoplatonic agenda ........................ 192
Appendix 1: To the greater glory of God ........................................................................... 198
Appendix 2: A proof from basics ........................................................................................ 202
Bibliography ......................................................................................................................... 204
Page 2
Page 3
Detailed contents
Contents ..................................................................................................................................... 1
Detailed contents ...................................................................................................................... 3
Preface ....................................................................................................................................... 9
Abbreviations .......................................................................................................................... 11
Terminology ............................................................................................................................ 14
Chapter 1: Introduction......................................................................................................... 23
Chapter 2: Neoplatonism and the awakening of science .................................................... 26
Introduction .......................................................................................................................... 26
The first Neoplatonists were “new Platonists” ..................................................................... 26
Arab philosophers ................................................................................................................ 27
Hermeticum and magic in Europe ........................................................................................ 30
Giordano Bruno ................................................................................................................ 31
Hermes in Europe by mistake .......................................................................................... 32
Natural magic ................................................................................................................... 33
The other kind of Neoplatonism, and operations ................................................................. 34
The Greeks as operators ....................................................................................................... 36
Mathematics in operations ................................................................................................... 37
Symbols ................................................................................................................................ 39
Conclusion ............................................................................................................................ 41
Chapter 3: Ushering in modernity ........................................................................................ 42
Introduction .......................................................................................................................... 42
Leibniz’s relationship to modernity ..................................................................................... 44
Laws that function without divine intervention ................................................................... 44
Hylarchic principle ............................................................................................................... 45
Modernity and discoveries achieved by reason .................................................................... 46
Hypothesizing a priori ......................................................................................................... 47
Broad-based influence of reason on culture ......................................................................... 49
Conflicting ideas on the nature of Reason ........................................................................... 50
Nicolaus of Cusa .................................................................................................................. 51
Cusa’s ideas on mind gain traction in science ...................................................................... 54
Vis viva or effect-producing force ........................................................................................ 56
Machines are indispensable to the concept of modernity .................................................... 58
Political influences on and of Leibniz .................................................................................. 61
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Seeking to understand Creation as a whole .......................................................................... 63
Geometry as a tool to understand Creation as a whole ........................................................ 65
Understanding Creation as a whole, and improving upon it ................................................ 66
Philosopher scientists and French nation-building ............................................................... 67
An era of pro-nuclear advocacy ........................................................................................... 70
Participating with God in Creation ....................................................................................... 71
Chapter 4: Idea of truth......................................................................................................... 72
Introduction .......................................................................................................................... 72
Minds can successfully search for truth ............................................................................... 72
Sense perception and Empiricism ........................................................................................ 74
Existence and attainability of truth: Rationalism and Neoplatonism, vs Empiricism .......... 76
Religious or moral truth vs scientific truth ........................................................................... 78
Neoplatonism, Empiricism and Atheism ............................................................................. 79
Summary of the ’isms .......................................................................................................... 80
Conclusion ............................................................................................................................ 80
Chapter 5: The Neoplatonist and Empiricist schools ......................................................... 82
Introduction .......................................................................................................................... 82
Schiller and the infinite ........................................................................................................ 84
Metaphysical calculations and a method of Analysis for concepts in physics ..................... 85
Invaluable medieval speculations ......................................................................................... 86
Leibniz saw the need for more powerful methods ............................................................... 86
Reason gives value to observations ...................................................................................... 87
Kepler, Neoplatonism and precision .................................................................................... 89
The Empiricist school and the influence of Paolo Sarpi ...................................................... 91
Sarpi and Galileo .................................................................................................................. 91
Sarpi’s influence in England ................................................................................................ 92
Sarpi’s closeness to Galileo .................................................................................................. 93
Sarpi and atheism ................................................................................................................. 94
Sarpi and Newton ................................................................................................................. 95
From Galileo to Newton ....................................................................................................... 95
Leibniz comments on the experimental program of the (British) Royal Society ................. 96
Newton’s prescribed method of discovery ........................................................................... 96
Evolution of Newton’s method of discovery ....................................................................... 98
Newton against hypothesis and towards mathematicization ................................................ 99
Knowledge without sensation .............................................................................................. 99
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Kepler and Leibniz in Newton’s method ........................................................................... 101
Consequences of mandating empirico-logic ...................................................................... 102
Empirico-logic in other domains ........................................................................................ 104
Reasoning other than empirico-logical .............................................................................. 105
The role of rigorists ............................................................................................................ 106
Conclusion .......................................................................................................................... 107
Chapter 6: Discovery and deduction .................................................................................. 109
Introduction ........................................................................................................................ 109
The art of discovery ............................................................................................................ 110
Discovery is often metaphysics .......................................................................................... 112
Un-rigorous use of rigor ..................................................................................................... 113
Archimedes had the same problem .................................................................................... 114
All things and no things – at the same time ....................................................................... 115
Does the infinite exist? It doesn’t really matter. ................................................................ 116
Do Platonic ideas exist? It doesn’t really matter. ............................................................... 118
Rigor is of service .............................................................................................................. 120
Science may use indemonstrables ...................................................................................... 120
Global architecture determines local mechanics ................................................................ 121
Intention in the large .......................................................................................................... 122
Conclusion .......................................................................................................................... 123
Chapter 7: Science by thought, and reality and substance .............................................. 124
Introduction ........................................................................................................................ 124
Science by thought ............................................................................................................. 125
Universal characteristic ...................................................................................................... 125
Dream vs reality, and substance vs phenomena ................................................................. 126
What is real? ....................................................................................................................... 127
Monads are real .................................................................................................................. 127
Immaterial souls exist ......................................................................................................... 129
The nature of ideas ............................................................................................................. 129
The applicability of ideas ................................................................................................... 131
Ideas and “the Best” are outside all that exists ................................................................... 133
The apparent governance over actual things by ideas ........................................................ 133
The relationship between ideas and actual things .............................................................. 134
Inexorable necessity created by the requirement of “the Best” ......................................... 135
Nominalism ........................................................................................................................ 136
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Interdependence of ideas .................................................................................................... 137
Simple monads and compound monads ............................................................................. 138
Passive vs active, or body vs soul ...................................................................................... 141
Actions of monads .............................................................................................................. 142
Immaterial realm is in harmony with material realm ......................................................... 143
Life ..................................................................................................................................... 143
Q&A ................................................................................................................................... 143
The Hermetic monad .......................................................................................................... 144
What survives death? ......................................................................................................... 145
Human souls differ from souls of beasts ............................................................................ 145
Monads are the same as one another .................................................................................. 146
Conclusion .......................................................................................................................... 147
Chapter 8: The ongoing role of thought and the ongoing creation of the Best .............. 148
Introduction ........................................................................................................................ 148
Choices God makes, continuously ..................................................................................... 148
First kind of choice: God mediates between monads ..................................................... 149
Second kind of choice: God chooses the best possible from what is possible ............... 150
Unanswered question re first kind of choice .................................................................. 151
Consideration re first kind of choice: God’s choices vs intelligent monads’ choices .... 151
Question re second kind of choice ................................................................................. 152
Answer to the question re second kind of choice: what God does from moment-to-
moment ........................................................................................................................... 152
Creating the universe anew in each moment ...................................................................... 153
Is it “Hello Humphrey Appleby” or, Does God control everything while appearing not to?
............................................................................................................................................ 154
Mystical theology ............................................................................................................... 155
Mathematical determinism ................................................................................................. 158
Is everything planned in advance? ..................................................................................... 159
A priorism ........................................................................................................................... 160
Is a priorism better than empirics? ..................................................................................... 160
For humans, ideas and empirics are interdependent .......................................................... 161
Free will .............................................................................................................................. 162
What is an evil person? ...................................................................................................... 163
The problem of evil ............................................................................................................ 164
Scientific comprehensibility of God’s plan ........................................................................ 166
Page 7
Intention and the Scholastic Leibniz project ...................................................................... 167
God’s thoughts are made real by his will ........................................................................... 171
Thoughts that soul monads have ........................................................................................ 172
Humankind as a subject of physical science ...................................................................... 173
“Ideas” (in the strict monadological sense) about humankind ........................................... 174
Foundations of a programme for discovery ....................................................................... 175
Conclusion .......................................................................................................................... 176
Chapter 9: A priorism in science ......................................................................................... 178
Introduction ........................................................................................................................ 178
Example 1: The infinitesimal and the calculus .................................................................. 180
Arithmetical progression ................................................................................................ 183
Eliminating terms ........................................................................................................... 183
Moments ......................................................................................................................... 183
Example 2: The Fifth Postulate and Non-Euclidean geometry .......................................... 184
Example 3: Elliptical orbits in a heliocentric solar system ................................................ 188
Alternative views of the Creator’s mind ............................................................................ 190
Conclusion .......................................................................................................................... 191
Chapter 10: Conclusion – Leibniz’s humanist and Neoplatonic agenda ........................ 192
Introduction ........................................................................................................................ 192
Leibniz and humanism ....................................................................................................... 192
Human progress .................................................................................................................. 192
Broadening the field of debate for Platonism ..................................................................... 193
Considering the big questions first ..................................................................................... 193
Human capability ............................................................................................................... 193
Human knowledge .............................................................................................................. 194
A new metaphysics as a blueprint for the good in human society ..................................... 195
Flexibility in scientific method, and the ultimate journey ................................................. 195
Leibniz’s policy for nations ................................................................................................ 197
Appendix 1: To the greater glory of God ........................................................................... 198
Appendix 2: A proof from basics ........................................................................................ 202
Bibliography ......................................................................................................................... 204
Page 8
Page 9
Preface
This publication is based on my MSc (Research) thesis awarded by the University of New
South Wales through the School of Mathematics and Statistics in 2012, supervised by
Professor Jim Franklin. There are minor changes from the original thesis, mostly in the last
page and section of the Conclusion.
The project began with the question of how Leibniz’s metaphysics influenced his
mathematical work. We found that Leibniz was just one of a number of thinkers who did
science by beginning with plausible and quasi-theological first principles regarding creation
as a whole. Scientific questions then act as pieces in a jigsaw puzzle in a quest for
understanding creation as a whole and the role of humanity in it. Mathematics is one tool
available to humans not merely to carry out calculations but to seek truth. Leibniz was one of
a line of thinkers who proceeded in this way. Significant forebears of Leibniz include
Nicolaus of Cusa and Johannes Kepler.
This orientation did not arise spontaneously but has ancient roots. In particular, there are
many parallels between Leibniz’s philosophy and the Hermetic corpus. Through the analysis
of Frances Yates, it is understood that the putative writings of Hermes Trismegistus were
central to the Neoplatonic revival and associated Humanist Renaissance. It seems that Leibniz
developed and systematised proto-science of figures such as John Dee, referred to by
Cornelius Agrippa as “real artificial magic”. The origin and potential of such so-called
“magic” which included proto-science and early engineering was appreciated by Giordano
Bruno and the circles he influenced. Further understanding the debt that Leibnizian thought
owes to Bruno and Egyptian “magic” is a promising direction for further work.
During the research, the dichotomy between thought and sense perception arose many times.
Of course, this has been discussed since Hermes and was addressed in detail in Plato’s The
Republic and by nearly every philosopher in Classical Athens and since. Generally speaking,
the Platonists are oriented towards a priori discovery – using the mind to construct
hypotheses – whose outcomes can be tested in controlled experiment. In the construction of
hypotheses, mathematical preconceptions of the universe play a central role because they
imply structure which becomes the foundation of predictive hypotheses which can be tested.
General sense perception is distrusted as not just unreliable but misleading. As an inventor
and pragmatist himself, Leibniz saw the necessity for experiment and described how he would
catalogue all desired experiments when trying to understand a particular issue in physics. The
distinction is that Leibniz did not expect answers to come from experiment. Rather,
experiments were merely an aid to the a priori thought process.
We found exciting open questions which if addressed would present the opportunity for leaps
in science. One example is a serious conception of “the Best” in the context of Leibniz’s best
of all possible universes doctrine. Another example is further detail on the pre-established
harmony, wherein God intervenes from moment-to-moment to ensure the best possible
correspondence between the thoughts of incorporeal substance (“mind”) and phenomena
exhibited by corporeal substance (“body”). Another is the question of Man as a physical
scientific force in the universe. Another is that which has received much attention but little
progress since Leibniz himself, which is the formulation of a Universal Characteristic, or
symbolic calculus whereby all kinds of problem can be solved in a systematic way.
Page 10
Page 11
Abbreviations
Copenhaver: Copenhaver, B.P. Hermetica Cambridge University Press 1992
Klemm: Klemm, F. (Singer, D. W. trans.) A History of Western Technology Allen and Unwin,
London 1959
Loemker: Loemker, L. E. Gottfried Wilhelm Leibniz: Philosophical Papers and Letters 2nd
ed. D. Reidel, Dordrecht Holland 1969
Monadology: Latta, R. (trans.), Leibniz, G.W. Monadology Oxford University Press 1898
edition
Theodicy: Huggard, E.M. (trans.), Leibniz, G.W. Theodicy §Project Gutenberg edition 2005
accessed at www.gutenberg.com
Wiener: Wiener, P.P. (ed.) G. W. Leibniz Selections New York: C. Scribner, 1951
Wootton, D. Paolo Sarpi: Between Renaissance and Enlightenment Cambridge University
Press, 2002
Yates 1964: Yates, F. Giordano Bruno and the Hermetic Tradition Routledge and Kegan Paul
Ltd London 1964
Yates 2009: Yates, F. Giordano Bruno and the Hermetic Tradition Routledge Classics, digital
reprint 2009
Page 12
Page 13
In fact, … final causes may be introduced with great
fruitfulness even into the special problems of physics,
not merely to increase our admiration for the most
beautiful works of the supreme Author, but also to help
make predictions by means of them which would not be
as apparent, except perhaps hypothetically, through the
use of efficient causes. Philosophers have in the past
perhaps not sufficiently observed this advantage of
final causes.
It must be maintained in general that all existent facts
can be explained in two ways – through a kingdom of
power or efficient causes and through a kingdom of
wisdom or final causes; that God regulates bodies as
machines in an architectural manner according to laws
of magnitude or of mathematics but does so for the
benefit of souls and that he rules over souls, on the
other hand, which are capable of wisdom, as over
citizens and members of the same society with himself,
in the manner of a prince or indeed of a father, ruling to
his own glory according to the laws of goodness or of
morality. Thus, these two kingdoms everywhere
permeate each other, yet their laws are never confused
and never disturbed, so that the maximum in the
kingdom of power, and the best in the kingdom of
wisdom, take place together.1
G. W. Leibniz, Specimen Dynamicum 1695
1 Loemker, p.442. Liberty has been taken to break the passage into two paragraphs.
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Terminology
We introduce some terminology and distinctions that are peculiar to the philosophy of
science, and which we use throughout the thesis. In the following list of terms, we group
associated terms rather than following a strict alphabetical order.
In the thesis, we will capitalise terms relating to new concepts introduced by Leibniz and
others which have not yet developed into methods in general use. Examples are Analysis as
Leibniz uses the term, versus the more general “analysis” in day-to-day usage. Another
example is Metaphysics as Leibniz understands it, versus mathematics which Leibniz
generally used in the same sense as everyone else. Another example is “the Best” where best
refers to the actuality of structures in the universe as designed by God as “the best possible
universe”. In any case, we hope that the reader does not set too much store by capitalisation.
A posteriori One is said to formulate a hypothesis or to gain understanding a posteriori if
one does so after and pursuant to or as a result of actual experience or
observation of physical events or empirical evidence.
A priori One is said to understand something a priori or prior to experience by working
it out in one’s mind.
Architectonic Leibniz’s adjective for the planned architecture and creation of the universe
and the laws by which it operates.
Averroes A Muslim thinker credited or discredited with extending Aristotle’s ideas into
positivism. St Thomas Aquinas who was active approximately a century later
was an ardent critic, and wrote that Averroes and extended Aristotle’s ideas far
beyond what Aristotle intended.
Averroism The doctrine that formal deduction is the only valid way to reason. It is named
after its leading proponent Averroes, a Muslim thinker, who adopted Aristotle’s
idea and extended them. Thomas Aquinas disagreed with Averroes’ reading of
Aristotle claiming that Averroes had commandeered Aristotle to support his
own agenda or outlook on the art of thought whether philosophical, scientific,
artistic, etc. Averroist reasoning or logic became increasingly influential. By
the 17th Century the University of Padua, formerly humanist, became a centre
for teaching and promoting Averroes’ method. Averroism was an attempt to
interpret Aristotelian teachings without concern for consistency with Christian
theology. The Latin Averroists at the University of Paris “seemed, at least to
their critics, to abandon the Scholastic attempt to reconcile the newly translated
texts of Aristotle with the dictates of Christian orthodoxy.”2
Baconian The doctrine that experimental effort to discover, record and categorise the
testimony of the senses is the way that science should be done. Early figures in
2 Petrarca relates in an entertaining way how his differences with an Averroist who came to visit him, with
emphasis on the gratuitous disrespect shown by the visitor to the Scriptures and Apostles. Petrarca “An
Averroist visits Petrarca” From a letter to Boccaccio from Venice 28 August 1364, pp.140-141 in Cassirer, E.,
Kristeller, P. O., Randall, J. H. Jr (eds and trans.) The Renaissance Philosophy of Man Phoenix Books,
University of Chicago Press 1948
Page 15
English scientific circles prior to the Royal Society, especially Francis Bacon,
promoted this outlook.
Empiricism The doctrine that what is knowable, and even what is worthy of consideration,
includes only what is perceptible. Galileo was perhaps the fork in the road, for
he had great success in science, especially mechanics, and apparently through
the extensive use of experiment. Galileo was both Kepler’s friend and,
financially, was supported by Paolo Sarpi who was supposedly the first to look
through Galileo’s telescope. Sarpi was an influential gentleman, politician and
statesman and wrote a rudimentary version of what later appeared as the rules
of Newtonian science. Arguably, it was in Galileo and Sarpi that the
empiricism of Newtonian science first took form. Today empiricism is
associated with Francis Bacon with whom Sarpi corresponded, and with Isaac
Newton and John Locke who were born after the deaths of Bacon and Sarpi.
Empiricism was pronounced in English scientific circles and is often associated
with Bacon, Isaac Newton and John Locke. The Royal Society of London for
Improving Natural Knowledge (“Royal Society”) was founded in 1660 out of
English scientific circles and, accordingly, empiricism is dominant in the
orientation of Royal Society. There are also strains in 17th Century English
science which are almost anti-empiricist. Henry More and Robert Boyle are
notable examples who will be discussed in this thesis.
Empiricist A scientist or philosopher thinker who believes in a more a posteriori approach
to science.
Conception, concept or notion (Leibniz’s usage) The content of a thought abstracted from
the thought, or the meaning of a thought in terms mathematical, symbolic,
logical or otherwise regarded with coherent and rational structure as far as
possible. Nearly all human thoughts are inexact, indistinct, confused or
defective in some way. Some thoughts are clearer, more coherent and less
confused than others.
Idea (Leibniz’s usage) A concept that is clear, distinct, self-consistent and consistent
with every other idea. Ideas do not exist independently of minds, but only exist
in minds, whether those of souls or of God. For Leibniz, ideas are not
subjective, are not social constructs and are not constructed but only
discovered. For Leibniz, ideas are concepts which are coherent. The ultimate
test of coherence is God’s design of the best possible universe which is the
actual universe that we are living in. Thus, an idea has explanatory capability.
Ideal A perfect or pure version of a thing or concept, including mathematical
concepts and geometrical constructions.
Perception (Leibniz’s usage) A thought or the way in which something is understood
by a mind. This is contrary to the standard use of the word to refer to
perceiving through the bodily senses.
Thought (noun, Leibniz’s usage) Used interchangeably with “perception”. It is a thought
is a mental image or, more generally, a mental impression.
Page 16
Rational thought (verb) The wilful process of creating mental impressions, and of
bringing structure to concepts. It might also be called “intellection” or “creative
mentation”. This is the process through which hypotheses are made, which are
often tantamount to discoveries though to earn the label “discovery”
confirmation via experiment or mathematical deduction or both is usually
needed. For Leibniz, holding all concept and information related to a problem
or circumstance in one’s mind in a single thought was a prerequisite to the
understanding which precedes a new subsuming higher hypothesis. For Kepler,
it was seeking after the manifestation of harmony in the context being analysed
that provided the superior or most fruitful impetus towards the higher
hypothesis. Each is an example of an a priori process.
Metaphysics The study of the relations, constraints and principles that underlie the physical
universe and that underlie physics. Metaphysics may also refer to philosophical
rules that govern one’s assumptions in and approach to physical science. The
Stanford Encyclopedia of Philosophy says that the term derives from Aristotle’s
Metaphysics.3 Whether or not this is true, Aristotle was not the first to deal with
metaphysics nor did subsequent thinkers, such as Leibniz, agree with
Aristotle’s Metaphysics on all points or even on fundamental points. The
Stanford Encyclopedia of Philosophy says that metaphysics is about things that
do not change, in contradistinction to physics which is about the physical world
which does change. However, it is not at all certain that metaphysics is not
dynamic or not evolving. In order to make progress in metaphysics, it is
generally necessary to step outside of any confinement by assumptions derived
from the physical universe. This may lead to considerations concerning the
origin of the physical universe which quickly leads to questions of theology
and matters of how the Creator or God works and, indeed, of what God is.
Such questions were addressed fearlessly by thinkers including Nicolas of
Cusa, Johannes Kepler, Leibniz and many others.
Nominalism The understanding that concepts and ideas are merely thoughts and do not
actually exist as the things that they are expressing. Rather, concept and ideas
exist only in the mind not as objects in the physical universe which are subject
to the laws of physics or of metaphysics.
Ontology Part of metaphysics that is more interested in what exists and does not exist,
with an emphasis on entities or “things” expressed by Platonic ideas.
Neoplatonism The outlook that Platonic ideas are valid, truth is discoverable and what is
perceptible via the senses is mere opinion, while truth is discoverable through
Reason. It has the connotation of emphasis on that which the senses cannot
perceive. As a result of this, in part, it has been confused with mysticism and
irrationality. This is correct only if stepwise deduction is considered the only
valid form of rational thought, and many - including Leibniz - would disagree.
3 Van Inwagen, P. “Metaphysics” 2007 in Stanford Encyclopedia of Philosophy
http://plato.stanford.edu/entries/metaphysics/ accessed 21 Feb 2012
Page 17
That is, Neoplatonism validates modes of thought other than stepwise
deduction under the umbrella term Reason.4
Neoplatonist A Neoplatonist is an adherent of variants of Plato’s ideas who appeared several
centuries after Plato, or a follower of one of these adherents of Plato.
Neoplatonism is often mixed with Hermetism, early Christian thought and even
Caballa. Leading Neoplatonists include Plotinus, Cardinal Nicolaus of Cusa
and even Cosimo de Medici.5 Some Neoplatonists, such as Plotinus, studied
the Corpus Hermeticum which was attributed to Hermes Trismegistus.
Newtonian The doctrine that empiricism and stepwise deduction from sensory
observations is the way that science should be done. This doctrine downplays
the role of forming hypotheses in science. The rules of Newtonian science
appear to be an extended version of rules of Averroist logic.6
Platonic idea To Kielkopf who undertook a thorough review of Wittgenstein’s foundations of
mathematics, an “absolute platonist [sic]” holds that “the domain of
mathematical objects themselves exist independently of minds.”7 The same can
be said of Platonic ideas more generally, even ones that are not mathematical.
Platonist A Platonist is a follower of Plato’s ideas in their pure form as expressed in
Plato’s own writings such as The Republic. This includes the ability to discover
truth through reason and dialogue, that universal truths exist and the doctrine of
reminiscence.
Pythagorean A follower of Pythagoras, the well-known philosopher. The significance of
being a Pythagorean would be greatly deepened if it were true that Pythagoras
had been a disciple of Hermes Trismegistus as some scholars argue.
Rationalism The doctrine that knowledge of the physical universe is primarily attained
through intellectual reasoning.
4 Petrarca denigrates dialectic saying it is appropriate for students but is immature and silly in a philosopher of
age and experience. Petrarca “A disapproval of an unreasonable use of the discipline of dialectic” Letter to
Tommaso Caloria from Avignon 12 March 1335, pp.134-139 in Cassirer, E., Kristeller, P. O., Randall, J. H. Jr
(eds, trans.) The Renaissance Philosophy of Man Phoenix Books, University of Chicago Press 1948, p.139 5 The historical importance of Nicolaus of Cusa cannot be understated. We quote the closing paragraph of the
introduction to a biography of Nicolaus of Cusa, “In the act of coming to the aid of Pius II and his crusade,
Nicholas of Cusa, the cardinal and longtime confidant of the pope, died three days before him on the road to
Ancona. It was August 11. By then Cusanus had succeeded in participating in, if not actually shaping, nearly
every major issue of the more than half-century that life had given him [1401-1464]. In addition to his career
as an energetic churchman, he also found time to leave a remarkable intellectual legacy that fascinates an
unusually broad range of people to this day, from astronomers and mathematicians to church historians, from
political theorists to theologians and philosophers.” Crowner, D., Christianson, G. (trans. and eds) Meuthen,
E. Nicholas of Cusa: A Sketch for a Biography The Catholic University of America Press, Washington, D.C.
2010 pp.xxv-xxvi. Accessed at http://cuapress.cua.edu/books/frontmatter/MENC.pdf 26 May 2011 6 It is said that Isaac Newton promoted this approach. However, the liveliness of the Neoplatonic environment in
which Newton studied at Cambridge, with people like Boyle around, and the evident mysticism in some of
Newton’s own writings cast doubt on how “Newtonian” he was himself. 7 Kielkopf, C.F. Strict Finitism: An examination of Ludwig Wittgenstein’s remarks on the foundations of
mathematics Mouton, The Hague and Paris 1970, p.32
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Reality (Leibniz’s usage) All that is. This comprises God and the universe. The universe
comprises the physical universe and all that is incorporeal. The physical
universe comprises “body monads” while the incorporeal comprises “soul
monads”. For Leibniz, “body” and objects as people perceive them are not real.
Rather, “coherence is the sign of truth, its cause is the will of God, and its
formal reason is the fact that God perceives that something is the best, i.e. most
harmonious, or, that something is pleasing to God. And so divine will is itself
the existence of things, so to speak.”8 For something to exist, it requires that
God willed it to exist. As far as Leibniz is concerned, the test for humans of
truth is coherence and the best test of coherence is the ability to predict future
phenomena.9
Monad (Leibniz’s concept) A part of reality. An indissoluble piece of substance. Monads are
all that exists, aside from God. Passive monads have perceptions or thoughts
but are sluggish in perceiving and thinking. Active monads have perceptions,
thoughts and memory. There is a continuum from the passive to the active,
with physical matter that is lifeless such as rocks being passive and called body
monads, amoebae being more active, animal souls being still more active, and
human souls being even more active. A collection of body monads controlled
by a mind, such as the body monads comprising the physique of a human
being, have a corresponding soul monad. The soul monad is the same thing as
the mind. The soul monad is linked to the collection of body monads through a
connection called the pre-established harmony. This is a pre-prepared
correlation between the soul monad and the body monads. This makes it appear
that the soul controls the body, when it fact the actions of the two have been
established in advance to ensure that they move in lockstep. This correlation of
actions between the incorporeal and corporeal domains is part of the design of
the universe as the best of all possible worlds.
Substance (Leibniz’s usage in the context of monads) A collective or aggregative
term for what is real. It includes physical and incorporeal substance. Physical
substance includes passive matter and all kinds of radiation. Incorporeal
substance comprises thinking (or “active”) souls, and these govern physical
substance.
“The best” (Leibniz’s usage) The most superior and good way of designing or doing
anything. It is in accord with ideas. It is the standard used by God to make
decisions. It is that which is capable of existing. The Best makes sense over
series events, not in a “snapshot”. Thus, hardship appears to be other than the
Best but in a wider context provides to be desirable and good in the impact on
the clarity and distinctness of the perceptions of minds, and the closeness of
mental conceptions to ideas. In a priori sense this must be the case. In the
context of passive matter such as the motion of planets or of an entire solar
system through a galaxy, and speaking simplistically, the Best manifests itself
as a concept of harmony.
8 Leibniz, G. W. “Body is not a substance” March 1689-March 1690 (?) Sämtliche schriften und briefe series VI
volume 4 Deutsche Akademie der Wissenschaften (ed) p 1637 Strickland, L. (trans.) Leibniz Translations
2009 accessed at http://www.leibniz-translations.com/bodysubstance.htm 6 June 2012 9 Leibniz, G.W. “On the method of distinguishing real from imaginary phenomena” Date unknown in Loemker,,
p.364
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Reminiscence (Leibniz’s usage) The doctrine that all knowledge has always been with
each person, and “acquiring” it is a matter of remembering. This was presented
in Plato’s Meno by a boy with no education solving a geometrical problem
purely by the aid of prompts. Leibniz thought that this made complete sense.
He wrote, “I am in no wise in favour of the Tabula rasa of Aristotle; and there
is something sound in what Plato called reminiscence. There is even something
more, for we have not only a reminiscence of all our past thoughts but also a
presentiment of all our future thoughts.”10 We will refer to a proponent of the
doctrine of Reminiscence as a “Recallist”.
Renaissance humanism The human-centred outlook of the historical period and
movement called the Renaissance. It was especially prominent in Italy but
found all over the Continent and the British Isles. This outlook touched
science, theology, art, literature and many – perhaps all – domains of human
endeavour. It is associated with a renewed interest in Roman and Greek
classics, as well as with an interest in the putative writings of Hermes
Trismegistus. The Hermetic humanist was typically also more inclined towards
classical Hellenic writings. Not all men of letters were interested in all three
categories. Those interested in Roman classics were considered “grammarian
pedants” by those who found truth in Hermetic writings. The Roman (Latin)
humanist hearkened back to the golden age of Cicero and regarded the Middle
Ages as barbarous. By contrast, there were Hermetic humanists such as Ficino
who revered Platonists from the Middle Ages such as Thomas Aquinas.11
Teleology The final or ultimate cause of a phenomenon or event, or the study which seeks
to understand the final or ultimate cause of a phenomenon or event.
Some concepts are best understood in terms of their relationship to other concepts, and some
concepts even exist primarily as a counter to other concepts. We introduce some pairs or
dichotomies, and some triples or trichotomies.
Dichotomy 1: Objectivism vs Subjectivism
An objectivist understands that there are truths that transcend time, space and social norms.
Indeed, the definition of truth is something that holds everywhere, at all times and for all
people. That is, truth is absolute truth. An objectivist may also be called an absolutist.
To a subjectivist, truth varies from person to person and, to an even greater degree, from
culture to culture. In particular, morals vary from person to person and culture to culture. This
variation is valid, and does not detract from the legitimacy of a truth for the person/s for
whom it holds. In short, truth is relative. Thus, a subjectivist may also be called a relativist.
10 Leibniz, G. W. New Essays on Human Understanding in Ladd, G. T. (trans.) The Philosophical Works of
Leibnitz Tuttle, Morehouse and Taylor Publishers, New Haven 1890, p.96 11 Yates 2009, pp.178-181
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Dichotomy 2: Theist vs Atheist
Does a person believe in God or not? A theist believes in a single God and an atheist does not
believe that God exists. Nearly all of the historical thinkers in this dissertation are theists.
What is more important than whether or not a thinker believes in God is their conception of
God. For example, do they consider the mind of God to be unknowable or is God a closed
book who, are far as human are concerned, acts arbitrarily. An atheist typically has to believe
that the universe has always existed, because once the creation of the universe is posited, then
there has to be a Creator. An atheist can concur with a theist who believes in a rational God on
many matters of science, if the atheist believes in a rational universe that has always existed.
Dichotomy 3: Gnostic vs Ignorantist
How close can we come to God? Gnostic means that we can have direct experience of God.
Ignorantist means that we can only ever indirectly understand God; while we can approach
God and get closer and closer, we will always be infinitely distant from God. We coin the
term ignorantist based on the title of Nicolaus of Cusa’s De Docta Ignorantia (“On learned
ignorance”). Petrarca before Cusa wrote on ignorance12 and it is known that Cusa had a copy
of Petrarca’s work.13
Trichotomy 1: hylarchic vs mechanist vs animist
How does God relate to or connect with the physical universe? The hylarchic principle is that
God is in the universe, giving the physical universe a real spiritual quality. God is intervening
in the functioning of the universe from moment-to-moment, perhaps in a structured or ordered
way. To a mechanist, God is not in the universe but made the universe so that it can function
without God’s ongoing intervention.
An animist believes that the physical universe contains divine entities; indeed, that everything
in the universe is a divine entity in itself. For example, even a rock is not only alive but is a
spiritual entity with divine qualities. Pantheism is similar to animism. A pantheist considers
the world as a whole to be a single, divine entity. There are no pantheists in this dissertation.
Trichotomy 2: Neoplatonist vs Rationalist vs Empiricist
A Neoplatonist or “new Platonist” considers that there is a world of ideals which can be
discovered and understood only by the mind. These ideals or perfect versions of things are
templates for what we find in the physical world. The physical world comprises combinations
of ideals so complex that it appears to not mirror the world of ideals at all. A Neoplatonist is
necessarily an absolutist.
Rationalism says that truth can be discovered by way of thought without any sensory input or
physical observation. Clearly, a Neoplatonist is a rationalist, but a rationalist is not necessarily
12 Petrarca “On his own ignorance and that of many others” c.1368, pp.47-133 in Cassirer, E., Kristeller, P. O.,
Randall, J. H. Jr (eds and trans.) The Renaissance Philosophy of Man Phoenix Books, University of Chicago
Press 1948 13 Brand, P. Cambridge History of Italian Literature Cambridge University Press 1999, p.134. Cusa continues
Petrarca’s programme of demonstrating that intellect as exercised by an intelligent commoner can be more
effective than the formal reasoning promoted by the scholastics.
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a Neoplatonist. An empiricist believes that what is known comes to a person via the physical
organs of their five senses.
An empiricist cannot be a Neoplatonist because a Neoplatonist believes that what the senses
perceive is no different from mere opinion and is therefore of no value in uncovering truth. A
Neoplatonist could be at least a quasi-empiricist because the sensory input might help prompt
or stimulate thoughts that help lead us to truth; however, a Neoplatonist cannot be entirely
empiricist because thought – not sense perception – must necessarily play the crucial
mediating role before truth may be uncovered.
The relationship between rationalism and empiricism is the same as the relationship between
Neoplatonism and empiricism.
Relationship of concepts to one another
Relations arising from Trichotomy 2 which fall out from the definitions are:
Neoplatonist ==> Rationalist
Empiricist ==> ¬ Neoplatonist
Empiricist ==> ¬ Rationalist
Neoplatonist ==> ¬ Empiricist
Rationalist ==> ¬ Empiricist
Leibniz’s “empiricism” is simply that sensory observation prompts thoughts and checks
reasoning, including the soundness of reasoning. Empiricism says that the senses give us
knowledge and that thoughts are only needed to organise the knowledge that the senses give
us. We see that this is at odds with Leibniz’s position as explained in his New Essays on
Human Understanding.
The doctrine of Reminiscence is connected to one’s conception of the soul, especially the
continuity, antuiqity and origin of the soul and its knowledge. Therefore, Reminiscence holds
a tenuous connection to the other concepts:
Rationalist =/=> Recallist
Recallist =/=> Rationalist, yet Leibniz is both Rationalist and Recallist.
Platonist ==> Recallist, from Meno and this is Leibniz’s position
Neoplatonist =/=> Recallist, though some were, like Leibniz.
Definitions do not do justice to the richness of the ideas expressed by the terms which are best
read in a context. However, the intention of this background is to ensure that the reader has
seen the terms once and has some familiarity with them when they are first encountered in
this dissertation.
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Chapter 1: Introduction
The thesis begins with the historical milieu in which Leibniz worked, and the earliest projects
he was engaged in after meeting Christian Huygens. Next we situate the goals of scientific
discovery which were being debated in Leibniz’s lifetime in terms that crossed over into the
theological. We then turn to the philosophical battles over how science should be conducted
into which Leibniz pitched. With that background, we will be ready to consider scientific
method including how philosophical positions influenced the way in which scientific
problems were approached. The questions addressed then broaden, quite naturally, into the
broadest questions such as the role of thought in science and which kinds of notions should be
encouraged in scientific endeavour and which not.
This argument in this thesis agrees with how Jürgen Lawrenz’s 2007 PhD thesis at the nearby
University of Sydney Leibniz: Double-Aspect Ontology and the Labyrinth of the Continuum
situates Leibniz among these three ’isms:
1. Realism: objects and events exist in the universe, and their properties and relations
obtain, irrespective of our beliefs about them and independently of our ability to
discern them;
2. Idealism: the world is comprehended wholly in the form of mental representations;
3. Phenomenalism: our knowledge of the world is mediated entirely by sense
impressions.
Lawrenz says that Leibniz is realist, idealist and not phenomenalist, and nothing in this thesis
contradicts those outcomes. That Leibniz is realist is almost immediate. That he is idealist
follows from the exclusive place given to mind in understanding and even – via the pre-
established harmony – in experience. That Leibniz cannot be phenomenalist also follows from
his conception of the role of mind.
Neoplatonism and the ideas that it encompassed is so important in understanding the context
in which Leibniz worked that it deserves its own chapter. So in Chapter 2, “Neoplatonism and
the awakening of science”, Neoplatonism is introduced along with the many ideas that are
encompassed by its contribution. This includes proto-science, “magic” and an interest in the
Hermetica or the writings of Hermes Trismegistus. Equally if not more important, but
secondary in this thesis, is the revival of interest in Aristotle, and the interpretation and
reinterpretation of Aristotle by people subject to such processes as a millennium of
Christianity and several centuries of Islam. What was significant about the renewed interest in
Hermetica and a serious approach to magic was a structured approach to what Yates calls
“operations” or human intervention in the physical universe.14 This leads to the next chapter.
In Chapter 3 “Ushering in modernity”, Leibniz’s role in nascent nation-building efforts of
Louis XIV are examined. It is argued that the expression of rationalism in machine-building
and inanimate power sources define the beginning of modernity. If this argument is correct,
then it follows that Leibniz was a central figure in the commencement of modernity. In
14 Yates 1964, pp.146-147
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modernity, the engagement in ever more efficient and powerful kinds of “operations” is
systemised and given a central place in society, policy and all aspects human endeavour. In
early modernity, the nature of Man as magus was first institutionalised like never previously
in history with the construction of machines and projects to “operate” on an unprecedented
scale for the benefit and ongoing realisation of Man’s purpose and role in the universe.
In Chapter 4 “Idea of truth”, the pursuit of truth is introduced with the reason versus
empiricism dichotomy which is indistinguishable from idealism versus phenomenalism. The
chapter introduces scientific personalities of decisive importance to the subsequent history of
science: Johannes Kepler, Galileo Galilei, Paolo Sarpi and Isaac Newton. It is explained that
for Leibniz the search for scientific truth is inseparable from the investigator’s ability to
understand the nature of the Creator of the universe. This is a Neoplatonic orientation.
Contrary to this, a junction in science is found in Paolo Sarpi in whom a stream separate from
the Neoplatonists that emphasises Empiricism, and which coincides with Atheism, is
introduced into Western science.
In Chapter 5 “The Neoplatonist and Empiricist schools”, the concept of calculation natural
evolution of the Neoplatonic school is introduced. This includes calculation with
metaphysical concepts. Lawrenz writes of Leibniz that one can, “Open any page of his
writings to see physics happily consorting with metaphysics – it is the pattern of his thinking.”
In this chapter, it is explained that Leibniz had no choice in this, for it is an inexorable
consequence of his integrated conception of how the capabilities of the human mind
understand and commandeer reality. Leibniz regarded mathematical calculation as but one
aspect of reasoned thought. Leibniz promoted the project to construct a Universal
Characteristic that would supply an answer to any question, not only in physical science but
also about moral differences, through unambiguous calculation.
In Chapter 6 “Discovery and deduction”, Leibniz’s recommended process for discovery is
described. The Leibnizian prescription is to hold all the considerations relating to a problem in
one’s mind in a single thought. It takes considerable time and thoroughness to be able to
achieve this. This is posited as a top-down method in contrast to stepwise reasoning also
known as discursive reasoning or, in some contexts, as dialectic. Much is owed to Nicolaus of
Cusa’s De Docta Ignorantia or “On Learned Ignorance” which argues that stepwise reasoning
does not lead to new knowledge. Rather, the approach that leads to breakthroughs is to seek
the higher hypothesis which subsumes a problem and its related sub-problems – such as all
previously understood phenomena and a new class of anomalous observations – in a new
whole with a simpler but more widely-encompassing explanation. To hold all considerations
on some topic in one’s mind in a single thought is the way Leibniz thinks that God works.
This is considered in the next chapter when God’s choosing of “the Best” of all possible
universes is dissected.
In Chapter 7 “Science by thought, reality and substance”, Leibniz’s theory of reality in toto –
physical and incorporeal – is put on the table. Its comprehensiveness is astounding and would
be more astounding had multiple attempts not been made in the past. Liberty is taken in many
footnotes to point out similarities between Leibniz’s theory and Hermes Trismegistus.
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Leibniz’s doctrine that this physical universe is the best of all possible worlds is part of his
scientific conception of the universe and of humanity’s role in it. It is shown that mathematics
is necessarily incomplete unless a rigorous conception of “the Best” can be formulated. The
chapter is left hanging with important open questions which suggest that the a priori method
is the only way to proceed when faced with really fundamental questions.
In Chapter 8, it is explained how the Leibnizian framework draws mind and humanity into the
domain of physical science. The thoughts of soul monads have a role in the physical universe.
The chasm between the perceptions of minds and the actions of body comes into view.
Leibniz spanned this divide with pre-established harmony, which requires God to intervene
every instant to mediate between mind and body in “the Best possible way”. The fact that God
mediates at all seems to be a glaring self-contradiction by Leibniz since he always held that
the Best possible universe cannot require God’s ongoing intervention. The contradiction is
resolved by the fact that the universe is re-created in every instant according to the thoughts of
soul monads, via the agency of the pre-established harmony which God shepherds along in
the best possible way.
Until science is expanded to subsume the pre-established harmony, science is – to a
significant degree – condemned to being an exercise in shadow boxing, or pursuing truths
lacking in context and lacking in metaphysical basis. Arguably, we cannot even understand
the mechanism of the physical universe without incorporating the pre-established harmony
into physics. The fact that the universe unfolds in “the Best possible way” implies that God
has built some kind of engine of intention into the physics of the universe. Of course, one
obvious intentional part of the universe is humankind itself. Thus, the status and role of
humankind as a physical force in the universe – as a law of physics, so to speak – warrants
attention in future research.
In Chapter 9 “A priorism in science”, three successful scientific projects that relied primarily
on a priori thought are described. The examples chosen are the infinitesimal, non-Euclidean
geometry and elliptical orbits in a heliocentric solar system. In all of these, but especially in
the first and third, the thinkers concerned pursued a priori methods deliberately, and were
self-conscious advocates for their own approach. In developing non-Euclidean geometry, the
thinkers were departing from the Fifth Postulate in a conscious effort to create a geometry that
is self-consistent and functional. Beyond that, Riemann in particular understood that any new
geometry must be viable in the physical universe. For Leibniz, the physical universe is the test
of truth because, as explained above, the physical universe expresses “the Best”.
In the concluding chapter, the humanist project in which Leibniz was engaged is discussed. It
is explained how his metaphysical and philosophical work, and his science and mathematics,
furthered the aims of this project. This project refines and furthers a conception of Man as
magus which was “modern” for the 17th and 18th Centuries. This survives today in the policies
of scientific-industrial nation state. Such policies were seen in early form under the leadership
of Louis XIV and Tsar Peter the Great. It is not a coincidence that Leibniz played an active
role in the nation-building efforts of Louis XIV’s France and Tsar Peter the Great’s Russia, as
described in this thesis.
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Chapter 2: Neoplatonism and the awakening of science
Introduction
It will be argued in the course of this thesis that Leibniz represented a continuation of the
influence of Neoplatonism as well as being a conscious referent to the original writings of
other movements and schools of thought, such as those of Plato, Aristotle and their followers.
When referring to actual figures such as Plato and Aristotle, the meaning is clear. What,
though, is Neoplatonism? The term encompasses a body of intellectual disciplines, mindsets
and networks of co-thinkers. I think of Neoplatonism as a movement that helped create much
larger shift, like lava flowing between two tectonic plates. To use phrases that might mean
little at present, Neoplatonism includes Hermetism, Cabala, medieval magic, memory
methods, the use of symbols and construction of languages, alchemy, proto-science, proto-
engineering and a bit of Platonism.
After the fall of Rome in 476 CE,15 intellectuals looked to Athens, Egypt and other ancient
cultures for wisdom and to increase their knowledge in almost any and all domains. Those
earlier cultures were regarded as more enlightened, virtuous and generally elevated than the
present. As Yates put it, the farther back in history a seeker explored, the purer and better the
minds found. The closer to the present, the more benighted and corrupt. Even the standout
philosopher of Rome, Cicero, leaned heavily on the philosophers of Athens. De Natura
Deorum reflects Plato’s The Republic in many ways. After Rome, the tendency and indeed the
imperative to look backwards in time increased.
The first Neoplatonists were “new Platonists”
The takeover of Athens by the Romans did not eliminate adherents of Plato. Indeed, Plato’s
Academy continued to teach students. The Greek thinkers Plotinus (c.204/5 – 270 C.E.),
Porphyry (c.234–305 C.E.) and Iamblichus (c.242-327 C.E. or 250-330 C.E.16) are regarded
as the founders of Neoplatonism. These can be regarded as the first kind of “Neoplatonist”.
The term has a literal meaning in the sense of a series of teachers known for their
interpretations of Plato. Indeed many if not all actually taught in Plato’s academy. This line of
teachers included Speusippus of Athens was the son of Plato's sister Potone; he became head
of the Academy when Plato died in c.348/347 BCE and remained its head for eight years.17 In
the early 4th century C.E., Iamblichus moved the major Neoplatonist school from Rome to
Syria. Thereafter, Neoplatonism flourished mainly in the Eastern Roman Empire, with centres
in Athens, Alexandria, and Pergamum (now in Turkey). In the 5th century C.E. one of its
greatest teachers was Proems, at Athens. This was also where the well-known Neoplatonist
Proclus (c.412–485 C.E.) taught. During the same century the writings of St. Augustine firmly
established its influence in Christian theology. St Augustine embraced some parts of 15 This is Edward Gibbon’s date which is conventionally accepted. 16 Accessed at http://www.goddess-athena.org/Encyclopedia/Friends/Iamblichus/index.htm on 5 April 2012 17 Hicks, R.D. (ed.) Diogenes Laertius Lives of Eminent Philosophers IV 1 accessed at
http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0258%3Abook%3D4%3Achapt
er%3D1 20 April 2012
Page 27
Neoplatonist teaching while filtering out parts considered to be contrary to his Christian
theology.
The Neoplatonists were not unmolested by the powers that were. While Constantine was the
first Christian emperor of Rome, his nephew Julian who succeeded him was a Neoplatonist, a
pupil of Aedesius, who had in turn been taught by Iamblichus.18 Emperor Julian was
sympathetic to Neoplatonism but his successor Emperor Theodosius was not. Under
Theodosius, the leading Neoplatonist teacher Hypatia was dragged from the academy and
murdered in 414/415 C.E. (there are differing accounts of the year) by a mob of Christian
monks under the control of Bishop Cyril of Alexandria. Ostensibly, Hypatia’s crime might
have been to attempt to publicly hold the light of reason to the “metaphysical allegories from
which Christianity had borrowed its dogmas”.19 It is likely that the threat posed by
Neoplatonism was much deeper, and that the real threat was not to Christianity but to the
Roman Empire and to the system of Empire in general. However, these ideas are speculative
and further discussion is beyond the scope of this thesis. Finally, under Emperor Justinian, the
last seven significant Neoplatonic teachers Hermias, Priscianus, Diogenes, Eulalius,
Damaskias, Simplicius and Isidorus were unable to continue the Neoplatonic school in
Athens.
The early Neoplatonists just mentioned were aware of the Egyptian writings especially those
of Hermes Trismegistus and there were Neoplatonists who absorbed Christian mysticism. The
first Christian Neoplatonist to write in Latin was Victorinus, who was converted to
Christianity in 360 C.E.20 In the 400s C.E., Pseudo-Dionysius, expounded a Christian
mysticism based on Neoplatonism that influenced later Christian mystics. Four centuries later,
a famed medieval Neoplatonist named John Scotus Erigena; he was the Irish theologian of the
9th century C.E. The Cambridge Neoplatonists are a group from the late 17th century C.E.
who sought to counteract the scientific materialism then prevalent.21
Arab philosophers
Following the first Neoplatonists, the Arab philosophers were active for at least a century or
so were also active in reviving ancient philosophy both Greek and Egyptian. This was an
intermezzo between the various wars and wrecking operations that beset the West between the
fall of the Roman Empire and the Renaissance.
Avicenna (Ibn Sina 980-1038) wrote on logic and philosophy, while also making original
discoveries in medicine. Avicenna is often cast as Platonist mediated by the
Neoplatonists,22,23,24 but he also adopted ideas from the Stoics such as Zeno and Chryssipus,
18 Accessed at http://www.wisdomworld.org/setting/hypatia.html on 5 April 2012 19 Ibid. 20 Inge, W.R. Christian Mysticism Plain Label Books, Chumley P. Grumley (series ed.) 1956 pp.153-154 based
on The Bampton Lectures 1899 delivered before the University of Oxford 21 Accessed at http://philosophos.hubpages.com/hub/neoplatonism on 7 April 2012 22 Wisnovsky, R. Avicenna’s Metaphysics in Context Cornell University Press 2003, p.64 23 Goodman, L.E. Avicenna Cornell University Press 2005, p.56 24 Harrison-Barbet, A. “Avicenna” in Philosophical Connections: Islamic Neoplatonism accessed at
http://www.philosophos.com/philosophical_connections/profile_036.html 20 April 2012
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and from Aristotle and his disciples.25,26 Avicenna also tried to reconcile the orthodox Islamic
doctrine of the “theologians” of his culture with the philosophers. Al Ghazali (1058-1111)
ridiculed attempts of writers such as Avicenna who tried to reconcile Islamic doctrine with
philosophy. However, he spared neither the Islamic doctrinal theologians nor the
philosophers. In talking about the philosophers, he was targeting contemporary proponents of
Aristotelian doctrine as they interpreted it. Platonic ideas seemed to be outside his purview.27
It was not only the “third force” of Plato that Al Ghazali let alone. Kilcullen at Macquarie
University says that the Muslims were “neo-Platonists” presumably that they nearly all were
neo-Platonists, though it is unclear where this means students of Plato or whether it means
Neoplatonists.28 There was also the “big influence of Hermetic and gnostic literature and
ideas on the Arabic world and particularly on the Arabs of Harran.”29 Further, “Talismanic
magic was practised by those Arabs, and the influence came through the Sabeans who were
immersed in Hermetism, in both its philosophical and its magical aspects.”30 The Picatrix was
probably written in the 12th Century, and while it lists magical images and procedures, it does
so in a philosophical setting similar to if not borrowed from the Corpus Hermeticum and the
Asclepius.31 The Picatrix circulated during the Italian Renaissance and references to it are
found in the writings of Pico the younger, among others. A copy was found in the library of
Pico Della Mirandola (1463-1494). It was included in a list of works “codex” on magic
copied in 1488 in Kraków, probably by a university student referred to as Egidius. The codex
includes Hermetic works and the Picatrix. From the commentary given in the codex, one
commentator concludes that “in all probability he [Egidius] knew the content of the
Picatrix”.32 Rabelais (c.1494-1553) came to its defence to it in one of his famous satires,
Pantagruel. Rabelais poked fun at those who shun the Picatrix by referring to “Father Devil
Picatrix, doctor of the faculty diabolical”. (Pantagruel III, 23)
While the Picatrix was written in the 1100s, we would expect it not to have arisen
spontaneously but to more likely have followed a century and probably more of thought,
teaching.
During the 1100s Averroes (Ibn Rushd, 1126-1198) which was guided or at least given initial
direction from Abu Bakr (Ibn Tufayl) chief physician to the monarch Abu Ya‘qub Yusuf
(c.1168-1169). Averroes was from an influential family possibly with a certain religio-
political orientation. For example, the distinctions of his grandfather Abul-Walid (1058-1126)
was that he was chief justice, wrote a definitive legal text, and leaned on the monarch Ali Ibn
25 Zabeeh, F. (ed. and trans.) Avicenna’s Treatise on Logic Martinus Nijhoff, The Hague 1971. For example, the
definition of “proposition” and the meaning of “indefinite” pp.20-21 nn.6-10, and p.23 n12 26 Supra nn.23-25 27 Bergh, S. van den Averroes’ Tahafut Al-Tahafut (The Incoherence of the Incoherence) Trustees of the E. J. W.
Gibb Memorial, London http://www.muslimphilosophy.com/ir/tt/index.html 7 April 2012. See the
introduction. 28 Kilcullen, R.J. Lectures for PHIL252 Medieval Philosophy, Tape 7 Al Ghazali and Averroes Macquarie
University, accessed at http://www.humanities.mq.edu.au/Ockham/x52t07.html 7 April 2012. See Kilcullen’s
full list of lectures here http://www.humanities.mq.edu.au/Ockham/kilcullen.html accessed 7 April 2012 29 Yates 1964, p.49 30 Ibid. 31 Ibid., p.47 32 Láng, B. Manuscripts of Learned Magic in the Medieval Libraries of Central Europe Magic in History Series,
The Pennsylvnia State University Press, 2008 pp.33, 35
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Yusuf to be tougher on the Christians of Andalus.33 Averroes ostensibly wrote in response to
Al Ghazali taking exception to his ridicule of Aristotelian philosophers saying that those
philosophers and Al Ghazali had not understood Aristotle. Averroes used persuasion not
attack. His main target seemed to be those engaged in the practice of law. Lawyers, judges
and clerics alike were wedded to Islamic doctrinal theology in the practice of law to the extent
that even the monarch Abu Ya‘qub Yusuf never publicly announced his support for
philosophy. Nonetheless, possibly under the persuasion or at least influence of Ibn Tufayl
himself, the monarch Abu Ya‘qub Yusuf allowed Ibn Tufayl to commission Averroes to write
commentaries on Aristotle that were accessible to the reading public.34
From the 9th Century up to Averroes there had been a cloud of suspicion alternating between
theology and philosophy. Ibn Masarra (883-931 C.E.) is regarded as the first Andalusian
philosopher. The zeitgeist was against them, and so he and his disciples only survived by
living as hermits. Hourani writes that Ibn Masarra introduced a pseudo-Empedoclean
pantheism.35 Empedocles (c.494/5-435 BCE) is one of the few pre-Socratic philosophers
whose original writings have been found.36 In the late 11th Century and into the 12th Century,
philosophy found increasing favour to the detriment of theology. In fact, the monarchs
encouraged the study of Malikite law and banned theology.37 Malikite law is a school of law
founded by Malik ibn Anas (c.710-795 CE) which recognizes a range of sources of law.38 Ibn
Bajja better known as Avempace (c.1095-1138 CE) was the first Andalusian philosopher to
make direct use of the works of Plato and Aristotle. Philosophers were still subject to
suspicion. For example, ibn Wahib a contemporary of Avempace, ceased to speak about
philosophy openly out of fear for their lives.39 It appears that ibn Sina also known as Avicenna
attempted to play a conciliatory role but to Al Ghazali both the theologians and philosophers
were wrong as well as being irreconcilable. Al Ghazali is best known for his Tahafut al
falasifa (“The Incoherence of the Philosophers”). While this sounds like a dogmatic and
politically-charged title, Al Ghazali was not a dogmatic contrarian. He wrote many other
works which describe, explain and grapple with both theology and philosophy.40 When
Averroes replied to Al Ghazali 80 years later, it was only to defend Aristotle not for the most
part any other philosopher and not (Islamic) theology.
It has been mentioned that Picatrix reached Europe and had some Hermetic effect on
intellectuals there centuries later. So too did Avicenna, Al Ghazali and Averroes. The reception
of Averroes was not entirely favourable in the increasingly Christianized Europe due in part to
Averroes’ advocacy of ideas such as that the universe has always existed and was not created,
33 Hourani, G.F. (ed,. trans.) On the harmony of religion and philosophy : a translation, with intro. and notes, of
Ibn Rushd's Kitab fasl al-maqal, with its appendix (Damima) and an extract from Kitab al-Kashf 'an manahij
al-adilla Luzac London 1961, p.14 34 Ibid., pp.12-19 35 Ibid., p.7 36 Accessed at http://history.hanover.edu/texts/presoc/emp.htm and http://plato.stanford.edu/entries/empedocles/
7 April 2012 37 Hourani, G.F. (ed,. trans.) On the harmony of religion and philosophy Luzac London 1961, p.7 38 Accessed at http://faculty-staff.ou.edu/V/David.R.Vishanoff-1/I-terms/Malikites.htm 7 April 2012 39 Hourani, G.F. (ed,. trans.) On the harmony of religion and philosophy Luzac London 1961, pp.8-9 40 A readable and apparently thorough summary of Al Ghazali’s works is at this page
http://www.ghazali.org/articles/gz1.htm accessed 7 April 2012
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and the impermanence of the individual human soul.41 In 1270, Saint Thomas Aquinas took
the trouble of writing “On there being only one intellect” in Paris while occupying a
Dominican chair of theology as Regent Master. Aquinas argued that intellect is a faculty of
the soul that animates the human body, and that there is not only a single separate intellect
that suffices for and furnishes all humans, but that each individual human soul has its own
intellect.42 Moreover, Aquinas argued, Aristotle would have agreed whereas Averroes had
misinterpreted Aristotle. Aquinas presented Aristotle as consistent with Christian metaphysics
on essential points.
Averroes is sometimes presented as having washed over Europe and fully infiltrated
institutions such as the University of Padua which became a “stronghold” of Averroism.
Martin explains that the truth is not so simple. (Martin, C. 2007) Intellectuals were as averse
to an Averroes dogma as they were to any other. Rather, Averroes was often quoted in order to
indirectly assert a secular orientation without inviting arrest, torture and death at the stake by
the Inquisition.43 Indeed, it is strange that no text by Averroes was placed on the Vatican’s
Index Librorum Prohibitorum (“List of Prohibited Books”) especially given Aquinas’
thorough case against the coldly argued anti-Christian position of Averroes’ Middle
Commentaries on Aristotle’s De Anima (“On the soul”).
Returning to the influence of pre-Athenian thought, we have already mentioned the Corpus
Hermeticum. This was and is ascribed to Hermes Trismegistus though the circuitous route
between Hermes Trismegistus and the documentation of Hermes’ teachings in the Corpus
Hermeticum is subject to debate. It is reported by some that Hermes was Egyptian,44 and he is
sometimes associated with the Egyptian god Thoth with the head of an Ibis. It has been
argued that there were three different Hermes Trismegistuses the third of whom was the
teacher of Pythagoras, while others dispute whether Hermes was ever a real person.
Hermeticum and magic in Europe
It has already been mentioned that the Arabs of Harran were steeped in Hermetic lore at least
from the 12th Century when the Picatrix was written.45 The Picatrix was in part a magical text
with symbols that can be used as talismans.46 In the 13th Century, Abulafia wrote a tract of
Cabalistic symbols and combinations of letters as an aid to understanding, meditation and
memory.47 In 1305, the Picatrix was translated into Latin by a scholar named Sloane.48 Yates
quotes at length from Book IV Chapter 3 of the Picatrix:
41 McInerney, R. Aquinas against the Averroists: on there being only one intellect West Lafayette, Purdue
University Press 1993, pp.71-99 42 Ibid., introduction p.ix 43 Martin, C. 2007 pp.6, 18-19 44 Barrett, F. The Magus Book III: Biographia Antiqua London 1801 pp.150-151 Accessed at http://www.sacred-
texts.com/grim/magus/ma255.htm 9 April 2012. However, as the Sacred Texts website itself notes, “The
biographical section has been deprecated by authorities such as Waite, and it's not even certain that it was
written by Barrett; it may have been added as filler by the printer.” Barrett’s book was re-published in 2003
by Kessinger Publishing under the title “The Magus a Complete System of Occult Philosophy”. 45 Yates 1964, p.49 46 Ibid., pp.51-52 47 Ibid., p.93 48 Ibid., p.50 n.1
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There are among the Chaldeans very perfect masters in this art they affirm that
Hermes was the first who constructed images by means of which he knew how to
regulate the Nile against the motion of the moon. This man also built a temple to the
Sun, and he knew how to hide himself from all so that no-one could see him, even
though he was inside it. It was he, too, who in the east of Egypt constructed a City
twelves miles long… Around the circumference of the City he placed engraved images
and ordered them in such a manner that by their virtue the inhabitants were made
virtuous and withdrawn from all wickedness and harm. The name of the City was
Adocentyn [in the Arabic original, al-Ašmūunain].
Giordano Bruno 279 years later wrote about the Greek origins of European civilisation
affirming that its origins were not Judaic but Chaldean. In Bruno’s work The Expulsion of the
Triumphant Beast published in 1548, he wrote, “Do not infer that the sufficiency of Chaldean
magic comes from Cabala; indeed, there has never been anyone who could pretend that the
Egyptians could have taken any principles from the Judaic corpus.”49
Giordano Bruno
Giordano Bruno was a philosopher, activist, teacher, Dominican friar and expert on the art of
memory. He was born in Nola Italy in 1548 and burned at the stake in 1600. He began a
period of almost 20 years as a fugitive from the Inquisition at the age of 28 when he fled the
Dominican monastery where he was studying following a tip-off that he would be
apprehended by the Inquisition for possessing forbidden books. He then lived as an itinerant
teacher and writer. Patronised by Henri III of France, he went to London as an organiser and
kind of emissary with a letter of introduction from Henri III to the French Ambassador. This
introduction afforded Bruno diplomatic protection. While in England, Bruno was active in
Hermetic and Neoplatonic circles, with known highlights including a lecture at Oxford and
strong relations with Fulk Greville a teacher of Shakespeare. It is believed that Bruno taught
his art of memory to Greville who passed it on to Shakespeare. Elizabethan actors were
required to have powerful memories due to the short times needed to learn lines and the fact
that a single actor would typically play many roles in a play. While acting circles were
necessarily full of skilled mnemonists, Shakespeare was respected even in acting circles for
his memory. It is not too much of a stretch to attribute this to Bruno’s memory methods.50
Bruno is credited with bringing a modern outlook on the universe, as full of stars which were
suns like our own. Each sun, according to Bruno, had many planets and many of those were
peopled with intelligent beings who probably worshipped their own gods. Like that of most of
the people mentioned in this thesis, the story of Bruno is an epic in itself. We introduce Bruno
because he is mentioned in this thesis in connection with the revival of the teaching of
Hermes, the role of a rational kind of magic, and the use of symbols as an aid to thought.
49 Paraphrasing the translation in Rowland, I.D. Giordano Bruno: Philosopher and Heretic University of
Chicago Press 2009, p.59 of a passage in Giordano Bruno’s Expulsion of the Triumphant Beast 1584 50 White, M. The Pope and the Heretic: The true story of Giordano Bruno, the Man Who Dared to Defy the
Roman Inquisition Harper Collins 2001
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Consistent with Bruno’s view is one circulating in the circles of “New Age spiritualism” that
the wisdom of the Cabala is merely what the Jews took with them of Egyptian wisdom when
they left Egypt.51
Magic, talismans, symbols, languages and the art of memory in the 16th Century were on
some occasions treated separately and at other times together. There are many studies that
show why it is artificial to separate these domains. Credit goes to Yates for showing how these
threads were drawn together by Bruno in Art of Memory.52 However, there were many figures
other than Bruno, such as Marsilio Ficino, Pico della Mirandola, perhaps nearly every
European thinker from the Neoplatonists up to Leibniz who would not have been able to
study one area without examining the others too. This does not mean that every contributor to
language such as Dante or to the art of memory such as Bruno was an aspiring magician.
Michael White writes that though Bruno “was fully cognizant of the power of magic ritual
and the occult tradition”, nonetheless Bruno “was convinced by very little of the occult
canon.” In fact, “he knew much of it was superstition, wild fantasy, and wishful thinking.”
However, it did indirectly play a role. “To Bruno, as to many great thinkers after him, the
occult was primarily a useful tool, a key that would open doors into arenas of thought and
hidden depths of the human psyche. Along the occult path he found tracks, roughly hewn, that
led to revelation and inspiration. Alchemy held no interest for Bruno; he was never motivated
by experiment and was not drawn by the search for the philosophers’ stone, the dream of
limitless wealth. Neither did he practice ritualistic magic or necromancy; indeed, he often
mocked practicing astrologers and many of the irrational precepts of witchcraft.”53
Hermes in Europe by mistake
In the first half of the 15th Century, Cosimo de Medici was financing expeditions for ancient
texts and the translation of those texts obtained. While Athenian writings such as those of
Plato or Aristotle or their disciples were valued, the earlier philosophers of Egypt, the
Chaldeans and others were even more highly valued. As mentioned above, the perception was
that more ancient meant purer and better. More than a hundred years after Medici, Bruno
continued to hold that the best of European philosophy emanated from Egypt, and not merely
because “older is better” was a catchy epithet. He understood that Ancient Egypt was the
source of Greek philosophy, implying that even the Greeks were a dilution of the best of
Egypt. In his The Expulsion of the Triumphant Beast published in 1584, Bruno wrote, “ We
Greeks recognize Egypt, the great monarchs of literature and nobility, as the parents of our
epics, metaphors and doctrines”.54,55
51 For example, http://www.amaluxherbal.com/bnewbooks/hermes%20trismegistus.html accessed 9 April 2012 52 Yates, F. Art of memory University of Chicago Press, 1966 53 White, M. The Pope and the Heretic: The true story of Giordano Bruno, the Man Who Dared to Defy the
Roman Inquisition Harper Collins 2001, pp.67-68 54 Giordano Bruno Nolano, Spaccio de la bestia trionfante. Stampato in Parigi MDLXXXIIII, in Dialoghi
filosofici italiani, a cura di Michele Ciliberto, Monciodadori, Milano 2000 55 Paraphrasing the translation from Rowland, I.D. Giordano Bruno: Philosopher and Heretic University of
Chicago Press, Chicago 2009, p.59, originally published in 2008 by Farrar, Straus and Giroux
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Returning to Cosimo de Medici, the monk Leonardo da Pistoia sent manuscripts of the
Corpus Hermeticum, which were putatively written by Hermes Trismegistus himself to
Medici. Medici sent them on to Ficino in c.1462 asking Ficino to pause on the translation of
Plato and to translate the Hermetic manuscripts as quickly as possible so that Medici could
read them for he was terminally ill.56 Ficino’s translations of the works were published in
1471. The works were the Asclepius or De voluntate divina “On Divine Will”, the Pimander
or De sapientia et potestate Dei “On God’s Wisdom and Power”, and the Asclepii
Definitiones.57 Ficino (1433-1499) holds a significant place in the history of theology,
religion, magic and philosophy.
In 1614, Isaac Casaubon proved that the Hermetic manuscripts were no older than the second
or third century C.E.58 However, by that time, they had already had 140 years to transform the
thinking of intellectual and influential circles throughout Europe.
To use Yates’ words, Hermes Trismegistus, through these proxy writings, authorised by his
antiquity the revival of forms of magic.59 That revival included the surrounding philosophy
including doctrine of the universe and the active place of humankind in it.
Natural magic
Ficino the translator of the manuscripts for Cosimo de Medici has an enthusiasm for “magna
naturalis” or natural magic or the magic of nature which, ultimately, can only mean the
science or, at least, proto-science albeit undoubtedly with a mystical overlay. Pico della
Mirandola began his career under Ficino’s influence and inherited Ficino’s enthusiasm for
magia naturalis.60 Pico perhaps partly due to the Judaic origins of the Old Testament account
of Creation in the Torah effected a marrying of Hermetic and Cabalistic magic. This, writes
Yates, “was to have momentous results, and the subsequent Hermetic-Cabalistic tradition,
ultimately stemming from him [Pico], was of most far-reaching importance.”61
Many have heard of a controversial 900 theses published in Rome centuries ago. It was Pico
who wrote them and in 1486 took them to Rome proclaiming that he was ready to prove in
public that they were all reconcilable with one another. Twenty-six of the theses were on
Cabalist and/or natural magic. One thesis says the old-style kinds of magic should be
forbidden, but that magia naturalis is a good, allowable magic. In another thesis, Pico says
good magic is in part the practical science of nature.62
In his survey of “learned magic” according to manuscripts from Central European medieval
libraries, Láng writes “It might sound surprising, but ‘necromancy’ in [one of its meanings]
was long a successful applicant for denoting a widely accepted part of science. The great
56 Yates 1964, p.58, Yates 2009, pp.13-14 57 Jaroszyński, P. Science in culture Rodopi B.V., Amsterdam, New York 2007, p.139 58 Ibid. 59 Yates 1964, p.58, Yates 2009, p.63 60 Ibid., original 1964 edition, pp.84, 87-88 61 Ibid., pp.86 62 Ibid., pp.87-88
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summary of Arabic magic, the Picatrix, defines it in a rather wide and naturalistic sense as the
science dealing with all the things that are hidden from the senses or from the intellect, the
functioning of which most people do not understand.”63
While Ficino obscured the Hermetic origins of the magia naturalis that he promoted, Pico
writes in his oration On the Dignity of Man, “It is really the magic of the Asclepius that I am
talking about, and I glory in Man the Magus as described by Hermes Trismegistus.”64 What
can Man the Magus do? He says, “The Magus the earth to heaven, that is to say the force of
inferior things to the gifts and properties of supernal things.” Indeed, Pico begins this oration
with the words of Hermes Trismegistus to Asclepius, “The Magus, oh Asclepi, is a miraculous
man.”
The other kind of Neoplatonism, and operations
It is at the point of Pico’s work that we have reached the Neoplatonic age. It is in this sense
that Neoplatonism is used in this thesis. It encompasses a confluence of ideas and
philosophies. Some might call the Pico era the Neoplatonic humanist Renaissance par
excellence at whose heart of which we find Hermes Trismegistus. The fact that we find
Hermes at the heart of that era raises warning bells about whether the Renaissance was in fact
Christian or even humanist. Thanks go to Yates for pointing this out.65 Ultimately, it was
about the power of humans acting individually and, later, in concert. Rather than Christian,
perhaps it was Hermetist with a Christian sheen. It was as pantheist as it was humanist, with
humans playing a special role in the matrix of the natural universe since humans could
understand, imitate and thence manipulate and bend the power of nature to the collective will
of humans.
The entire current of “magic” led into the domain of “operations” as Yates calls it or the post-
Renaissance idea of Man consciously intervening in the physical universe to change it or
amend it for Man’s benefit or for Man’s beneficial ends. An intermediate kind of magic which
Yates refers to as “astral magic” is unmistakeably tending towards operations. It seeks “escape
from astrological determinism by gaining power over the stars, guiding their influences the
direction the operator desires.”66 Walker explains Ficino’s theory that is Stoic in origin of a
spiritus mundi (“cosmic spirit”) which provides a channel of influence between planets and
stars, and the world in which humans live. Methods of influencing the stars and planets were
detailed by Ficino.67
63 Láng, B. Unlocked Books: Manuscripts of Learned Magic in the Medieval Libraries of Central Europe Penn
State Press 2008, pp.41-42 64 Ibid., pp.90-91 65 Yates 2009, p.185 n.25 66 Ibid., p.60 67 Ficino “De Vita coelitùs comparanda” in Opera Omnia Basileae 1576 p.493, as analysed and discussed in
Walker, D.P. Spiritual and Demonic Magic: From Ficino to Campanella University of Notre Dame Press,
Notre Dame and London 1975 (first published by the Warburg Institute, University of London 1958), pp.12-
15
Page 35
In 1510, Henry Cornelius Agrippa wrote his De occulta philosophia which he published in
1533. In fact, this work was a survey of renaissance magic including Natural Magic which
was the subject of Book I.
Agrippa did the same as Ficino did in the sense of explaining methods for influencing the
stars. However, while Ficino tries to force his magic in to a Christian framework, Agrippa
does not.68 Rather, arguing the converse, Agrippa says explains that some Christian practices
including prayer itself were forms of magic, some of which were legitimate and effective. Of
these, some were effective in their own right while others were effective in influencing people
and, though people, events, but only because people were emotionally affected by the rituals
and practices.
While the perception of the cosmic order of the Agrippan Magus is almost identical to the
medieval perception, Man has now changed and so also has Man’s role in the cosmic order.
Man is “no longer the pious spectator of God’s wonders in the creation, and the worshipper of
God himself above the creation.”69 Man is now an “operator” who “seeks to draw power from
the divine and natural order.”70 In an image by Robert Fludd of Man’s Art is shown as a
monkey which imitates Nature.71 The apparent loss in dignity is more than exceeded by the
gain in power by “becoming the clever ape of nature, who has found the way nature works
and by imitating it, has obtained her powers.”72 Man has “learned how to use the chain linking
earth to heaven.”73
Contemporaneously with Agrippa, John Dee had his career as perhaps the consummate
Renaissance Magus. Dee was interested in the applied sciences, and built devices such as a
flying crab for a college stage play.74 He was certainly a mathematician as far as was possible
in the 16th Century, having written the preface to the first English translation of Euclid, by
Billingsley.75 The book included pop-up figures of 3D geometry.76
Nearly a century after Agrippa, Tommaso Campanella wrote Magia e Grazia which is mostly
on religious magic. Perhaps this is one of the reasons why Campanella was imprisoned in
68 Walker, D.P. Spiritual and Demonic Magic: From Ficino to Campanella University of Notre Dame Press,
Notre Dame and London 1975 (first published by the Warburg Institute, University of London 1958), p.93 69 Ibid., p.144 70 Ibid. 71 It is on the cover of Fludd’s Tratactus secundus de naturae simia seu technica macrocosm historia “Second
Treatise on the Imitation of Nature or the Technical History of the Macrocosm” published in 1618 with the
ape sitting in the middle of a circle depicting the sciences and engineering, including music, building
construction, surveying and geography. It is also in Utriusque cosmi historia “The history of both worlds”
published between 1617 and 1621 in which the ape is sitting on top of the world again in the centre of the
arts and sciences, but above the ape a woman is standing atop the world clearly with spiritual power with her
wrist chained to heaven above. The image is given the title “Integra naturae” or “Integration of nature”. Both
works were first published by Theodore de Bry in Oppenheim. 72 Yates 1964, pp.58, 145 73 Ibid. 74 Ibid., p.148, n.2 with primary sources Smith, C.F. John Dee London 1909 and Calder, I.R.F. John Dee, studied
as an English Neoplatonist unpublished Ph.D. thesis London University 1952 75 Billingsley, H. (trans.) Euclid The Elements of Geometrie London 1570, reprinted by Ann Arbor 1967 76 Franklin, J. “Diagrammatic reasoning and modelling in the imagination: the secret weapons of the Scientific
Revolution” in Freeland, G. and Corones, A. (eds.) 1543 and All That: Image and Word, Change and
Continuity in the Proto-Scientific Revolution Dordrecht 1999, p.82
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Rome by the Inquisition for twenty-five years suffering both torture and solitary
confinement.77 In Magia e Grazia, Campanella classifies different kinds of magic one of
which he calls “real artificial magic”. He enumerates examples such as when “Architas [sic]
made a flying dove of wood” or “when Daedalus made statues which moved through the
action of weights or of mercury” or “to make a head which speaks with a human voice”.78
Dampening the enthusiasm of those engaged in artificial intelligence in today’s computer
science, Campanella also writes, “But such forces and materials can never be such as to
capture a human soul.”
We have attempted to show how Neoplatonism and the magic talked about – if softly – during
the Neoplatonic age led to the rise of mechanical devices which are not far from the art of
invention, engineering and the age of mechanisation. This is one point where Leibniz enters
our story, as he shall do in the next chapter.
The Greeks as operators
Yates contrasts the operational role of Man with the Greek position. Yates argues that the
Greeks were not particularly interested in operations:79
The Greeks with their first class mathematical and scientific brains made many
discoveries in mechanics and other applied sciences but they never took whole-
heartedly, with all their powers, the momentous step which western man took at the
beginning of the modern period of crossing the bridge between the theoretical and the
practical, of going all out to apply knowledge to produce operations. Why was this? It
was basically a matter of the will. Fundamentally, the Greeks did not want to operate.
They regarded operations as base and mechanical, a degeneration from the only
occupation worthy of the dignity of man, pure rational and philosophical speculation.
The Middle Ages carried on this attitude in the form that theology is the crown of
philosophy and the true end of man is contemplation; any wish to operate can only be
inspired by the devil. Quite apart from the question of whether Renaissance magic
could, or could not, lead on to genuinely scientific procedures, the real function of the
Renaissance Magus in relation to the modern period (or so I see it) is that he changed
the will. It was now dignified and important for man to operate; it was also religious
and not contrary to the will of God that man, the great miracle, should exert his
powers. It was this basic psychological reorientation towards a direction of the will
which was neither Greek nor mediaeval in spirit, which made all the difference.
[emphasis in original]
Even Zeno the putative founder of the Stoics saw the aim of attaining full understanding of
the universe as a self-governing system was to “rise to complete wisdom and attain perfect
77 White, M. The Pope and the Heretic: The true story of Giordano Bruno, the Man Who Dared to Defy the
Roman Inquisition Harper Collins 2001, p.158 78 Amerio, R. (ed.) Magia e grazia Fratelli Bocca, Rome 1957, p.180 79 Yates 1964, pp.155-156
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ethical conduct.”80 At least per Cicero’s interpretation of Zeno’s doctrine – according to Hunt
– there was an element of stasis in the Stoic doctrine and conception of the universe which
might have tended away from operations. Surprisingly, it may be in the domain of what is
called magic that we see the earliest signs of the orientation towards operations in any culture.
On the other hand, an argument for the operational Greeks can be put. Greeks such as
Archytas of Tarentum (400-350 BCE) and Archimedes of Syracuse (c.287-212 BCE) were
impressive as inventors, and these two are that we know about which readily come to mind.
Could it have been that the Greeks were so beset by war from the Persians and Romans,
and/or distracted by foolish policy such as that which led to the Peloponnesian Wars, to
achieve their potential in operations?
A modern parallel can be drawn. Consider the once highly “operational” USA with the Apollo
landing, plans for the colonisation of Mars and dam-building projects on a grand scale.
Contrast that history with the veering of American policy away from operations from the
Vietnam War onwards.
It is has been written that Archimedes preferred contemplation in abstract geometry to
inventing machines and was only compelled to invent machines by his monarch, King Hero,
to defend Syracuse against the Roman army led by Marcellus (214-212 BCE) during the
Second Punic War. Nonetheless, for every potential operator like Archimedes who had to be
imposed upon to engage in operations, there might have been ten who were able and who
preferred to operate rather than to meditate. Arguably, had the Greek civilisation not been
prevented from flourishing by institutions such as the Persian and Roman Empires, then they
would have realised and expressed an intention to “operate”.
This leads to a discussion of nations or “the state” as operator, Magus or magician
commandeering the capabilities of entire populations for operations in grand projects.81
Further discussion on this topic is beyond the scope of this thesis, though the topic is worthy
of further effort.
Mathematics in operations
The related current also intertwined with magic but also running in to the ocean of “Man as
operator” is number or mathematics. Many of those who took “magic” seriously as a means
whereby Man may “operate” also took mathematics seriously. Indeed, inventors from
Archytas of Athens to Archimedes of Syracuse to John Dee of England to Huygens and
Leibniz were equally well-known for their contribution to mathematics.
80 Hunt, H. A. K. A physical interpretation of the universe: the doctrines of Zeno the Stoic Melbourne University
Press 1976, p.20 81 Couliano, I.P. translated by Cook, M. Eros and Magic in the Renaissance University of Chicago Press,
Chicago and London 1987, p.105. However, the premise of Couliano’s discussion is the as magician in the
sense of controlling education, religion and every other aspect of the lives of citizens to ensure uniformity.
Couliano does not consider that such uniformity may merely be a by-product of the organisation needed for
operations via grand projects of the kind described in the text above.
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Johannes Trithemius, Abbott of Sponheim, was a friend and teacher of Agrippa. Trithemius’
book the “Steganographia” was printed in 1606. It is a tract about cryptography and Cabalist
angel magic. It presents methods for summoning networks of angels to enable the
transmission of messages using telepathy. Part of the process of summoning these angels
involved many pages of calculations involving the numbers that represented the angels.82
It is consistent with the central role of number in the discipline of magic that Ficino includes
Pythagoras amongst the prisci magi given the mathematical achievements of the
Pythagoreans. Pico’s 900 theses include fourteen that, according to Pico, followed from the
mathematics of Pythagoras. Yates concludes that “Renaissance magic was turning towards
number as a possible key to operations”83 meaning, in our language, that without mathematics
there would be no science or engineering.
Leibniz is famous for his contribution in creating the infinitesimal calculus. The relevance to
this chapter is that Leibniz’s calculus was part of his larger vision for a Characteristica
Universalis which would be a language that serves not only for numerical calculation but for
reasoning in general.
Ramón Lull (1232/5 – 1316) sought a language with which truths could be communicated to
infidels who did not know Latin and with illiterates who knew no written language. Lull
considered such questions as how many anagrams could be formed from a finite alphabet. For
example, there are n! permutations of n letters. Lull called it the ars combinatoria or “art of
combinations” which 400 years later became one of Leibniz’s first sources.84 In 1660, Leibniz
wrote his Dissertatio de arte combinatoria which used concepts from Lull.85 Eco explains that
aside from Leibniz’s vision for a universal characteristic, his ars combinatoria had a great deal
in common with the many project for universal language undertaken over centuries up to
Leibniz’s career. However, Leibniz’s search for a characteristic emphatically was not a search
for a universal language any more than the infinitesimal calculus can be used for day-to-day
communication. Peckhaus asks the question whether Leibniz envisioned a universal language
for reasoning or a universal characteristic.86
Leibniz thought about what would be the best way of providing a list of primitives and
ultimately an alphabet of thoughts. Leibniz described an encyclopaedia as an inventory of
human knowledge which might provide the basic material for the art of combination. Leibniz
wrote that the greatest aid for the mind could be to discover a small set of thoughts from
which an infinity of other thoughts might issue in order, just as the symbols for all numbers
are obtained from the symbols 0-9.87
82 Yates 1964, p.145 83 Ibid., p.146 84 Eco, U. The search for the perfect language Blackwell 1995, p.272 85 Gerhardt, C.I. (ed.) Die philosophischen Schriften von Gottfried Wilhelm Leibniz 7 vols., 1875-1890. reprinted
by Georg Olms, Hildesheim 1978, volume IV, pp.27-102 86 Peckhaus, V. “Calculus Ratiocinator Versus Characteristica Universalis? The Two Traditions in Logic,
Revisited” History and Philosophy of Logic 2004 Vol.25, No.1, pp.3-14 87 Eco, U. 1995, p.275. Eco’s source is Leibniz’s Elementa in Couturat, L. Opuscules et Fragments Inedits de
Leibniz: Extraits des manuscrits de la Bibliothèque royale de Hanovre F.Alcan, Paris 1903, reprinted George
Olms, Hildesheim 1961, pp.42-92
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Leibniz’s work in this area was a continuation of a long line of philosophers and philologists
who had dreams akin to those of Leibniz of finding a tool for human thought in the form of
language. We do not wish to paraphrase the comprehensive work of Eco but merely to provide
some signposts.
Symbols
Symbols that putatively had a magical purpose were also used in the art of memory with only
the pragmatic purpose of assisting in memory. Certain symbols stood out for their magical
purpose and also because they were striking as well as forbidden and, therefore, memorable.
Bruno is of course the standout figure among these. Ficino is not a good example because,
unlike Bruno, he actually believed in the magical power of the images.88 Such images were
regarded by the clergy as demonic whether officially or otherwise. Augustine had already
banned the Egyptian gods whether presented as gods or recast in modern costume.89
The art of memory was not only about memory but about thought. Similarly, Leibniz’s
encyclopaedia project was about recording knowledge, but it could only be useful if
undertaken in tandem with a scientia generalis or “general science” from which the whole
encyclopaedia could be derived.90 Leibniz equated the scientia generalis with the scientia
felicitatis or the search for wisdom.91 A tool for this science was the Leibniz proposed a
battery of related and indispensable disciplines such as created about creating it with the aid
of a tool for reasoning which he called the “universal characteristic” about which more below.
Leibniz’s programme had some similarity to George Dalgarno’s bifurcated project for a
universal language, culminating in his Ars signorum of 1661. Eco says that Leibniz was
“perhaps the only scholar who considered Dalgarno respectfully”.92 For Dalgarno, a universal
language had first to accommodate the plane of content, that is, a classification of all
knowledge. Second, it required an expression level or a grammar to denote the content.93 John
Wilkins ran a similar project on behalf of the Royal Society which started in 1668. Eco’s
assessment is that while Wilkins “accomplished what Dalgarno only promised to do”,
Wilkins’ achievement was limited to an image of the universe “designed by the Oxonian
culture of his time.”94
Bruno’s memory methods converged with his pitch into the quest for the perfect language
which was at one time thought to be the language of Adam. The perfect language was
believed to allow metaphysical and scientific secrets to be unlocked. Bruno thought that the
88 Yates 1964, pp.56-57 89 Ibid., p.57, second paragraph 90 Leibniz, G.W. “Studia Felicitatem Dirigenda” winter 1678/9 in Academy of Sciences of Berlin (ed.) Sämtliche
Schriften und Briefe Series VI Philosophische Schriften, no. 4 pp.137-138 as quoted and translated in
Antognazza, M.R. Leibniz: An intellectual biography Cambridge University Press 2009, p.237 91 These ideas on the Leibnizian orientation are from Antognazza, M.R. Leibniz: An intellectual biography
Cambridge University Press 2009, pp.237-238 92 Eco, U. The search for the perfect language Blackwell 1995, p.229 93 Ibid. 94 Ibid., p.239
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hieroglyphic language of the Egyptians was superior to the alphabetic languages. This is
contrasted with Leibniz who thought that alphabetic language were superior to pictographic
languages like Mandarin because one need not be highly educated to know and alphabetical
language but need only learn the 26 or so letters to get started. No doubt, Bruno’s knowledge
of the usefulness of symbols as markers that are pregnant with meaning and his active use of
them to great effect in his memory techniques was to Bruno a provable demonstration of the
power of symbols. Moreover, Yates says that Bruno was steeped in understanding and belief
in the magical power of certain symbols. To the extent that this is correct, the magical power
of symbols will have added a new raft of power to symbols for Bruno. Leibniz’s argument
makes sense as far as getting started in a language is concerned, but moving past the
rudiments one does need a certain amount of education to understand the meaning associated
with words beyond a minimal vocabulary. In Bruno’s defence, a well-formed symbol can
convey meaning even to an illiterate person. Further, if symbols are formed in a sensible and
intuitive way, then someone trained in the rudiments of a symbolic language should be able to
make some sense of “advanced” symbols even without advanced training.
Plotinus and Iamblichus seem to imply that they too, like Bruno, prefer the hieroglyphic
language of the Egyptians.95 The search for the best language to aid the process of revelation
or as an aid to uncovering metaphysical truths “the perfect language” was on. Those who
studied the art of memory along with occult symbology with or without magical content were
also involved the search for the perfect language. The study of the accessibility of symbolic
versus alphabetic languages is another topic that is beyond the scope of this thesis but worthy
of further study.96
As explained above, the “magical” disciplines led into the domain of “operations” as Yates
calls it or the post-Renaissance ideas of Man consciously intervening in the physical universe
through science and engineering. This current in magic was really magic minus mysticism
and minus occultism. The search for the universal characteristic was part of this. That is,
rather than symbolism with occult meaning and purpose, the idea was a structure of symbols –
indeed, a language – that would not only be an aid to reasoning or to rational thought, but that
would in effect as a gateway to reason. For a moment, consider Reason as an abstract body of
“correct answers” or “Leibnizian ideas” provided. The right answers would be calculated by
the universal characteristic, or – more correctly – would be forced out by the relationship of
the structure and design of the language to Reason. The imperative “let us calculate” using
Leibniz’s envisioned characteristic was more a charge to sit down and use the characteristic to
obtain the correct answer.97 The reason was built into the system, whereas “calculating” using
the system was relatively mechanical; one would merely “crank the handle” to get the answer
out. This would not be a stretch for Leibniz since he developed and designed a mechanical
calculating machine that was eventually built. Leibniz certainly did not think that such
automatic calculation methods had to be restricted to questions involving numbers.
95 Eco, U. The search for the perfect language Blackwell 1995, pp.144-145 96 For an introduction and historical overview of this domain, see Franklin, J. “Diagrammatic reasoning and
modelling in the imagination: the secret weapons of the Scientific Revolution” in Freeland, G. and Corones,
A. (eds.) 1543 and All That: Image and Word, Change and Continuity in the Proto-Scientific Revolution
Dordrecht 1999, pp.53-115 and Eco, U. The search for the perfect language Blackwell 1995. 97 Eco, U. The search for the perfect language Blackwell 1995, p.277
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Franklin gives an overview more specific to diagrammatic reasoning.98 It may be that the
universal characteristic would be a diagrammatic language. Frege’s work towards Leibniz’s
vision is for a diagrammatic language.99 Category theory also uses diagrams, but is only
intended to apply to specifically to algebra.
Conclusion
To conclude, Neoplatonism largely encompasses what is discussed above regarding symbols,
magic and Egyptian thought. The last item largely means Hermetism, Platonism itself, and
more. Proto-science grew out of Neoplatonism, from which science grew shaking off its
mystical magical roots while certain aspects of its roots remained such as the interest in
language and the desire to create the best possible language or characteristic that aids
reasoning and acts as a gateway to propositions concerning theology, metaphysics and science
that are correct. This is about acquiring the ability to discover a principle before the effect of
that principle has been discovered or observed. This is in effect the idea of an a priori
approach to science. Further to that approach, tools of thought aid purposeful creative thought
and structured reasoning. Many will disagree with an attempt to encapsulate Neoplatonic
revolution in a single sentence. However, it is arguable that the many currents which meet in
Neoplatonism effected recognition of the goal, purpose and capability of humanity to
influence, affect, build in, improve and – indeed – control the physical universe. That is, to
“operate”.
98 Franklin, J. “Diagrammatic reasoning and modelling in the imagination: the secret weapons of the Scientific
Revolution” in Freeland, G. and Corones, A. (eds.) 1543 and All That: Image and Word, Change and
Continuity in the Proto-Scientific Revolution Dordrecht 1999, p.82 99 Smith, B. Characteristica Universalis in K. Mulligan, ed., Language, Truth and Ontology (Philosophical
Studies Series) Kluwer, Dordrecht/Boston/Lancaster 1990, pp.50–81
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Chapter 3: Ushering in modernity
Introduction
Leibniz appears at the tail-end of the Renaissance, born shortly before the end of the Thirty
Years War which was concluded with the 1648 Peace of Westphalia. The Thirty Years War
was not an historically isolated eruption. Bloodshed fuelled by misguided religious passion
marred much of the 16th Century. That the senseless slaughter above all else required a
revolutionary change in thinking had already been noticed by one born nearly one hundred
years before Leibniz, named Giordano Bruno whom we met in the previous chapter. As
Michael White wrote, “The Wars of Religion provided a harsh backdrop to Bruno’s entire
adult life and added further turmoil to the usual privations and struggles of sixteenth-century
common folk. Wherever Bruno travelled within Europe, doctrinal intolerance and endemic
slaughter in the name of God reassured him that only a spiritual and intellectual revolution
could ever disassociate religion from murder, horror, and endless pain.”100
Moving into the 17th Century, the Thirty Years War and the peace which ensued affected
intellectual circles, which included Leibniz’s teachers. In particular, it was apparent that an
intellectual order was needed which would ensure tolerance and an enduring peace. This
certainly affected the philosophical and political outlook of Jakob Thomasius (1622-1684)
the jurist, historian and philosopher whom Leibniz says is “the most celebrated German
Peripatetic”.101 Indeed, Thomasius was known as an important conciliator.102 Thomasius is
particularly relevant because Leibniz was taught and mentored by Thomasius from when
Leibniz entered the University of Leipzig at the age of 14. The need to conciliate recurs so
frequently in Leibniz’s writings that Christia Mercer coined the phrase “conciliatory
eclecticism” to describe one of the main drivers of Leibniz’s philosophy.103 In a letter to
Thomasius in April 1669, Leibniz explained that the Aristotelian and the mechanical
philosophies could be reconciled, and suggested a conception of substance to this end.
Averroism was growing in strength. Leibniz was a reader of classical Greek texts in his youth,
and also read medieval writers and Renaissance Humanists.
European-centred civilisation was extending its knowledge of the world beyond Europe.
Columbus had landed in the New World 150 years before and the Republic of Letters had
access to educated persons far beyond Europe. Leibniz was a diplomat by profession. He had
international influence, and corresponded with leaders and men of letters across Europe and
100 White, M. The Pope and the Heretic: The true story of Giordano Bruno, the Man Who Dared to Defy the
Roman Inquisition Harper Collins 2001, p.29 101 Mercer, C. “Leibniz and His Master: The Correspondence with Jakob Thomasiu” Chapter 2 in Lodge, P. ed.
Leibniz and his Correspondents CUP 2004 102 Sturm, J.C. Philosophia eclectica Altdorf, 1686, pp.72-73 Jolley writes, “Sturm’s works were widely read.
Leibniz refers to them throughout his life, although he does not refer specifically to Philosophia eclectica.”
Jolley, N. The Cambridge Companion to Leibniz CUP 1995, p.116 103 Mercer, C. Leibniz’s Metaphysics: Its Origins and Development Cambridge University Press: Cambridge,
2001, Chapter 1, pp.23ff, and Mercer, C. “The Platonism at the Core of Leibniz’s Philosophy” Chapter 15 in
Studies on Platonism and Early Modern Philosophy in the series Hutton, S. ed. International Archives of the
History of Ideas Springer Volume 196, 2007, pp.225ff
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Eurasia, and in China.104 That Leibniz corresponded with Peter the Great is not surprising
considering that a month after Leibniz met Peter the Great, Leibniz in 1712 was appointed by
Peter to the Russian Justizrat.105 Leibniz was in such frequent correspondence with Jesuits in
China that he wrote in a letter to Princess Sophia Charlotte to whom he was close in 1697, “I
will thus have a sign placed at my door, with these words: bureau of address for China,
because everyone knows that one has only address me in order to learn some news.”106
Science had recently made strides through Kepler and Galileo. Galileo was affected by the
Reason-oriented mould of Kepler and the empirico-deductive mould of Averroism. With the
success of Galileo and the influence of his patron Paolo Sarpi, who was Atheist and an
Averroist, the empirico-deductive approach to science was being promoted in opposition to
Kepler’s Reason-oriented approach which was also represented by Leibniz’s second (after
Jakob Thomasius) mentor Christian Huygens. Invention of machinery was burgeoning in
Leibniz’s day, and Leibniz took an active interest in the design and building of machines.
We see that while Leibniz never announced his scientific programme, he drip fed it over his
career and developed his method over his career.107 While his mathematics and physics were
developed over time, the seeds of certain aspects of Leibniz’s metaphysics such as the
doctrine of the best of all possible worlds and the principle of sufficient reason were present
in his earliest writings. Arguably, those earliest ideas were animated by Leibniz’s passion to
advance humanity by pursuing knowledge. Since Leibniz was born into “the century of
genius”,108 the seventeenth century, he found many debates and programmes to contribute to
as well as starting a few of his own. If one is looking for a declaration of Leibniz’s scientific
agenda, it is set forth most cohesively in summary form in his letters.
While the empirico-deductive approach was rising on the Continent and in Britain, Leibniz
was a torch-bearer for the Keplerian approach to science. Continental Platonism was imbued
with science, and scientific discovery was its central concern. Their definition of the
Discovery process is part of their definition of the relationship between God, Man and the
physical universe, which is not reconcilable with empirico-deductivism.
104 Cook, D. J. and Rosemont, H. Jr. (trans.), Leibniz, G. W. Writings on China Open Court Publishing Company,
Chicago and La Salle, Illinois 1994 accessed at
http://www.strongwindpress.com/word/TuiJian/LeibnizWritings%20on%20China.doc on 29 Apr 2011 105 Smith, J. E. H. An archive of philosophy news, notes, and academic work in progress
http://www.jehsmith.com/philosophy/2009/01/peter-the-greats-decree-appointing-leibniz-to-the-russian-
justizrat.html accessed 26 Feb 2012 106 Perkins, F. Leibniz and China: a commerce of light Cambridge University Press 2004, p.115. Also see pp.113-
115. 107 See Mercer and Antognazza’s detailed diachronic studies of Leibniz’s development. Mercer focusses on
Leibniz’s metaphysics while Antognazza traces Leibniz’s overall intellectual development including his
professional responsibilities as a diplomat. Mercer, C. Leibniz’s Metaphysics: Its Origins and Development
CUP 2001 and Antognazza, M.R. Leibniz: An Intellectual Biography CUP 2008 108 Koyré, A. Newtonian Studies Chapman & Hall, London 1965, p.53
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Leibniz’s relationship to modernity
Popularly, modernity encompasses a lessening of importance of forms of tradition, and
increasing importance of new abstractions such as nation states and corporations.109 The
popularly promoted idea of modernity encompasses capitalism, industrialisation, the nation
state, scientific experiment, secularization and rationalization.110 The Thirty Years War which
ended two years after Leibniz was born is regarded as being at the beginning of modernity.111
This might be because it ushered in the Westphalian system of nation states with the 1648
Treaty of Westphalia.
Modernity is also regarded as encompassing rationality, an absence of excessive religiosity,
the use of logical deduction and reliance on precision in taking observations especially with
sensitive instrumentation. Leibniz promoted all of these things. He also promoted industry,
science, the nation state and rationality. At the same time, he recognised the Neoplatonic roots
of science. However, he was no alchemist or magician. Leibniz was motivated by the power
of rationality when applied to pragmatic scientific and industrial operations. He saw even
greater power in the application of lucid rationality to metaphysics and philosophy, because it
is in these intellectual acts that the largest questions that trouble humans can be dealt with
thereby allowing great leaps in science, thence in industry and thence in the human condition.
The rise of modernity saw a decline in the sway of religious fundamentalism and this went
hand-in-hand with the rise in the use of and high regard for rational thought. This does not
mean that people believed in God any less. Rather, what changed was the conception of God.
The rising conception of God was of a rational God whose every act is for good reason.
Leibniz tirelessly promoted the idea that the universe is the best possible universe, because it
was made by a Creator who designed it so that it would unfold in the best possible way.
Laws that function without divine intervention
Leibniz believed that God acted through objective laws to create the universe, and these
objective laws can be understood whereby the universe is accessible to human understanding
and modification. In criticism of the philosophers Scaliger, Sennert and Sperling, Leibniz
wrote to his former teacher Thomasius, “…they conclude that God produces creatures rather
from his own active power than from the objective and, so to speak, passive power of nothing.
In their opinion, therefore, God produces things out of Himself and is thus the primary matter
of things. But you will judge more correctly on this subject.”112
109 Hooker, R. “World Civilizations” http://www.wsu.edu/~dee/GLOSSARY/MODERN.HTM Washington State
University 1996, accessed on 7 August 2010 110 Barker, C. Cultural Studies: Theory and Practice SAGE Publications, London 2008 pp.188ff and pp.211-212 111 University of Sydney Library http://www.library.usyd.edu.au/libraries/rare/modernity/ accessed on 7 August
2010 112 Loemker, p.94 first paragraph (Letter to Thomasius, April 20/30, 1669). In this, Leibniz at least approximates
Hermes Trismegistus, “…god [sic] is father of the cosmos, but the cosmos is the father of the things in the
cosmos; the cosmos is the son of god, and the things in the cosmos are made by the cosmos.” Corpus
Hermeticum XI §8 in Copenhaver, B.P. Hermetica Cambridge University Press 1992, p.29
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In his “Discourse on metaphysics” of 1686, Leibniz writes, “when we say that things are not
good by any rule of excellence but solely by the will of God, we unknowingly destroy, I think,
all the love of God and all his glory. For why praise him for what he has done if he would be
equally praiseworthy in doing exactly the opposite?” Rather, “the eternal truths of
metaphysics and geometry, and consequently also the rules of goodness, justice, and
perfection” are the consequences of God’s understanding rather than of his will, and God’s
understanding does not depend upon his will.113 Essentially, geometry and harmony are
beautiful in themselves not because God has chosen them and, moreover, God chose them as
they are because of his perfect understanding.
Hylarchic principle
Leibniz’s opposition to Henry More’s hylarchic animism includes, among other arguments,
that the idea is superfluous,114 is not distinctly conceived and leaves many things to be
explained unlike the commonly believed notion of individual souls.115 Leibniz says that space
is not a real absolute being and is certainly not God himself. Moreover, since space has parts,
it does not belong to God. Leibniz then says that space is merely relative as time is, and that it
is merely an order of coexistences as time is an order of successions.116
Newton expresses a kind of hylarchic animism similar to Henry More’s. By contrast, Leibniz
has said that God is not in the universe and the universe exists independently of God. Newton,
on the other hand, writes that God is omnipresent, not only virtually but substantially. Newton
says that God constitutes space and time. Leibniz says that space and time do not even exist,
but are creations of the human mind.117 Of God, Newton writes, “He endures for ever, and is
everywhere present; and by existing always and everywhere, he constitutes duration and
space. … He is omnipresent, not virtually only, but also substantially, for virtue cannot subsist
without substance. In him are all things contained and moved; yet neither affects the other:
God suffers nothing from the motion of bodies; bodies find no resistance from the
omnipresence of God.”118 Of course, Newton is self-contradictory because he also posits a
mechanical clockwork universe,119 and Leibniz disagrees with this side of Newton too:120
113 Loemker, p.304 114 Leibniz, G.W. 1698 in Wiener, pp.138-9 115 Leibniz, G.W. “Reflections on the doctrine of a single universal spirit” 1702 in Loemker, , pp.554-560 116 Wiener, p.223 §3; also see Leibniz’s response to Dr Clarke’s second letter, pp.222-228 117 Incidentally, the concept that space is infinite is core to Henry More’s conflating God with space. More is
quoted saying this many times in Koyré, A. From the Closed World to the Infinite Universe Harper and
Brothers, New York 1957. See Chapter 6 “God and Space, Spirit and Matter” pp.125-154. By contrast,
Leibniz does not think that the universe is infinite. 118 General Scholium in the 1713 (second) edition of Principia II, 311ff quoted in Burtt, E.A. 1932 2nd ed,
reprinted 1950, p.257. Burtt suggests that Newton added the General Scholium to counter the disapprobation
from the Godlessness of the pure positivism and empiricism of the first edition of the Principia. Was the
General Scholium a populist sop to the churchgoing masses? The General Scholium appears to be
irreconcilable with the positivism and empiricism of the Principia. In any case, even considering each
outlook on its own, neither is supportable. 119 Burtt, E.A. The Metaphysical Foundations of Modern Physical Science Routledge London 1932 2nd ed.
reprinted 1950, p.256 120 Leibniz, G.W. Letter to Samuel Clark November 1715 §4 Wiener, p.216
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Sir Isaac Newton, and his followers, have also a very odd opinion concerning the work
of God. According to their doctrine, God Almighty needs to wind up his watch from
time to time: otherwise it would cease to move. He had not, it seems, sufficient
foresight to make it a perpetual motion. Nay, the machine of God’s making is so
imperfect, according to these gentlemen, that he is obliged to clean it now and then by
an extraordinary concourse, and even to mend it, as a clockmaker mends his work…
The idea that God acts through objective laws is fundamental to modernity, and a defining
feature of it. The objective action of God’s mind through universal principles that are
discoverable by human minds makes the universe the legitimate and natural locus of proactive
intervention, change and improvement by humans. This marks the onset of modernity. Of
course machines (modes of mechanised work and transport) and increasingly powerful energy
sources are media for such human proactive action.
Modernity and discoveries achieved by reason
Modernity often refers to lifestyle or amenities of modern life being made available to as wide
a section of the population as possible. It can also mean a departure from superstition and
religious fundamentalism, and an embrace of rationality. This implies the ability to answer
questions using a thought process rather than dogma, which is a rather Platonic-dialogue
oriented technique. It is a rational process whose pursuit is open to any human mind, and that
process can lead if not to a final answer then at least to greater clarity and an incomplete
answer. Leibniz took this further with his quest for a Universal Characteristic which would
allow questions to be answered using a calculating procedure. This, however, was distinct
from the “art of discovery”.121 The Universal Characteristic would make a process of
reasoning available to all people who had a basic education and were armed with pen and
paper.122 It would be a calculus which depends on the analysis of ideas, and which is more
important than the calculi of arithmetic and geometry. He says, “its formation seems to me
one of the most important things that can be undertaken.”123,124
To a 21st Century mind, the idea that the earth revolves around the Sun rather than vice-versa
seems a “modern” one. However, what is perhaps the feature that more powerfully gives it the
distinction of “modern-ness” is the fact that a human mind discovered it through reasoning
and by considering the paradoxes in the previously prevailing belief system. Praise for this
goes to Kepler. However, one of the most salient features of Kepler’s reasoning is hypothesis.
121 Leibniz, G.W. c.1693 in Wiener, , pp. 77-80 122 Leibniz, G.W. 1677, Ibid., pp.23-24 123 Leibniz’s final words in “Response to Bayle’s Dictionary article Rorarius” 1702, Loemker, p.585 124 Richard Brown notes that Leibniz did not seem to have made much real progress towards the Universal
Characteristic. As regards stating more than has actually been delivered, “this is especially the case in his
writings on the Universal Characteristic or Lingua Philosophica. It is true that this project motivated his
unique approach to calculus and logic, which in itself is a singular achievement. Yet as far as the general
project is concerned little had been accomplished except the production of a vast amount of propaganda.
Over and over again he describes the wonderful things he expects from the Characteristic, while strongly
hinting that it is already is in his possession.” Brown, R. C. Leibniz, unpublished, Chapter 11 “Epilogue”
p.167 Judgement may be reserved, however, for there may be much more to find in the vast trove of Leibniz
manuscripts.
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Of course, Kepler was an able mathematician. Yet, fundamental to Kepler’s process of
reasoning was a presumption of harmony, the formulation of a hypothesis, and then the
undertaking to test the hypothesis and check its complete conformance with observations.
Kepler and Galileo’s outlook typified the “new mechanical philosophy”125 (“NMP”) which
was part of the development of Rationalism which is to be discussed below in Chapter 3.
Brown reports that Erhard Weigel who taught some mathematics to Leibniz at the University
of Jena shared with Leibniz a desire to reconcile Christianity, Aristotle and the NMP. While
Weigel regarded the NMP of Kepler, Galileo and such like as an extension of Aristotle,126 we
will later argue that Rationalism is better regarded as an evolution out of Neoplatonism rather
than as an extension of Aristotle.
Kepler and Galileo’s outlook typified the “new mechanical philosophy”127 which developed
hand-in-hand with Rationalism, which will be discussed further in Chapter 3. Brown reports
that Erhard Weigel who taught Leibniz mathematics at the University of Jena shared with
Leibniz a desire to reconcile Christianity, Aristotle and the new mechanical philosophy.
Weigel might have seen the new mechanical philosophy of Kepler, Galileo and such like as an
extension of Aristotle. However, we will later argue that Rationalism was for the most part an
evolution of ideas from Neoplatonism.
Hypothesizing a priori
What is critical is the pre-thought that went into his hypothesis formulation; clearly, Kepler
did not pick a hypothesis “out of the air”. Kepler had metaphysical presumptions in how the
universe must be designed, and in what the Creator’s intention must have been in designing
the universe arising from his overall understanding of God, geometry, Man and Man’s place
in the universe. An interconnection between God and geometry was present in Kepler’s
thoughts, and to that he owed a debt to the Pythagoreans and Plato. What did Kepler add to
the traditions of Pythagoras and Plato? Why could Pythagoras and Plato not have completed
Kepler’s work? Perhaps they “only” lacked the observational data that Kepler had access to,
and the numerical methods work of Tycho Brahe and Kepler himself. Of course, the data and
methods involve enormous suites of resources. However, arguably, the Pythagoreans and
Plato worked with a similar metaphysical orientation and set of assumptions about the nature
of the universe to Kepler.
Leibniz recommended Kepler’s a priori method, “The most perfect method involves the
discovery of the interior constitution of bodies a priori from a contemplation of God, the
author of things. But this method is a difficult one and not to be undertaken by anyone
whatever.” Further, “Some hypotheses can satisfy so many phenomena, and so easily, that
125 A term used by Richard Brown in Brown, R. C. Leibniz, unpublished, Chapter 4 “A Young Central European
Polymath Between the Scholastics and the Moderns” p.33 126 Ibid. 127 Ibid.
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they can be taken for certain. Among other hypotheses, those are to be chosen which are the
simpler; these are to be presented, in the interim, in place of the true causes.”128
We cannot say that the ancient Greeks were not used to formulating hypotheses. Archimedes
might have done so mentally without committing it to writing. Archytas might have done so
in working out how to double the cube.129 Most of all, Plato in his dialogues makes an art and
game of proposing hypotheses for testing.130
It is clear in Leibniz’s work on the mine pump, simply through his multiple attempts, that he
must have followed a process of hypothesis testing. Similarly with Papin in his letters to
Leibniz, and his several steam engine design proposals to various persons of significance.
Leibniz was familiar with hypothesis formulation in Astronomy, perhaps directly from
reading Kepler.131
A cohesive reason-based metaphysics is associated with the foundations of modernity –
counterposed against, say, a literalist reading of religious scriptures. Such a metaphysics
provided preconceptions about the structure of the universe. The current discussion
surrounding cosmic radiation and the lack of empty space is forcing re-examination of the
foundations of physics.132 Yet Leibniz himself had much to say on the non-emptiness of
space,133 and much of it was derived from the principle of sufficient reason which emanated
from a quasi-theological position on how God necessarily must think.
128 Loemker, p.283 129 While the cube cannot be doubled with straight-edge and compass, Archytas who studied at Plato’s Academy
worked this out a solution in the 4th Century BCE. We quote Rivest, F. and Zafirov, S. “Duplication of the
cube” accessed at http://www.cs.mcgill.ca/~cs507/projects/1998/zafiroff/Duplication of the Cube.htm 27
April 2011. Archytas used “a bold construction in three dimensions, determining a certain point as the
intersection of three surfaces of revolution, (1) a right cone, (2) a cylinder, (3) a tore [sic] or anchor-ring with
inner diameter nil. The intersection of the two later surfaces gives (says Archytas) a certain curve (which is in
fact a curve of double curvature), and the point required is found as the point in which the cone meets this
curve.” (italics in original) 130 It is not confirmed whether Gauss performed the experiment on the angle sum of a large triangle by placing
fires on two distant mountains to check whether the triangle formed by the path travelled by the light had an
angle sum of 180 degrees. Mathematics Illuminated Geometries Beyond Euclid published by Annenberg
Learner §8.4 Spherical and Hyperbolic Geometry accessed at
http://www.learner.org/courses/mathilluminated/units/8/textbook/04.php 4 June 2011 131 1686, Wiener p.296 132 Shaviv, N. J. “Cosmic Ray Diffusion from the Galactic Spiral Arms, Iron Meteorites, and a Possible Climatic
Connection” Physical Review Letters 2002 Vol. 89, Iss. 5. The work of Dayton Miller casts doubt on the
negative result of the Michelson-Morley ether experiments. For an overview, see Swenson, L. S. “The
Michelson-Morley-Miller Experiments before and after 1905” Journal for the History of Astronomy Vol. 1,
p.56 accessed at http://articles.adsabs.harvard.edu//full/1970JHA.....1...56S/0000056.000.html 12 May 2011.
Also see Miller, D. “The Ether-Drift Experiment and the Determination of the Absolute Motion of the Earth”
Reviews of Modern Physics 1933 Vol. 5, Iss. 3, 203-242 in which Miller calculates an absolute cosmic motion
of the earth and of the solar system using the results of the Morley-Miller experiments 1902-1906 and
subsequent experiment at Mount Wilson 1921-1924. 133 Leibniz, G.W., White, C. (trans.) An Essay on the Causes of Celestial Motion Fusion Energy Foundation
1986, esp. pp.6-7
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Broad-based influence of reason on culture
An absence of superstition was seen in many cultures prior to extended European civilisation.
Thus, modernity cannot be defined by a lack of superstition or even by a lack of reason. Even
the modern era’s broad-based facility of reason among the populace is not definitive because
there have been entire civilisations prior to the present in which a large proportion of the
population used methods which today could be described as scientific or at least with
ingenuity, with regard to astrogation, navigation, ship-building or boat-building, fishing,
domestication of animals, etc. Thus, modernity is simply this civilisation’s manifestation of
reason/rationality. There are parts of the world in which superstition can be found even today.
However, there were, to varying degrees, parts of the world – national cultures, say – certainly
through the 19th and 20
th Centuries in which reason was dominant. Of course, on the other
hand, irrationality even exists in the Western world in the 21st Century even in professional
and social circles with a high-level of formal education.
Leibniz had something to do with the establishment of a dynamic which saw the broad-based
influence of reason on culture and the infusing of post-17th Century culture with reason.
Leibniz’s letters to national leaders, such as the Czar of Russia, indicate a concern to uplift the
minds of the population as a whole. Leibniz’s personal drive in this direction is illustrated by
Kutateladze when he describes the debt owed by Russian science to Leibniz:134
Science in Russia had started with the foundation of the Academy of Sciences and Arts
which then evolved into the Russian Academy of Sciences of these days. The turn of
the sixteenth and seventeenth centuries is a signpost of the history of the mankind, the
onset of the organized science. The time of the birth of scientific societies and
academies accompanied the revolution in the natural sciences which rested upon the
discovery of differential and integral calculus. The new language of mathematics
brought about an opportunity to make impeccably precise predictions of future events.
To the patriotism of Peter the Great and the cosmopolitanism of Leibniz we owe the
foundation of the Saint Petersburg Academy of Sciences as the center of Russian
science. Peter and Leibniz stood at the cradle of Russian science in much the same
way as Catherine I and Euler are the persons from whom we count the history of the
national mathematical school in Russia. We must also acclaim the outstanding role of
Leibniz who prepared for Peter a detailed plan of organizing academies in Russia.
Leibniz viewed Russia as a bridge for connecting Europe with China whose
Confucianism would inoculate some necessary ethical principles for bringing moral
health to Europe. Peter wanted to see Leibniz as an active organizer of the Saint
Petersburg Academy, he persuaded Leibniz in person and appointed Leibniz a
Justizrat with a lavish salary.
This is an early conception of the General Welfare of the population of a nation which was
also found in the American Declaration of Independence and Constitution. Leibniz’s New
Essays Concerning Human Understanding in response to John Locke’s Essays on Human
Understanding indicates this contention of “life, liberty and the pursuit of happiness” versus
134 Kutateladze, S. “The Mathematical Background of Lomonosov's Contribution” Journal of Applied Industrial
Mathematics, 2011, V. 5, No. 2, 155–162. Accessed at
http://www.math.nsc.ru/LBRT/g2/english/ssk/mvl_e.html 4 Dec 2011
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“life, liberty and property”.135,136 (In the next chapter, we shall meet Paolo Sarpi whose
influence Leibniz opposed. Robertson writes that “there can be no doubt” that Sarpi
“anticipated Locke in the sphere of metaphysics” and that “there is reason for thinking” that
Sarpi “supplied Locke with the germs of many of the ideas which we find expanded” in
Locke’s writings.137) In this way, Leibniz’s thought was instrumental in the creation of
national cultures which fostered the pursuit of knowledge, invention and an expanding
scientific understanding by the participation in these activities of as many members of the
population as possible. Leibniz’s idea was not only that society and nations should be
organised to promote these things but that these pursuits were the main and perhaps only
reason for the existence of society and nations, and are the end around which all political
activity and legal principles for organising people should be directed.
Conflicting ideas on the nature of Reason
Let us explore the idea that Reason and its application are at the heart of modernity. Before
doing so, a digression on what is meant by “reason” is needed. In particular, reason is
counterposed to deduction, and more will be said on the distinction in Chapter 6 “Discovery
and deduction”.
Reason as Plato exhibited it in The Republic is not logical deduction for the character Socrates
raises new points from “left field” that could not have been deduced and thereby commence
entirely new lines of enquiry and, often, confounds his interlocutors.
135 In John Locke’s Essays on Human Understanding, §18 Chapter III Book IV Volume 2, Locke exhibits a
purist positivist stance by writing, “‘Where there is no property there is no injustice,’ is a proposition as
certain as any demonstration in Euclid: for the idea of property being a right to anything, and the idea to
which the name ‘injustice’ is given being the invasion or violation of that right.” That is, without property
rights, it is not possible to do anyone any harm, because all concepts of justice arise from property. This is
opposed to confluence of Justice with Goodness in Plato’s The Republic. If we are in doubt as to Locke’s
positivist position, Locke then writes, “Again: ‘No government allows absolute liberty.’ The idea of
government being the establishment of society upon certain rules or laws which require conformity to them;
and the idea of absolute liberty being for any one to do whatever he pleases; I am as capable of being certain
of the truth of this proposition as of any in the mathematics. saying that there where is no property there can
be no injustice, just as where there is no government there is absolute liberty.” This indicates that liberty is
not the freedom to do what is good or right, but “freedom” to do what one pleases or liberty in the sense of
libertinism. The very title of Book I Volume 1 “Neither principles nor ideas are innate” is a bold positivist
statement with which any concept of happiness distinct from pleasure would find it difficult to coexist.
Project Gutenberg edition Volume 1 at http://www.gutenberg.org/cache/epub/10616/pg10616.txt and Volume
2 at http://www.gutenberg.org/cache/epub/10615/pg10615.txt accessed 7 May 2011. 136 Contrarily, Wikipedia credits John Locke with the “pursuit of happiness” part of the Declaration of
Independence. While Wikipedia may not be relevant as an academic or research source, it is a common
popular source, and are taking this opportunity to correct in a research context what is being promoted among
the public at large. See http://en.wikipedia.org/wiki/Life,_liberty_and_the_pursuit_of_happiness accessed 7
May 2011. 137 Robertson, A. Fra Paolo Sarpi: The Greatest of the Venetians George Allen & Company, London 1911,
pp.84-85
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Nicolaus of Cusa
We first introduce Cardinal Nicolaus of Cusa (1401-1464) because he revolutionized the
understanding of human reason, the pursuit of knowledge by humans, and helped define the
domain of what is knowable or in which humans seek knowledge.
Cusa was educated in a region of Germany where the Brotherhood of the Common Life was
active though possibly not in a school directly run by the Brotherhood. An overview of the
Brotherhood and its circles will help place the reader in the milieu in which Cusa worked.
The Brotherhood was a religious order founded by Gerard Groote (1340-1384). In 1374,
Groote, at the age of 34 having hitherto lived in relative luxury, experienced religious
conversion following a sickness, and entered the Carthusian monastery as a guest to
participate in their severe regimen of prayer, fasting and manual labour. Ultimately he
departed from the Carthusians but thenceforth lived an austere life of prayer, study and
preaching. Groote was from a wealthy merchant family. After his parents died as victims of
the Black Plague, he left one of the estates he inherited to the Carthusians. He ceded part of
the house in which he lived to poor women so establishing a community known as the Sisters
of the Common Life. The sisters supported themselves through agricultural and artisan
pursuits, and gained great expertise in agriculture and sewing. They built a flourishing dairy
business and earned an impressive income from sewing and knitting. Discipline and
obedience to the two matrons was expected. The Sisters became a powerful centre of Church
reform.138
Groote also established a monastery for the Brotherhood of the Common Life, which
provided superior education to capable children regardless of family background. The
Brotherhood was dedicated to educating capable children in the Greek classic and in the
Christian tradition of St Augustine. The Brotherhood ultimately established over a hundred
schools across Europe and influenced many more. The order sought to provide education
regardless of background and sought to transform the potential of the population of Europe in
the wake of the Black Plague.
Erasmus of Rotterdam also had an education heavily influenced by the Brotherhood. Erasmus
along with others created the Devotio Moderna movement which catalysed a revolution in the
Church. This “New Devotion” or “Modern Piety” ultimately swept up such personalities as
Luther, Calvin and Loyola.139 While Nicolaus of Cusa was only peripherally connected with
the Devotio Moderna,140 he played an equally important role in history.
138 Jalas, J. (ed.) Gilbert, W. Renaissance and Reformation Lawrence Books, Kansas 1997: Carrie E-Books,
1998, Chapter 9: The Northern Renaissance and the background of the Reformation accessed at
http://vlib.iue.it/carrie/texts/carrie_books/gilbert/09.html 24 Dec 2011 139 Broekhuysen, A. Gerard Groote and the Brethren of the Common Life Wisdom’s Goldenrod Center for
Philosophic Studies, Hector New York accessed at
http://wisdomsgoldenrod.org/publications/misc/gerard_groote.html 24 Dec 2011 140 Bocken, I. “The Language of the Layman: The Meaning of Imitatio Christi for a Theory of Spirituality”
Studies in Spirituality vol. 15, 2005, pp.217-249
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In 1418 the Brotherhood was charged with heresy. Jean Gerson the former chancellor of the
University of Paris defended the Brotherhood, making appeal to the Christian concept of
imago viva Dei and the duty of Christians to act in the imitation of Christ. Gerson’s further
significance is underlined by the fact that he later wrote the educational curriculum for the
young Louis, the future King Louis XI of France, with emphasis on the study of St
Augustine’s City of God. Thomas á Kempis (1380-1471) was a follower of Groote, and
expressed the ideals of the Brethren in The Imitation of Christ (c.1418-1427). Nicolaus of
Cusa’s theology has been called that of Thomas á Kempis but in philosophical language.141
The Brotherhood and its supporters were significant in the movement that supported the
independence of France. It might not be an accident that Jeanne d’Arc who was so central in
liberating France from England grew up in Domremy adjacent to German towns where the
Brotherhood was active.
Nicolaus of Cusa, usually referred to as “Cusa”, was a very significant reformer and activist
from his position within the church hierarchy. Cusa wrote the Concordata Catholica in 1433
during the Council of Basel which ultimately failed. In the Concordata, the concepts of
human rights and national sovereignty can be found in seminal form. More than 200 years
later, the concept of national sovereignty became the basis of the 1648 Treaty of Westphalia.
What is now known as the “Westphalian principle of national sovereignty” is the foundation
for international relations in the 21st Century.
Cusa was active in initiating the Council of Florence which brought together the Roman and
Eastern Orthodox Churches, and concluded in 1439. It was due to Cusa that the doctrine of
the Filioque (i.e. “and from the Son”) was agreed at the Council of Florence. This is the
doctrine that the Holy Spirit proceeds from the Father and from the Son, not only from the
Father. This ensures a greater accessibility of the Holy Spirit to humanity.
While God an infinite being created the universe, humans have access to ideas concerning the
infinite. Humans can get closer to God, and the human mind can always get a better grasp of
Creation. This is the same as the consequence of the doctrine of the Filioque, that humans are
able to come closer to God. Indeed, the Holy Spirit proceeds from the Son to reach us. The
spiritual relevance might be more obvious. However, in the intellectual and scientific sense,
the human mind can progressively come to better understand the universe, and this is nothing
but the process of scientific discovery. Cusa explained the competence of human thought for
scientific understanding even of God’s creation as a whole though his work De Docta
Ignorantia which he defended in correspondence with Johannes Wenck against attacks by
Wenck.
Most significant for this thesis, Cusa conceived the role and position of mind in relation to
Creation or the physical universe, so establishing an epistemological foundation for science.
The premise of De Docta Ignorantia is the relationship of the finite with the infinite. The
human mind is finite compared with the infinity of God. The physical universe, which is
God’s creation, is also infinite. This might not mean that the physical universe is infinite in 141 Scharpff, F.A. Der Kardinal und Bischof Nikolaus von Cusa als Reformator in Kirche, Reich und Philosophie
des fünfzehnten Jahrhunderts (“The Cardinal and Bishop Nicolaus of Cusa as Reformer in the Church,
Empire and Philosophy of the Fifteenth Century”), Tubingen 1871
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size, but in possibilities. Thus, the human mind can never fully understand the universe nor
can it fully grasp God. However, this does not mean that there is no point in trying. On the
contrary, there is every point in trying. Since a circle is like an infinite-sided polygon, a
polygon with a finite number of sides can never match it but it can get progressively closer.142
This is an analogy only, but one that is as useful as it is simple to fight back against those who
argue that humanity is condemned or destined to be passive or ineffective in a universe which
is governed by the diktat of a God who can never be understand. It is a powerful argument
against regarding God and the forces of nature as playing with human in an arbitrary or
whimsical way.
In explaining the human ability to formulate a “higher hypothesis”, the analogy of the circle is
used again. By remaining within the framework of existing thinking, we behave like a
geometer who restricts himself to a regular polygon. How can he achieve the smooth circle?
The circle is not an infinite-sided polygon but a qualitatively different object. A higher power,
so to speak. Humans have the ability to transcend the existing “system of regular polygons”
and move up to the circle thus solving the conundrum of the day, be it political, artistic, social,
cultural or – what is most relevant to this thesis – scientific.
How does one find or move to that higher hypothesis? The ability to do so is fundamental to
progress of all kinds. Thus, renaissances are typically renaissances of human thought, and
progress explodes not in one but in all domains: political, artistic, social, cultural, scientific.
Indeed, the separation between these domains is artificial, for all are refined and advanced by
the capabilities of the human mind.
Cusa might have been the first to say that inspiration is a valid source of scientific insight and,
ultimately, of new knowledge. Indeed, Cusa argued that it is the only source of new
knowledge, and he counterposed it to discursive logic. Translations of Cusa, such as Jasper
Hopkins’ translation, use “discursive reasoning” but the context makes clear that “discursive
logic” or even “deductive logic” was his meaning.143
Kepler consciously used Cusa’s prescription in formulating his hypotheses for the working of
the solar system yet we do not see inspiration in Kepler so much as an a priorist belief in
universal harmony combined with impressive rigor or discipline in his calculations.144 The
testing of the hypotheses was conducted largely mathematically. Leibniz’s method in
discovering the calculus was very different, but he was equally supportive of the use of
hypothesis. Meli defends Cassirer’s thesis of 1902, that “the legitimacy of hypotheses in
natural philosophy and mathematics was defended by Leibniz exactly as Kepler had done in
142 Grosholtz refers to “Leibniz’s infinite-sided polygon” several times Gillies (ed.) “Was Leibniz a
Mathematical Revolutionary?” pp.125, 126, 133 However, Cusa deserves credit for the concept. 143 Available online at http://jasper-hopkins.info/ Accessed 1 Jan 2010 144 Kepler wrote of Cusa others divinus mihi Cusanus, i.e. “Cusa and others seem to me divine” in drawing the
analogy of the circle compared with polygon to God compared with his creatures. Duncan, A.M. (trans.)
Kepler, J. Mysterium Cosmographicum Abaris Books, Janus Series Norwalk USA 1981, Chapter II “Outlines
of the primary derivation” p.93
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astronomy. In their philosophical systems phenomena assume a new dignity and the true
hypothesis becomes the instrument for binding them to the laws of knowledge.”145
Cusa was the first to say that the universe is infinite.146 Bruno who was born 84 years after
Cusa died also proclaimed the infinity of the universe. Fifty years after Bruno’s death, Henry
More was gripped by the idea of the infinity of the universe, not necessarily adopted from
Cusa or Bruno directly, and in particular the infinity of space. As mentioned elsewhere in this
thesis, the infinity of space inexorably led Henry More to the idea that God is in space.
Newton was in partial agreement with these conceptions of More.
Cusa’s ideas on mind gain traction in science
Kepler consciously used Cusa’s prescription in formulating his hypotheses for the working of
the solar system yet we do not see inspiration in Kepler so much as an a priorist belief in
universal harmony combined with impressive rigor in his calculations. The testing of the
hypotheses was conducted largely by calculation. Leibniz’s method in discovering the
calculus was very different.
The gaining of understanding and following it up with the potent exercise of mind on the
physical universe brings us into the domain of modernity. This certainly was not something
that had never been thought before. For example, Plotinus (3rd Century C.E.) wrote that since
Intelligence is the base of all, understanding is a prerequisite to the exercise of noös (spirit) on
söma (body).147 The gaining of understanding and following it up with the potent exercise of
noös on söma brings us into the domain of modernity. Central to modernity was insight into
and consequent control over the physical universe. Kepler’s purpose was to better understand
Creation. For a human to even consider that it was for them to understand the universe could
have been regarded as blasphemous arrogance that put him/her on the same footing as God.
Kepler’s theological justification, perhaps to keep himself out of trouble with church
authorities, was that by better understanding God’s creation we can better glorify God.148
Similarly, Leibniz wrote that “the greatest usefulness of theoretical natural science, which
deals with the causes and purposes of things, is for the perfection of the mind and the worship
of God.”149 What Kepler did not say was that scientific development contributes to the public
good, perhaps because Kepler’s work in understanding the heavens did not have a clear
pragmatic application in his time. Leibniz could see that science and engineering contribute to
the public good, and he united Kepler’s position with the public good, saying, “To contribute
to the public good and to the glory of God is the same thing.”150
145 Meli, pp.19-21 referring to Cassirer, E. Leibniz’ System in seinin wissenschaftlichen Grundlagen Marburg
1902, pp.362-3 and 503, and other places 146 Koyré, A. From the Closed World to the Infinite Universe Harper and Brothers, New York 1957 p.6 147 Mead, G.R.S. Plotinus Theosophical Society London 1895, pp.24, 26, 28 148 Duncan, A.M. (trans.) Kepler, J. Mysterium Cosmographicum Abaris Books, Janus Series Norwalk USA
1981, p.53 149 Leibniz, G.W. c.1682-4, Loemker, p.280 150 Draft of a letter by Leibniz to Thomas Burnett 1699, quoted in Antognazza, M.R. Leibniz: An intellectual
biography Cambridge University Press 2009, p.233
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Understanding principles of nature precedes surpassing or controlling nature, and the
motivation for gaining that understanding is to surpass and control nature. As mentioned in
the previous chapter, it allows us to ape nature and thence master it. That it is natural for
humankind to do so was also a consequence of the Egyptian/Hermetic conception of Man as
magus who can control even the stars. That it is humankind’s right to do so was the message
of the Renaissance bound up with Christian theology, resulting from the conception that
humanity is made in the image of the Creator.151,152 Leibniz explained, “It seems to me that
the aim of all humankind should chiefly be nothing other than the knowledge and
development of the wonders of God and that it is for this reason that God has given
humankind dominion over this globe.”153 Humanity consciously using its superior status over
all of nature is the defining characteristic of modernity and crystallises what the Renaissance
worked towards.
Speculation and contemplation, including the exercise of reason, do not on their own define
modernity. Whittaker says “the happiness involved in [the speculative life] ... is regarded as
something that necessarily goes with mere thinking and understanding” in reference to “a
self-conscious theory of [the speculative life as set forth] ... as at the opening of Aristotle’s
Metaphysics”. However, there is no mere thinking or understanding, for action does not stop
there.154 Understanding is a prerequisite to:155
(a) Organise men justly and accordingly arranging government in the best way possible
which is no small thing because it can mean the difference between violent tyranny,
for example, and enlightened republic, and
(b) Build machines, ships and cities, undertake agriculture, and modify the physical
universe in significant ways and even on a vast scale.
There is a “difference between nature and art, that is to say, between the divine art and ours”.
Machines built by God or nature are machines in their smallest parts ad infinitum. By
contrast, the constituents of, say, a wheel give no indication of the use for which the wheel is
151 “Man is made to be in the visible universe an image and likeness of God himself (Cf. Genesis 1:26.), and he
is placed in it in order to subdue the earth (Cf. Genesis 1:26).” (references in original) Opening statement of
Laborem Exercens Papal Encyclical of Pope John Paul II, 14 Sep 1981 accessed at
http://www.vatican.va/holy_father/john_paul_ii/encyclicals/documents/hf_jp-ii_enc_14091981_laborem-
exercens_en.html 8 May 2011 152 Cassirer, E., Kristeller, P. O., Randall, J. H. Jr (eds and trans.) The Renaissance Philosophy of Man Phoenix
Books, University of Chicago Press 1948; for example, see Mirandola pp.223-5 and Pompanazzi pp. 282-3. 153 Ibid. 154 See also Cusa Idiota de Mente, p.63 on the precise role of understanding in human thought vis-à-vis God’s
thought. Also see the footnote in Merlan, P. From Platonism to Neoplatonism Martinus Nijhoff, The Hague
1968 at p.5, “Understanding (knowledge) is not the only form of significant mental activity of man. We may
enjoy something esthetically [sic]; we may be in empathy with an animal or our fellow-man; any mood is
some kind of mental engagement. But none of these activities is of the order of understanding – they are
attitudes, reactions, modes of being.” 155 “Through work man must earn his daily bread (Cf. Ps 127(128):2; cf. also Gen 3:17-19; Prov. 10:22; Ex 1:8-
14; Jer 22:13) and contribute to the continual advance of science and technology and, above all, to elevating
unceasingly the cultural and moral level of the society within which he lives in community with those who
belong to the same family.” (references in original) Opening statement of Laborem Exercens Papal Encyclical
of Pope John Paul II, 14 Sep 1981 accessed at
http://www.vatican.va/holy_father/john_paul_ii/encyclicals/documents/hf_jp-ii_enc_14091981_laborem-
exercens_en.html 8 May 2011
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intended and indeed are not machines – and certainly are not alive – in any real sense.156
Modernity has seen increasing – and in some cases exponentially increasing – power of
“human art”. Leibniz assisted this process with his concept of vis viva.
Vis viva or effect-producing force
Normally, vis viva belongs in a discussion on the history of physics. It is known in the context
of the debate between Leibniz and the Cartesians on the kinetic energy formula mv2 versus the
Cartesian momentum mv (i.e. mass times speed, or m|v|) which was corrected to mv (mass
times velocity) in 1668 by the non-Cartesians John Wallis, Christopher Wren and Christian
Huygens.157 Vis viva which literally means “living force” is today regarded as kinetic energy
quantified by mv2. It will be seen below that vis viva meant more Leibniz than this formula.
The concept certainly was the subject of debate, and Leibniz fired the opening salvo by
attacking the Cartesian concept of mv in his 1686 essay entitled “Brief Demonstration of a
Notable Error of Descartes”.158 The discussion fits in this chapter, “Ushering in modernity”,
because for Leibniz the effect that a projectile or explosion could produce was what mattered
and that, in turn, was more a result of velocity than of mass though mass plays a role.
Leibniz is not only concerned velocity but also with acceleration and the effect produced by
sudden deceleration. Leibniz writes, “by effect I mean here not any effect whatever but that
for which force is expended or consumed and which may therefore be called violent.”159
(emphasis in original) Leibniz then explains what kind of effect he does not mean and which
machines are, by comparison, “harmless”. Leibniz says, “the ancients had a knowledge of
dead force only, and it is this which is commonly called mechanics, which deals with the
lever, the pulley, the inclined plane (…), the equilibrium of liquids, and similar matters
concerned only with the primary conatus of bodies in itself, before they take on an impetus
through action. Although the laws of dead force can be carried over, in a certain way, to living
force, yet great caution is necessary, for it is at this point that those who confused force in
general with the quantity resulting from the product of mass by velocity were misled because
they saw that dead force is proportional to these factors.”160 According to Meli’s
interpretation, “for Leibniz living force could be represented as the integral of dead force
times an infinitestimal distance”.161
Leibniz agrees that mv is a valid concept with many uses and is only explaining that mv2 is
something qualitatively different. To compare the two, Leibniz writes, “The force which a
heavy body exercises in moving along a perfectly horizontal plane is not of this kind [living
force], because however far such an effect is prolonged, it always retains the same force, and
156 Latta, pp.254-5, §§64-5 157 Iltis, C. “Leibniz and the Vis viva Controversy” Isis Spring 1971, Vol. 62, No.1, pp.21-35 at p.22 158 Hankins, T.L. “Eighteenth-Century Attempts to Resolve the Vis viva Controversy” Isis 1965, Vol. 56, No.185,
pp.281-297 at p.281 159 Leibniz, G.W. Specimen Dynamicum 1695 in Loemker, p.442 160 Ibid., p.439 161 Unfortunately, a translation of De Motu Gravium was not able to be obtained for this thesis, so we are reliant
on Meli’s reading. Meli’s quote is in reference to Leibniz’s essay De Motu Gravium lines 95-101. Meli, D.B.
Equivalence and Priority: Newton versus Leibniz Clarendon Press, Oxford 1993, p.154
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though we use the same principle in calculating this effect also, which we call harmless, we
now exclude it from consideration.”162 (emphasis in original) Leibniz then goes on to discuss
how the “force”, or kinetic energy as we would call it today, of a tiny projectile is absorbed by
and affects even the largest of bodies.
The debate between conservation of momentum or mv and conservation of kinetic energy or
mv2 was an important aspect of the vis viva discussion. However, arguably, to Leibniz’s mind
the more powerful “take away” was the power of explosive force versus the relative
“harmlessness” of “dead force” exemplified by the uniform motion of a heavy body along a
perfectly horizontal plane. The question then becomes – how do we generate and make us of
explosive force? This will lead us into the below discussion of steam power.
First, however, understanding the principles of “effect” is prerequisite to machine-building or
mechanical engineering, since a machine harnesses that effect in a deliberate way. In turn,
modernity embraced the task of consciously harnessing the laws of physics through machines
or otherwise to do useful work in the service of human ends. This is in the same way that the
calculus is the use of symbols as tools of thought in the service of human ends.
Leibniz’s concern with the ability to do useful work is shown where he argues that the
difference between mv and mv2 was:,163
not worthless to consider, nor are they quibblings over words, for they are of the
greatest importance in comparing machines and motions. For example, if power is
obtained from water or animals or from some other cause, by which a weight of 100
pounds is kept in constant motion so that within a fourth of a minute it can be made to
complete a circle of 30 feet diameter, but someone else maintains that a weight of 200
pounds can in the same time complete half the circle with less expenditure of power,
his calculation seems to yield a gain; but you ought to know that you are being
deceived and getting only half the power.
While this argument is straightforward today, it pitched into an ongoing discussion among
leading scientists and engineers of Europe. Experiments by Mariotte, Poleni and ’sGravesende
confirmed that the effect of a moving object varies in proportion to the square of velocity in
the early 18th Century. These experiments were performed with balls accelerated to different
speeds being fired into clay or wax, with the depths of the impressions made being measured.
Of these, ’sGravesende took the most active role in the vis viva debate. In 1720 in his book
Mathematical Elements, ’sGravesende had taken up the question of how to measure the ability
of an “action of power” to overcome obstacles.164 This is the Leibnizian concern with the
effect of a force or “power”. In 1722, ’sGravesende proceeded to defend Leibniz’s idea of vis
viva.165 In 1733, Poleni had published an article reporting the results of an experiment in
which balls were dropped onto tallow to compare the depths of the impressions made. His
analysis showed that the “force in motion” was proportional to the square of the velocity. A 162 Ibid., p.442 163 From Part I of Specimen Dynamicum 1695 in Wiener, p.137 164 Hankins, T.L. “Eighteenth-Century Attempts to Resolve the Vis viva Controversy” Isis 1965, Vol. 56, No.185,
pp.281-297 at p.287 165 Ibid.
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Swiss mathematician Calandrin published an anonymous article in the same issue of the same
journal saying that since the resistance of the tallow was constant, equal force should be
consumed in equal times. Feeling compelled to respond to the Calandrin article, ’sGravesende
replied disagreeing that equal amounts of “force” are consumed in equal times. The question
of why the impressions left in clay by a cylinder in motion were not proportional to vis viva
was taken up, as this seemed to contradict the theory. ’sGravesende argued that the early
deceleration split seconds after striking the clay and making the initial impression reduced the
ability of the cylinder to drive deeper into the clay. This was confirmed by experiments with a
series of parallel adjacent taut strings that are struck by a projectile. More strings are broken
per unit time while the object is moving rapidly than when it has slowed down.166
’sGravesende noted that collisions are not instantaneous but proceed in a continuous fashion
causing the gradual deformation and deceleration of the bodies upon and after collision or,
during collision. ’sGravesende explained that, likewise, in a simple machine a small mass
may counterbalance a larger provided it is moving proportionately faster at the instant the
machine starts. According to Hankins, the most aggressive detractors of Leibniz’s concept of
vis viva d’Alembert and Boscovich missed key points and, in retrospect, made little
progress.167
The purpose here is to introduce the vis viva concept and suggest how it fit in Leibniz’s
overall interests and unfolding agenda.
Machines are indispensable to the concept of modernity
Machines increasingly were being invented and used during Leibniz’s lifetime. Essentially,
control of man over nature was on the rise. A machine systematises a physical process like a
Universal Characteristic would systematise a mental or intellectual process, and Leibniz was
interested in both. The two intersected in the calculating machine that Leibniz designed. In a
sense, the calculus is an example of a specific kind of Universal Characteristic because a
mechanical algorithm is availed to answer questions about a huge class of (non-linear)
functions.
Leibniz collaborated with Denis Papin on his invention of the steam engine, providing input
including relating Papin’s work back to Leibniz’s concept of vis viva and also providing
pragmatic mechanical suggestions.
Christian Huygens had mentored Papin168 and Leibniz.169 Papin had worked as Huygens’
ammenuensis (“scribe”) and had built machines under his direction.170 Huygens had invented
166 Ibid. 167 Ibid., pp.283-6, 291ff 168 Klemm writes, “Near the end of the century Huygens’s assistant Denis Papin who, through working on
problems with gunpowder, hit on the steam engine in 1690.” Klemm, F. A History of Western Technology.
History of Science and Technology reprint series Iowa State University Press, Ames (1954), 1991 accessed at
http://www.cmpense.org/worksinprogress/summary/Klemm.html 17 May 2012 169 For example in 1672 it was Huygens who suggested Leibniz summing the series 1/1 + 1/3 + 1/6 + … in
which the denominators are the triangular numbers. Bos, H. J. M. “Newton, Leibniz and the Leibnizian
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a steam pressure cooker powered by gunpowder. Leibniz then worked with Denis Papin to
invent the steam engine. This may have helped Leibniz formulate and crystallise his idea of
vis viva. Though there seemed later to have been a falling out between Papin and Leibniz,
apparently due to resentment on Papin’s side for unknown reasons.171 Leibniz’s participation
represents his involvement in systematising methods for modifying or improving the human
condition by effecting change in the physical world, whether by hauling earth or ore from a
mine shaft, pumping water from a mine shaft, or accelerating a transportation vessel rapidly
over the sea.
The dawn of modernity was characterised by the rise of human commandeering of machine
power, which was a result of contemplation and experiment to uncover physical principles on
the one hand, and deliberate design as well as trial-and-error with machines on the other. The
power of machines would hardly exceed what Archimedes’ machines were able to do without
the vis viva or concentrated, violent or explosive force which Leibniz’s advocacy forces us to
consider as distinct from passive force.172
The association between modernity and the rise of machines is also represented in Leibniz.
Leibniz’s role in designing machines receives little mention in the Leibniz literature, in favour
of his mathematical work, especially the calculus, and his metaphysical thought, especially
monads. Yet, Leibniz himself wrote that if he could spend time on whatever he wished, then
he would design machines full-time.
Papin’s steam engine design was published in the Leipzig Acta Eruditorum, a journal edited
by Leibniz, of August 1690 with the diagram in Figure 1.
Papin was using the idea of explosive force as a continuation of Huygens’ experiments with
gunpowder. Leibniz concept of vis viva was developed after Leibniz was aware of Huygens’
work, and was an effort to give the different kinds of force a theoretical basis. There are
letters from Leibniz to Papin on the vis viva idea.173 The question of priority is not of interest,
only the fact that Leibniz was involved in the earliest experiments with steam power.
tradition” in Grattan-Guinness, I. (ed.) From the Calculus to Set Theory, 1630-1920: An Introductory History
Duckworth: London 1980, p.61 from Leibniz, G. W. Writings (Mathematische Schriften) Gerhardt, C. I. (ed.)
1849-1864 Berlin and Halle, vol. 5, p.405 170 Freudenthal, G. “Perpetuum mobile: the Leibniz-Papin controversy” Studies in the History and Philosophy of
Science Vol. 33 (2002) 573-637, p.599 171 Ibid. 172 According to Klemm, Papin’s contribution to the steam engine which further developed Huygens’ design –
and Leibniz’s collaboration with Papin – was an important marker of the end of the Baroque Period and the
beginning of the Age of Rationalism early in the 18th Century. Klemm, Chapter 3 “The struggle for a new
prime mover” pp.208ff. C.f. Giedion, S. Mechanization takes command: a conttribution to anonymous
history Norton & Co. New York 1969 first published by Oxford University Press 1948. The book covers the
mid-15th Century to the early 20th Century. 173 Referring to August 1690 issue of the Acta Eruditorum which explained his steam machine, Papin wrote to
Count Philipp Ludwig von Sinzendorff in the early 1690s, “As water has the property that it is converted by
fire to steam … and can then be easily condensed again by cold, I thought that it should not be too difficult to
build engines in which, by means of moderate heat and the use of only a little water, that complete vacuum
could be produced which had been sought in vain by the use of gunpowder. … I have ascertained by
experiment that the piston, raised by the heat to the upper part of the cylinder, will immediately return again
to the bottom, and this happens several times in succession so that one might suppose that there was no air at
all exerting pressure from below or offering any obstacle to the descent of the piston. Now my tube, whose
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Figure 1: Denis Papin’s Atmospheric Steam Engine from Acta Eruditorum August 1690
with caption from Klemm, p.221
Leibniz’s mathematical work is usually if not always considered separately from his interest
in mechanics. Leibniz was interested in the problems of mechanics (time, distance and
motion) precisely due to his interest in machines. He had pragmatic ends in mind. In fact, the
demarcation between mathematics and mechanics is artificial and probably was not known to
Leibniz in the way that it is presumed by modern readers.
The idea that Leibniz started with abstract curves and then found these to be helpful in
problems in the domain of mechanics has some truth, 174 but for the most part Leibniz was
thinking about mechanics all the time and from the start. Leibniz from the outset regarded
geometry and his calculus as worthwhile pursuits because of what they add to human
capability in society and in the physical world. Grosholz refers to Leibniz’s synthesis of
different domains, “Because his [Leibniz’s] analysis deals with magnitude in general, it can
also apply to mechanics – to distance, time velocity, and force; because it does not eschew the
infinitary, it can apply to continuous motion.”175 Later on the same page, Grosholz says, “A
fuller integration of mechanics and mathematics depends on precisely the reorganization of
mathematical domains”. Further, “Leibniz’s synthesis does not take place merely because he
diameter is only 2 ½ inches is nevertheless able to to raise 60 lb. the whole distance through which the piston
falls. And the tube does not even weigh 5 oz. … I have also proved that one minute is sufficient time for a
moderate fire to force the piston to the top of the tube. … If consideration is given to the magnitude of the
force that can be generated by this means, and the small cost of the wood that has to be used, it must certainly
be admitted that this method is far preferable to the use of gunpowder. … It would lead us too far afield to
discuss how this discovery could be applied to extract water from the mines, to throw bombs, to sail against
the wind or other similar applications which might arise.” (Klemm, p.221-222) 174 Grosholz, E. “Was Leibniz a mathematical revolutionary?” in Gillies, D. Revolutions in Mathematics Oxford
University Press, New York, 1992, pp.117-133 175 Ibid., p.126
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employs abstract algorithms that can be instantiated in different domains, but also because the
synthesis engenders hybrids that exist simultaneously at the overlap of different domains.”176
In fact, Leibniz is always working in both domains, and does not really see the difference
between them. Grosholz says, “Transcendental curves like the tractrix and the catenary, and
the ellipse re-imagined as a trajectory, will illustrate my point.”177 However, these curves also
illustrate the opposite point – that Leibniz’s work in mathematics was work in mechanics,
which was ultimately motivated by his interest in machines, at the same time.
What are regarded as different domains in the late 20th and early 21st Centuries may not have
been in Leibniz’s time, when it was natural for thinkers to be across multiple “domains” most
of the time, as indeed most were.178
In describing the constellation of scientists influenced by Mersenne, which included Rene
Descartes, Pierre Gassendi, Gerard Desargues, Pierre de Fermat, Gilles Personne de Roberval
and Galileo Galilei, Cohen wrote that they were trained in a “liberal approach to study that
embraced many fields” and “were, in fact, philosopher-scientists whose attitudes pervaded the
times and helped provide a very special basis for the establishment of the Paris Academy”.179
In a similar vein, Bell describes the intellectual environment of Christian Huygens,180 the first
president of the (French) Royal Academy.
Political influences on and of Leibniz
The breadth of scientists in Leibniz’s time went even further than Cohen and Bell have said.
Scientific and philosophical thought was connected with the necessities and potentialities of
social and political life. Indeed, Mercer constructed the theory of “conciliatory eclecticism” to
explain the philosophy of both Leibniz and his teacher Jakob Thomasius as emanating from
the religious conflict that gave rise to the Thirty Years War (1618-1648) which Thomasius
lived through. Leibniz was born towards the end of that war, in 1646.181 In a sense, the Thirty
Years War was a continuation of European religious strife in the vein of the foregoing century.
That religious conflict affected intellectuals throughout Europe and gave a greater sense of
mission and urgency to their deliberations and activism. It certainly was a motivation for
Bruno in the 16th Century, many decades even before the Thirty Years War. “Wherever Bruno
travelled within Europe, doctrinal intolerance and endemic slaughter in the name of God
reassured him that only a spiritual and intellectual revolution could ever disassociate religion
176 Ibid., pp.126-127 177 Ibid., p.127 178 In the same vein, an information-theoretic approach might not be helpful when studying 17th century thinkers
as Grosholz attempted to do. See Grosholz, E. “Partial Unification of Domains, Hybrids and the Growth of
Mathematical Knowledge” in Grosholz, E. and Breger, H. (eds.) The Growth of Mathematical Knowledge
Kluwer, Dordrecht 2000 179 Cohen, A. Music in the French Royal Academy of Sciences: A study in the evolution musical thought,
University Press, Princeton NJ 1981, pp.3-6 180 Bell, A. E. Christian Huygens and the Development of Science in the Seventeenth Century E. Arnold, London
1947, p.47 181 Mercer, C. Leibniz’s Metaphysics: Its Origins and Development Cambridge University Press, Cambridge
2001, pp. 80-110
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from murder, horror, and endless pain.”182 Yates has written extensively on Catherine de
Medici’s use of talismans and astrological rituals for the “pacification of the wars of
religion”.183 In any case, as far as Leibniz’s contemporaries were concerned, the Thirty Years
War had a major influence on the political and theological thinking of the time, much as WW
II influenced political and economic thought for the early post-WW II decades.
In fact, Leibniz in Theodicy did not refrain from addressing the protestant side of the Thirty
Years conflict:184
Luther’s book against Erasmus is full of vigorous comments hostile to those who
desire to submit revealed truths to the tribunal of our reason. Calvin often speaks in the
same tone, against the inquisitive daring of those who seek to penetrate into the
counsels of God. He declares in his treatise on predestination that God had just causes
for damning some men, but causes unknown to us. Finally M. Bayle [Pierre Bayle,
1647-1706] quotes sundry modern writers who have spoken to the same effect (Reply
to the Questions of a Provincial, ch. 161 et seq.).
Leibniz refers to Pierre Bayle (1647-1706). Bayle typified some aspects of the time and is
regarded as one of the most prominent men of letters of the 17th Century.185 As a Protestant in
France, Bayle suffered and fled religious persecution living much of his life as a refugee in
Holland.186 Bayle’s best-known work, the Dictionnarie historique et critique (“Historical and
Critical Dictionary”), published in 1696, comprises over 2500 articles on people from before
Christ to Zeno to Hobbes. The Dictionnarie is sometimes called the “arsenal of the
Enlightenment” because it was used by activists and writers for material for their respective
arguments. It is estimated to be the “single most popular work of the eighteenth century”. The
issues of religious tolerance, respect for the conclusions of the consciences of others, and the
problem of evil were central concerns in Bayle’s writings.187 Unfortunately, we do not have
the space or time to examine Bayle’s ideas further here.
Johnson had this same thought:188
Leibniz lived during a period of intense crisis. Dynamic individualism, in Church and
State, threatened to reduce the life of Europe to chaos. There was a desperate need to
achieve a harmony which would not destroy the fruitful forces of individualism.
Leibniz set himself to formulate a comprehensive philosophy which would serve as
182 White, M. The Pope and the Heretic: The true story of Giordano Bruno, the Man Who Dared to Defy the
Roman Inquisition Harper Collins 2001, p.29 183 Yates 1964, p.176; Yates, F. The French Academies of the Sixteenth Century Warburg Institute, University of
London 1947, pp. 236 ff.; and Yates, F. The Valois Tapestries Warburg Institute, University of London 1959,
pp. 82 ff. 184 Leibniz, G.W., Huggard, E.M. (trans.) Theodicy §49 Project Gutenberg edition 2005 accessed at
www.gutenberg.com, p.101 185 ARTFL Project, Project for American and French Research on the Treasury of the French Language, of the
Department of Romance Languages and Literatures, University of Chicago. Accessed at http://artfl-
project.uchicago.edu/node/60 on 22 May 2012 186 Lennon, T. M. and Hickson, M. “Pierre Bayle” Stanford Encyclopedia of Philosophy article first published
2003 revised 2008. Accessed at http://plato.stanford.edu/entries/bayle/ on 22 May 2012 187 Ibid. 188 Johnson, A.H. “Leibniz’s method and the basis of his metaphysics” Philosophy 1960 vol. 35, p.60
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the intellectual foundation for a new age which would facilitate fulfilment of the best
dreams of scientists and practical men, saints and sages. Leibniz did not philosophize
in a logical vacuum.
Loemker says that Leibniz had hoped that his metaphysics “would be adopted and made a
blueprint, so to speak, by men of good will (honestas) for the restoration of European order”
and it may “be regarded as the intellectual high point of the century’s efforts toward a
renovation of Europe through the ideal of loyalty and obedience to a universal nature and
moral order.”189 Loemker points out that Descartes and Spinoza, like Locke, saw no
imperative to particular social rules. Descartes separated the objective from the subjective in
his two-level order of, on the one hand, creator and, on the other hand, created “which must
interact, yet cannot interact since they have no common nature which could make this
possible”. Spinoza separated God from “the everyday evaluations of man” and is “accessible
only to the mind which is completely disciplined”. However, life in society “is recommended
on the commonsense ground of utility and conservation [preservation?], and such active
affections which contribute to them. There is no great overarching order of commands
whereby the social order is to be regulated”.190 We will see below that Leibniz distinguished
salutary social influences from detrimental ones not purely on the grounds of utility and
preservation. This was bound up with Leibniz’s conception of the nature of Man and the
unique role and position in the universe allocated to humanity by God.
Seeking to understand Creation as a whole
In discussing the analysis of Grosholz, we review the thread from the pre-moderns to
modernity.
Grosholz finds it impressive that Leibniz’s calculus could be applied to enrich Kepler’s work
on the solar system. Yet, it would make sense for Leibniz to have considered the problems
involved in Kepler’s grappling with the structure and harmony of the solar system as a proxy
for other problems in science and machine design. Planetary motion was one of the active
“research areas” of the period, a bit like “climate change” was in the early 21st Century. Thus,
it is unlikely that Leibniz simply stumbled on the fact that his calculus was useful for
questions relating to planetary motion; Leibniz would have been aware of Kepler’s
contribution and of open problems left by Kepler. Huygens would surely have mentioned the
problem of planetary motion and Kepler’s work to Leibniz. Further, the fact that Leibniz read
so many of the Greek classics in his teens makes it almost certain that he’d have been aware
of the problems in understanding the solar system which could have drawn him to Kepler who
represented the state of the art on the question of planetary motion in Leibniz’s time.
189 Loemker, L. E. Struggle for Synthesis: The Seventeenth Century Background of Leibniz’s Synthesis of Order
and Freedom Harvard University Press: Cambridge USA 1972, p.127 190 Ibid., p.130
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Grosholz raises a supposed contradiction at the end of a paper on Leibniz191 as a conundrum.
Grosholz says that modern problems in physics such as relativity theory and quantum theory,
pose grave philosophical problems because if we call them part of mathematics, and so mere
patterns and relational structures, we cannot explain how they can play the role of the
furniture of the universe, as in contemporary physical theory they surely do. If we call them
part of the description of nature, we must wonder how nature has come to look so exact and
immaterial. If we call them hybrids, they seem to contain an internal contradiction.192
Grosholz has raised nothing less than the ancient problem of duality. That is, what is the
relationship between abstract mathematical structure which is outside and separate from the
physical universe, and the physical universe which seems to realise or embody abstract
mathematical structures. This problem was first dealt with no later than Plato (c.427-c.347
BCE). According to Plato, arithmetic comes first whence plane geometry is derived, whence
solid geometry. On these foundations sit astronomy and harmonics.193 The fourfold structure
of arithmetic, geometry, astronomy and music is known as the quadrivium.
The primacy of the quadrivium was maintained by Neoplatonists Iamblichus (c.242-327 CE)
and Proclus of Athens (412–485 CE). While the quadrivium was part of the curriculum of
medieval universities, it should not be regarded as naïve, arcane or quaint for that reason.
Given the sequence set forth by the quadrivium, Kepler’s hypotheses regarding the
relationship between the Platonic solids and the structure of the solar system, and between
musical harmony and the design of the solar system, are natural and even canonical.
The quadrivium resolves the problem of duality within itself. In the transition from geometry
to astronomy, we cross from the abstract to the physical, but all within the framework of the
quadrivium. In short, it need only be said that the quadrivium defines structure and structure
comes first. Thus, the quadrivium precedes anything perceived, including by measurement, to
be in the physical universe. It forms part of the (imperceptible) rules governing the universe.
It is connected with the best of all possible worlds doctrine, because to adhere to the structure
of the quadrivium provides the best design for, at least, the non-living part of the universe. We
can see the effects of the quadrivium not the quadrivium itself. Is it “part of” the universe? In
the same way, the rules of cricket are part of a cricket game, though those abstract rules are
not perceptible in the very physical acts of catching, throwing and hitting a cricket ball.
Unlike cricket, the universe has no definite rule book we can open as the final word or arbiter;
we have to work it out for ourselves. Within the confines of the quadrivium, “Geometry is
prior to astronomy,” as Proclus said. Failure to note Kepler’s Neoplatonic mindset results in a
191 Grosholz, E. “Was Leibniz a mathematical revolutionary?” in Gillies, D. Revolutions in mathematics Oxford
University Press: New York, 1992, p.117-133 192 Ibid., p.133 193 The literature on the quadrivium is vast. See http://scienceworld.wolfram.com/biography/Plato.html accessed
26 Feb 2012 for a summary. For Plato’s advocacy, see The Republic Book VII in Rouse, W.H.D. (trans.)
Great Dialogues of Plato Signet Classic, New American Library a division of Penguin, New York 1999,
pp.324-331 culminating at p.332 with the name “dialectic” being given to the progress of thought through the
quadrivium, likening it to a turning away from the shadows and progression through a tunnel upward to the
sun in the context of the allegory of The Cave. Plato says much more about music in Book III of The
Republic.
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misunderstanding of his method.194 After the quadrivium comes the hybrid between
mathematical reasoning and observation. Last and least is pure observation.
The quadrivium was core to learning at least as late as the 17th Century. 195 Leibniz, at least
earlier in his career, said that most problems in physics can be resolved into problems of pure
geometry.196,197 However, Johnson interprets Leibniz as having gone farther than he did, in
regarding Leibniz’s ten maxims on the Art of Discovery as being tantamount to rules of
geometrical reasoning.198
Much of 20th Century, and 21
st Century, science has a tendency to start with the observations.
Further, there is a pre-occupation with things that we can physically observe and visualise in
the mind. There are exceptions, such as string theory. As a civilisation, we have forgotten how
we reached the starting line of 20th Century science. Franklin and Newstead’s defence of the
reality of indefinable real numbers is a sally against “empiricist idealism”.199 Of course,
empiricist idealism did not begin in the Twentieth Century and perhaps existed in more
strident form in certain circles in earlier centuries.
Geometry as a tool to understand Creation as a whole
To Kepler, the beauty and perfection of mathematics – especially geometry – reflected the
perfection of God’s methods. Astronomy, being representative of Creation on the larger scale,
had to reflect this beauty and perfection.200 Such a contention tends to be associated with
Renaissance Platonism and its supposed commitment to vitalism and natural magic. While
Stuart Brown has done much to dispel this misconception,201 it persists in abundance. For
194 For example, Lawrenz refers to Kepler’s Mysterium Cosmographicum and Harmonice Mundi as “luxuriant
cosmological fantasias” Lawrenz, J. Leibniz: Double-aspect ontology and the labyrinth of the continuum
2007 PhD thesis, University of Sydney, p.23, note 5 195 Cohen, A. Music in the French Royal Academy of Sciences: A study in the evolution musical thought,
University Press: Princeton NJ 1981, pp.3-5. Cohen writes of Mersenne who was influential in the
intellectual life of Paris in the early 17th Century, and the parenthetical comments are in the original:
“Having derived his thinking from the platonic traditions of the Renaissance (in which music was considered
‘the image of the whole encyclopedia’), Mersenne assigned a special role to music in his own system, where
it formed an essential part of mathematics, ‘utile à toutes le sciences.’ Certainly, this view of music reflects
its place in the quadrivium of the medieval artes liberales (the four mathematical disciplines that had their
common basis in numerical ratio and proportion: “arithmetic - pure number, music - applied number,
geometry - stationary number, astronomy - number in motion”), which continued to form the foundation for
university studies through the Renaissance and into early modern times.” 196 Johnson, A. H. “Leibniz’s method and the basis of his metaphysics” Philosophy 1960 vol. 35, 51-61, pp.52-
53 197 Letter to Ferrault, 1676 “All problems in gravitation, magnetism, electricity and light are explicable by the
resolution of a few problems of pure geometry” in Wiener, pp.xxi-xxii 198 Johnson, A. H. “Leibniz’s method and the basis of his metaphysics” Philosophy 1960 vol. 35, 51-61, p.53 199 Franklin, J. and Newstead, A. “On the Reality of the Continuum; Discussion Note: A reply to Ormell,
‘Russell’s Moment of Candour’, Philosophy” Philosophy (published by the Royal Institute of Philosophy) 83
2008, p.121 200 Duncan, A.M. trans. Kepler, J. Mysterium Cosmographicum Abaris Books, Janus Series Norwalk USA 1981,
p.57 201 Brown, S. “Leibniz and Berkeley: Platonic Metaphysics and ‘The Mechanical Philosophy’” Chapter 16,
pp.239-253 in Hedley, D. and Hutton, S. (eds) Platonism at the Origins of Modernity: Studies on Platonism
and Early Modern Philosophy Springer 2008
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example, Lawrenz wrote that the Mysterium Cosmographicum and Harmonice Mundi were
“luxuriant cosmological fantasias integrating mysticism, music, mathematics, Pythagorean
number lore, the doctrines of the Timaeus”.202 Lawrenz then says that Leibniz distanced
himself from Kepler’s work, “having recognised during his Paris sojourn that its mathematics
was woefully inadequate to the task”. On the contrary, we can see Kepler in many places in
Leibniz’s metaphysics including in the “best of all possible worlds” doctrine. Also, the
inadequacy of Kepler’s mathematics had nothing to do with his selection of the Platonic
solids hypothesis. Rather, that hypothesis was a natural result of a priorist thinking. It would
not have been possible to be an a priorist without a familiarity with standard Pythagorean and
Platonic ideas such as the quadrivium and platonic solids.
Let us try to draw an analogy close to the heart of 21st Century readers. The art of ordering
human affairs is increasingly being addressed through the formal sciences, “‘the sciences of
complexity’ or ‘sciences of the artificial’” – typified by optimisation – amongst which
Franklin has attempted to find a unifying thread.203 Belief in a Creator of the universe who
has certain qualities of beneficence leads one to conclude that there must be structure
underlying the physical universe whose creation was itself an “artificial” act. If well-
organised humans structure their activities using formal sciences, what science of the artificial
did the Creator use? The first principles of structure, the quadrivium, must be at or near the
foundation, or so it would have been reasonable for the Renaissance and early post-
Renaissance scientist to believe.
Understanding Creation as a whole, and improving upon it
The burgeoning of modernity, as we argue here that modernity should be defined, is seen in
the correspondence between Leibniz and Papin. Their collaboration did not arise in a vacuum
but in the drive towards national development for the benefit of the General Welfare of the
nation of France. As minister of the young French King Louis XIV, Jean Baptiste Colbert
initiated a project to discover and make effective a source of power capable of enabling a
dramatic human advance. The project was put into motion as a national effort.
The project was conceived when the drive for development had outstripped the power sources
available. For example, Klemm explains that the:204
...vast and costly establishment at Marly [14 water wheels in the river Seine, each 40
feet in diameter driving 442 pumps], and its relatively small output [80 horsepower]-
shows us clearly the limits of the old power-machine technology. To seek a more
efficient and reliable prime mover than the traditional wind and water wheels was in
fact to become the urgent task of the technology of that period.
202 Lawrenz, J. Leibniz: Double-Aspect Ontology and the Labyrinth of the Continuum of Sydney PhD thesis
2007, p.23, footnote 5 203 Franklin, J. “The Formal Sciences discover the Philosopher’s Stone” Studies in History and Philosophy of
Science Vol. 25 No. 4 1994 pp.513-533, p.18 204 Klemm, p.208
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Philosopher scientists and French nation-building
In 1666, Colbert established the Academy of Sciences in Paris to further this purpose, and
Christian Huygens was recruited as the first president of the new Academy. Thus, Colbert was
following the prompting of the ‘useful science’ idea. We do not know whether Colbert
acquired this idea from Francis Bacon.205 Huygens proposed a program including research
into the power of gunpowder of which a small portion is enclosed in a very thick iron or
copper case, research into the power of water converted by fire into steam, and experiments
with vacuum pumps, wind-powered engines, and the communication of force by the collision
of bodies.206
The experiments carried out on the basis of these suggestions were the prelude to the
development of a new power-engine. Huygens’ suggestions followed the work of Guericke
who had shown in 1661 that a piston, forced by air pressure into an evacuated cylinder, can be
utilized to perform work. Guericke’s cylindrical vessel was 15 inches in diameter and just
over 21 inches long; it was evacuated by an air-pump invented by Guericke. In 1664 Caspar
Schott announced Guericke’s experiments on obtaining power by pistons forced through a
metal cylinder by air pressure.
In 1672, Huygens acquired two students: Leibniz the 26-year old diplomat, and Denis Papin a
25-year old French medical doctor who was introduced by Madame Colbert. Within 12
months, Huygens, Leibniz and Papin had modified the Guericke air pump into an engine that
could transform the force of exploding gunpowder into useful work.207 Huygens demonstrated
a model “gunpowder engine” to Colbert. In 1673 Huygens wrote this about the model:208
The violent action of the powder is by this discovery restricted to a movement which
limits itself as does that of a great weight. And not only can it serve all purposes to
which weight is applied but also in most cases where man or animal power is needed,
such as that it could be applied to raise great stones for building, to erect obelisks, to
raise water for fountains or to work mills to grind grain …
It can also be used as a very powerful projector of such a nature that it would be
possible by this means to construct weapons which would discharge cannon balls,
great arrows, and bomb shells … And, unlike the artillery of today these engines
would be easy to transport, because in this discovery lightness is combined with
power. This last characteristic is very important and by this means permits the
discovery of new kinds of vehicles on land and water.
And although it may sound contradictory it seems not impossible to devise some
vehicle to move through the air …
205 C.f. Franklin, J. “‘Useful science’, as an idea, is Baconian, but science as an accepted route to profit, military
superiority is an eighteenth-century development.” in “Artifice and the Natural World” in Cambridge History
of Eighteenth Century Philosophy CUP p.842 206 Klemm, p.212 207 Ibid., p.212 208 Ibid., p.213
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Leibniz had invented a mechanical computer in 1671, a year before Leibniz was introduced to
Huygens. The computer was built in 1673.209 Working alongside Huygens and Papin, Leibniz
was active as an inventor of machines that could perform physical work. Brown and Calvör
describe Leibniz’s work between 1678 and 1679 on his own design for a wind-powered pump
to keep the mines in the Harz mountains dry.210,211 Leibniz personally supervised construction
of his mine pump and personally paid for part of it.212 Klemm speculates that the failure of the
pump was possibly due to Leibniz’s frequent absence from the site and the poor skills of the
workers. Despite the difficulties, Leibniz then designed a machine which sought to overcome
the limitations that had plagued earlier attempts. Leibniz’s hand drawn diagram is included
here as Figure 2. In the end, the Harz mine pump project was unsuccessful and very expensive
to his employer in lost potential income.213
209 Calvör, H. Historisch-Chronologische Nachricht … des Maschinenwesens … auf dem OberharzeBrunswick,
1763 translated by Dorothea Waley Singer in Klemm, pp.208-9 210 Ibid. 211 Brown, R. C. Leibniz unpublished, Chapter 11 “Epilogue”, pp.163-4 212 Klemm writes that Leibniz personally paid for the pump. Richard Brown qualifies this, writing that it was
originally agreed with the Leibniz’s employer the Duke Johann Friedrich that Leibniz would personally paid
for the prototype mine pump for test purposes, in September 1679. However, in October 1680 Leibniz’s share
was reduced to one third with the balance to be provided by the Duke and the Mining Office. Brown, R. C.
Leibniz unpublished, Chapter 11 “Epilogue”, p.163 213 Brown, R. C. Leibniz unpublished, Chapter 11 “Epilogue”, pp.163-4
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Figure 2: Leibniz’s design for a wind-driven water pump (Klemm, p.211)
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An era of pro-nuclear advocacy
An anonymous article from the late 17th Century entitled “To the greater glory of God” refers
to a letter by Huygens on the potential of gunpowder. The promise and danger in the use of
gunpowder in that period mirrors today’s debate about nuclear energy. The writer is on the
“pro” side, and he cites Renaissance-like arguments similar to those used by today’s
proponents of nuclear energy. The letter defends Huygens’ experiments with gunpowder.214
The letter illustrates the optimistic spirit of technological progress for the benefit of
humankind which defines Leibniz’s outlook. This optimism, in spite of the Thirty Years War,
characterized the period of Leibniz’s career. The letter makes the transition from the
philosophical regard for the primacy of Man to its practical implications and realization in
machines. The letter also relies most heavily on the concept of vis viva without using the term.
The letter is anonymous, and given Leibniz’s diplomatic employment, young age (late 20s) at
the time the letter was published and association with Huygens, one wonders whether it was
written by Leibniz himself or one of his friends perhaps at Leibniz’s instigation.215
Leibniz’s optimism about human nature and rejection of the contention that humans cannot be
trusted is more explicit when he explains why the health of the mind and body are too often
neglected. He says, “those who reflect find more reason to admire the excellence of human
nature than to despise it.”216 Leibniz writes that “empirical physics is useful for human life
and should be cultivated in the state” and that “new experiments are to be undertaken at
public expense, and only men outstanding not merely in science but in virtue are to be placed
in charge.”217
Despite the failure of Leibniz’s water pump initiatives, his collaboration with Huygens and
Papin continued. Leibniz supplied a number of practical suggestions, right up to Papin’s
invention of steam locomotion and Papin’s proposal to use it to power a steamboat.
Klemm indicates that Thomas Newcomen, often credited with the invention of steam-
powered locomotion,218 was really riding on the back of Papin and Leibniz’s work.219
Arguably, Newcomen’s engine was a scaled-up version of Papin’s atmospheric steam pump
based on a combination of two of Papin’s earlier ideas: to use a lever to transmit power from
one pump to another (from 1687) and to use steam to create a vacuum and drive a piston
(from 1690).
214 Klemm, pp. 218-220 from Ad majorem Dei gloriam published in Nouvelles de la Republique des Lettres
Amsterdam 1695. The article is extracted in Appendix 1. 215 Brown, R. C. Leibniz unpublished, Chapter 11 “Epilogue”, pp.175, 179 216 “Essay on a new plan of a certain science” in Wiener, p.583 217 “On the elements of natural sciences” ca. 1682-4, Loemker, pp.281, 282 218 E.g. see Franklin, J. “Artifice and the Natural World” in Cambridge History of Eighteenth Century
Philosophy Cambridge University Press, p.843 219 Klemm, pp.226-227
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Participating with God in Creation
Leibniz ties physical action to the mind of God, analogously to how he participates in the
design of machines to the mind of humans. “For we have now seen, from the pre-established
harmony, that God has ordered all things so wonderfully that corporeal machines serve minds
and that what is providence in a mind is fate in a body.”220 That is, minds are in the driver’s
seat. This has the end-implication that laziness of mind in an individual, a nation or a
civilisation is self-correcting for it leads, inevitably, to conditions which will force that mind
of the individual or the collective, perhaps in a future generation, to reconsider.221
Arguably, the power of Man over nature defines modernity. Man could not have this power
were the design of the universe not rational for otherwise there would not be sufficient order
and predictability in the universe for Man to effect change in it. A rational universe, in turn,
requires a rational conception of the Creator of the universe. The power of Man to effect
change in the universe is summed up at a higher order of magnitude in Leibniz’s conception
of dynamics. The concept is addressed in the correspondence between Leibniz and Papin on
the power of steam to lift a column of air and more. It is indirectly brought out by Leibniz in
his Specimen Dynamicum (1695). However, the argument in Specimen Dynamicum mainly
goes to why the ancients only had half the story with dead force, versus vis viva, and why
Leibniz believed Cartesian doctrines on matter, force and velocity to be wrong.
220 Loemker, p.593 (Letter to Hansch, July 25, 1707) 221 Thus, Leibniz would agree that the divine mind does “design through us” albeit that we have free will. We
disagree with Austin Farrer in his commentary on Theodicy (November 24, 2005 [EBook #17147], pp.32-
33):
And perhaps some such element enters into all our choices, since our life is to some extent freely
designed by ourselves. If so, our minds are even more akin to the divine mind than Leibniz realized. For
the sort of choice we are now referring to seems to be an intuitive turning away from an infinite, or at
least indefinite, range of less attractive possibility. And such is the nature of the divine creative choice.
The consequence of such a line of speculation would be, that the divine mind designs more through us,
and less simply for us, than Leibniz allowed: the ‘harmony’ into which we enter would be no longer
simply ‘pre-established’. Leibniz, in fact, could have nothing to do with such a suggestion, and he
would have found it easy to be ironical about it if his contemporaries had proposed it.
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Chapter 4: Idea of truth
Introduction
We begin with the nature of the pursuit of truth, which touches on mathematics, the
limitations that Leibniz found in mathematics, and the relationship of mathematics to
metaphysics. This will take us into the domain of Neoplatonic exactitude which anticipates
the next chapter. We will be required to introduce Paolo Sarpi and the limitation on deductive
reasoning when it is constrained by the empiricist ideology. This will take us into a bridge
from Paolo Sarpi to the English Royal Society. Both Galileo and Newton had a doctrine of
how discovery takes place, and we see the continuum on increasing empiricist constraint
stepping upwards from Galileo to Sarpi to Newton. Relevant to this is a correspondence
between Sanderson’s 17th Century textbook on logic and Newton’s doctrine of the discovery
process, and especially Newton’s bias against hypothesis, both explored in the next chapter.
Contrary to a desire to make science appear to be “objective”, we argue that pure positivism is
not possible in either science or mathematics, nor can metaphysics be removed from science.
While having a Neoplatonist orientation in many matters of science and philosophy, Leibniz
saw the necessity for formal reasoning in science. Rather than being an innovation that
Leibniz added to Neoplatonism, rigor is found in Platonism itself.
Minds can successfully search for truth
In the previous chapter, we explained that modernity was about the accessibility of scientific
principles crystallising in pragmatic tools or machinery for physical influence by humans in
the universe. In this chapter, we attempt to lift these considerations to the concept of truth
itself.
That truth is discoverable via the exercise of reason was a presumption maintained by
Leibniz. This includes physical and metaphysical principles, and principles of human purpose
and conduct. At least, if truth does not exist or is not discoverable then Leibniz makes no
sense. More importantly, since we never fully know truth, it is the process to reach towards
truth that is of greatest concern and, when it comes to science, the only concern.
The ability of the human mind to access truth or parts of truth gives the mind special place in
the universe. Leibniz saw a relationship between the human mind and the divine, which
appears to be akin to how Nicolaus of Cusa understood the relation between the mind and the
divine. Leibniz begins this passage in a letter with reference to the Neoplatonist Plotinus:222
as Plotinus rightly said, every mind contains a kind of intelligible world within itself;
indeed, in my opinion it also represents this sensible world to itself. But there is an
infinite difference between our intellect and the divine, for God sees all things
adequately and at once, while very few things are known distinctly by us; the rest lie
222 Letter to Hansch, July 25, 1707 in Loemker, p.592-3
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hidden confusedly, as it were, in the chaos of our perceptions. Yet the seeds of the
things we learn are within us [doctrine of Reminiscence] - the ideas and the eternal
truths which arise from them. … Although our mind depends continuously on God in
its existence and action, as does every other creature, I do not think that it needs his
particular concourse over and above the laws of nature for its perceptions, but rather it
deduces its later thoughts from its earlier ones by its internal force and in an order
prescribed by God, as Roelius, whom you quote, rightly says.
The passage raises the question of the relationship of the finite to the infinite, which Nicolaus
of Cusa referred to time and again, which we will discuss further in Chapter 4 “Discovery and
Deduction”. Leibniz also reiterates that human minds can delve into the universe and make
discoveries about it without intervention by God. Rather, minds have their own “internal
force”. The “chaos of our perceptions” implies that our perceptions do not get us very far.
What we “learn” are ideas, and eternal truths arise from those. Clearly, the mind’s access to
what is universal and eternal deserves special consideration. In one of his essays, Leibniz
refers to the mind as the divine in miniature.223 Therefore, it is not surprising that Leibniz is
more Rationalist than Empiricist. He observed that a geometrical argument made in a dream is
as accurate as one made when awake, while a perception in a dream is illusory. Of course, our
perceptions usually dominate our thoughts and there is no thought so abstract that it does not
involve the object of some perception.224 However, just as no-one would deny that air is
essential to life, yet life is something different from and higher than air.
For Leibniz, the nature of transcendental curves is part of the domain of truth, but
transcendental curves are real not only because they or their effects are realisable. Of course,
they are of immediate interest because their effects are real and realisable. As he says, “we
must free the human mind from arbitrary contingencies, in order to bring out the underlying
nature of the thing [the curve] itself.”225 It is difficult to say whether Leibniz would go further
than the structuralist school, which requires a mathematical to be able to manifest in the
physical universe for it to be called real.226 In any case, the point is moot because Leibniz
never spent time on anything for which he could not see significance or repercussions for the
physical universe, or even potential useful to humankind, and regarded such pursuits as a
frivolous waste of time. The point would not be moot if the discussion led to an overturning
of our conception of how God thinks or from where he got the raw ideas from which to design
the universe.
223 Leibniz, G.W. Theodicy §147. There is also an epigraph to Theodicy which says the same, though this author
has been unable to find an edition which includes this epigraph. The epigraph is quoted in Jorgensen, L.M.
“Leibniz’s Theodicy” PHIL 375: Advanced History of Philosophy course notes Fall 2008 p.10 accessed at
http://faculty.valpo.edu/ljorgens/teachres/Theodicy%20Packet.pdf 12 May 2011. Jorgensen states that the
edition being used is an unpublished manuscript of W.H. Warren (trans.), Mulvaney, R.J. (ed.) Leibniz, G.W.
Theodicy. 224 Loemker, p.556 225 Leibniz, G.W. Beaudry, P. (trans.) “The String Whose Curve Is Described by Bending Under Its Own Weight,
and the Remarkable Resources That Can Be Discovered from It by However Many Proportional Means and
Logarithms” Acta Eruditorum, Leipzig, June 1691, last sentence accessed at
http://www.schillerinstitute.org/fid_97-01/011_catenary.html#1 on 9 June 2012 226 See Franklin, J. “Mathematical Necessity and Reality” Australasian Journal of Philosophy Vol. 67 No. 3,
September 1989
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Truths such as the nature of the catenary curve can be discovered systematically. “That is, the
case of developing methods is always more crucial, than particular problems, although it is
the latter which usually bring applause.”227 Further, such methods lead to real benefits. It
extends the science of discovery, “in other words the science of Analysis, which up to now
has been incapable of tackling such questions. Second, it extends the progress of construction
techniques. In point of fact, I have come to realize that the resourcefulness of this curve is
only equal to the simplicity of its construction, which makes it the primary one among all the
transcendental curves.”228
Sense perception and Empiricism
In The Republic, Plato said that all sense perception is opinion and thus is not of great
assistance in discovering truth.229 Leibniz largely concurred. Today, to the layman and
scientist alike, it is common for the testimony of the senses is regarded as the final or, at least,
the most reliable arbiter.
Yet sense perception had already gained primacy through circles associated with Galileo, and
had gained a foothold in the British Isles, especially in the circles of what became the Royal
Society. Fulgenzio Micanzio was the closest friend of Galileo’s patron, Paolo Sarpi. Micanzio
corresponded with William Cavendish the English scientist. Those letters were translated by
the secretary of Francis Bacon, Thomas Hobbes, for wider circulation.230 It is possible that
when Hobbes and Cavendish visited Venice, they met with Sarpi and there is evidence in
Hobbes’ translations that he was familiar with the personality of Sarpi’s personality.231 Sarpi
advocated a presumption that what is perceived with the senses is real, as we will discuss later
in more detail. In fact, Sarpi may have believed that excessive reflection was a weakness and
instinctive action superior leading to the conclusion that in some ways Man is weaker than the
animals which primarily act on instinct based on their sense perception.232 Contrary to these
conclusions reached by Sarpi, he was a regular attendee in meetings of occult circles in
Venice. Giordano Bruno was warmly welcomed by these circles when he was in Venice, and
through this connection Michael White says that Sarpi “knew Bruno well”.233 It would seem
227 Leibniz, G.W., Beaudry, P. (trans.) “The String Whose Curve Is Described by Bending Under Its Own Weight,
and the Remarkable Resources That Can Be Discovered from It by However Many Proportional Means and
Logarithms” Acta Eruditorum, Leipzig, June 1691, second-last paragraph accessed at
http://www.schillerinstitute.org/fid_97-01/011_catenary.html#1 on 9 June 2012 228 Ibid., first paragraph 229 In The Republic (602 d), Plato says, “A stick will look bent if you put it in the water, straight when you take it
out, and deceptive differences of shading can make the same surface seem to the eye concave or convex; and
our minds are clearly liable to all sorts of confusions of this kind.” 230 Malcolm, N. “A summary biography of Hobbes” in Sorell, T. (ed.) The Cambridge Companion to Hobbes
CUP Cambridge 1996, p.22. On the same page, Malcolm speculates that “Hobbes must have gained a special
interest in the writings and political actions of Sarpi, who had defended Venice against the papal interdict of
1606 and developed a strongly anti-papal theory of Church and State”. 231 Kainulainen, J. “Paolo Sarpi between Jean Bodin and Thomas Hobbes: a study on ‘political animal’ in early
modern Europe” Thesis at the European University Institute, Department of History and Civilisation.
Accessed at http://www.eui.eu/Personal/VanGelderen/Theses/JaskaKainulainenThesis.shtml 24 April 2012 232 Ibid. 233 White, M. The Pope and the Heretic: The true story of Giordano Bruno, the Man Who Dared to Defy the
Roman Inquisition Harper Collins 2001, pp.38-39
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that Sarpi was not persuaded by Bruno. Returning to the British Isles, the primacy of sense
perception became much more firmly entrenched after Newton had passed away and the
Royal Society had passed into the hands of its second-generation membership.
Another name of the ideology that places sense perception first is Empiricism or a
posteriorism as against a priorism. Of course, even a priori thinking uses the testimony of the
senses but only to confirm or deny hypotheses formulated by the use of reason.
To maintain that the testimony of the senses is the only thing that has validity in science
removes the role of thought except for undertaking deductive exercises from what is
observed. The dangers of empiricist idealism are pointed out by Franklin and Newstead when
discussing Ormell’s desire to remove indefinable real numbers from mathematics.234 First of
all, what we observe with our senses is subjective. The testimony of one’s own senses can
change; the use of different measuring instruments causes observations to differ.
Observational data requires thought to give it meaning and order; for example, there was
nothing directly discernible in the observational data that Kepler pored over that indicated
elliptical orbits. To give the senses sole authority is also to maintain that the senses are the
only reliable reference in determining what we can be sure of, that is, in determining what we
know. Such authority would be misplaced since, as explained just now, the senses are not
reliable determinants of anything.
As sense perception is mere opinion, to demand that only the senses be relied upon is to desire
that nothing be known. The opponent of this who uses Platonic dialogue to help discover
truth, however, also denies knowability. If there is one thing that Plato succeeds in doing, it is
that answers given always lead to new questions.
Whether we use sense perception or intellectual enquiry, we get nowhere. This catch-22 might
drive anyone with an interest in science to pessimism as to their prospects for success and
depression, for it seems that humans are excluded from science.
Humans are excluded from science in another way. Burtt’s celebrated thesis The Metaphysical
Foundations of Modern Physical Science considered the human-free conception of the
universe adopted by positivist physics. That is, the laws which physics seeks to understand are
unaffected by humans and impact humans only in a physical kinetic sense. This stands in
contrast to the Renaissance view of Dante – for example – that humans are not only part of
Creation but are at its pinnacle. Burtt knew that he did not like the cold Newtonian-Russellian
clockwork universe in which Man is irrelevant, but he purported to supplant it by substituting
an extreme empiricism. That is, Burtt sought to bring Man back into science by declaring that
perceptions of the universe are close enough to reality and therefore should be taken to be
correct, in preference to a mathematical model, say. Burtt seems to think that the only way
234 Franklin, J. and Newstead, A. “On the Reality of the Continuum; Discussion Note: A reply to Ormell,
‘Russell’s Moment of Candour’, Philosophy” Philosophy (published by the Royal Institute of Philosophy) 83
2008 especially p.124. In an earlier article, Franklin discusses the impossibility of the formal sciences were
we to reject imperceptible relations between intangible mathematical objections. Franklin, J. “The Formal
Sciences discover the Philosopher’s Stone” Studies in History and Philosophy of Science Vol. 25 No. 4 1994 ,
p.26 of version at http://www.maths.unsw.edu.au/~jim/philosophersstone.pdf accessed 1 Nov 2010
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humans can be brought back into the universe is through sensory experience, neglecting the
role of mind:235
...the space of perception is too much like the space of real objects to reveal any
essential difference from it. All it needs is to be freed from illusions, private images,
and other experiences lacking social objectivity, to function quite acceptably as real
space. And once this point has been reached there seems no longer any excuse for
maintaining the distinction between sensed qualities and the real characters to which
they correspond.
…
There is simply no science possible of the realm of sensible phenomena unless the
trustworthiness of our immediate perception of spatial directions and relations be
taken for granted.
Not detracting from Burtt’s many and deep insights into the evolution of science, his
empiricist proposal was only possible because he did not give due consideration to the idea of
truth. Leibniz in §§27-29 of The Monadology addressed this directly by contrasting empirics
with reason calling Reason an act of “the rational soul or mind [esprit]” whereas empirics are
a result of perception which even the beasts have.236
As we explained above, to maintain that the senses, including deductions from their
testimony, are the only thing we can be sure of is to say that we can be sure of nothing. To
Cusa, humans could never know the truth as to know it would be to know God and,
metaphorically, such a light would be too bright for any human intellect to gaze upon directly.
Leibniz agreed that there was nothing but a theoretical possibility of anything but that “the
number of all truths which all men together can know is quite mediocre, even if there were an
infinity of men who for all eternity should exalt themselves in the advancement of
sciences.”237
Existence and attainability of truth: Rationalism and Neoplatonism, vs Empiricism
We will refer to promoters of the Primacy of Reason as Rationalists, which is a standard term.
Upon investigations, many Rationalists, up to and including Leibniz, turn out to be
Neoplatonists and vice-versa. The difference between the terms is in their historical baggage
and connotations. Both believe in the primacy of reason, or the inferiority of the senses to the
mind. Now, there are Aristotelian thinkers who give Reason a high position. However, in the
end, the Aristotelian does not give as much primacy as the Neoplatonist, and certainly not
relative to the sense perception, for the Neoplatonist is almost purely Rationalist giving no
value to sense perception except to check the conclusions of Reason.
235 Burtt, E.A. The Metaphysical Foundations of Modern Physical Science Routledge and Kegan Paul, London
2nd ed. 1932, reprinted 1950 pp. 315-6 236 Latta, p.233 237 post-1690, Wiener,p.76
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The “new mechanical philosophy” (“NMP”) of the 17th Century, partly due to thinkers like
Kepler, was an understanding that structure was discoverable in the universe, pursuant to the
belief that the universe has a rational design, and its achievements seemed to confirm that the
universe has a rational design. However, the “mechanism” or order is not necessarily
discoverable by sense perception and certainly not by the senses unaided by artificial (human-
made) instruments. Thus, the NMP has its roots in – and so is more closely allied with –
Rationalism and, hence, is closer to Neoplatonism than it is to Empiricism.238
Empiricists clearly differ from Rationalists on the path to truth, though they agree that final
truth – whatever their brand of it might be – is not unattainable by humans. The nature of the
two positions also impacts questions relating to the existence of truth. Cusa regards truth as
having a reality, but considers access to it as being exclusive to God. Humans can make gains
in its direction and approach it, merely. The Empiricist position allows judgement on whether
truth exists at all to be reserved, for the Empiricist does not need to answer the question, even
though an individual Empiricist may accept particular truths.
Empiricism allows – but does not imply – denial of the existence of truth. Once the Empiricist
tries to get into the question, he can only affirm truth’s existence if truth, like observation, is
subjective, which is no truth at all. Burtt made the error of asking what truth is while wearing
an Empiricist hat. The perennial trick of the Empiricist is to not ask the question, and to say
that the question of truth is irrelevant or to say that, as far as they are concerned, what is true
is what we can perceive (see, hear, touch, taste, smell) which can only be an appeal to a
popular kind of commonsense. This approach omits a significant role for the human mind
except in collating and making deductions from observations, which a computer can do quite
well.
If we allow their different conceptions of truth, the Empiricists and Rationalists agree that the
final truth is unattainable by humans. However, there is a key difference. Empiricists say that
all we can know is what the senses tell us, mediated by our prejudices and what we think we
know. Rationalists led by Cusa generally say that we can know what the senses indicate but
we need to understand the very restricted meaning of sensual testimony. However, using our
reason we can come to know what are the causes or causal principles P1 whose effects S1 the
senses detect. By understanding those causes well enough, perhaps we can design and build
an apparatus that allows us to observe them. Thus, through human ingenuity, P1 gives rise to
S2. We now perceive at a deeper level because “higher order” causes have been flushed out to
our senses. In turn, we shall find that sensory effects S2 are caused by principles P2. In
general, our Pn gives rise to Sn+1 through an artificial human intervention to generate the
sensory effect using the principles Pn. In the process of human ingenuity generating Sn+1, it
may also find a method or technique (“C”) of controlling those principles within certain
boundaries, so we get Cn+1 also. In symbols, we could write:
Pn → Sn+1 or Pn → Cn+1
238 This is contrary to the view of Leibniz’s mathematics teacher at the University of Jena, Weigel, who took the
view that the NMP was an extension of Aristotle. See Brown, R. C. Leibniz unpublished, Chapter 4 “A
Young Central European Polymath Between the Scholastics and the Moderns”, p.33 and Mercer, C. Leibniz’s
metaphysics: its origins and development Cambridge University Press 2001
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with each of the arrows representing an active role played by human ingenuity. If observation
were the only capability at our disposal, then neither of these two steps could occur.
This question is all the less frivolous when we note that, for the Empiricists, the causal
connection between the phenomena behind the perception and the perception itself:
Sn = effects(Pn)
is restricted territory in which the human mind has no business because any enquiry in that
domain is pure speculation and perhaps even fantasy. It is difficult to find a pure Empiricist.
For example, Newton, who will be discussed below in more detail, did permit some
consideration of the “effects( )” part of the above formula.
Religious or moral truth vs scientific truth
We now turn to the question of whether religious or moral truth differs from anything
discoverable in science through whatever means. The current Pope suggests that there is no
such difference, while acknowledging that there have been attempts to separate religious faith
from universal truths. In September 2006, Pope Benedict XVI said:239
Dehellenization first emerges in connection with the postulates of the Reformation in
the sixteenth century. Looking at the tradition of scholastic theology, the Reformers
thought they were confronted with a faith system totally conditioned by philosophy,
that is to say an articulation of the faith based on an alien system of thought. As a
result, faith no longer appeared as a living historical Word but as one element of an
overarching philosophical system. The principle of sola scriptura, on the other hand,
sought faith in its pure, primordial form, as originally found in the biblical Word.
Metaphysics appeared as a premise derived from another source, from which faith had
to be liberated in order to become once more fully itself. When Kant stated that he
needed to set thinking aside in order to make room for faith, he carried this programme
forward with a radicalism that the Reformers could never have foreseen. He thus
anchored faith exclusively in practical reason, denying it access to reality as a whole.
Similarly, Leibniz stated that truth must be consistent across all domains. In particular, he
included the toughest example of science and religion, saying that we should not accept for
religion what we would not accept in science. He was not referring to a standard of proof but
was saying that the actual content of ideas and doctrine should be consistent across science
and religion.
239 Papal Speech, Apostolic Journey of His Holiness Benedict XVI to München, Altotting and Regensburg 9-14
September 2006, Meeting with the representatives of science, Lecture of the Holy Father, Aula Magna of the
University of Regensburg 12 September 2006 “Faith, Reason and the University: Memories and Reflections”
accessed at http://www.vatican.va/holy_father/benedict_xvi/speeches/2006/september/documents/hf_ben-
xvi_spe_20060912_university-regensburg_en.html 1 May 2009
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Neoplatonism, Empiricism and Atheism
The scientific work of Kepler and Cusa was bound up with their theology; any inconsistency
between science and theology was anathema to them. Indeed, there was not really any
separation between the two for the discoveries of science were but a manifestation of true
theology. They would have been forced to revise their science or theology or both were an
inconsistency to arise; at least, they would have known that one or the other or both must be
wrong.
The Empiricist position neither confirms nor denies the existence of God. Thus, the Empiricist
is free to choose on the basis of personal preference whether or not they believe in the
existence of God.240 It might not be a coincidence that Empiricism was promoted by Venetian
gentlemen, such as Paolo Sarpi, Galileo’s patron, who were also inclined towards Atheism.
The consummate Neoplatonist is an a priorist who strives for a “God’s eye view”. That
Neoplatonist assumes his latest and best version of the “God’s eye view” in formulating
hypotheses within that pan-Creation context.
Many in the Empiricist school, even today, not only deny the necessity of a God or Creator in
science, but also deny that metaphysics has any relevance to science and some even claim that
metaphysics is meaningless word play.241 This is in itself a kind of metaphysics though its
advocates refuse to label it so. To say that science is the study of things that can be observed
and that nothing outside of the field of observation is a valid domain of enquiry at its simplest
level denies the validity of any metaphysical enquiry. How does one justify this position
without something like a metaphysical argument? Burtt points out that an attempt to exclude
metaphysics and not hold any metaphysical position itself carries a raft of embarrassing
metaphysical assumptions.
If sensory observation is all that we have, then we can only be aware of correlations and
coincidence of events. No pan-Creation view is possible; it would not make sense to ask what
is happening on the other side of the universe. Thus, the Empiricists differ from the
Rationalists in the nature of truth. For the former, truth is an ever-expanding basket of
correlations. Ultimate truth is all the correlations that exist. For the latter, truth is a principle
which is implied and necessarily so in the context of philosophy, metaphysics and theology. If
it is a principle of natural science, then it must have been confirmed by experiment whether
by direct observation or by deduction or calculation from observations. Ultimate truth is the
mind of God or the Creator of the universe.
Atheists had fewer difficulties in adopting an Empiricist outlook. The theology of Cusa and
Kepler did not permit an Empiricist outlook. However, even though the British Royal Society
240 By adopting the “practical reason” of the Empiricist, Kant declared that he needed to set thinking aside in
order to make room for faith. Kant is beyond the scope of this dissertation. However, we are here nudging up
against Pope Benedict XVI’s objection to Kant’s outlook on faith. Ibid. 241 For example, this is precisely the outlook of Ian Bryce who is on the executive committee of the Humanist
Society and is the founder of the Australian Secular Party. From conversation with Dr Ian Bryce of the NSW
Humanist Society at Eastside Radio 89.7 FM studio in Paddington on Friday 5 Feb 2010
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was generally Empiricist in its orientation, British thinkers such as Henry More, Robert Boyle
and Isaac Newton spent some effort in ensuring that their scientific outlook had some basis in
a theology they could believe in. Indeed, Newton wrote, “One principle in Philosophy is the
being of a God or Spirit infinite eternal omniscient omnipotent.”242 However, McGuire also
explains that the meaning Newton ascribes to the term “principle” in this context is different
from when he uses the term as having implications or repercussions for science.243 Boyle in
particular drew inspiration for his belief in God from his scientific insights though in a rather
gnostic way seeing the universe as divinely animated. Leibniz regarded a universe of that kind
as imperfect – and therefore impossible – because it would require ongoing participation by
its Creator. So “there is no soul of the universe” and “I hold … the hylarchic principle of
Henry More, as being either impossible or superfluous; and it is enough for me that the
mechanism of things [the universe] is constructed with so much wisdom that all these marvels
come to pass through its very development, organised beings being evolved, I think,
according to a preconceived plan.”244 Leibniz explained that he did not think that proponents
of the hylarchic principle had a distinct idea of their conception.245
Summary of the ’isms
We are now considering the converse of the earlier discussion on Empiricism neither
confirming nor denying Theism. Assuming the theology of Kepler and Leibniz, say, we have
said enough to formulate this summary:
Theism ==> ¬ Empiricism
Empiricism =/=> Theism
Empiricism =/=> ¬ Theism
Theism ==> Rationalism246
Rationalism ==> Theism
The diagram becomes more complicated when we consider different kinds of Theism, i.e.
different theologies. It would be interesting to consider the God of Newton, and how it differs
from the God of Neoplatonists such as Kepler. For example, is it hylarchic like the God of
Henry More? That discussion is beyond the scope of this thesis.
Conclusion
We have situated the idea of truth in each of the Rationalist, Neoplatonist and Empiricist
schools. Those of the Rationalist and Neoplatonist orientations (which are often found
together in any given thinker) have a better structured conception of the existence and
attainability of truth. This might be a result of their Theism. Empiricism does not necessitate a
242 McGuire, J.E. “Newton’s ‘Principles of Philosophy’: An intended preface for the 1704 Opticks and a related
draft fragment” The British Journal for the History of Science 1970 vol. 5 no. 2 pp. 178-186, at p.183 243 Ibid., p.180 244 1698, Wiener pp.138-9 245 “Reflections on the doctrine of a single universal spirit” 1702, Loemker, p.555 246 c.f. Wiener p.292 that God’s justice is not arbitrary, that God governs Himself by reason
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developed conception of a Creator or the Creator’s relationship to the physical universe and
so does not derive clarity on the whole of Creation as a single conception that the Rationalist
and Neoplatonist do. We do not say that such clarity would be impossible for an Empiricist to
attain. However, it would only be possible after further development of ideas that can only
arise from Rationalism and Neoplatonism which are not as tightly bound by the constraints of
sense perception or sense perception enhanced by scientific instruments. To be sure,
understanding how the Creator thinks is not easy but in making the attempt leaps of
understanding can occur. This sets forth the situation as it was until the early 18th Century.
Whether the attempt to understand how the Creator thinks is also required today may require
further enquiry. However, since the considerations and principles raised in this chapter are
universal and timeless, we would expect the same conclusions.
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Chapter 5: The Neoplatonist and Empiricist schools
Introduction
In the previous chapter, the understanding that structure is discoverable in the universe was
discussed. It was called the “new mechanical philosophy” (“NMP”) though it was often
anything but “mechanical” in the modern sense. Indeed, Kepler epitomized the origins of
NMP and his search, indeed expectation, was to find harmony and beauty in the structure of
the universe. Leibniz was a continuation of this new mechanical philosophy. Today, the idea
that the universe has a rational design is associated with physical experiments and observable
evidence in the tradition of Baconian Empiricism. On the other hand, Neoplatonism is
regarded as a mystical quasi-religious movement that lacks scientific credibility. The
orientation and methods of Neoplatonism were not always regarded in this way before
Leibniz. It was in Leibniz’s lifetime that the lines between Neoplatonism and Empiricism
started to be drawn. Leibniz adopted many of the methods of Neoplatonism but certainly
accepted the importance of experiment. In effect, Leibniz saw the pendulum swinging too far
towards what experiments can indicate to the senses, rather than using intellectual tools
informed but not dominated by observations. Because Leibniz saw the fork in the road and
made his position clear, much can be learned about Leibniz by considering the Neoplatonist
and Empiricist orientations and Leibniz’s relationship to them.
The reality and intellectual accessibility of concepts and “things” in a non-visible domain is
the nub of many debates between Neoplatonists and non-Neoplatonists. Concepts such as the
infinite exemplify what is unobservable. Friedrich Schiller (1709-1805) is one of Germany’s
most famous poets and playwrights. Schiller took the idea of the infinite seriously, and we
begin the chapter with Schiller’s thoughts on the topic as an example of the accessibility of
metaphysical ideas. Leibniz regarded mathematics as a subdomain of a domain of calculation
on metaphysical concepts. Any domain of mathematics that relies on the infinitesimal
presents itself as an example. To the layman, it might be paradoxical that “medieval
speculations” were instrumental in bringing about the concept of the infinitesimal used by
Barrow, Leibniz and Newton in their respective and evolving versions of the calculus.
As far as calculations of any kind go, they are tedious when they represent stepwise reasoning
using symbols to ensure precision. Leibniz saw the limitations and understood that such a
mode of reasoning was retarding the development of physics and engineering, because
available mathematical tools were time-consuming to use. Leibniz made reference to his idea
for a Universal Characteristic to help mathematics escape its limitations, but left it to future
generations to further build the idea on which he placed so much hope. Leibniz also short-
circuited stepwise reasoning with his maxim that perfect knowledge of a domain requires us
to see the whole in a single act of the mind or in a single thought.247 From there, one can lever
up to a higher conception that subsumes and transcends the existing understanding in a single
bound.
247 Wiener pp.77-78
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The distinguishing features of Neoplatonism have not been mainstream since the 17th Century.
Further, significant developments are rare adding to the obscurity. It is true that forming
hypotheses is a distinguishing feature of Neoplatonism. However, Neoplatonists tend to
hypothesise based on their conception of the Creator’s mind in relation to the entire universe,
rather than on considerations attained through observation alone. Once an initial hypothesis is
formed and work to verify it begins, the process of observation, calculation and number-
crunching is indistinguishable from any other mode of science. Thus, the Neoplatonic
scientist must be as concerned about precision as any other scientist. We will discuss Kepler
and his Neoplatonist orientation. Franklin notes that Kepler’s refusal to compromise with
precision is as impressive as that of any scientist of his age and perhaps ours.248
The rest of the chapter will be spent discussing the development of the Empiricist school prior
to Leibniz’s lifetime. The British Royal Society typifies the Empiricist school today but we
argue that this is not where Empiricism began. The ideological takeover of the University of
Padua – which was once a centre of Neoplatonism – by Averroism may have been
instrumental in nurturing the early stages of what became the “Empiricist revolution”.249
However, we will not go back that far. We start with Paolo Sarpi of Venice, a contemporary of
Galileo, though 10 years elder to Galileo. England was heavily influenced by Italy, and it was
standard for the scions of wealthy English families to undertake a “grand tour” of the
Continent including Italy and especially Venice. Sarpi’s writings were often translated into
English and published in England, though we have no evidence of how widely read they
were. We discuss how Sarpi’s atheism was naturally allied to his tendency to Empiricism.
Empiricism leads to rigorous deductivism, hence the double-barrelled term “empirico-logic”.
Any discussion of influences on science in the British Isles in that period leads to Newton, so
we will discuss Mamiani’s discovery of the close parallel between Newton’s prescribed
method of discovery and a standard logic text of the period. Newton’s opposition to forming
hypotheses is a natural part of an empirico-logical stance, but it contrasts with the approach of
Neoplatonic scientists like Kepler.
Next, we shall discuss the consequences of mandating empirico-logic as a scientific method.
There were members of the Republic of Letters who criticised Leibniz’s calculus on the
grounds that it lacked precision. Leibniz openly disagreed with them and used the label
“rigorists” for such writers. However, the fact that there were such critics is not surprising
248 Franklin, J. The Science of Conjecture: Evidence and probability before Pascal J. H. Press Baltimore and
London 2002, pp.150-1 249 “In the course of the 14th Century, European primacy in Averroan studies passed from the University of Paris
… to the great Italian schools: Bologna and especially Padua.” Carboni, S. Venice and the Islamic world,
828-1797 Yale Univesity Press 2007 p.153. The influence of Padua on English intellectual life is explained in
Woolfson, J. Padua and the Tudors: English students in Italy, 1485-1603 University of Toronto Press 1998. A
Princeton University webpage for second year undergraduate history says, “The University of Padua was one
of the most prominent universities in early modern Europe, known particularly for the rigor of its
Aristotelean logic and science.” accessed at http://www.princeton.edu/~his291/Padua.html 10 May 2011
While Thomas Aquinas would disagree with Averroes calling himself an Aristotelean, Averroan studies are
regarded as a development of Aristotelean logic, and Averroes was regarded as a Neo-Aristotelean.
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given the dominance of Averroism in such eminent institutions as the University of Padua.
Finally, we close with a couple of open questions that would benefit from future research.
Schiller and the infinite
Friedrich von Schiller (1759 – 1805) is one of Germany’s most famous poets and playwrights.
Seventy-seven years after Leibniz’s death, Friedrich Schiller distinguished the logical
estimation of magnitude from the aesthetic.250 Estimating a magnitude in a “logical fashion”
is to relate it to the “cognitive faculty”, experience something about the object and behold
something outside oneself. To estimate something aesthetically is to relate it to the faculty of
“sensibility”. Contrary to the logical fashion, in aesthetic estimation one experiences
something within themselves caused by the imagined magnitude of the object. In this case, the
thinker is neither measuring nor estimating magnitude, but themselves become for the
moment an infinite magnitude to themselves. Schiller writes that anything that evokes the
(intellectual) experience of being an infinite magnitude to the intellect is sublime.251
Schiller’s distinction between logical versus aesthetic estimation is the same as that between
logical deduction and Neoplatonic reasoning. This “aesthetic” is not merely a quality of being
subjectively pleasing to the mind, but is a universal quality. While Schiller uses the concept of
the infinite with in the individual thinker as the yardstick, Schiller’s central point is that that
measure is itself universal.252 While Schiller did not use the term “Neoplatonic”, the
Neoplatonist encourages abstraction “in the sphere of the infinite”.253 This is the sphere of
truth, whether it is classified as metaphysical, mathematical, moral, artistic, musical or
otherwise; after all, these are human categories and Schiller would argue that truth does not
know the separations between these categories. The corporeal human condition inevitably
“resumes her rights to give an imperious reality to our existence, to give it contents,
substance, knowledge, and an aim for our activity.” This “concretisation” of
realisations/discoveries brings new understanding out of the domain of the infinite and into
the temporal; the result cannot be universal or eternal though it may be useful for a few
centuries or millenia.
The Neoplatonist acknowledges that the domain of the infinite is accessible to the human
mind. The Neoplatonist goes further and, like Schiller, says that discoveries are only worthy
of the name if they do pluck something new from that domain. Such a discovery is generally
useful to the bulk of humankind precisely because it brings a new principle into the corporeal
domain where “the rest of us” can work with it.
250 Friedrich Schiller, Johnson, S. trans. Of the Aesthetic Estimation of Magnitude (1793) Accessed at
http://www.schillerinstitute.org/transl/Schiller_essays/magnitude.html 1 Jan 2011 251 Schiller was responding to Kant’s The Critique of Pure Reason first published 12 years before Schiller’s
essay. Accessed at http://www.gutenberg.org/cache/epub/4280/pg4280.html 1 Jan 2011 252 Schiller uses this to posit a universal and objective quality of beauty which is equated to absolute truth, in
opposition to Kant’s position that beauty is subjective. In Schiller’s Letter XII, he refers to a “unity of idea”
according to which “We are no longer individuals but a species”. See next footnote. 253 Friedrich Schiller, edited by Riikonen, T. and Widger, D. Aesthetical and Philosophical Essays, Letter XII
Accessed at http://www.gutenberg.org/files/6798/6798-h/6798-h.htm on 1 Jan 2011
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Metaphysical calculations and a method of Analysis for concepts in physics
The calculus is founded on the concept of the infinitesimal which might have an ontological
reality but has no physical reality. Therefore, the calculus is a tool of precision in metaphysics
before it is a mathematical tool. We may call it a mathematical tool because it brings ideas
from the metaphysical domain into a form which humans, or computers, can systematically
work with. However, the crux of the calculus is metaphysical before anything else. It is also
outside the domain of any possible empirical enquiry, for it allows the expression of a
principle about a curve which cannot be seen from the curve itself. It embodies the idea of a
hidden underlying principle and provides a method for dealing with such principles in a
systematic way. Today, we take the idea of instantaneous rates of change and all that flows
from them for granted, but in Leibniz’s time none of it was obvious.
Leibniz regards the relations of forms and abstract quantities as belonging to Metaphysics,
which can be structured into a calculus or Universal Characteristic. He regards Combinatorial
Science as an application of such Universal Characteristic. In turn, he says “all of Algebra is
merely an application of a Combinatorial Science of quantities.”254 According to Leibniz,
Algebra is a particular class of methods of calculation within Metaphysics.255
If Algebra is just a subdomain of a subdomain of Metaphysics, and if all of Metaphysics is
subject to structured Analysis, we would expect physics – a consequence of Metaphysics – to
be subject – or at least amenable – to structured (a priori) reasoning. We will see below that
this is exactly what Leibniz believed. It is not surprising that Leibniz placed experiment a
long way below Reason in importance.
Decades before Leibniz was born, Tycho Brahe and Johannes Kepler had done significant
work with mathematics in the sense of “number-crunching” to test hypotheses. We know that
Kepler was as ambivalent as Leibniz about the infallibility or ultimacy of mathematical
methods; that is, Kepler did not come close to claiming that the entire universe could be
reduced to, or understood via, mathematical calculation. For Leibniz, the domain of structured
Reasoning was much broader than mathematics. Leibniz says that “there is an art of Analysis
which is more inclusive than Mathematics” and that he has a proof that such an art or method
exists.256 This method turns out to be a framework for reasoning about matters metaphysical
and physical based on premises which are, at least in part, a result of a priori metaphysical
thought.
254 Wiener, p.210 255 It is this distinction between mathematics and a more general Analysis that Johnson does not recognise.
Johnson seems to think that where the usefulness of Mathematics ends for Leibniz, experiment begins.
However, for Leibniz, mathematics is only one of many tools for Analysis. In turn, Analysis is one of many
modes of human Reason. Johnson, A. H. “Leibniz’s method and the basis of his metaphysics” Philosophy
1960 vol. 35, 51-61. See page 54. 256 1715, Wiener, p.201
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Invaluable medieval speculations
Prior to Brahe and Kepler, Boyer says that the medieval period added little to the classical
Greek works in geometry or algebra. Rather, it contributed, chiefly, “speculations, largely
from the philosophical point of view, on the infinite, infinitesimal, and continuity” as well as
“new points of view with reference to the study of motion and variability. Such disquisitions
were to play a not insignificant part in the development of the methods and concepts of the
calculus”.257
One of the most significant of the “medieval speculators” was Nicolaus of Cusa of whom
Kepler was a disciple although a century separated them. “The fullest expression of Nicolaus
of Cusa’s mathematical thoughts on the infinite and the infinitesimal, however, are found in
the work of Johannes Kepler, who was strongly influenced by the cardinal’s ideas … and who
was likewise deeply imbued with Platonic and Pythagorean mysticism. It was probably the
imaginative use by Cusa of the concept of infinity which led Kepler to his principle of
continuity.”258 Kepler wrote of Cusa others divinus mihi Cusanus, i.e. “Cusa and others seem
to me divine” in drawing the analogy of the circle compared with polygon to God compared
with his creatures.259 It was Cusa, says Boyer – albeit himself rigorist in his sympathies – who
led Kepler to include normal and limiting forms of curves under a single definition of
continuity encompassing conic sections as a single family of curves.260
Leibniz saw the need for more powerful methods
To Leibniz, mathematics was the art of reasoning using symbols to represent quantities.
Symbols could be used for many other purposes.261 Leibniz’s recognition that attention to
detail sufficed for calculations to be carried out indicates that he did not think particularly
highly of the creative or intellectual content of the calculation process.262 “For it is unworthy
of excellent men to lose hours like slaves in the labour of calculation which would safely be
relegated to anyone else if machines were used.”263 Rather, the real achievement was in
257 Boyer, C. The History of the Calculus and its Conceptual Development Dover Publications, New York 1949,
p.94 258 Ibid., p.93 259 Duncan, A.M. (trans.) Kepler, J. Mysterium Cosmographicum Abaris Books, Janus Series Norwalk USA
1981, Chapter II “Outlines of the primary derivation” p.93 260 Boyer quotes from Kepler’s Opera omnia II 595, translated from the quoted Latin by this author, “we find
that a straight line is an hyperbola obtuse in the extreme. And from Cusanus we learn that a circle is an
infinite linear thing. They are several things simultaneously, not discrete alternatives, whose different faces
are turned to light by the use of analogy.” Ibid. 261 post-1690 in Wiener, pp.74-5 262 This is pretty close to the heart of Leibniz’s critique of Descartes’ subjectivism. Quite simply, “Yes, humans
make mistakes, but that does not mean that reality is subjective.” As Raynaud wrote, “Methodological doubt
[for Leibniz] does not have the weight which Descartes accords to it.” in McCarthy 1998, p.152 referring to
Leibniz’s response to Article 5 of Descartes’ Principles of Philosophy. This response by Leibniz is in
Loemker 1969, p.384 263 Leibniz, G.W. 1685 “Machina arithmetica in qua non additio tantum et subtractio sed et multiplicatio nullo,
divisio vero paene nullo animi labore peragantur” Kormes, M. (trans.) “Leibniz on his calculating machine”
in Smith, D. E. A Source Book in Mathematics Dover Publications, Mineola N.Y. 1959 (first published by
McGraw-Hill 1929), p.181
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devising the systems and methods for calculation, which could then be exercised in a
relatively straightforward way and perhaps even automated in the future. Leibniz had made a
contribution to such automation by designing the “Step Reckoner” calculating machine in
1671 which was built in 1673.264
Reason gives value to observations
Without effective reasoning, even plentiful observations are worth little.265 Thus, Leibniz saw
the need for an ability to comprehend “the whole” with mathematical precision in order to
derive advantage from new observational technologies. Leibniz wrote:266
I have no hope that we can get very far in physics until we have found some such
method of abridgement to lighten its burden of imagination. For example, we see that
a series of geometrical reasoning is necessary merely to explain the rainbow, one of
the simplest effects of nature; so we can infer what a chain of conclusions would be
necessary to penetrate into the inner nature of complex effects whose structure is so
subtle that the microscope, which can reveal more than the hundred-thousandth part,267
does not explain it enough to help us much. Yet there would be some hope of
achieving this goal, at least in part, if this truly geometrical analysis were established.
Leibniz is referring to the absence of a tool that would allow us to calculate the inner
workings of nature. For Leibniz, too much imagination is required and far too many detailed
experiments. Since nature was constructed rationally, such a tool must be available. This
returns us to Leibniz’s high hopes for the concept of the Universal Characteristic. The idea is
that all concepts in science are representable by symbols which can then be manipulated to
locate new truths.268 The use of symbols helps enforce precision. “[T]he best advantage of
algebra are only samples of the art of characters whose use is not limited to numbers or
magnitudes.”269 It is clear that the emphasis is on introducing precision and adding tools to the
arsenal of methods available to the reasoning process. Leibniz’s Universal Characteristic
might today go some way towards “a unified theory that covers mathematics, pure and
applied, as well as the formal sciences.”270
264 Leibnitia webpage http://www.gwleibniz.com/calculator/calculator.html accessed 7 August 2010. At the IBM
Archives it is described as “a major advance in mechanical calculating. The Leibniz calculator incorporated a
new mechanical feature, the stepped drum — a cylinder bearing nine teeth of different lengths which increase
in equal amounts around the drum. Although the Leibniz calculator was not developed for commercial
production, the stepped drum principle survived for 300 years and was used in many later calculating
systems. ” http://www-03.ibm.com/ibm/history/exhibits/attic3/attic3_037.html accessed 7 August 2010 265 Wiener, last para. p. xxiii and first para. p.xxiv. Also Book IV, Chapter XII, §13 New Essays on Human
Understanding in Wiener, p.479 266 Loemker, p.250 (Letter to Huygens, 1679 around two years after Leibniz’s initiation into Huygens’
experiments with the [French] Royal Academy of Sciences) 267 The microscope was central to the work of Hooke in his role as demonstrator at the British Royal Society.
Arnol’d, V.I. Huygens and Barrow, Newton and Hooke Birkhauser Verlag, Basel 1990, pp.11-14. 268 Wiener, pp. 73-4 269 Ibid., p.74 270 Franklin, J. “The Formal Sciences discover the Philosopher’s Stone” Studies in History and Philosophy of
Science Vol. 25 No. 4 1994 pp.513-533, p.25 of online pdf
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Leibniz affirmed that ultimately the universe could be explained through analytical methods
of some kind.271 Yet Leibniz saw too great store being set by formal reasoning using the
inadequate tools of logic which were then available. Penetrating the “inner nature of complex
effects whose structure is so subtle that the microscope, which can reveal more than the
hundred-thousandth part, does not explain it enough to help us much” would require
something far more powerful than the syllogism. Indeed, it might need to be a “truly
geometric analysis”. These cumbersome methods were represented by the positivism through
formalist logic which had been gaining primacy since Averroes in the 12th Century.272 Thomas
Aquinas wrote a critique of Averroes’ interpretation of Aristotle, disputing metaphysical
fundamentals such as the relationship between the intellect and the soul rather than logical
methods. Whether Averroes sought to expound on Aristotle faithfully or whether he
deliberately added a little too much positivism of his own is beyond the scope of this thesis.
That mathematics is an indispensable tool in science is self-evident to 21st Century readers.
But if we are saying that some mathematical tools are really just a kind of metaphysics, then
more explanation is needed.
The prevailing view that science stands independent of metaphysics contrasts with the outlook
of Leibniz and his forebears in the Humanist Renaissance. For Leibniz, science as it is done
by humans is a temporary understanding to be replaced by a future, richer understanding.273
This is somewhat like Cusa’s doctrine of docta ignorantia or “learned ignorance”. Physical
phenomena are a manifestation of an underlying metaphysical reality that is dynamically in
progress, and is only discoverable by the mind. What is the nature of the underlying
metaphysical reality? We will never know it completely, but the little we can know about its
abiding nature flows from our reasoning about God and God’s expression in Creation. Thus,
the Neoplatonic scientist necessarily is an a priorist.
This thesis is not alone in pointing out and supporting Leibniz’s preference for a priori
explanation merely provides description. In the science of motion, Leibniz regarded rational
analysis as superior to experiment. Indeed, “experiment must be eliminated from the science
of the abstract reasons for motion, just as they should be eliminated from geometrical
reasonings. For they are demonstrated not from fact and sense, but from the definition of the
271 “nor do I know why you should consider as most absurd the view that everything happens mechanically in
nature, that is, according to certain mathematical laws prescribed by God.” Leibniz, G. W. “Letter to Herman
Conring” 19 March 1678, in Loemker, L. G. W. Leibniz Philosophical Papers and Letters 2nd ed. Reidel
Publishing Company 1969, p.189 272 Petrarca was an outspoken and influential critic of Averroism as well as of stepwise reasoning as practised
under the heading of “dialectic”. Petrarca “On his own ignorance and that of many others” c.1368, pp.47-133
in Cassirer, E., Kristeller, P. O., Randall, J. H. Jr (eds and trans.) The Renaissance Philosophy of Man
Phoenix Books, University of Chicago Press 1948 and Brand, P. Cambridge History of Italian Literature
Cambridge University Press 1999, p.134. Cusa continues Petrarca’s programme of demonstrating that
intellect as exercised by an intelligent commoner can be more effective than the formal reasoning promoted
by the scholastics.. 273 Post-1690 “On the horizon of human doctrine”, Wiener pp.76-77
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terms.”274 Brown mentions this without criticism. Indeed, he calls Leibniz’s quasi-a priori
explanation of elasticity in collisions ingenious.275
Kepler, Neoplatonism and precision
Much of this was said by Kepler in his introduction to Mysterium Cosmographicum in which
his statements on science make it clear that he is the consummate Neoplatonist. Neoplatonism
is – among other things – a mindset, and there were areas with greater concentrations of
“practitioners” and sympathisers than others. Burtt tells us that Copernicus sought the
“southern” intellectual influences.276 Perhaps this was because he was already interested in
the Neoplatonic outlook which had many adherents in that region of Europe.
There is a misconception that Pythagoreanism and Neoplatonism are synonymous with
mysticism, particular interpretations of Cabala and occult sects.277 There may have been such
offshoots, but as far as leading Neoplatonists from the 15th Century onwards are concerned,
nothing could be less correct. Their theology caused them to shun “mystical arts” as sophistry
at best and Satanic at worst.278
Copernicus himself was decisively Neoplatonic.279 In the decades after Copernicus lived, it
was only arch Neoplatonists who carried his standard. An example is Giordano Bruno whom
no-one could disagree was an arch Neoplatonist:280
In his work The Ash Wednesday Supper, a story of a private dinner, being entertained
by English guests, Bruno spreads the Copernican doctrine. A new astronomy had been
offered the world at which people were laughing heartily, because it was at variance
with the teachings of Aristotle. Bruno was carrying on a spirited propaganda in a
274 From a preliminary study towards his Theoria Motus Abstracti. Leibniz, G.W. Sämtliche Schriften und Briefe
Deutsche Akademie der Wissenschaften zu Berlin (eds) Akademie-Verlag, Berlin, 1923 Series 6, volume 1,
p.160, translated in Garber, D. Leibniz: body, substance, monad Oxford University Press 2009, p.15 275 Brown, R. C. Leibniz unpublished, Chapter 4 “A Young Central European Polymath Between the Scholastics
and the Moderns”, pp.41-42 276 Burtt, Chapter II “Copernicus and Kepler” p.22ff 277 Stuart Brown addresses this misconception. Brown, S. “Leibniz and Berkeley: Platonic Metaphysics and ‘The
Mechanical Philosophy’” Chapter 16 in Platonism at the Origins of Modernity: Studies on Platonism and
Early Modern Philosophy Springer 2008, pp.239-253. Edward Rosen’s writing has done something to help
perpetuate the misconception. See Rosen, E. “Was Copernicus a Neoplatonist?” Journal of the History of
Ideas Vol.44, No.4 (Oct-Dec 1993), pp.667-9. Rosen’s argument that Copernicus was not a Neoplatonist is
based on the misconception of Neoplatonism as a mystical doctrine that “all reality has its source in the
transcendent One, which produces a series of less unified levels of being, down to the last and lowest, the
physical universe, a living creature endowed with a divine soul; at our highest, we humans can join the One
in a mystical union.” This sounds more like the doctrine taught in a New Age yoga class. We wonder whether
Rosen merely has an innocent misconception, because apparent bias – not only against Plato but against
Christianity – is reflected by his opening sentence, “Christian thought was profoundly influenced by the
ancient pagan Greek philosopher Plato.” 278 Leibniz wrote, “There are some who imagine a world of light in their brains.” However, “this is not the light
but only a heating of their blood.” Loemker p.367. Also see “On the general characteristic” in Loemker 1969,
p.221. 279 Kuhn, T. S. The Copernican Revolution Harvard University Press 1957 280 Kessler, J. J. PhD “Giordano Bruno: The Forgotten Philosopher” accessed at
http://www.infidels.org/library/historical/john_kessler/giordano_bruno.html 10 May 2011
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fighting mood. Between the year[s] 1582 and 1592 there was hardly a teacher in
Europe who was persistently, openly and actively spreading the news about the
‘universe which Copernicus had charted’, except Giordano Bruno.
How could a work so precise and ultimately so successful in describing physical phenomena
as that of Copernicus have been carried out by a Neoplatonist? This question only needs to be
asked if one has a preconception that Neoplatonism is a mystical and “unscientific” set of
doctrines. It would make no sense to even ask whether precision and calculation figures in
such a set of doctrines.281 As mentioned, this is a misconception.
The need for exactness is outlined by Burtt:282
As with Kepler, so with Galileo, this mathematical explanation of nature must be in
exact terms; it is no vague Pythagorean mysticism that the founder of dynamics has in
mind.
As an aside, Leibniz addressed this directly in c.1679, “Men have been convinced ever since
Pythagoras that the deepest mysteries lie concealed in numbers. It is possible that Pythagoras
brought over this opinion, like many others, from the Orient to Greece. But, because the true
key to the mystery was unknown, more inquisitive minds fell into futilities and superstitions,
from which there finally arose a kind of popular Cabala, far removed from the true one, and
that multitude of follies which is falsely called a kind of magic and with which books have
been filled.”283 This confirms Stuart Brown’s analysis in that Leibniz attacks what are
popularly – but incorrectly – thought to be elements of Platonism even today.
Galileo says that the peripatetics, especially, have written great volumes on the problem of
falling bodies and yet have never made their understanding exact. Indeed, the idea that
quantification is the only way of defining a phenomenon precisely is Neoplatonic. This is not
surprising considering that the reality of numbers was maintained by the Pythagoreans against
opposition. Thus, the idea of measurement ultimately is Neoplatonic.284 Galileo himself, a
pioneer of precise experimental physics and who is not normally classified as a Neoplatonist,
is said to have been “supported by the onrushing Pythagorean tide”.285,286 Leibniz’s search for
precision and the idea of a Universal Characteristic for precise and systematic reasoning is
consistent with this.
281 Rosen, E. “Was Copernicus a Neoplatonist?” Journal of the History of Ideas Vol.44, No.4 (Oct-Dec 1993),
pp.667-669 282 Burtt, E.A. The Metaphysical Foundations of Modern Physical Science Routledge London 1932 2nd ed.
reprinted 1950, p.70 283 “On the general characteristic” in Loemker , L. E. Gottfried Wilhelm Leibniz: Philosophical Papers and
Letters 2nd ed. D. Reidel, Dordrecht Holland 1969 1969, p.221 284 Hence, today’s heavily quantitative society and culture reflect a heavy influence of Neoplatonism in our
heritage. 285 Burtt, E.A. The Metaphysical Foundations of Modern Physical Science Routledge London 1932 2nd ed.
reprinted 1950, p.197 286 Galileo’s debt to Pythagoreanism is explained in Demarco, D. “The dispute between Galileo and the Catholic
Church” Homiletic and Pastoral Review 2002 accessed at
http://www.catholiceducation.org/articles/science/sc0043.htm 26 May 2011
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The Empiricist school and the influence of Paolo Sarpi
Paradoxically, it is with the non-supporters of Renaissance Neoplatonism that precision and
numerical methods are usually associated – that is, with the Empiricists. We introduce Paolo
Sarpi, Chief Theologian to the Republic of Venice. Paolo Sarpi is an example of the
importance of the intentional view of history: that it is difficult or impossible to make sense of
historical processes and the decisions of historical figures without having some grasp of their
intentions and the intentions of the cultural and political framework in whose thrall they
acted.
Contrasting with Kepler, in Sarpi’s statements on science he declares himself a pure
Empiricist. Sarpi sat at the centre of a network of influence with the blessing of the Republic
of Venice whose power was waning but which was a significant financial and political force
in Europe.
Sarpi and Galileo
Galileo was personally close to both Kepler and Sarpi. Galileo was friends with Kepler as a
student and they continued their correspondence following graduation. Eventually, Sarpi
became Galileo’s friend and patron.287 Rather than being a disinterested source of funds for
budding scientists, Sarpi had his own philosophy of science and made a clear statement which
sounds a little like Newton who was not to enter the scene until nearly 75 years later. Sarpi
says that potentially there are four possible ways of gaining new knowledge:
1. pure reason
2. pure perception
3. perception followed by reason
4. reason followed by perception.288
Sarpi rejects 1 (pure reason) as pure speculation/fantasy. He rejects 2 (pure perception) as
preventing the obtaining of meaning. He rejects 4 (reason followed by perception) as
prejudicing perception with advance speculation based on air. Kepler addressed option 4 by
saying that hypotheses should not be formulated by idle speculation. Meli explains that
Kepler imposed constrains on how hypotheses are formed. In Epitome Astronomie
Copernicae Kepler wrote, “astronomers should not have absolute freedom to think up
anything they please without reason; on the contrary, you should give causae probabiles for
your hypotheses which you propose as the true cause of the appearances, and thus establish in
advance the principles of your astronomy in a higher science, namely physics or
metaphysics.”289
287 Burtt, E.A. The Metaphysical Foundations of Modern Physical Science Routledge London 1932 2nd ed.
reprinted 1950, Chapter III “Galileo” p.63 and Robertson, A. Fra Paolo Sarpi: The Greatest of the Venetians
George Allen & Company, London 1911, pp.73-75 288 Wootton, D. quoting from Sarpi’s Pensiero no.146 in Paolo Sarpi: Between Renaissance and Enlightenment
Cambridge University Press, 2002, p.37 289 Quoted in Meli, D.B. Equivalence and Priority: Newton versus Leibniz Clarendon Press, Oxford 1993, pp.22-
23
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Sarpi prefers 3 (perception followed by reason) because it starts with what is real and then
allows the derivation of meaning from it. We must ask what Sarpi meant by “reason” in 3;
was it the same as Newton’s prescription of induction from experimental observation or
testimony of the senses?
We also need to ask why Sarpi was choosing between these four options. If he was aware of
Galileo’s ongoing correspondence with Kepler, then he would certainly have had been
interested to look into the work of the energetic and capable Johannes Kepler. Kepler’s
Mysterium Cosmographicum was first published in 1596 while Paolo Sarpi became state
theologian to Venice 10 years later. Given Kepler’s tactful questioning of the church’s
orthodoxy on the movement of the sun, and Sarpi’s own atheism (curious for a theologian),
Sarpi must have been able to understand that the Neoplatonic school itself had no respect for
church dogma even with its mindset of a divine Creator which Sarpi found necessary to reject.
Sarpi wrote of his fears for Galileo in being called to Rome by certain cardinals to explain his
support for the Copernican theory.290 At the same time, Venice was seeking to neutralise the
influence of the Catholic Church.291 (Wootton refers to Sarpi as a “Protestant conspirator”.292)
Thus, Sarpi saw the opportunity to stoke conflict between forces that arose from rationality
(represented in part by the cultural tide that Kepler merely typified) – which he opposed – and
the worldly authority and wealth of the Roman Catholic Church which also represented the
Hellenic cultural current albeit largely corrupted by ambitious or weak-minded sections of its
officers (i.e. clergy, etc).
We are seeing that cultural currents were at play independent of the influence of any particular
figure, but which could be influenced or capitalised upon by shrewd, committed or capable
persons. Kepler, Sarpi and Leibniz are examples of such persons; chronologically, Sarpi sat
between the other two. Sarpi was a contemporary of, though elder to, Kepler while Leibniz
was born after Sarpi had passed on. Sarpi lived amidst the conditions which gave rise to the
Thirty Years War, while Leibniz’s educators (particularly Jakob Thomasius), wider network
and Leibniz himself dealt with that war’s aftermath both politically and philosophically.
Sarpi’s influence in England
Sarpi corresponded with Francis Bacon who acted as an inspiration for the founders of the
English Royal Society. Indeed, “[l]etters are still extant showing his [Sarpi’s] friendship with
Lord Bacon”.293 Perhaps some of the lost letters would shed light on the origins of the Royal
Society, an organisation of political as well as scientific importance to this day. The
connection is worth looking into because the Venetian Party was an established force in
English politics at the time, and was instrumental in what became the British East India
290 “Per mia memoriae” (“From my memory”) Schedae Sarpianae (Writings/Manuscripts of Sarpi) in Robertson,
A. Fra Paolo Sarpi: The Greatest of the Venetians George Allen & Company, London 1911, pp.73-75 291 Sarpi himself was excommunicated in 1607 by the Inquisition. Robertson, A. Fra Paolo Sarpi: The Greatest
of the Venetians George Allen & Company, London 1911, p.152 292 Wootton, p.46 quoting from Sarpi’s Pensiero no.146 293 Robertson, A. Fra Paolo Sarpi: The Greatest of the Venetians George Allen & Company, London 1911 p. 84
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Company. Paolo Sarpi was the most frequently translated Italian writer in 17th Century
England,294 which indicates quite an influence given that the English looked to Italians for
intellectual leadership in that period.
Given Sarpi’s relationship to Galileo and indirect relationship to Kepler and his stance on how
science should proceed, it would be consistent that he should try to influence the course of
science. It might not be coincidental that the Baconian approach to science which came to
dominate the Royal Society is so closely allied to Sarpi’s empiricist outlook.
Leibniz was picking up the pieces of Kepler’s approach to science and carried that standard,
even while Empiricism rose in prominence particularly in the circles of the Royal Society
perhaps in the wake of the influence of Sarpi.
Sarpi’s closeness to Galileo
We talk about Sarpi because scientific and political battles often go together. Venice was a
political and financial centre of Europe during the 15th through 17th Centuries. Sarpi was
Galileo’s patron and was said to be the first to look through Galileo’s telescope. Robertson
says that Galileo and Sarpi actually constructed the telescope together, and that that telescope
was then presented to the Doge of Venice, Leonardo Donato, as a gift.295 Indeed, Sarpi was, at
least initially, a mentor to Galileo in astronomy “as he [Sarpi] had had the start ... of his friend
[Galileo] in the study of astronomy and its cognate sciences, the advantage lay with him
[Sarpi]”. However, “Galileo, instead of being jealous of Fra Paolo, was jealous only for his
honour and pre-eminence, calling him ‘Il mio pare e maestro’ – ‘My father and my
master.’”296
Sarpi is said to have gathered a group of young noblemen around him and secretly
communicated with them his conception of Atheism. Further, patrons are rarely dispassionate
about or detached from the work they finance, including Venetian patrons. The fact that Sarpi
was chief theological advisor to the Venetian republic while maintaining a secret Atheism
makes him particularly curious. Wootton argues that Sarpi did try “to give practical
implementation to ideas he had expressed in private: his political activities were intended to
subvert religious authority in general, his published works were intended to undermine the
foundations of religious argument.”297
We must ask whether Sarpi was privy to any of the correspondence between Kepler and
Galileo. We suggest only a communication of methods and mindset. Given the fundamental
differences between Kepler’s outlook on science and God, and Sarpi’s, and the Venetian
294 Tedeschi, J. (review) “Venetian Phoenix: Paolo Sarpi and Some of His English Friends (1606-1700) by John
L. Lievsay” Modern Philology Vol. 75, No. 2 (Nov., 1977), pp. 191-194 295 Robertson, A. Fra Paolo Sarpi: The Greatest of the Venetians George Allen & Company, London 1911 pp.
73-74 296 Ibid., p.73 297 Wootton, D. Paolo Sarpi: Between Renaissance and Enlightenment Cambridge University Press, 2002, p.46
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Republic’s reputation for intrigue,298 the closeness of Sarpi to Galileo and the friendship
between Galileo and Kepler warrants more research.
Sarpi and atheism
Pietro Pompanazzi is an important intellectual figure of the 16th Century. For our purposes
here, Pompanazzi’s is relevant to one particular issue. In The Treatise on the Immortality of
the Soul (1516) Pompanazzi argued that, to a degree, the soul is inseparable from the body
and to that extent the soul is mortal.299 This was at odds with Thomas Aquinas’ reading of
Aristotle as consistent with Christian theology.
Wootton explains that Sarpi’s Pensiero no. 4 reproduces Pompanazzi’s argument to
demonstrate the mortality of the soul.300,301 Averroes argues that humans participate in an
immortal reason and so the rational was essentially, or behaved as if it was, immortal although
it is actually mortal. Pompanazzi said that man is a mean between mortal and immaterial
things. However, his essay Immortalitate Animae refutes every argument it puts in favour of
the immortality of the soul302 and ends with an almost patronising sop to the Platonic and
Christian belief in the immortality of the soul.303
Wootton says that Sarpi’s Pensieri adopted Pompanazzi’s arguments against the immortality
of the soul, and then makes the case that Sarpi was relatively isolated in holding his views as
outlined in the Pensieri.304 Wootton misunderstands the meaning and role of “political views”.
It is not necessary for many/any to adopt those views overtly for them to be used and even
provide a steering role for the Council, particularly if Sarpi was important in Venetian
political circles and especially if the views expressed were of value to Venice.
Some of Sarpi’s contemporaries were awake to what he was doing and were unafraid to speak
out. For example, Tommaso Campanella was an anti-Venetian polemicist who argued that the
inevitable outcome Venetian policy was a society of moral atheists. Wootton explains that
Campanella’s warning against Sarpi’s methods provides a direct link between Sarpi’s Pensieri
and the principles adopted in/by the Venetian policy during the Interdict.305
298 For example, see any English edition of The Ghost Seer by Friedrich Schiller and The Bravo by James
Fenimore Cooper. Also refer to Othello: Moor of Venice by William Shakespeare wherein Venetian methods
are represented by Iago. 299 Perfetti, S. “Pietro Pomponazzi” 2004 in Stanford Encyclopedia of Philosophy. Accessed at
http://plato.stanford.edu/entries/pomponazzi/ on 28 April 2012 300 Wootton, p.41 301 See de Immortalitate Animae by Pompanazzi, chapter 4 302 Pompanazzi, P. Immortalitate Animae “On the immortality of the soul” in Cassirer, E., Kristeller, P. O.,
Randall, J. H. Jr (eds and trans.) The Renaissance Philosophy of Man Phoenix Books, University of Chicago
Press 1948, pp.280-380 303 Ibid., pp.380-381 304 Wootton, p.47 305 Ibid., p.46
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Sarpi and Newton
Whatever Sarpi’s influence on Galileo may have been, we certainly find much of Sarpi in
Newton. Reading the account of Galileo’s conception of simplicity in nature following the
account of Kepler, we begin to see that the core of Newton’s scientific method was not new.
However, its Averroism and its Atheism were new. We say Averroism because Newton
required that each conclusion be based strictly on observations without leaps of faith or leaps
of understanding even if the intention is to test that leap by experiment. No conclusion is
allowed unless it strictly and evidently follows from what is observed. This is a kind of
Averroist rigor, and it was a long way from Galileo’s scientific method. Burtt explains that
Galileo said that logic is no tool for discovery and, indeed, Galileo is not a priori to the
degree that Kepler was.306 Burtt explains that Galileo is close to Kepler in scientific
mindset.307 However, there may have been differences in their conceptions of God.
David Wootton explains that Sarpi espoused an empiricism similar to Newton’s: the best way
of philosophising, Sarpi says, is to begin with the evidence/testimony of the senses and to use
reason to build upon that. Wootton does not explain what kind of reason (clinically inductive
or otherwise) Sarpi advocates. Sarpi explicitly rejects perception following reason because a
priori reason can only be speculation.308 All of this, of course, is similar to Newton’s position.
From Galileo to Newton
Burtt presents Galileo’s understanding that sensory perception, including experimental
observation, often delivers misleading results.309 Application of reason guided Aristarchus and
Copernicus to conclusions contrary to observations. To say that it was purely “simplicity” that
guided their reason would be to ignore their wider and richer intellectual and cultural
environment. In any case, what is missing from the simplicity doctrine is the higher
hypothesis which, to be sure, is often more beautiful and harmonious than the view “in the
small” that had been being taken. That shift in thinking may not at first sight appear to be
simple. Perhaps a circle is simpler than a polygon, but not when one has been reared on
polygons and is trying to regard a circle as an infinite-sided polygon.
In any case, it appears that Newton lifted the “simplicity” idea from Neoplatonists such as
Copernicus, and added it as a qualification to a purist Empiricism.
Galileo follows these steps:
1. Intuition or resolution (hypothesis)
2. Demonstration
306 Burtt, E.A. The Metaphysical Foundations of Modern Physical Science Routledge London 1932 2nd ed.
reprinted 1950, p.65 307 Ibid., pp.65-8 308 Wootton, p.37 309 Burtt, E.A. The Metaphysical Foundations of Modern Physical Science Routledge London 1932 2nd ed.
reprinted 1950, pp.68-9
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3. Experiment.
The Royal Society’s “British” approach to science involves repeating experiments, finding a
pattern, expressing the pattern mathematically and calling that a law. This might be called a
“statistical” approach. It has the benefit of discouraging thinking that is inexact or too
speculative.
The more usual approach, closer to Galileo’s three step and taught in Australian high schools
in the 1980s, is to form a hypothesis, design an experiment to test the hypothesis, and conduct
the experiment. Alternatively, one could check data from experiment or observation against
the hypothesis. Australian high school curricula omit the question of how hypotheses are
formed. Kepler’s a priori understanding of how God thinks and acts would be relevant and
useful in filling that gap.
Leibniz comments on the experimental program of the (British) Royal Society
In New Essays on Human Understanding, Leibniz called the causes of phenomena “true
hypotheses”, and says that if the art of discovering true hypotheses is not joined to Bacon and
Boyle’s art of experimenting, “we shall never with utmost cost attain [from experimenting] …
what a man of great penetration might discover at first sight.” Leibniz notes that Descartes
made a similar remark regarding “the method of the Chancellor of England [i.e. Sir Francis
Bacon].” He then notes Spinoza’s observation to the Secretary of the Royal Society,
Oldenburg, that Sir Robert Boyle “stops a little too long to draw from a great number of fine
experiments no other conclusion than this which he might take for a principle, namely, that
everything takes place in nature mechanically; a principle which can be rendered certain by
reason alone, and never by experiments however numerous they may be”.310
Newton’s prescribed method of discovery
Contrast this Newton, from an article by McGuire:311
1. The Method of Analysis should always precede the Method of Composition when
investigating difficult problems in Nature and in Mathematics.
2. This “Analysis” is making observations and drawing general conclusions from them
by induction.
3. Hypotheses are not to be regarded in experimental philosophy.
4. Admittedly, arguing from experiments and observations by induction does not
demonstrate general conclusions, yet it is the best method available as admitted by the
Nature of Things. Further, the argument may be regarded as stronger by the level of
generality of the Induction. I.e. a more general Induction is a stronger argument.
310 Leibniz, G.W. New Essays on Human Understanding Book IV Chapter 12 §13 in Wiener, p.479-480 311 McGuire, J.E. “Newton’s ‘Principles of Philosophy’: An intended preface for the 1704 Opticks and a related
draft fragment” The British Journal for the History of Science 1970 vol. 5 no. 2 pp. 178-186, at pp.184-5
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5. If no exception/contradiction arises in phenomena, then the conclusion may be
accepted generally.
6. If at any future time an exception is found in experiments, then the conclusions are to
be pronounced with the exceptions.
7. Thus, we proceed:
From Compounds to Ingredients,
From Motions to the Forces producing them in general, from Effects to Causes
and from particular causes to more general ones,
…until the argument ends in the most general causes.
This is the method of Analysis. The Synthesis comprises (i) assuming the causes discovered
and established as principles, and (ii) using the causes to explain the phenomena proceeding
from the causes. Further, the causes/principles are used to prove the explanations.
This is the method that Copernicus used, as Kepler explains.312 Newton’s summary starting
“This is the method of Analysis” is a restatement of Kepler’s claim that Copernicus’
conclusions could be proved with Euclidean exactitude, and a priori too. At that same page,
Kepler is clearly working entirely within the domain of hypotheses. He first refers to the
“customary hypotheses” with an excessive number of circles, and then to Copernicus’
hypothesis with far fewer circles.
In point 7 above, in light of point 4, when Newton refers to “causes”, it appears that he is
referring to correlations. If Motion B always follows Force A, then A causes B. What if there
is an undetected phenomenon that is causing both A and B? Newton’s concept of Cause
requires more examination.
In point 3 above, Newton says Hypotheses are not to be regarded in experimental philosophy,
i.e. we must always argue by induction from the observational data. But does Newton posit a
method other than experimental philosophy in which Hypotheses are permitted?
In the next section, a thesis on the origin of Newton’s method of discovery that was first
presented by Mamiani in 2001 is described. It is argued that Newton’s source was a logic text
by Robert Sanderson published in 1618 which is essentially a Ramist logic text, meaning in
the tradition of Petrus Ramus who lived in Europe from 1515 to 1572. Ramus’ influence was
largely through his logic texts rather than on logic itself. While Ramus energetically promoted
the need for reform in the Aristotelian curriculum of the time, it was not so much that he
thought that the content of the curriculum was incorrect as the way in which it was taught. For
example, he thought that students were required to spend too many years learning material
which ultimately would not be useful to them. Also, Ramus argued, the many years required
for study meant that education was too expensive for children from poor families.313 An
extended discussion of Ramus is beyond the scope of this thesis.
312 Duncan, A.M. trans. Kepler, J. Mysterium Cosmographicum Abaris Books, Janus Series Norwalk USA 1981,
pp.77-79 313 Selberg, E. “Petrus Ramus” 2006 revised 2011, in Stanford Encyclopedia of Philosophy Zalta, E.N. (ed.).
Accessed at http://plato.stanford.edu/entries/ramus/ 11 Aug 2012
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Evolution of Newton’s method of discovery
Mamiani in Isaac Newton’s Natural Philosophy (“INNP”) describes the origin of Newton’s
regulae philosophandi as this transformation:314
1. Robert Sanderson’s Logicae artis
compendium (Oxoniae 1618)
which Newton read c.1661-4 and which
follows the Ramists rather than the
Scholastics, in that it emphasizes the theory
of method
2. Newton’s Treatise on the apocalypse 16 rules
3. Newton’s Principia 1st ed. 1687 2 rules
4. Newton’s Principia 2nd ed. 1713 3 rules
Noteworthy points regarding the Treatise on the Apocalypse are:
1. The rules are ordered from most to least general, as Sanderson ordered his rules.
2. Their literary style is similar to that in Descartes’ Discours de la méthode.
3. The 12th rule is from Descartes’ Discours: “Every truth I found is the rule that I need
afterwards to find other truths”.
Mamiani is intentionally seeking a transformation process of these rules of philosophy,
emphasizing the inductive nature of the process.
The original of Sanderson gives the method of invention as having no law, unlike methods of
resolution and composition which are stated as having five laws in common. Rather, the
method of invention is given as having means-cum-steps:
1. Sense
2. Observation or history (which includes “data”)
3. Experience
4. Induction.
Sanderson says that the method of invention has nothing in common with the method of
resolution or analysis.
True to the statement of Analysis by Newton, Newton writes to Oldenburg with a plan to
simplify the rules of optics to the most general form.
314 Mamiani, M. “To twist the meaning: Newton’s regulae philosophandi revisited” in Buchwal, J. and Cohen I.
eds. Isaac Newton’s Natural Philosophy MIT Press, Cambridge MA 2001
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Newton lists laws which, says Mamiani, start with these from Sanderson’s Logicae artis
compendium or Compendium of the Art of Logic:
The law of brevity
The law of harmony
which are then transformed by Newton to:
(a) Observe the consent of the Scriptures
(b) Choose constructions which reduce contemporary visions to the greatest harmony of
their parts.
Newton against hypothesis and towards mathematicization
Newton’s stance against Hypothesis is affirmed and elaborated by Burtt.315 Koyré writes, “one
can interpret the Newtonian view [of hypotheses] as the end-product of the English tradition
of empiricism, that of Bacon and of Boyle”.316 Newton’s four laws of scientific enquiry are
then given:317
1. We are to admit no more causes of natural things than such as are both true and
sufficient to explain their appearances.
2. To the same natural effects we must, as far as possible, assign the same causes.
3. The qualities of bodies, which admit neither intension nor remission of degrees, and
which are found to belong to all bodies within the reach of our experiments, are to be
esteemed the universal qualities of all bodies whatsoever.
4. In experimental philosophy we are to look upon propositions collected by general
induction from phenomena as accurately or very nearly true, notwithstanding any
contrary hypotheses that may be imagined, till such time as other phenomena occur,
by which they may either be made more accurate, or liable to exceptions. This rule we
must follow, that the argument of induction may not be evaded by hypotheses.
Knowledge without sensation
The fourth of these affirms that empirico-deductive interpretation to be given to the first three.
Newton goes beyond Galileo and Sarpi’s empiricism by asserting that it is not only
methodological limits that prohibit hypotheses but also the nature of what is knowable.
Indeed, the upshot of the fourth law is that nothing is known or can be known beyond
experimental results (p.216). The exclusion of hypotheses acts as a prohibition on creative
mentation. This repeats one of Pompanazzi’s arguments against the immortality of the soul:
that “the intellect is inseparable from matter”318 which might imply – as Wootton says – that 315 Burtt, E.A. The Metaphysical Foundations of Modern Physical Science Routledge London 1932 2nd ed.
reprinted 1950, pp.211-213 316 Koyré, A. Newtonian Studies Chapman & Hall, London 1965 p.40 317 Ibid., pp.214-216 318 Pompanazzi, P. Immortalitate Animae “On the immortality of the soul” in Cassirer, E., Kristeller, P. O.,
Randall, J. H. Jr (eds and trans.) The Renaissance Philosophy of Man Phoenix Books, University of Chicago
Press 1948, p.293
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“there is no knowledge without sensations”.319 Sarpi adopted the position that “there is no
knowledge without sensations” in his Pensiero No. 4.320 Nor did this idea stop at Sarpi. John
Locke adopted it saying, “The knowledge of the existence of any other thing we can have
only by sensation: for there being no necessary connexion of real existence with any idea a
man has in his memory.” Further, “the having the idea of anything in our mind, no more
proves the existence of that thing, than the picture of a man evidences his being in the
world.”321 (emphasis in original) Leibniz argues the contrary in New Essays giving the
example that we are certain that Julius Caesar lived without having had direct sensation of the
phenomenon. Leibniz refers to knowledge arising from “immediate internal experience of an
immediateness of feeling” as “primitive truths of fact”.322 We add that Leibniz – knowingly or
otherwise – was reiterating Hermes Trismegistus.323 In Bruno’s De umbris idearum published
in Paris in 1582, the character of Hermes introduces a philosophy in which “The intellect
stands certain of what it instructs whereas the senses are falsely moved”.324
For Newton, mathematics had a role in describing observed phenomena precisely. Using the
mathematical description, one then simplifies that description as far as possible.325 From
Newton’s laws of scientific enquiry, one’s observations in mathematical expressions are
reduced to something like a simplest “normal form”. Thus, mathematical symbols are
descriptive of observations, and do not represent Platonic ideas or anything incorporeal.
Newton said that reducing motion to his three laws of motion was an achievement, but if a
single cause of all three can be “found out” so that they can be reduced to a single equation
then that will be an even greater achievement. It must be asked how a “cause” can ever be
found when all one has is descriptions of observations.326 Only correlations are observable,
not causes. Newton says that deductions from observations are allowed, so that better
experiments can be designed. Yet one wonders how experiments could have been designed in
Kepler’s time that would have allowed Kepler to “deduce” the inverse square law of gravity.
Koyré argues that Newton’s hostile attitude to hypotheses can be detected throughout
Newton’s career and, indeed, “the antihypothetical attitude is present – though in a much less
rigid form – in the very first works of Newton.”327
319 Wootton, p.41 320 Ibid. 321 Locke, J. Essays on Human Understanding Vol. 2 Book IV Chapter XI, Project Gutenberg edition accessed at
www.gutenberg.org 322 Leibniz, G.W. New Essays on Human Understanding Vol. 2 Book IV Chapter II §1, Bennett, J. (trans.) 1st ed.
Feb 2005, amended April 2008, p.166 Accessed at www.earlymoderntexts.com/jfb/leibne.pdf 10 May 2011 323 To Hermes Trismegistus, there is for humans a symbiosis between understanding and sensation, but
understanding and sensation are distinct because understanding comes to be especially by the agency of the
mind. Corpus Hermeticum IX §2 in Copenhaver, B.P. Hermetica Cambridge University Press 1992, p.27 324 “Hunc intellectus non errans stare docet: Sensus autem fallax suadet moueri.” Giordano Bruno De Umbris
Idearum Paris 1582 digital edition Peterson, J.H. (trans.) 1997 accessed at
http://www.esotericarchives.com/bruno/umbris.htm 27 April 2012. 325 Burtt, E.A. The Metaphysical Foundations of Modern Physical Science Routledge London 1932 2nd ed.
reprinted 1950, pp.216-8 326 Ibid., p.219 327 Koyré, A. Newtonian Studies Chapman & Hall, London 1965, p.40
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Kepler and Leibniz in Newton’s method
Mamiani says that in the 9th rule of Newton’s Treatise on the Apocalypse, Newton makes clear
that simplicity is a consequence of the law of harmony, and then quotes from the 9th rule:328
To choose those constructions which without straining reduce things to the greatest
simplicity. ... Truth is ever to be found in simplicity, and not in the multiplicity and
confusion of things. As the world, which to the naked eye exhibits the greatest variety
of objects, appears very simple in its internal constitution when surveyed by a
philosophic understanding, and so much the simpler by how much the better it is
understood...
Kepler’s dedicatory epistle to Mysterium Cosmographicum emphasizes the beauty, brilliance
and harmony of creation more than its simplicity. Later in Mysterium Cosmographicum when
considering hypotheses, Kepler explicitly prefers the simple to the convoluted. Leibniz in
1686 reiterated Kepler, “Reason wishes to avoid multiplicity in hypotheses or principles very
much as the simplest system is always preferred in Astronomy.”329,330
Newton seeks simplicity by combining, transforming, resolving rules generated by
observations. Importantly, we see that Newton adopts a mechanical process of discursive
logic to “derive” the laws of nature from observations. For Leibniz, this was nothing but the
“art of reasoning well”, quite distinct from the “art of discovery”.331 In 1677, Leibniz
elucidated on his boyhood ambition of creating a Characteristic whereby reasoning could be
conducted in a symbolic language comprehensible to people of all classes and nations so that
disputes could readily be settled.332 The important thing is that what for Newton was
appropriate for discovery, for Leibniz was extremely useful for reasoning and teasing out the
implications of existing knowledge.
As far as discovery is concerned, Kepler had gone the opposite way to Newton by forming a
hypothesis that is harmonious and beautiful, and then going about calculating the connection
between it and the observations.
Regarding Newton’s approach in Treatise on the Apocalypse, which was written prior to
Newton’s Principia, Mamiani writes:333
328 From Yahuda Manuscript 1.1a in Mamiani, M. “To twist the meaning: Newton’s regulae philosophandi
revisited” in Buchwal, J. and Cohen I. eds. Isaac Newton’s Natural Philosophy MIT Press, Cambridge MA
2001, p.6 329 Wiener, p.296 330 It may be stretch to draw a connection. However, the seeking after simplicity goes back to Hermes
Trismegistus. He writes, “The many make philosophy obscure in the multiplicity of their reasoning.” He says
that they (the many) combine philosophy with arithmetic, music and geometry, so making it
incomprehensible. Instead, “Pure philosophy that depends only on reverence for god should attend to these
other matters [arithmetic, music and geometry]” leaving only problems in astronomy unsolved. “Asclepius”
§§12-13 in Copenhaver, B.P. Hermetica Cambridge University Press 1992, p.74 331 c.1693, Wiener, pp.77-78 332 1677, Wiener, p.18 333 Mamiani, M. “To twist the meaning: Newton’s regulae philosophandi revisited” in Buchwal, J. and Cohen I.
eds. Isaac Newton’s Natural Philosophy MIT Press, Cambridge MA 2001, p.7, middle paragraph
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What is new in Newton’s rules? Neither the content nor the expression. It is true that
Sanderson’s laws are very concise whereas Newton’s rules are verbose and redundant.
We must wait for the rules of the Principia in order to find a conciseness equivalent to
the laws of Sanderson’s Logic. There is, however, a great difference among Newton’s
rules of interpreting the Apocalypse and Sanderson’s laws. Sanderson is repeating the
precepts of a dead tradition for presenting or teaching acquired knowledge. Newton
[on the other hand] is proposing rules to be used in discovering new knowledge.
This suggests the source of the current view of:334
the English “Baconian” concept of how to do science, followed by the Royal Society.
You’re supposed to collect lots of observations and reports, do experiments, then
generalise to laws, then (optional) think up causal hypotheses (e.g. atomism) that
might explain them. That’s an OK method to do science, especially if the science you
want to do is very empirical, e.g. chemistry. And very British and practical.
Leibniz was not “anti”-discursive logic or propositional logic. Indeed, Leibniz has been called
the most original logician of the 17th century and credited with many original discoveries
including the beginnings of what is now regarded as standard classical propositional logic
decades if not centuries before others.335 The difference is that Leibniz would only have
regarded this as a way of interpreting existing knowledge, or to answer questions which could
be answered from the existing body of human knowledge. This is no small thing, and Leibniz
aimed with his universal/general characteristic even to allow disputes between people and
perhaps nations to be resolved. This coming from Leibniz the diplomat was quite an
ambition.336 However, it is different from making a new discovery. We will see in a later
chapter that when Leibniz considers the principles and process of new discovery or “the art of
discovery”, a different field of considerations comes into play.
Consequences of mandating empirico-logic
What if experiments cannot be designed but only hypothesising is possible? An example is the
quest to understand the solar system using observational data collected from earth. It can
surely be demonstrated that restricting ourselves to Sanderson’s induction/deduction
precludes leaps of understanding, inspiration and anything that smacks of human creativity
under most definitions. No a priori presumption of harmony is permitted. Newton has
certainly excluded Cusa’s “learned ignorance” in which leaps of knowledge arise from
inspiration rather than from “discursive logic”. Science and mathematics would cease. For
example, consider the famous case of Gauss answering his teacher in adding the integers from
1 to 99. No doubt there would have been an admixture of some kind of deduction alongside
an “aha!” sense of inspiration. We are now as much in the domain of psychology as science.
Leibniz, in his essay “How to Reason Well”, named the holding of all aspects of a body of 334 Private communication from Prof. Jim Franklin, UNSW, 23 Aug 2009 335 Cambridge History of 17th Century Philosophy C.U.P. 1987, pp.141-3 336 Loemker 1969 p.224 and Wiener 1951, pp.23-24. Loemker ascribes this writing to c.1679 and calls it “On the
general characteristic”; Wiener dates it 1677 and calls it “Towards a Universal Characteristic”. This writer
thinks both translations/interpretations are fair.
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knowledge in one’s mind at once as a prerequisite to full understanding and formulating a
hypothesis as to the solution to a problem.
There are serious consequences of Newton’s mandate for science:
1. Painfully slow progress in scientific thought and understanding which suggests that
Newton himself – perhaps without being conscious of it – did not follow his precepts
in his own scientific work;337
2. Acceptance that causes are not known unless observed, and may not be speculated
upon;338
3. Resignation to the unknowability of nearly everything; and
4. Prohibition of ideas, which would have prevented Leibniz’s calculus, complex
numbers and much of Plato’s own reasoning.
Leibniz actually noted that an overemphasis on amassing experimental data and an
underemphasis on reasoning was retarding scientific progress at the Royal Society. Some
years after his 1673 visit to the Royal Society in London, Leibniz wrote, “they confessed to
me in England that the great number of experiments they have amassed gives them no less
difficulty than the lack of experiment gave the ancients”.339 At the same time, Leibniz tried to
reconcile the pragmatic need for experiment with a priori reasoning.340
Burtt sums up Newton’s approach in, “Science is the exact mathematical formulation of the
processes of the natural world. Speculation is at a discount, but motion has unconditionally
surrendered to the conquering mind of man.”341 (emphasis in original) Actually, the
Newtonian method says that science is the exact mathematical description of the observable
processes of the natural world, and this stems from Newton’s failure to embrace the method of
337 Ihmig is similarly sceptical. He writes, “The strict rejection of hypotheses in experimental philosophy,
however, hardly seemed to agree with how Newton practiced science. As hypotheses can be found in all
editions of the Principia, some authors have tried to determine the meaning of the concept of hypotheses and
its various forms more precisely. This has made it possible to separate ‘good’ from ‘bad’ hypotheses, thus
confining Newton’s rejection of hypotheses merely to the bad ones.” Ihmig, K.-N. “Newton's Program of
Mathematicizing Nature” in Hoffmann, M.H.G., Lenhard, J., Seeger, F. (eds) Activity and sign: grounding
mathematics education Springer 2005, p.242 338 In New Theory about Light and Colours 1672, Newton had already rejected the use of hypotheses. In the
Principia 2nd ed. 1713, Newton admits the difficulties that avoiding hypotheses is causing him, but reiterates
the validity of avoiding hypotheses: “But hitherto I have not been able to discover the cause of those
properties of gravity from phenomena, and I frame no hypotheses; for whatever is not deduced from the
phenomena is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of
occult qualities or mechanical, have no place in experimental philosophy.” 339 Letter to Herzog Friedrichm, 1679 in Wiener, p.xxiii. Wiener also writes, “It is in his correspondence
concerning the experimentalists of the Royal Society of England (whom he had visited in 1673) that Leibniz
reveals most clearly what he thought his logical instrument could do, and what the plain empiricists were not
doing, namely, to elicit all the knowledge deducible from a given number of presuppositions. Lack of a
proper art of demonstration had made it necessary, in Leibniz’s opinion, for Baconian experimental
philosophers like Boyle to resort to many observations in order to find out what Galileo and Descartes were
able to know by reasoning.” Wiener, p.xxii. Wiener does not say whether the “logical instrument” is the
Universal Characteristic. 340 Letter to Berthey, 1677, in Wiener, pp.xxiii-xxiv 341 Burtt, E.A. The Metaphysical Foundations of Modern Physical Science Routledge London 1932 2nd ed.
reprinted 1950, p.223
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hypothesis even if Newton did not outright reject the use of hypotheses. Further, motion has
not unconditionally surrendered to the mind but, rather, the mind of man is surrendering to the
bodily senses’ observation of motion, which enslaves the mind by denying its capability to
reason or create.
Burtt rebuts the contention that Newton is free of metaphysical concerns or crutches. In fact,
the “objective rigor” of Newtonian science is replete with metaphysical assumptions which
we outline below. Newtonian positivism is the idea that truth can be acquired without
presupposing any theory of their ultimate nature, such as of the Creator. Burtt says that
anyone who claims to be free of metaphysical assumptions actually does adopt metaphysical
notions in these ways:342
1. “He will share the ideas of his age on ultimate questions, so far as such ideas do not
run counter to his interests or awaken his criticism”;
2. “If he be a man engaged in an important enquiry, he must have a method, and he will
be under a strong and constant temptation to make a metaphysics out of this method”;
3. “Since human nature demands metaphysics for its full intellectual satisfaction, no
great mind can wholly avoid playing with ultimate questions, especially where they
are powerfully thrust upon it by considerations arising from its positivistic
investigations, or by certain vigorous extra-scientific interests, such as religion.”
The way Newton said science should be done and the way that Newton actually did science
were two different things. Newton himself did not proceed by pure induction in his own work.
Burtt goes on to explain how and in what way Newton adopts metaphysical assumptions of all
of the above three kinds. Koyré disputes that Newton followed his own prescription on
hypotheses. Koyré writes, “The expression ‘hypothesis thus seems to have become, for
Newton, toward the end of his life, one of those curious terms, such as ‘heresy,’ that we never
apply to ourselves, but only to others. As for us, we do not feign hypotheses, we are not
heretics. It is they, the Baconians, the Cartesians, Leibniz, Hooke, Cheyne, and others – they
feign hypotheses and they are the heretics.”343
We now make some comments on logical deduction which will usher in the next chapter.
Empirico-logic in other domains
Empirico-logic was promoted under a different guise by the contention that a computer
equipped with sensory apparatus and programmed with the rules of logic can replicate the
human mind. Thus von Neumann’s viewpoint is at its heart the same as the rules of
Newtonian science. This dissertation will not investigate this question further.344 Despite the
342 Burtt, E.A. The Metaphysical Foundations of Modern Physical Science Routledge London 1932 2nd ed.
reprinted 1950, p.226 343 Koyré, A. Newtonian Studies Chapman & Hall, London 1965, p.52 344 This author’s MPhil thesis “On object Petri nets” (University of Warwick, Department of Computer Science,
2004) dealt with Petri nets which seek to model complex systems using enhanced graph-like models which
carry tokens under deterministic rules, like a pinball machine in which the balls run through tubes. Nested
Petri nets use nets themselves as tokens and so can represent greater complexity. Some nets can be reduced to
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fact that he was a designer of machines and even of a calculating machine, Leibniz could not
have shared von Neumann’s view. For a start, the human soul is a monad which is connected
with every other monad in the universe and has full knowledge of everything that is occurring
in the universe, which is quite unlike any computer. If Leibniz regarded the action of the mind
of man as God in the miniature,345 then there is another raft of reasons why Leibniz would
disagree, since few would argue that God can be modelled by a computer.
Euclidean geometry presents a system of formal reasoning, essentially embodying the idea of
the syllogism. Plato embraces such reasoning with a geometrical problem in Meno. Plato also
demonstrated how reasoning can be conducted on non-mathematical ideas, such as justice and
goodness. There was a raft of mathematical thinkers in Ancient Greece who were able to
reason both in the abstract and in the concrete. Their work raises the question of the
relationship between abstract formal reasoning and the real world. It also raises the question
of the relationship between abstract ideas and the physical world.
Reasoning other than empirico-logical
We become aware that there is a mode of thought that is not formal reasoning, as such, but
which is structured and is worthy of the name reasoning. Indeed, Archimedes performed work
in his own way, but when he recorded or transmitted what he had found, he used formal
reasoning of a kind which was not what he used to reach the discovery in the first place. At
the same time, the actual method he used was not to simply “throw things together and watch
what happened” either. There was a searching for an order that was indefinable because it had
not yet been found. He would recognise it when he found it, though he might be misled many
times along the way. The search for order or harmony was undertaken with a sense of
necessity or passion and that search amounts to the search for a viable hypothesis. Leibniz
wrote, “The art of discovering the causes of phenomena, or true hypotheses, is like the art of
deciphering, where an ingenious conjecture often shortens the road very much.”346 The formal
reasoning was often not part of the search and was only added afterwards. Of course,
confirmation by formal reasoning or by experiment must follow.
Boyer notes that Archimedes of Syracuse “tempered the strong transcendental imagination of
Plato with the meticulously correct procedure of Euclid.”347 Boyer suggests how Archimedes
did this in one case: that Archimedes used infinitesimal considerations as an heuristic method
“simply as an investigation preliminary to the rigorous demonstration by the method of
exhaustion.”348 It could be argued that the heuristic method represents the transcendental
a linear logic proposition. Net-based models are essential for systems such as telecom networks, computer
chips and circuit boards. Attempts to use nets to model processes with a human element have seen a
resurgence with the modelling of social networks such as those found on Facebook or LinkedIn. As yet, there
is no convincing argument that net-based models bear any qualitative resemblance to the mind or to thought
processes. 345 Leibniz, G.W. Theodicy §147 346 Leibniz, G.W. New Essays on Human Understanding Book IV Chapter II §13 in Wiener, p.479 347 Boyer, C. The History of the Calculus and its Conceptual Development Dover Publications, New York 1949,
p.48 348 Ibid., p.51
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imagination of Plato while the method of exhaustion represents Euclidean rigor. However,
must there be an exclusive choice between transcendental imagination and rigor?
Platonists were never opposed to rigor, and not even opposed to the kind of work that Euclid,
for example, did. The difference was that the Platonists insisted that there was an indefinable
something that was the quarry of any process of discovery or creativity, and it is always in
that that the crux of any question in science or philosophy lies. The Pythagoreans insisted that
such concepts were valid, and number was the key example in their time. Such concepts
should be the focus of science but, in doing so, the human mind becomes the centre of
science. We conclude that the human mind must be intertwined with what science is. This is
not as Burtt understood things in opposing the Russellian pessimism and the Newtonian
empiricism. Burtt argued that human experience was intertwined with science. On the
contrary, we would argue that the Leibnizian view is that the pure ideations of the human
mind are where science actually takes place. These are tested in the world, as Archimedes did
in his bath tub or as Huygens and Papin did with their work towards building powered
engines. Adopting Leibniz’s view would resurrect what Burtt instinctively knew to be right
but could not quite understand – that Dante’s conception of Man as the jewel of Creation due
to the uniqueness of Man’s mind was correct and the Newtonian-cum-Russellian notion is
unsustainable. The Russellian notion is that Man is not only insignificant in Creation but is
not even part of Creation, except for Man’s biological body. The inexorable conclusion from
the Russellian notion is that the mind has no role in science.
Burtt purports to bring humanity back into the universe by saying that human experience is a
scientifically “good enough” representation of the physical universe. This is essentially
Cartesian subjectivism. As Plato said in The Republic, all experience is opinion, which flies in
the face of Burtt’s resolution. We prefer to solve the paradox that Burtt correctly posed using
Leibniz’s essays. Leibniz’s metaphor is that reason is akin to life while experience is the air
essential to “life” in that it gives context and subject matter to reason about. An alternative
analogy is that the mind is the potter and the experience is the clay which gives subject matter
to the mind.
The role of rigorists
Where did mathematics of the kind we now know originate? For a history of mathematics in
toto, we can refer to Ball, Boyer and many others. Suffice it to say that the Pythagoreans and
Neoplatonists and their descendants made strides in mathematics, and it was generally the role
of the rigorists, as Leibniz called them, to complain that a certain innovation could not be
done or was not allowed. Such critics still played the role of mastering the art of calculation
within any given domain of mathematics as it stood during any particular historical milieu.
Further, Leibniz was not opposed to rigor per se.
Who were the “rigorists”? There were and are many. Leibniz named a number in his letters
when defending his methods against their attacks. It is too simplistic to associate such critics
with any particular philosopher though it is common to associate such attempts with adherents
of Aristotle or Averroes. Leibniz agreed with Aristotle on many points and sometimes
explicitly resorted to Aristotle to rebut Descartes. Leibniz says that Aristotle was the first to
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give demonstrations in Logic, “and we may say he was successful, but he was far from being
successful in the other sciences he treated.”349 Scholasticism owed its roots to Aristotle and
owed something to Averroes, but it also owed much to Thomas Aquinas’ purported defence of
Aristotle against Averroes’ alleged misinterpretation or misuse of Aristotle’s work and name.
By Leibniz’s time, mathematics included so much more than Averroes ever addressed
centuries prior. Further, there was no recognisable school of “Averroists” active in Leibniz’s
time.
Conclusion
Did Newton combine rigor such as that of Euclid and the scholastics with the wave of
Neoplatonic numerical mathematicisation? It might seem that Newton had the effect of
constraining the new Neoplatonic mathematics with an incarnation of Aristotelian reasoning,
namely that of Petrus Ramus as presented in Robert Sanderson’s Compendium of the Art of
Logic. However, we must deny such credit to Newton due to Kepler and Leibniz’s own rigor.
What, then, is different about the Neoplatonists?
Nikulin quotes Manin, “A good physicist uses formalism as a poet uses language.”350 Nikulin
explains, “since the language in poetry often says and suggests more than is intended, and, in
fact, negates and suspends the language by means of language itself, it may become
prescriptive, opening new possibilities for experiencing the world.” We would amend this
slightly to say that the good poet intends more than the language denotes, and already knows
what new possibilities they are seeking to describe and prescribe for further investigation.
Similarly, the physicist is not constrained by the limitations of formalism, but is required by
pragmatic circumstances to fit their ideas in formal language. What Nikulin touches on with
respect to the physicist and formalism, we would also say about the Neoplatonic
mathematician and formalism. In particular, a Leibnizian communicates the
metaphysical/ontological using the constraints of formal symbology.
Leibniz took exception to the rigorists demanding more from mathematical formalism than it
is fit to deliver. That is, he took exception those who criticised his calculus because it relied
on the infinitesimal which is a thing that might not exist. However, this may have been
because the ontological concept of the infinitesimal had not as yet been made sufficiently
precise. It may also have been because the symbology was not sufficiently advanced or
appropriate.
Nikulin may provide the key to an ontologically real mathematical object.351 However,
Leibniz was not concerned whether the infinitesimal was even ontologically real; to Leibniz,
it was merely a thought tool. In any case, the question of what an ontologically real
mathematical object is might tell us what is different about Neoplatonic mathematicisation
349 1685, Wiener, p.52 350 Manin, Y. I. Mathematics and Physics Trans. by A. and N. Koblitz, Boston-Basel-Stuttgart 1981, p.5 in
Nikulin, D. V. Matter, imagination and geometry: ontology, natural philosophy and mathematics in Plotinus,
Proclus and Descartes Ashgate, Aldershot and Burlington 2002, p.xii, n.17 351 Ibid., pp.70-74ff
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versus Newtonian mathematicisation. Nikulin was grappling with the question of how
mathematical objects which do not have physical existence can be used in physics.352 It is the
duality issue discussed in Chapter 3 in the context of Grosholz’s analysis of Leibniz’s
mathematics. To Leibniz, this is the distinction between the incorporeal domain of ideas and
active souls, and the passive domain of matter. Leibniz tied them together with his doctrine of
pre-established harmony.353
To Kielkopf who undertook a thorough review of Wittgenstein’s foundations of mathematics,
the mark of an “absolute platonist [sic]” is that they hold that “the domain of mathematical
objects themselves exist independently of minds.”354 As we will see in Chapter 7, Leibniz is
not this kind of platonist because Leibniz does not consider ideas to be real and only regards
them as able to exist in minds. However, he also regards them as having an objective truth so
that their qualities and properties are independent of minds although they have no existence
independent of minds. Further, he finds an actual correspondence between ideas and the
physical universe which is neither subjective nor dependent on sense perception implying that
there is a universe parallel to the physical universe in which ideas exist. Thus, we find a
mixedquality in Leibniz’s conception of mathematicals (a subset of “ideas”) such that in
certain discussions, he is platonist in the sense of believing there to be a non-physical universe
in which ideas exist, in other contexts he analyses and plays with ideas as if they were real
things, while in other cases still and especially in the context of monads Leibniz regards ideas
as never existing anywhere except in minds and usually only in God’s mind.
Are group theory, category theory and other modern forms of algebra Neoplatonic
mathematics or are they better described as positivist disciplines? Lie groups are
indispensable to modern physics, so they should be given at least as much credit as the
infinitesimal.355 We know that group theory has utility for the study of symmetry and pattern,
and has been useful in the practical sciences such as chemistry. Leibniz would have to regard
them as useful and therefore legitimate domains of mathematics. It is likely that Leibniz
would embrace them. In fact, it is hard to see how those who criticised the idea of the
infinitesimal could not direct the same criticism at Lie groups or any part of group theory.
What would Leibniz say about category theory which is an all-encompassing form of algebra
or meta-algebra?356 Any discussion of the philosophy of category theory must consider
Bourbaki, for the founding fathers of category theory (Eilenberg and Moore) were among the
handful of founders of Bourbaki. The contribution of category theory to diagrammatic
representation of abstract concepts would surely be admired alongside its contribution to the
understanding of mathematical structure.
352 Ibid., pp.x-xi 353 Leibniz, G.W. New Essays on Human Understanding Chapter X Book IV §§9-10 in Wiener pp.473-474 354 Kielkopf, C.F. Strict Finitism: An examination of Ludwig Wittgenstein’s remarks on the foundations of
mathematics Mouton, The Hague and Paris 1970, p.32 355 Stetz, A. Lie Groups in Modern Physics 1996, freely available online p.5 “There can be no question that the
modern theory of elementary particles rests heavily on the formalism of Lie groups.” accessed at
www.physics.orst.edu/~stetza/Lie.pdf 7 May 2011 356 This author’s Honours thesis was entitled “A categorical approach to universal algebra” Department of Pure
Mathematics, University of Sydney 2000 supervised by Dr Steve Lack. It is a relatively easy-to-read
introduction to category theory and universal algebra. Universal algebra was an even earlier form of meta-
algebra pioneered by mathematicians such as Whitehead.
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Chapter 6: Discovery and deduction
Introduction
There was a raft of mathematical thinkers in Ancient Greece who were able to reason both in
the abstract and in the concrete. Their work raises the question of the relationship between
formal reasoning and the physical world, and between abstract ideas and the physical world.
We become aware that there is a mode of thought that is not formal reasoning, but which is
structured and is worthy of the name reasoning. There was a searching for an order,
indefinable because it had not yet been found. It is recognised only when found, though the
thinker or scientist might be misled many times along the way. Formal reasoning is often not
the part of the search when breakthroughs occur, but is often added afterwards. We argue that
this searching or exploration is what Leibniz called Reason, and it is the primary tool in what
he called the Art of Discovery.
The “from where to which” of discursive reasoning is limited compared with learned
ignorance (docta ignorantia). This larger “God’s eye view” transcends discursive reasoning
and also allows us to see reason and harmony in the large. Because we can do this, we can
approach God’s mind and should seek to do so. It is from such a level that hypotheses are
formed ultimately to be tested by the structured observation of experiment.
Petrarca considers “learned ignorance” in the sense that Augustine uses it. To Petrarca,
“learned ignorance” is ignorance which is blessed and to possess it is to be enlightened by the
teachings of the Holy Spirit. To Petrarca, it is the highest kind of knowledge. To put it more
accurately, since it is more a mindset or an ideological and intellectual disposition, it is the
most amenable to (a) gaining new understanding of anything concrete and even (b) to
thinking.
Nicolaus of Cusa crystallised the Augustinian idea which had been picked up by Petrarca, and
we know that Cusa possessed a copy of Petrarca’s De Ignorantia. The book by Cusa applies
specifically to science and reason. In the debate with Wenck, Cusa sets learned ignorance
against discursive logic and explains why it is superior. Through De Docta Ignorantia, Cusa
explained why number is so important and guessed at why the Pythagoreans thought it to be
so. Cusa claimed that even comparative relation cannot be understood independently of
number, which means that number perhaps subsumes even the syllogism.
Since, relative to all that can be known, unknowing or ignorance is our state, and since as
Cusa says the highest knowledge we can attain is that of our own ignorance, then no original
knowledge can emanate from deduction because deduction uses only what we already know.
To have knowledge of our ignorance is to enter the larger, and vast, domain, of what we do
not know. Deduction does not work in this domain, but only inspiration and leaps of insight
can make any sense.
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Leibniz understood that deduction allows us to clarify points of contention using what we
already know. He regarded this as something that the common man could do when arguing
with his colleague over any mundane matter. This is very different, however, from the Art of
Discovery as Leibniz called it.
The Sarpian-Newtonian idea of conducting experiments and deducing from those experiments
is all very well too. This is not inconsistent with “physical Platonism” explained by Davies:357
Most theoretical physicists are by temperament Platonists. They envisage the laws of
physics too [like the principles of mathematics] as perfect idealized mathematical
relationships and operations that really exist, located in an abstract realm transcending
the physical universe. I shall call this viewpoint physical Platonism.
We do not advocate this brand of Platonism as metaphysically correct, or even as a
perspective of physics that Plato himself would have adopted. The question here is about how
those laws are discovered understood. Overemphasis on experiment does not allow the
investigator to address ironies or contradictions in what is observed, which can only be found
through thought and by entering the domain of our ignorance in science, mathematics and
metaphysics. In his learned ignorance discourse, Cusa is not describing a mystical process but
carrying on a conversation about human thought. We argue that Leibniz worked in this
tradition.
The art of discovery
In a letter, Leibniz poetically prefaces his recent discoveries with a gunpowder metaphor on
the light of discovery: “I am also sending you a little of the corporeal fire, which can well be
called a perpetual light, for when properly protected, it lasts many years without being
consumed. … It is easy to ignite gunpowder, either by the sun or through friction, after a little
of this phosphorous is mixed with it.”358
How does Leibniz come by this “phosphorous”? He gave us 10 maxims on the art of
discovery.359 These are to be distinguished from the art of reasoning for which Leibniz in the
same essay gave three maxims.360
To discover something new, we must – broadly speaking – follow these broad steps:361
1. Prerequisites: Consider all of its prerequisites or everything that distinguish it from
every other thing; this is its definition, nature or essential property.
2. Prerequisites of the prerequisites: Consider all of the prerequisites of each of its
prerequisites. This is true analysis or distributing the problem into parts.
357 Davies, P. C. W. “The Implications of a Cosmological Information Bound for Complexity, Quantum
Information and the Nature of Physical Law” Arxiv Preprint, p.2 Accessed at http://arxiv.org/ftp/quant-
ph/papers/0703/0703041.pdf on 21 July 2012 358 Loemker, p.249 (Letter to Huygens, 1679 around two years after Leibniz’s initiation into Huygens’
experiments with the [French] Royal Academy of Sciences) 359 c.1693, Wiener, pp.78-80 360 Ibid., pp.77-78 361 c.1693, Ibid., pp.78-80
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3. Perfect knowledge: When we have come to considering a few natures understood only
by themselves without prerequisites, we have reached perfect knowledge of the thing.
4. Single thought: Try to have this perfect knowledge in the mind so that we see it as a
whole in a single act of the mind. (This is “intellection”, the highest kind of the four
kinds cognition proposed by the Platonists, with which Leibniz agrees.362)
5. Prove our proposition: Now, nothing appears in the thing that cannot be accounted for
and nothing occurs which cannot be predicted in advance. This sounds difficult, but to
prove the single proposition that concerns us, which is usually all we need to do, then
it is readily done with the thing we are now acquainted with.
6. Start with the easiest: Always begin enquiries with the easiest things, or the most
general and simple things, like numbers, lines, motions, etc. depending on the kind of
thing we are trying to understand.
7. Find a natural progression: Proceed from easy to difficult things, attempting to find
some natural progression so we do not miss anything out.
8. Omit nothing: Try to omit nothing in our distributions and enumerations. Dichotomies
with opposites can help.
9. Catalogue of simple thoughts: The result should be a catalogue of simple thoughts or,
at least, of not very difficult thoughts.
10. Build up from the start: From the catalogue of simple thoughts, explain the origin of
things starting from their source in a perfect order and using a synthesis that is
absolutely coherent.
Leibniz may be referring to this process when nine years later he says that “my own views
have become fixed only after my considering all sides and weighing them well” summing it
up as “I have anticipated everything and gone through it in my mind”.363
According to the Platonists as Leibniz describes them, they were “not far wrong” in listing the
four kinds of cognition of the mind as:364
Level Called Or, in other words...
1 sense experience
2 opinion conjecture
3 knowledge demonstration which is reasoning by which
some proposition is made certain, and this is
nothing but the analysis of a truth into other
truths which are already known365
4 understanding intellection which looks into the connections
of truth in a single act of the mind
362 Leibniz, G. W. “Letter to Hansch on the Platonic Philosophy or Platonic Enthusiasm” 25 July 1707, in
Loemker, L. G. W. Leibniz Philosophical Papers and Letters 2nd ed. Reidel Publishing Company 1969, p.593 363 “Reply to Bayle’s Dictionary article Rorarius” 1702, Loemker, L. G. W. Leibniz Philosophical Papers and
Letters 2nd ed. Reidel Publishing Company 1969, p.582 364 Leibniz, G. W. “Letter to Hansch on the Platonic Philosophy or Platonic Enthusiasm” 25 July 1707. Ibid.,
p.593 365 Leibniz, G. W. “Letter to Herman Conring” 19 March 1678. Ibid., p.187
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Of these, the third bears some resemblance to item 2 “prerequisites of the prerequisites” and
the fourth (“intellection”) appears to be item 3 or the goal of “perfect knowledge” of the
thing. Further, to have perfect knowledge of a thing, everything arising from experience and
that can reasonably arise from conjecture must be consistent or explicable in terms of the
knowledge that has been attained, otherwise that knowledge is not yet perfect.
Intellection “belongs to God in all things but is given to us in simple matters only”.
The most striking thing is how much hard work is involved. The generality of the Leibniz’s
ten points indicates that the same rules apply whether one is designing a water pump,
addressing a legal problem, or working out the metaphysical foundations of science.
Neoplatonism is about having worked through these steps with the fundamental questions of
theology and metaphysics before working in the natural sciences. Having done that
background work in the largest context for natural science, one has a ready source of plausible
a priori hypotheses which are consistent with one’s theology and metaphysics with which to
start.
Leibniz explains in greater detail what is involved in science in a paper An introduction on the
value and method of natural science.366 Section subheadings in the paper include these which
are descriptive enough to be useful to list here among a number of others:
1. “Empirical physics is useful for human life and should be cultivated in the state”,
2. “A catalogue of experiments is to be compiled”,
3. “New experiments are to be undertaken at public expense, and only men outstanding
not merely in science but in virtue are to be placed in charge”,
4. “With the experiments are to be combined accurate and thoroughly extended
reasonings after the manner of geometry, for only in this way can causes be
discovered”,
5. “The most perfect method involves the discovery of the interior constitution of bodies
a priori from a contemplation of God, the author of things. But this method is a
difficult one and not to be undertaken by anyone whatever”,
6. “Some hypotheses can satisfy so many phenomena, and so easily, that they can be
taken for certain. Among other hypotheses, those are to be chosen which are the
simpler; these are to be presented, in the interim, in place of the true causes”,
7. “Analogies are useful in guessing at causes and in making predictions” and
8. “The method of reasoning from experiments resolves the phenomenon into its
attributes and seeks the causes and effects of each attributes”.
Discovery is often metaphysics
Discovery often takes place in the domain of metaphysics, and sometimes even in theology.
Leibniz wrote, “The a priori method is certain if we can demonstrate from the known nature
of God that structure of the world which is in agreement with the divine reasons and from this
structure, can finally arrive at the principles of sensible things. This method is of all the most
366 cc.1682-4 in Loemker, pp.281-9
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excellent and hence does not seem to be entirely impossible. For our mind is endowed with
the concept of perfection, and we know that God works in the most perfect way.” Due to the
difficulty of this method, Leibniz says that we must retain the a posteriori method, which is
the second method.367
The 15th to 17
th Centuries saw a Neoplatonic revival in the sciences. That revival brought with
it a conviction that “mere patterns and relational structures” (to use Grosholz’ words) are the
metaphysical “furniture of the universe”. To a Neoplatonist, there is no contradiction but,
rather, it is the order of things as derivable from Book VI of Plato’s The Republic.368 That
book argues that the philosopher has a method for finding answers which resolve existing
contradictions, though there is no guarantee that new contradictions will not be found. Book
VI is famous for introducing the “divided line” which is a classification of intellectual
exercise from the lowest level of conjecture and up to belief. Both conjecture and belief are
mere opinion, not knowledge. Next comes understanding and finally there is exercise of
reason. The latter two deserve the label “knowledge”. Opinions derive from the exercise of
the sense. Understanding includes mathematical thought, but is defined by Plato as images
and concepts of thought such as ideal squares and cubes. The highest level of the exercise of
reason is called dialectical thought, and it deals with ideas or ideals such as perfect beauty,
justice and goodness.369
Leibniz was conscious of the metaphysical dependencies of his infinitesimal calculus and of
its ancient roots. He recognised that Archimedes faced similar problems of political
correctness that he himself was facing:370
When my infinitesimal calculus, which includes the calculus of differences and sums,
had appeared and spread, certain over-precise veterans began to make trouble; just as
once long ago the Sceptics opposed the Dogmatics, … and such as Francisco Sanchez,
the author of the book Quod nihil scitur, brought against Clavius; and his opponents to
Cavalieri, and Thomas Hobbes to all geometers, and just lately such objections as are
made against Archimedes by that renowned man, Dethlevus Cluver.
Un-rigorous use of rigor
Leibniz notes a pattern through the republic of letters. He objects in particular to the use of a
requirement of rigor as a particular set of restrictions to criticise his method, as he appears to
regard such requirements as arbitrary or at least unjustified. Leibniz was well aware of the
opposing personalities:371
367 Ibid., p.283 368 Rouse, W.H.D. (trans.) Great Dialogues of Plato Signet Classic, New American Library a division of
Penguin, New York 1999, pp.281-311 369 Ibid., p.309 370 Child, J.M. (ed. and trans.) The early mathematical manuscripts of Leibniz; tr. from the Latin texts by Carl
Immanuel Gerhardt with critical and historical notes by J. M. Child The Open Court Publishing Company
1920, reprinted by the University of Michigan Library, p.145 371 Ibid., pp.145-146
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When then our method of infinitesimals, which had become known by the name of the
calculus of differences, began to be spread abroad by several examples of its use … a
certain erudite mathematician, writing under an assumed name in the scientific
Journal de Trevoux, appeared to find fault with this method. But to mention one of
them by name, even before this there arose against me in Holland Bernard
Nieuwentijt, one indeed really well equipped both in learning and ability.
Leibniz discusses the concept of infinitesimals and explains that, for example, “those things
that are found to be true about a parabola [treated in such a way] are in no way different, for
any construction, from those which can be stated by treating the parabola rigorously”.372
Nieuwentijt’s disagreements with Leibniz were not over rigor, but rather over the use of the
infinitesimal. First of all, Nieuwentijt believed that the infinitesimal was zero so any quantity
multiplied by it gave a product of zero. Second of all, there could certainly be no second order
differentials because these require the multiplication of an infinitesimal with another
infinitesimal. When opposing Leibniz on second order differentials, Nieuwentijt argues that
“there are no gradations in the infinite”.373 Nieuwentijt’s proofs assume a mathematics that is
based in what can be visualised. The fact that an infinitesimal could have a geometrical
meaning and could even be multiplied by another infinitesimal thereby obtaining a new
geometrical meaning is beyond the pale if one is trying to visualize the calculation. Of course,
Nieuwentijt’s objections to “gradations in the infinite” pre-empted Cantor’s theory of
transfinite numbers.
Archimedes had the same problem
Leibniz suggests that Archimedes faced similar criticisms, “Truly it is very likely that
Archimedes, and one who seems to have surpassed him, Conon, found out their wonderfully
elegant theorems by the help of such ideas [infinitesimals]; these theorems they completed
with reductio ad absurdam proofs, by which they at the same time provided rigorous
[Euclidean] demonstrations and also concealed their methods.”374 It may be that Gauss did the
same.375
Leibniz notes that Descartes too was aware of this. “Descartes very appropriately remarked in
one of his writings that Archimedes used as it were a kind of metaphysical reasoning
(Caramuel would call it metageometry), the method being scarcely used by any of the
ancients (except those who deal with quadratrices); in our time Cavalieri has revived the
372 Ibid., p.149 373 Vermij, R.H. “Bernard Nieuwentijt and the Leibnizian Calculus” Studia Leibnitiana 1989 Vol.21 No.1 69-86,
p.77 374 Ibid., p.149 375 “It is interesting to note that Gauss did not publish many of his ideas. It is commonly thought that this was
because he was a perfectionist and would only make his views known if they were above criticism. To that
end, he would not provide the intuitions behind his proofs, preferring instead to give the impression that they
came ‘out of thin air’” which, of course, they did not. Mathematics Illuminated Geometries Beyond Euclid
published by Annenberg Learner §8.4 Spherical and Hyperbolic Geometry accessed at
http://www.learner.org/courses/mathilluminated/units/8/textbook/04.php 4 June 2011
Page 115
method of Archimedes, and afforded an opportunity for others to advance still further.376
Indeed Descartes himself did so, since at one time he imagined a circle to be a regular
polygon with an infinite number of sides, and used the same idea in treating the cycloid; and
Huygens too, in his work on the pendulum, since he was accustomed to confirm his theorems
by rigorous demonstrations; yet at other times, in order to avoid too great prolixity, he made
use of infinitesimals; as also quite lately did the renowned La Hire.”377
All things and no things – at the same time
This two millenia context surpasses Grosholz’s interpretation of Leibniz.378 For Grosholz
referred to “the contradictory being of Leibniz’s infinite-sided polygons, at once continuous
and discrete, geometric and combinatorial, infinitary and finite”. Yet the “infinite-sided
polygons” were treated by Nicolaus of Cusa two centuries before Leibniz wrote about
mathematics. Cusa explained that a circle or “infinite-sided polygon” is of a higher order than
a polygon. It was relevant to the discovery process because the human mind is like a polygon
while God is the circle. Kepler wrote “in this one respect Nicholas of Cusa and others seem to
me divine, that they attached so much importance to the relationship between a straight and a
curved line and dared to liken a curve to God, a straight line to his creatures; and those who
tried to compare the Creator to his creatures, God to Man, and divine judgements to human
judgements did not perform much more valuable a service than those who tried to compare a
curve with a straight line, a circle with a square.”379
With God’s mind being of a higher order to human understanding, as a circle compared with a
square, so we must delve into our ignorance, and understand its quality in order to step closer
to the unreachable magnitude and quality of God’s intelligence.
The contradictions of “infinite-sided polygons, at once continuous and discrete, geometric and
combinatorial, infinitary and finite” were addressed by Cusa in his argument as to what God
is and is not, in considering God’s enfolding and unfolding. Cusa might as well be speaking to
Grosholz as “the Adversary” who was actually Johannes Wenck a leading Aristotelian
cleric.380 Cusa says that God is “absolutely, everything which is at all possible; and in this
376 Bonaventura Cavalieri (1598-1647) carried out seminal work with infinitesimals. It was often referenced
merely out of a sense of obligation by subsequent writers rather than because Cavalieri’s work bore any
relation to the later work. Cavalieri understood the infinitesimal as a fixed but extremely small quantity, quite
opposed to that of Leibniz. That is, Cavalieri’s infinitesimal appealed to common sense of the physical rather
than the ontological. Beeley, P. “Infinity, Infinitesimals, and the Reform of Cavalieri: John Wallis and his
Critics” in Goldenbaum, U. Infinitesimal Differences: Controversies between Leibniz and his
Contemporaries Walter de Gruyter, Berlin 2008, p.33 377 Child, J. M. (ed. and trans.) The early mathematical manuscripts of Leibniz; tr. from the Latin texts published
by Carl Immanuel Gerhardt with critical and historical notes Open Court Publishing Company, Chicago and
London, 1920 reprinted by University of Michigan Library, p.149 378 Grosholz, E. “Was Leibniz a mathematical revolutionary?” in Gillies, D. Revolutions in Mathematics Oxford
University Press, New York, 1992, pp.117-133 379 Duncan, A.M. trans. Kepler, J. Mysterium Cosmographicum Abaris Books, Janus Series Norwalk USA 1981,
p.93 380 Hopkins, J. trans. Nicolaus of Cusa Apologia De Docta Ignorantia 3rd ed., J. Arthur Banning Press,
Minneapolis, pp.481-2
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coincidence is hidden all apprehensible theology.” In particular, “in the mode of enfolding
[God] is all things but that in the mode of unfolding He is not any of these things.”
In De Ignota Litteratura, Wenck responds to Cusa:381
…if God (as he supposes the essence of the unqualifiedly Maximum to be) is the
precise measure of every essence, then how will it be the case that He exceeds,
incomparably, every essence? And how will the following [doctrine from]
Metaphysics X [i.e. book 10 of Aristotle’s Metaphysics] remain standing?: “In each
genus a first thing is the measure (metrum et mensura) of the subsequent things of that
genus; hence, in each genus there is a proper and precise measure.”
Cusa’s counter response is “the infinite is the most adequate measure of finite things—even
though the finite is altogether disproportional to the infinite.”382 Though “the impossibility of
there actually being an infinite line is shown in many ways in On Learned Ignorance” yet “by
the positing of an infinite line the intellect is helped to make headway toward the
unqualifiedly Infinite, which is Absolute Necessity of being.”
On a different track, Leibniz argues that if a proposition is not true in the infinitely small, then
it is questionable as to whether it is true in the finite. He explains that all of the propositions
that hold for the ellipse should hold for the parabola because regarded merely as an ellipse
with one focus infinitely far from the other. Similarly, if rest is regarded as infinitely small
velocity or infinite slowness, then whatever is true of motion should be true of rest regarded
as infinite slowness. Formulated laws starting to break down or show contradictions in the
case of rest when it is regarded as infinite slowness is a sign that those laws were wrongly
formulated.383
Does the infinite exist? It doesn’t really matter.
There are many relationships between Cusa and Leibniz, which have been explained by
historians of science.384 The purpose here is not to explore all of those relationships. However,
as regards the infinite and intellectual constructs in general, there are relationships that are
381 Hopkins, J. trans. Johannes Wenck De Ignota Litteratura 3rd ed., J. Arthur Banning Press, Minneapolis,
p.439
382 Hopkins, J. trans. Nicolaus of Cusa Apologia De Docta Ignorantia 3rd ed., J. Arthur Banning Press,
Minneapolis, p.482
383 Leibniz, G.W. “Letter on a general principle useful in explaining the laws of nature through a consideration of
the divine wisdom: to serve as a reply to the response of Rev. Father Malebranche” Nouvelles de la
république des lettres July 1687 in Loemker, L.E. Gottfried Wilhelm Leibniz: Philosophical Papers and
Letters D. Reidel Publishing Company, Dordrecht 1969 p.352 384 Hopkins says that Cusa’s likening of human minds to “living mirrors that mirror one another and all of
reality” was also adopted by Leibniz though Hopkins does not say where. The concept seems to be
encapsulated in Monadology since every monad perceives every other monad. Hopkins notes that in De
Docta Ignoranta II, 1 (97), Cusa says that the world is as perfect as it can be, which prefigures Leibniz’s
doctrine that this is the best of all possible worlds. Hopkins, J. “Nicholas of Cusa (1401-1464):
first modern philosopher?" in French, P. A., Wettstein, H. K., and Silver, B. (eds) Midwest Studies in
Philosophy Volume XXVI (2002): Renaissance and Early Modern Philosophy Blackwell Publishing, Boston
and Oxford, pp.16, 20
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relevant here. If the infinite and the infinitesimal are both infinites, then the method Leibniz
and others are using with infinitesimal reasoning are subsumed by Cusa’s understanding that
“positing an infinite line helps the intellect make headway” towards the truth. It is the positing
of possibilities or the musing about scenarios that pulls us forward, not necessarily the truth or
falsity of such hypotheticals.
We see that in Leibniz’s mathematics, the aim is pragmatism in the service of truth and utility,
not truth itself. For example, due to the facility of arithmetic progressions in simplifying
equations, to use them “is an exceedingly remarkable method”.385 Extending this to dy as it
depends on x, “by this artifice many excellent theorems with regard to curves that are
otherwise intractable will be capable of being investigated, namely, by combining several
equations of the same kind.”386 At the same time, it is not a game with an unimportant
outcome. If the method of causing a secant to approach the tangential did not yield an
understanding of the tangent, then an array of contradictions would have emerged and Leibniz
would have had to find another method.
Leibniz is frank on his pragmatism in the use of the infinitesimal itself, “whether such a state
of instantaneous transition from equality to inequality, from motion to rest, from convergence
to parallelism, or anything of the sort, can be sustained in a rigorous or metaphysical sense, or
whether infinite extensions successively greater and greater, or infinitely small ones
successively less and less, are legitimate considerations, is a matter that I own to be possibly
open to question”.387 He then goes on to address his “rigorist” critics directly and explain
what the infinite or infinitesimal are, as far as the question needs to be addressed, saying that
a quantity is such when it is “as great as you please, or as small as you please, so that the error
that any one may assign may be less than a certain assigned quantity”.388 In technical terms,
this does not contradict Cusa’s maintaining “the impossibility of there actually being an
infinite line”.
Leibniz declares that impossibility or otherwise of the infinite or infinitesimal is not
important:389
If any one wishes to understand these as the ultimate things, or as truly infinite, it can
be done, and that too without falling back upon a controversy about the reality of
extensions, or of infinite continua in general or of the infinitely small, ay, even if he
think that such things are utterly impossible; it will be sufficient to make use of them
as a tool that has advantages for the purpose of calculation, just as the algebraists
retain imaginary roots with great profit.390 For they contain a handy means of
385 Child, J. M. (ed. and trans.) The early mathematical manuscripts of Leibniz; tr. from the Latin texts published
by Carl Immanuel Gerhardt with critical and historical notes Open Court Publishing Company, Chicago and
London, 1920 reprinted by University of Michigan Library, p.97 386 Ibid., p.98 387 Ibid., p.149 388 Ibid., p.150 389 Ibid. 390 John Wallis geometrically explained negative numbers and square roots of negative numbers, treating
negative as the opposite direction of positive. Wallis geometrically constructs the square root of a negative
number by applying Pythagoras’ theorem to a right-triangle in mirror-image of another sharing the same
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reckoning, as can manifestly be verified in every case in a rigorous manner by the
method already stated.
Possibly in a continuation of the rigorist attacks on the infinitesimal, many of which stem
from its apparent non-existence, Boyer criticised Leibniz’s infinitesimal as lacking a precise
definition and said that it required Cauchy to make it precise. Boyer said that both Nicolaus of
Cusa and Nieuwentijdt, a critic of Leibniz, defined the infinite and the infinitesimal
unsatisfactorily.391 Leibniz maintained that it did not need to be any more precise than it was.
Moreover, is it possible that everything Cauchy used to make the infinitesimal precise,
purportedly, would several centuries before have required the same “leaps of faith” that
Leibniz’s calculus did?
Do Platonic ideas exist? It doesn’t really matter.
The method stated above (of arithmetic progressions extended to dy as it depends on x)
translates into the / method which was once taught in high school. Arguably, such “tools”
are examples of Platonic ideas, and Platonic ideas have given rise to much controversy. In his
statement about the calculus being “a handy means of reckoning” Leibniz stays outside that
controversy. For example, Cheyne and Pigden wrote that, “Mathematical platonists claim that
at least some of the objects which are the subject matter of pure mathematics (e.g. numbers,
sets, groups) actually exist.” They argue, “Either the dispensability of mathematical objects to
science can be demonstrated and, hence, there is no good reason for believing in the existence
of platonic objects, or their dispensability cannot be demonstrated and, hence, there is no
good reason for believing in the existence of mathematical objects which are genuinely
platonic.”392 However, Leibniz makes clear that it is irrelevant whether the infinite and
infinitesimal actually exist and, indeed, science can derive great benefit from them even if we
believe that it is “utterly impossible” that they exist. In any case, more than a century later,
Riemann proposed unboundedness as a postulate more general than infinitude.393
Cheyne and Pigden have implied that there is a distinction between platonic objects and the
“reality” of science. For Leibniz there is no distinction between the status of mathematical
objects and scientific theories which are putative scientific realities. This is not because
Leibniz regards mathematical objects as real but, rather, because scientific theories are as non-
real as mathematical objects even though scientific theories seem to the human mind to bear a
closer relationship to reality. At the same time, it must be noted that Leibniz’s conception of
vertical side. Incidentally, Wallis’ argument leads naturally to construction and use of x- and y-axes, which
bear Descartes’ name “Cartesian” today. Wallis, J. Algebra 1673, cap. LXVI (Vol. II, p.286) in Smith, D. E. A
Source Book in Mathematics McGraw-Hill: New York, 1929 p. 46ff 391 Boyer, C. The History of the Calculus and its Conceptual Development Dover Publications: New York, 1949
p.214 392 Cheyne, C. and Pigden, C. “Pythagorean Powers or a Challenge to Platonism” Australasian Journal of
Philosophy (1996) 74, 639-645. Leibniz may respond in the same way to Franklin, J. “Mathematical
Necessity and Reality” Australasian Journal of Philosophy 67(3), Sept 1989, 11-17. 393 Smith, D. E, A Source Book in Mathematics Dover Publications, New York 1959, quotes from Riemann’s
1854 dissertation Über die Hypothesen welche der Geometrie zu Grunde liegen
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platonic objects is not the same as those of mathematical Platonists because, for Leibniz,
platonic objects like all ideas are certain and inexorable but are not real.
What if everything we understood, or think we understood, about science were itself a
platonic object or corresponded to one? This is the obvious hypothesis for Nikulin to test in
his quest to understand how it is that, “Mathematical entities are to be associated with
empirical objects.”394 To do so, Nikulin must bridge the divide between the “platonic domain”
and the physical universe. On the one hand, Cantor appears to have thrown a spanner in the
works because his theory of infinite sets purported to understand the concept of an infinite
God. Chaitin calls it “mathematical theology”.395 On the other hand, such an investigation is
perfectly natural and potentially fruitful in Leibniz’s conception of science.
Metaphysical truths are at the heart of final causes which Leibniz emphasized are one of the
most powerful ways of understanding efficient causes. Recall that final causes are essentially
the workings of God’s mind in designing the universe.396 While the metaphysical tools used to
understand final causes do not exist, they are indispensable to minds that put themselves to
the task of understanding reality including scientific reality. For example, like the
Pythagoreans, Leibniz gives number the highest place when he says, “But there is nothing
which is not subordinate to number. Number is thus a basic metaphysical figure, as it were,
and arithmetic is a kind of statics of the universe by which the powers of things are
discovered.”397 (emphasis added) Given Leibniz’s outlining of the role of the concepts of the
infinite and infinitesimal, we can only conclude that he agrees completely with Cheyne and
Pigden when they say, “Numbers are needed to underwrite any conceivable causal order but
they themselves play no part in the proceedings. They provide a sort of metaphysical
framework for any possible physics — an indispensable, indeed, a necessary backdrop for the
causal show. But though there could be no causal structure without numbers, numbers are not
implicated in the causal shenanigans described by any science whether actual or merely
possible.”398
Thus, “the objects which are the subject matter of pure mathematics” are just as real as the
science to which they may or may not be dispensable. That “reality” is not corporeality or an
Empirical reality. This is an easy conclusion to reach if, like Leibniz, one is not prepared to
embrace Empiricism. This does not mean that Leibniz believed that mathematical objects
actually exists as mathematical platonists claim. Mathematical objects have the same
existential status as any other idea: they are not real and exist only as intellectual impressions.
394 Nikulin, D. V. Matter, imagination and geometry: ontology, natural philosophy and mathematics in Plotinus,
Proclus and Descartes Ashgate, Aldershot and Burlington 2002, p.xi 395 Chaitin, G.J. “The Search for the Perfect Language” a talk delivered at The Perimeter Institute for Theoretical
Physics, Monday, September 21, 2009. Text accessed at http://www.umcs.maine.edu/~chaitin/pi.html 15
April 2012 396 Leibniz, G. W., Specimen Dynamicum 1695 in Loemker, p.442 397 Loemker 1969, p.221 398 Cheyne, C. and Pigden, C. “Pythagorean Powers or a Challenge to Platonism” Australasian Journal of
Philosophy (1996) 74, 639-645, p.645
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Rigor is of service
Leibniz even grants that the rigorists have done a useful service as they have “discovered an
art of advancing and of deriving so many things from a few. If they had tried to put off the
discovery of theorems and problems until all the axioms and postulates had been proved, we
should perhaps have no geometry today.”399 It is the process that they have given us, even
though their axioms and postulates, and so the conclusions, are not true. Leibniz cites Euclid’s
unproven axiom that two straight lines can meet no more than once which project was
hampered by Euclid’s lack of a good definition of a straight line.400 Ironically, one of the
fathers of rigor, Euclid, assumed the infinitude of the line.401 However, writes Leibniz, “I
blame Euclid much less [than Descartes] for assuming certain things without proof, for he at
least established the fact that if we assume a few hypotheses, we can be sure that what follows
is equal in certainty, at least, to the hypotheses themselves.”402 Leibniz sees a broader
application of this lesson, for he writes, “if some careful and meditative mind were to take the
trouble to clarify and direct their [theologians’ and scholastic philosophers’] thoughts in the
manner of analytic geometers, he would find a great treasure of very important truths, wholly
demonstrable.”403
Science may use indemonstrables
Raynaud’s reading of Leibniz leads him to the same endpoint as we are reaching here. In
particular, that according to Leibniz “truth admits of degrees”, and “its discovery passes by
the way of logical analysis, and that even in theoretical matters it may sometimes be
reasonable to employ ‘indemonstrables’. Logic thus becomes once again the model for
science, because the first principle of necessary truths is the principle of non-contradiction. …
By the same token, he establishes a new continuity between science and the active life,
between knowledge and practical judgement, and between reason and faith. For if science
may employ indemonstrables, that also means that the initial absence of such principles does
not impede one from progressing on the path of reason, without having to ‘cast into doubt’
common beliefs.”404
Science and all thought worthy of the name Reason is about hypothesization, it seems, and
active use of hypotheses, until it becomes clear that a hypothesis contradicts something that
we know to be true through other means.
We now move away from considerations of pragmatism, and onto leaps of logic and the use
of metaphysics in physics.
399 Loemker, p.384 400 Wolfe, H. E. Introduction to Non-Euclidean Geometry Holt, Rinehart and Winston New York 1945, p.17ff.
See esp. pp.20-21 regarding Wolfe’s objections to Playfair’s Axiom. 401 Ibid., pp. 6, 8 402 Loemker, p.384 403 1686, Wiener, p.304 404 Raynaud, P. “Leibniz, Reason and Evil” in McCarthy, J. C., (ed. and trans.) Modern Enlightenment and the
Rule of Reason CUA Press Washington D.C. 1998, p.152
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Global architecture determines local mechanics
Franklin wrote that Kepler “aims to explain everything about the system of the heavens,
including facts of kinds that modern astronomy makes no attempt to explain.”405 So did
Leibniz, and like Kepler thought that it was not only perfectly reasonable to pursue such an
ambition but that ultimately that is the only valid quest for the scientist and metaphysicist
who, to Leibniz, were in almost identical professions.
Aside from all else, a kind of universal or physical morality, or morality of physics, is posited.
In c.1696, Leibniz wrote, “those minds in whom the imaginative faculty predominates …
believe that they need to use only mathematical principles, without having any need either for
metaphysical principles, which they treat as illusory, or for principles of the good, which they
reduce to human morals; as if perfection and the good were only a particular result of our
thinking and not to be found in universal nature.”406
The expression of that perfection and good is labelled by Leibniz as the architectonic nature
of things, or the way in which God acted in creating the universe. Leibniz is unambiguous on
the relation between the purpose of God in the very large, the mechanics of corpuscles, and
the relationship between religion and natural science. The dichotomy is from God as architect
to God as legislator, and there is really no dichotomy because they are both the same God.407
We quote from Leibniz’s Tentamen anagogicum which is extracted at greater length in
Appendix 2:408
...the smallest parts of the universe are ruled in accordance with the order of greatest
perfection; otherwise the whole would not be so ruled. It is for this reason that I
usually say that there are, so to speak, two kingdoms even in corporeal nature, which
interpenetrate without confusing or interfering with each other - the realm of power,
according to which everything can be explained mechanically by efficient causes when
we have sufficiently penetrated into its interior, and the realm of wisdom, according to
which everything can be explained architectonically, so to speak, or by final causes
when we understand its ways sufficiently. In this sense one can say with Lucretius not
only that animals see because they have eyes but also that eyes have been given them
in order to see...
The architectonic view is almost synonymous with a presumption of intention in the design of
everything in the physical universe.
405 Franklin, J. The Science of Conjecture: Evidence and probability before Pascal J. H. Press Baltimore and
London 2002, pp.150-1 406 Loemker, p.478 407 Wiener (a) efficient and final causes, first half of p.524 from “The Principles of Nature and of Grace, based
on Reason” 1714 second half of § 3, and (b) regarding God as architect, or maker of efficient causes, and
God as legislator, or maker of final causes first half of p.551-2 from Monadology 1714, §§ 89-90. 408 c.1696, Loemker, pp.477-9
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Intention in the large
Thus not only is the Creator most powerful and intelligent, but he acts with intention. Since
God created the universe and the universe “runs” without divine intervention, it would be
more accurate to say that God has designed an intention capability – indeed, imperative – into
the universe. That is, the universe itself “intends” for certain outcomes or, at least, for certain
tendencies or trajectories to be followed in the unfolding of events. At this stage, we refer to
this trajectory as “towards the Best”. For Leibniz, discoveries in natural science are bound up
with this presumption.
When doing science, philosophy or even when thinking about something to try to understand
it, Leibniz advises that we are greatly aided by first identifying whether we are considering
efficient causes or final causes. The former are the laws of nature or physics whereas the latter
are the appetition of souls. The appetition of souls are the tendencies or “desires” of the more
active part of the universe; these are the (incorporeal) monads with clearer perceptions to be
explained in detail in the next chapter. Neither the efficient nor final cause is adequate without
the other.409 We ultimately can better understand an efficient cause or how a process works in
the small if we also look to its final cause which, effectively, really is the intention of the
Creator in putting that process there or where that process fits in the Creator’s big picture.
The architectonic foundations of reality manifest in the ongoing functioning of the universe
including humankind within it moreso than in the initial act of creation. It is an anti-entropic
outlook on everything in the universe, including humanity.410 On the contrary, the usual
corollary of Clausius’ Second Law of Thermodynamics, often referred to as “the Second Law
of Thermodynamics”,411 is that the universe tends towards increasing entropy. The Leibnizian
corpus is therefore opposed to the Second Law of Thermodynamics.412 In any case, Clausius’
Second Law of Thermodynamics was only formulated in the context of particular heat
409 Ibid., Loemker p.588 middle of first paragraph 410 A detailed study of the Second Law of Thermodynamics is beyond the scope of this dissertation. However,
Hans Reichenbach’s explanation and proof is entirely statistical at the microscopic level. At the macroscopic
level it is entirely Empirical except where he is reasoning inductively from the microscopic. See
Reichenbach, H. The Direction of Time University of California Press, Berkeley and Los Angeles 1956,
pp.49ff and 145ff. Note that Leibniz was of the mind that time does not exist as an independent physical
reality; it is merely a creation by humans for ordering events. Similarly, pure space is merely an invention by
humans for giving objects position. Leibniz writes, “Time is the order non-contemporaneous things” and,
“Extension is the quantity of space” but “It is false to confound extension, as is commonly done, with
extended things, and to view it as substance.” 1715 “Metaphysical foundations of mathematics” in Wiener,
p.202 411 The literature on this topic is vast. For a summary, see http://hyperphysics.phy-
astr.gsu.edu/hbase/thermo/seclaw.html accessed 26 Feb 2012 412 It is not only the Leibnizian corpus that casts doubt on the Second Law of Thermodynamics. Prigogine says,
“...it must be recognized that the formulation of the second law seems to us today to be more a program than
a well-defined statement, because no recipe was formulated by either Thomson or Clausius to express the
entropy change in terms of observable quantities. This lack of clarity in its formulation was probably one of
the reasons why the application of thermodynamics became rapidly restricted to equilibrium, the end state of
thermodynamic evolution. For example, the classic work of Gibbs, which was so influential in the history of
thermodynamics, carefully avoids every incursion into the field of nonequilibrium processes (Gibbs 1975).”
Moreover, and noting that only irreversible processes contribute to entropy production, recently “a complete
change in perspective has arisen, and we begin to understand the constructive role played by reversible
processes in the physical world.” (emphasis in original) Prigogine, I. From Being to Becoming W. H.
Freeman and Company, San Fransisco 1980, p.78
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engines assumed to be closed systems. Whether there is any such thing as a closed system is
subject to debate. However, this question is well beyond the scope of this thesis.
This is seen in Leibniz’s “best of all possible worlds” doctrine – “everything in the whole
wide world proceeds mathematically, that is, infallibly” – as clearly as anywhere else in
Leibniz.413
Conclusion
A mathematics MSc thesis in the early 21st Century might seem an unlikely place in which to
debate some of the considerations addressed debated in this chapter. We are not undertaking
mathematical calculations, true. However, we are discussing in what context and to what end
mathematical work should be done. Mathematical work is always done with philosophical
and metaphysical assumptions. Scientific work, especially basic research, can only benefit
from at least being aware of what one’s assumptions are so that they can be questioned. The
entire direction of a scientific or mathematical career could be changed by an awareness of
the metaphysical foundations on which a researcher is building their work.
413 “On destiny or mutual dependence” in Wiener 1951, p. 571
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Chapter 7: Science by thought, and reality and substance
Introduction
It is often thought that modernity is synonymous with an aversion to the supernatural. Further,
the common stance is that belief in God is unscientific or, at least, something that bears no
connection to science. For Cusa, Kepler and Leibniz, this was not the case. For them, doing
science was coming to understand God. This was not just a matter of labelling their
discoveries as a deepening of their understanding of God’s Reason. The process of scientific
enquiry was benefited by seeking to better understand God. In fact, the act of scientific
enquiry could not be undertaken without thinking about God and God’s relationship to the
universe. Leibniz held that sense perception has limited usefulness in discovering truth and
that truth is discoverable only by the mind. Moreover, Platonic ideas when known can be used
as tools by the mind.
Were we only able to “know” what our senses tell us, God would not exist for us since God is
not manifest to the senses. In that case, there could be no harmony or meaning, only
coincidences. But our reason tells us that there is harmony which cannot be accidental. Cicero
argued this in De natura deorum with the analogy of the barbarians observing a ship of a size
and complexity they have never before seen and slowly realising that it was made by
intelligent minds. In the analogy, the barbarians represent thoughtful humans and the ship
represents the physical universe. If there are only coincidences, we must give up hope that
reason can ever be of use, and rather deduce only from sensory observations. Platonic ideas
cannot be discovered within such a framework. Furthermore, we only discover the harmony –
that is actually there – using reason. Once we allow that reason, then we see a great deal else.
It does not make sense that there would be harmony in some things but randomness
elsewhere. From here we get the principle that if only we could see things “in the large” we
would see the harmony everywhere. Leibniz took Kepler’s harmony further, beyond the study
of the solar system into all things, philosophical and metaphysical, and into all fields of
enquiry.
The empirico-deductive line of thought was largely atheist, though some thinkers – like
Newton adopted a conception of God consistent with the positivist science they were
constructing. Theistic metaphysics gives rise to a scientific method that is more permissive in
the range of thought allowed and more powerful in pushing the envelope of human
understanding through progressive grasping of principles that govern the universe. Leibniz
would argue that it is not only more permissive but necessarily more correct. As we shall see
in Chapter 7, Kepler argued the same.
In this chapter we will describe and discuss Leibniz’s framework for reality, including God,
substance, the universe as a whole, the components of the universe and how these components
interact. There are many references to Leibniz’s Monadology.
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Science by thought
Leibniz’s writings lead us to the conclusion that all science can be undertaken in the mind.
Parkinson asks, “Is it true to say that Leibniz, in his science, relies wholly on a priori
arguments?”414 answering “Yes” for the most part, but not exclusively as Leibniz leaves room
for observation and induction from observations.415 We hope to flesh out Parkinson’s answer.
Science is done by investigation and reasoning about metaphysics, mathematical rules and
other ideals. Science done in this way seeks to understand the mind of God, ultimately. Thus,
theology is prominent in Leibniz’s framework.416 Farrer says, “If we condemn Leibniz for
writing physical theology, we condemn not him but his age.”417 However, Leibniz makes it
clear that it is a conscious contention of his own that God must be a central consideration,
rather than something he has imbibed from the zeitgeist.
It has already been mentioned that in Theodicy Leibniz defended the daring endeavour of
seeking to understand the mind of God against detractors, and supported others who
encouraged and joined the quest. While Leibniz accepted Cusa’s doctrine that in ignorance the
soul finds its repose, nonetheless what Christian faith commands does have reason at the back
of it which is comprehensible by men.418
While experiment is useful in checking our reasoning, it is not necessary by force of reason. It
is practically necessary because our conceptions and thoughts are generally indistinct and
confused. Occasionally, a human mind comes along which is clear and distinct, and teaches us
how to think and in so doing gives an entirely new set of ideas. Plato and Jesus Christ would
be examples. The “believe it when I see it” mentality is nothing but a popular choice for
perception over thought.419 Indeed, the soul itself and its function of thought are not furnished
by the external senses.420 That Einstein was thinking in this vein would be evidenced by the
quote anecdotally attributed to Einstein, “I want to understand his [God’s] thoughts. The rest
are details.”421
Universal characteristic
The power of a priori reasoning is why the Universal Characteristic, or calculus of ideas or
Lingua Philosophica (“Language of Philosophy/Logic”), is the one of the most important
414 Parkinson, G. H. R. “Science and metaphysics in Leibniz’s ‘Specimen Inventorum’” Studia Leibnitiana Vol.
6, 1974 p.4 415 Ibid., p.15 416 Leibniz writes, “Herr Dreier of Königsberg has aptly observed that the true metaphysics which Aristotle
sought, and which he called [Greek: tên zêtoumenên], his _desideratum_, was theology.” Theodicy PG
edition 2005, p.244 §184 417 Leibniz, G. W. Theodicy Project Gutenberg edition 2005, p.28 accessed at www.gutenberg.org 418 Ibid., §§48-51, pp.101-102 419 §5 in Loemker,p.638 420 Loemker, p.556 start of 2nd last paragraph 421 At http://web.ceu.hu/yehuda_einstein_and_god.pdf accessed 30 Oct 2010, §9, pp.4-5. This is quoted as a
statement by Albert Einstein in many locations, but we have been unable to find the primary source.
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undertakings.422 While Brown notes that Leibniz made little apparent progress towards it,423
Leibniz had a clear conception of its nature, what it would be able to achieve and why it was
possible. However, it may be that Leibniz’s work on the calculus and especially in logic were
themselves strides achieved toward the universal characteristic.
Antognazza says that the characteristica universalis was a part – though a large part – of a
scheme for all areas of human thought and endeavour from science and technology to law and
religion which Leibniz called the scientia generalis. Leibniz wrote an array of articles towards
this goal. For example, in a series of logical texts written over 1678-1684, Leibniz outlined a
logical calculus whose structure remains valid independent of the contents assigned to the
letters used in the argument such as, to use Antognazza’s example, a for “animal” and b for
“rational”.424 With it, as well as many day-to-day uses, we could conduct a priori science.
This is not “discovering the mind of God” per se, which is impossible according to Cusa,
since truths, physics and all else but God are distinct and independent from God, as Leibniz
explains in “Reflections on the doctrine of a single universal spirit”.425 It is also impossible
according to Leibniz.426 Nonetheless, the more clarity we can achieve, the better off we are; in
particular, the more useful our hypotheses will be.
Leibniz lamented that the art of discovery was little known outside of mathematics and said
that it should exist in systematic form for all domains of knowledge.427 Developing a calculus
of ideas might start with the Platonic dialogues. Leibniz’s own development of the
infinitesimal calculus might provide ideas on how to start, albeit there would necessarily be
important differences.
Dream vs reality, and substance vs phenomena
Leibniz distinguishes body from a phenomenon in a coherent dream.428 On the contrary,
Leibniz also said there is value in the Platonic perspective that we wake up on death implying
that death is more real than life.429 Confirming Leibniz’s dim view of how much truth is
directly discernible from what is experienced in life, he explains that he can demonstrate
experimentally that all the perceivable properties of bodies are apparent only. This does not
mean that there are no real bodies. It is just that what we perceive about them is misleading.
Bodies are real when they have substance. Such bodies may act or be acted upon, whereas
bodies that are not real cannot act on other bodies and cannot be acted upon. Bodies that are
not substantial are merely phenomena or, at most, are aggregates of true bodies. A coherent
dream has phenomena but not substantial bodies, and a phenomenon is defined by the act of
observing it not by the substance that it may or may not include. A substance has active
422 Leibniz, G.W. “Towards a Universal Characteristic” 1677, Wiener, p.17 423 Brown, R. C. Leibniz, unpublished, Chapter 11 “Epilogue” p.167 424 Antognazza, M.R. Leibniz: An intellectual biography Cambridge University Press 2009, pp.241-243 425 1702, Loemker, pp.554-560 426 Ibid., §14 p.640, and p.641 5th, 4th and 3rd last lines 427 “Precepts for advancing the sciences and arts” 1680, Wiener, p.49 428 At http://plato.stanford.edu/entries/leibniz-mind/ section 1 second last paragraph, accessed 30 Oct 2010 429 Loemker, pp.364-5, i.e. last paragraph of p.364 spilling over to first paragraph p.365
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metaphysical power insofar as it expresses something distinctly, and passive power – i.e. the
ability to suffer being acted upon – insofar as it merely expresses something confusedly.
There are infinite degrees between. Here we see that Leibniz has defined the power of a thing
by its clarity of expression. To Leibniz, as we will see in his definition of ideas, clarity
includes consistency with all else that is real in the universe.
What is real?
The mind is part of reality, but it also sees reality imperfectly. By contrast, God’s mind
perfectly sees all of reality, i.e. all soul and body monads, and all ideas and their implications
on reality.
The “really real” are the simple substances or created monads which once existing endure
always, as well as God himself.430 So: (i) God and (ii) simple substances also known as
created monads also known as souls comprise all that is real. These do not exist in the mind of
God. That is, the “verities” (universal truths or ideas) are not the result of God’s will, contrary
to Descartes.431 Harmony itself is a thing separate from God, but harmony is perceived –
albeit that it is an idea – so it is not really real either. God creates things in conformity with
such because to do so is the highest and best way of doing things. Beauty is something
separate from God but only God makes it possible because everything that gives rise to it in
reality (viz. monads) comes from God.
Monads are real
With the monad concept, Leibniz is following his own advice in the principles of discovery in
breaking things down to their most fundamental parts. However, monads are not just atoms or
quarks, for they have knowledge of one another and of the entire universe. Thus, they each
express the whole as well as being the parts. If we regard the entire universe as an organism,
then monads are like cells in that each cell has DNA with an encoding of the entire organism
(i.e. the entire universe) while also “knowing” its role in the organism.
A “created thing” is a created monad.432 Monads are actual things, i.e. they are “existing”.433
The monad is nothing but a simple substance, or a substance without parts, which enters into
compounds.434 The soul of humans is simple.435 The rational soul is the “substantial” part of
man.436 The physical form of man comprises trillions or more of body monads. Those body
monads exist and are substantial, but the physical form is passing, temporary and, in any state,
430 Ibid., p.592 last para. 4th line 431 Theodicy, p.244 §186, p.428 §21 accessed at www.gutenberg.org 432 Monadology §49 in Latta, p.245 433 Ibid., §8 p.220 434 Ibid., §1, p.217 435 Ibid., §16, p.226 436 1705, Loemker , L. E. Gottfried Wilhelm Leibniz: Philosophical Papers and Letters 2nd ed. D. Reidel,
Dordrecht Holland 1969 p.586
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is an idea only. Thus, it is insubstantial. Even as an idea, it can accurately only be conceived
in the infinitesimal and this is only an idea. In fact, it is never existent or being, and is only
ever becoming.
In explaining monads in Monadology, Leibniz referred to substance in a particular way suited
to his conception of monads. This is sensible because, as Leibniz explains it, all substance
comprises monads; there is no substance that is not “made” of monads, and this includes
substance that is not intangible such as the soul and even such as God. Passive substance,
which includes the constituent simple body monads of a human’s physical form, is simply in a
prolonged unconsciousness.437 Yes, the soul is a substance and no substance is perishable in
nature.438
Most monads come in pairs – body and soul – wherein either or both elements of the pair may
be a compound monad.439 They cannot be the same monad because one is a compound monad
and thus destructible whereas the other is not. One is relatively passive, the other active. Yet
they act in harmony. This is a “pre-established harmony”.440
Each monad is aware of every other. There is an inter-communication between all monads and
thus between all things, so each body feels the effect of all that takes place in the universe.
Thus, it would be possible to “read in each what is happening everywhere, and what has
happened and shall happen, observing in the present that which is far off in time as well as in
place.”441 Latta writes that this inter-communication is a symbol of pre-established
harmony.442 It is not clear why Latta uses the term “symbol”. Arguably this is not what
Leibniz means. In formulating the pre-established harmony, we do not know whether Leibniz
had any purpose other than to bridge the gap between the realm of final causes and that of
efficient causes, or between the immaterial and the material. The universal awareness of each
monad is a property of monads. A harmony, on the other hand, is an idea which governs
things from outside of themselves.
There must be body monads without a soul which are entirely passive, and spirit monads
without a body. However, the question of being soulless or bodyless is one of degree for
Leibniz posited a continuum in most metaphysical matters he considered. Neither generally
exists in the natural order of things and Leibniz does not discuss them in detail.
The soul (part) is not divisible into an active eternal part and a passive temporal part as
Averroes believed.443
437 Monadology §14, p.224 438 Ibid., p.225 mid-footnote 25 439 See esp. the 9th and 10th last lines of the 2nd para. p.590 of Loemker, and the overall 1702 essay on universal
harmony Loemker, pp.574-591 440 Ibid., p.587 441 Monadology §14, p.251 §61 442 Latta writes, “the material action and re-action throughout the universe, such that a change at any one point
affects every other, is a symbol of the Pre-Established Harmony among the Monads.” Ibid., p.251 footnote 96 443 1702, Loemker p.554 2nd paragraph
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Immaterial souls exist
Leibniz calls Epicurus and Hobbes’ doctrine of material souls an evil one. Leibniz also wrote
that they have extended to humans what the Cartesians have held for animals, which is also
incorrect.444 Leibniz also says that it is pernicious to deny the immortality of souls.445 This
brings Leibniz into opposition with Pompanazzi and Sarpi.
We will see later that active or more perfect monads (souls) and passive or more imperfect
monads (bodies) are all that exist aside from God. It is consistent with this that Leibniz often
wrote that the purpose of the universe is to help intelligent souls get closer to perfection and
felicity. According to Leibniz, this is the paramount goal of God.
The nature of ideas
Ideas exist in our memory, indistinctly, and can be brought to the fore as Plato showed in
Meno. God too has many ideas, but perceives/thinks them distinctly and all at once as Leibniz
encourages humans to do in “the art of reasoning well”.446
God Body monad Soul monad
Perfect Imperfect Less imperfect
Active Passive Less passive
Immaterial Material Immaterial
Memory No memory Memory Together, Memory and
Perception are Sentiment. Knowledge Perception Perception
Will Appetite Appetite
Figure 3: Some qualities of God and substances
Time, distance or length (i.e. “extension”), motion, the continuum and numbers express only
possibilities.447 They are ideas at the simplest level; they are primary. Leibniz recalls that
Hobbes described space as a “phantasm of the existent”448 without disagreeing. A phantasm is
an image as if in a dream. Dreams do not exist anywhere except in the mind of the dreamer;
dreams are not real. The definition of real is that which conforms to metaphysical principles
and mathematical rules.449 It is only through such that we know that we are not dreaming; that
is, that our experience is not a mere figment of (or a mere image in) our mind. This is because
our perceptions are imperfect and indistinct; if our current reality was a creation of our minds,
then there would be inconsistencies everywhere and there would be measurable discrepancies
from ideals. In any case, there is only one reality and it is not just a “coherent dream”.450
444 Ibid., p.577 445 Ibid., §4 Loemker p.638 446 Inference from Monadology, p.247 §53; “On wisdom” c.1693, Wiener p.77 447 Loemker p.583 448 Loemker p.583 449 Loemker p.583 last 20 lines 450 At http://plato.stanford.edu/entries/leibniz-mind/ section 1 second last paragraph, accessed 30 Oct 2010
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More accurately and specifically, extension is the order of possible coexistence whereas time
is the order of possibilities which are inconsistent (i.e. cannot coexist) but which have a
connection through (continuous) events. However, space and time themselves are indifferent
to what content is put into them, just as numbers are indifferent to what they enumerate. Here,
Leibniz is distinguishing the real from the non-real. An ideal is an idea all the same.
Leibniz consistently fails to enlighten us on where ideas exist probably because he would
maintain that an idea is a potentiality at most. It does not exist anywhere. In his characteristic
style, he would likely dismiss the question as nonsensical. For a truth does not exist
anywhere, it simply has the potential to be used in governing the behaviour of passive
substance because it is the best pattern of behaviour. An idea is not real. For Leibniz, the non-
reality of concepts and ideas dates back to his bachelor’s dissertation published when he was
17 and written under the direction of Thomasius.451 While the intelligible world in the divine
mind is the region of ideas, the ideas are not real. The divine mind knows them, and it does
not need to work ideas out (i.e. find them) or general them for it sees all at once.
In a book entitled Strict Finitism, Kielkopf undertakes a thorough analysis of Wittgenstein’s
foundations of mathematics. There, Kielkopf says that an absolute platonist (sic, small “p”)
believes that mathematical objects have their own independent existence. Strictly speaking,
Leibniz is not platonist according to this definition because, to Leibniz, ideas are not real and
do not “exist” in any domain nor even in God’s mind or our own minds. Ideas might be
perceived in our minds, but they do not exist there. An idea does not exist to be perceived but,
rather, the idea is itself a perception. When the mind perceives an idea, it is, in part, creating
an impression on itself and, in part, receiving impressions from the physical universe. Note,
however, that the only mind which ever perceives an idea is God’s mind. Humans can only
ever approximate ideas.
To Leibniz, “real” includes physical and metaphysical. Though he says ideas are not real, it is
clear that ideas played a role in God’s mind during the design of the universe and they
certainly occupy the activities of human minds. This does not mean that ideas exist in God’s
mind or human minds. Concepts, too, play a role in human minds, though concepts have
internal inconsistencies causing them to fail to qualify as ideas. Nor do concepts exist in
human minds. This is because to be a mental impression alone is not to exist.
There will now be some discussion involving ideas, hypotheticals and hypotheses.
Hypotheticals are treated as atomistic propositions or concepts, such as the existence of a
perfect square or just the concept of a perfect square. A hypothesis is a set of concepts that are
more widely encompassing of a process, formulated with the intent of explaining that process.
The usual meaning of hypothesis as a precursor to a theory of some physical process, force or
phenomena is useful too. There is usually some unifying thread between the concepts in the
hypothesis. This high level of abstraction is sufficient for what follows.
451 “In the course of his analysis he shows himself a Nominalist. To Leibniz nothing exists independently of
individuals. Common properties shared by individuals do not actually exist in re; they are purely creations of
the mind.” Brown, R. C. Leibniz unpublished, Chapter 4, pp.32-33
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An idea has the same reality as a hypothetical or a concept. None of them exists in the
physical universe. The geometrical concept of a perfect square is a hypothetical. Yet a perfect
square can never be realised in the physical universe and so it is not an idea. Nonetheless, it is
useful in understanding the universe because the universe is designed in the best possible way
and so there is an order to it. Thus, understanding the best possible involves the use of ideals
such as geometrical ideals and other mathematical ideals. Mathematics as the study of
structure presumes that there is structure in the universe or underlying the universe. Really, a
hypothetical can be regarded simply as a concept formulated with the purpose of aiding in
understanding the universe and therefore it probably should and usually does have some
structure to it.
A hypothetical is just an intellectual impression in the mind, and the mind is capable of
formulating nonsense which could never be realised in the physical universe as well as
concepts which can never be realised but which are nonetheless useful to us. A thought of a
thing that can be realised is no more real than a thought of an impossible thing – both are
merely a thought or mental perception. A concept is promoted to the status of an idea if it is
internally consistent and consistent with all else that must be realised in the universe for the
universe to be the best possible. Ideas do not exist in the immaterial domain of souls, but they
do define the best possible – indeed, the only – way of organising the universe. So where do
ideas exist? Again, ideas are only perceived by minds, yet they are perceptions that can be
manifested in the physical universe. Were hypotheticals and thus hypotheses prohibited in the
conduct of science, we would not be able to attempt to work towards ideas and thus predict
and control how the physical universe works, except in a haphazard and accidental or
serendipitous way. The benefits of mathematics and physics would be denied to us.
The applicability of ideas
We will use the term ideals and ideas interchangeably. However, the term “ideals” will tend to
emphasise metaphysical principles and mathematical rules all of which are ideas themselves.
Leibniz sometimes regards the laws of physics as comprising metaphysical principles and
mathematical rules. On the other hand, the term “ideas” encompasses all truths including
ideals.
Ideas or ideal things – since an idea only qualifies with that name if it is clear and non-
contradictory and is therefore a derivative of ideals – express possibilities. They are not
actually real.452 However, they are imprinted on or remembered by our soul’s memory which
may or may not be infinite though it can comprehend the infinite though in fact it only ever
does so in an indistinct way.453 X being remembered by the memory does not mean that X
exists there. “Something mental” only designates “the possibility of parts, not something
actual”.454
452 1704, Loemker pp.535-6 453 1702, Loemker p.583 454 1704, Loemker p.536
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Considering numbers, an idea is different from a possibility but it is a characteristic or
denotation of an aspect of a possibility since no number can ever really exist. Yet ideals
govern passive substance and the passive part of substance that has active and passive parts,
for everything is composite and nothing actual is purely active or passive. The most active
souls and God choose to create in conformity with ideals because that leads to the greatest
harmony and leads to the greatest felicity for all souls. Deluded souls (including humans
limited in understanding) create in a way that contradicts ideals and, thus, in a temporary way.
However, God has designed things so that something good comes from perceptions and
appetite based on delusion or incomplete understanding (or “silliness”). Relatively passive
souls like those of ants are mostly governed by laws and cannot choose. Humans with
understanding can choose even better than nature can because nature passively unfolds matter
in accord with ideals, which is different from the proactive creation undertaken by an active
or less passive monad.
It is only through the impact of principles/ideals on passive substance that we can ever really
“see” them. Yet the active/experienced mind can think things through and “see” what is right
without it having to manifest in the world of substance. Note that the “world of forms” is
nothing but mental images or perceptions in the interior of monads. Thus, ideals are
significant although they can never be touched even by God. Ideals can only subsist as
images/impressions in the mind, including in God’s mind. At the same time, they only have
any effect due to the universe of passive substance. The world provides the opportunity for
minds to test their notions and to draw ideas out.
How can it be that a geometrical straight line or triangle do not exist anywhere? They are
impressions in the mind which we can describe unambiguously and so can convey to another
mind which will then also have the impression in their mind. The impressions will likely not
be identical though they will be very close, particularly between the minds of, say, geometers
who conduct proofs with them and so force themselves to coincide almost entirely in their
conceptions or notions of these ideas. Testing our notions in the physical world also aids.
The Aristotelian view of ideals seems to agree with the Leibnizian view because (Platonic)
ideas are apparent only in the created (physical) universe. However, a mind can find and
“know” ideas whether or not they seem to be available to sense perception by their
correspondence to phenomena in the universe.455
God does not need to “see” ideas and probably does not perceive them one at a time like
human minds do. God can grasp not only the entire actual universe in one act of his mind, but
also all possible universes in one act of the mind. His “perception” is indistinguishable from
his mind in perspicacity. The concept of sense perception makes little sense for God.
For humans to perceive something via the senses usually generates a more distinct mental
impression than does conceiving something purely through an act of the mind. For humans,
455 For example, geometry would exist without the universe, but it would have no object. As Leibniz puts it, “For
it is, in my judgement, the divine understanding which gives reality to the eternal verities, albeit God's will
have no part therein. All reality must be founded on something existent. It is true that an atheist may be a
geometrician: but if there were no God, geometry would have no object.” Theodicy p.243 §184
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no act of the mind is pure but is always bound up with a memory of a sense perception to a
greater or lesser degree. But God has created souls anyway from which the physical universe
results because it is his nature to create and to do so in a way that is the highest and best and,
to a mind that can grasp it (and only God can do so fully), the most beautiful.
Ideas and “the Best” are outside all that exists
If ideas required the world to be perceived, by being realised in objects, substance, radiation,
etc. then God would need the world to guide divine decisions which is patently absurd since
the Creator of the very laws of the universe does not need to refer to the Creation that results
from his thoughts. For example, harmony is an idea. The fact that there may be harmony
among ideas or a “harmony of ideas” suggests that ideas are independent of the world.
However, on the other hand, the meaning of the Best is intertwined with the universe which
God has designed to be the Best.
There is no domain of principles by which, if we were to grasp it, we would understand the
universe. The universe is what it is, and our minds’ development of ideas is nothing but us
(humans) trying to comprehend it. Ideas are indispensable as intellectual tools because the
universe is perfect and ideas have the necessary condition of perfection that they are internally
consistent with themselves and externally consistent with all other ideas and, indeed, with all
of reality. It could almost be said that ideas describe the real universe. There is a
correspondence can be seen between mathematical frameworks and the physical universe,
which Figure 4 presents in schematic form.
The apparent governance over actual things by ideas
The utility of mathematical thinking is not diminished by the fact that it is not real. This is
because actual things cannot escape its rules as Leibniz put it.456 While ideas allow us to
understand how phenomena must have happened and how phenomena will happen, they are
only useful because they are non-contradictory. Where a conception is confused then its
power is merely of passive or primary matter, otherwise it is active.
Distinctness of perceptions gives active power, and confusedness gives passive power which
is attributed to metaphysical matter or materia prima.457 So distinctness of ideas is paramount
in monads. The universe comprises only monads, but most are effectively asleep, indeed
comatose, or passive.
For example, though extension is an idea, it is of great importance in physics. Leibniz’s entire
discussion of Aristotle’s primary matter takes place in the realm of ideas not once referring to
monads (which alone are real, aside from God) or their equivalent.458
456 Loemker p.583 457 Loemker, p.365 last paragraph 458 Loemker, pp.536-7
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Figure 4: Ascendence in the geometrical part of idea systems of improving perfection towards
one that corresponds to “the Best”
The relationship between ideas and actual things
Rather than saying that actual things cannot escape the rules of mathematical thinking, it
would be more accurate to say that minds attempting to understand the universe cannot escape
its perfection, and so will be drawn to mathematics. So the converse is true, that the rules of
mathematics cannot escape actual things.
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Creation of additional mathematical structures that bear no relation to observed phenomena in
the physical universe is useful and beneficial. In particular, if the conceptions so developed
bear no self-contradictions or contradictions with other truths – i.e. are ideas – then they
furnish readymade hypotheticals which may help close gaps in understanding that may arise
from paradoxes in future empirical observations.
If Leibniz is so adamant that ideas are not real, then why do we need them? After all, in
becoming a well-known mathematician, Leibniz spent a great deal of time with ideas, or
putative ideas. It is certain that the human race would little more than an advanced mammal if
we were to abandon our quest for ideas. Ideas are an intellectual abstraction and resolution
that are essential for these among many other reasons:
1. Prediction of future events/phenomena
2. Committing to a record and communicating from one person or group of people to
another, and from one generation to another
3. Connections between ideas connect seemingly unrelated phenomena
4. Machine-building
5. Computer programming, which is a specific case of machine-building.
Indeed, this discussion could quickly expand beyond ideas into concepts and into thought
itself. In fact, the above five points relate primarily to concepts because most or all or what
we think we know to be ideas will in future generations be found to be incorrect and to be
merely concepts. A successful hypothesis is an idea whereas an unsuccessful hypothesis is a
conception. Both ideas and conceptions are examples of thoughts.
Inexorable necessity created by the requirement of “the Best”
Necessity arises in deductive processes from presumption, axioms or postulates. Necessity in
what is manifested in the physical universe is a result of something being part of “the Best”.
The concept of the Best does not arise merely as a thing in isolation (like a triangle with angle
sum 180°) but as a process of events over time which – by the definition of an idea – connects
with everything else in the universe. It is hard to see how an idea could ever be conceived
without having the ability to perceive the entire universe and all of its processes over all of
time in a single thought. This is not even equal to the ability to perceive all ideas in one’s
mind at once in a single thought, and it is this which would be needed for the design of the
universe as a whole. One would also need to know what the Best is, and either create that (in
one’s mind, first) or choose it from multiple possibilities or options. On the other hand, rather
than this “brute force processing power”, which would require no effort for God in any case,
it may be that the Keplerian search for harmony would be more accessible and more fruitful.
That is, start with what we know has harmony and beauty, and expect to find that in the
physical universe.
It is in these considerations that the relation between God’s mental universe, or the domain of
ideas, and the physical universe is found. The connection with our intellectual/mental world is
similar, but is complicated by the fact that we do not know any ideas only concepts and those
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confusedly and indistinctly. Nonetheless, over time, we can get closer to ideas and thus to an
understanding of the physical universe.
Nominalism
Leibniz does not regard ideas as things that exist. Nonetheless, he sees a certain necessity to
them. He regards ideas as being able to be instantiated in the physical universe and, indeed, as
the only things that can be instantiated in phenomena and processes in the physical universe.
Richard Brown writes, “To Leibniz nothing exists independently of individuals. Common
properties shared by individuals do not actually exist in re; they are purely creations of the
mind.”459 Once an idea has been instantiated, the instance exists in re, but the idea does not
exist in re.
Having regard to the standard definition of Nominalism,460 Leibniz denies both the existence
of universals and also of abstract objects.
For Leibniz, not all universals are abstract objects. An abstract object is a mental artifice,
which is either a concept or an idea. A concept is a confused intellectual creation which might
be useful notwithstanding its confusedness. An idea is a refined concept, that is consistent
with every other idea. A concept that is not an idea cannot be instantiated in the physical
universe but an idea must instantiate at some point.
“Universals” like whiteness (the example at the Stanford Encyclopedia of Philosophy), or
strength or humanity (the examples from the less formal and more popular Wikipedia), are
nothing but a concept laced with opinion derived from sense perception and useful for day-to-
day conversation. It is a particularly vague kind of concept, and a kind of concept which we –
at least in Western culture at the time of writing – are not motivated to evolve upwards into an
idea because it a serves a role in its vague form. Other kinds of concepts, like mathematical
and metaphysical ones, should be developed into ideas so that they may be of use – or of
greater use – in physics and in helping to understand the universe as a whole. The evolution of
Euclidean geometry to non-Euclidean geometry is an example.
Thus, we have the paradoxical position that ideas are discoverable a priori, but anything
found a priori does not exist and is only a thought with some status or other, whether pure
idea, pure concept or something between. At the same time, a priori discoveries are useful in
understanding that which has been instantiated in the universe because the universe adheres to
the concept of the Best, and we know that God uses ideas to design and choose the Best for
the design and unfolding of the universe. Hence, a priorism – particularly with a Nominalist
foundation – can only make sense when we are consciously attempting to retrace God’s
thoughts. Were one not Nominalist, a priorism could conceivably be used to directly discover
the universe. As a Nominalist, however, thoughts are useful in understanding the physical 459 Brown, R. C. Leibniz, unpublished, Chapter 4 “A Young Central European Polymath Between the Scholastics
and the Moderns” p.33 460 Rodriguez-Pereyra, G. “Nominalism in Metaphysics” 2008 revised 2001, in Stanford Encyclopedia of
Philosophy http://plato.stanford.edu/entries/nominalism-metaphysics/ accessed 21 Feb 2012, with which the
popular source Wikipedia agrees at http://en.wikipedia.org/wiki/Nominalism accessed 20 April 2011
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universe to the extent that they are similar to – or help us understand – the modus operandi of
the Creator.
Interdependence of ideas
If you grasp one idea, you grasp them all. But you cannot grasp any one idea without grasping
them all. We explain why.
1. We start with a question: Is to truly understand an idea to grasp all of its consequences,
which are – by the principle of plenitude – all of its manifestations in the physical
universe? The answer is yes, both by Leibniz’s “Art of Reasoning Well” and by
Leibniz’s conclusion on the qualities of the thoroughness with which God perceives
and understands.
2. No idea can be brought to bear in phenomena independently of any other idea because
the entire universe is needed for each and everything in it. For example, the rest of the
universe would make no sense if Mars were removed from it, and Mars is a necessary
consequence of everything in the universe other than Mars as well as of Mars itself.
The same argument holds when we replace “Mars” with anything else in the universe
including “humanity”, “John Smith” or any individual human being, and any grain of
sand. There is thus a connection between any two ideas, and therefore between all
ideas. It might be easiest for human minds to find that connection using the chain of
dependencies that can be found in the physical universe rather than relying on the
distinctness of perceptions to do the same with ideas in the mind. Since there is a
perfect harmony between soul and body, what we can say about necessary
dependencies in the domain of efficient causes (the physical universe) we can
immediately say about the domain of final causes (the immaterial universe). (See the
above heading “Immaterial realm is in harmony with material realm”.)
3. Thus, no idea can be fully understood independently of any other idea. This means
that we cannot fully grasp the truth or the consequences of the angle sum of a triangle
unless we perceive all ideas at once, which no human will ever do. This has enormous
consequences even for “closed systems” of ideas like Euclidean geometry. In effect,
we cannot fully understand the necessity or consequences of any part of Euclid’s
system without understanding all ideas. Now that some kinds of non-Euclidean
geometry are understood, we understand that Euclidean geometry is just a particular
limiting case of a broader geometry. We know that Euclidean geometry could not
manifest in the physical universe expect in extreme circumstances.
4. Thus, to understand any idea in the domain of ideas is to understand all ideas – i.e.
everything – in that domain. To have such understanding, in turn, is to understand
every possible manifestation of those ideas in physical form.
5. Since every possible manifestation of those ideas has been realised in physical form,
according to the principle of plenitude, and in the most minimal way, according to the
principle of sufficient reason, it thus follows that to understand ideas is to understand
the entire physical universe, and vice-versa.
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6. One cannot understand the physical universe without necessarily and automatically
having grasped the ideas which govern it, or which it accords to because for the
universe to so accord makes it the best possible.461
7. The conclusion is that understanding any idea is to understand the entire universe.462
The conclusion is consistent with Leibniz’s view that the entire universe is contained in a
single grain of sand. This is just a physical analogy of the metaphysical fact that every monad
perceives every other monad albeit with varying degrees of distinctness.
The corollary of the above conclusion is that humans do not understand any single idea and
never will. Even now, for example, all the consequences of the Euclidean concept that the
angle sum of a triangle is 180° are not known. Of course, that concept is not even an idea
because it is not consistent with every other idea. It acted as an hypothesis for many centuries,
but ultimately was done away with.
Do we know any idea? We know some such as “God is creative” though few humans ever to
have lived would know the meaning that would need to be ascribed to “God” and to
“creative” in order for that statement to be correct, i.e. for it to be an idea and so consistent
with every other idea. Similarly with the statement “God is good”.
The large class of statements we think we know, the mathematicals, are not as evident as they
first seem to be. For example, the statement A B A B requires qualification to avoid
nonsense, such as when A and B . The statement A A A does not hold in linear
logic in which propositions act like resources.
Simple monads and compound monads
Another word for monads is “unities”, and the words are similarly connected to the idea of an
indivisible unit or a whole or a One. Leibniz explains that monas means unity or one, in his
preface to Monadology.463 Physical bodies or objects as we experience them, on the other
hand, are multitudes/compound monads which can be destroyed through dissolution of their
units. By inference, this answers the question of how compound monads are formed. Monads
aggregate to form a compound as part of the divine choices made for the best in the way the
universe unfolds from the infinity of possibilities in conformity with ideals.
461 The “least time” principle of the refraction of light is an example. This “idea” makes sense when we embed
the concept of light in the physical world. Theoretically, we may have been able to derive it a priori, but it is
unlikely that it ever would have been. Nonetheless, the principle is a consequence of the Best-ness of the
universe. 462 The Corpus Hermeticum implies this, because it posits the concept of “a nature that enables a thing to come
to be” and “a nature that prevents a thing from ever existing”. (This also relates to the concept of “the good”
as the final cause in Plato’s Timaeus and Phaedo.) This is a unifying concept across all things in the universe
and all things that do not exist. If we understand why one thing exists, then we will understand why every
thing exists. If there is something missing in our understanding of one thing, then – because the existence of
each thing is based on a foundation common to all things – our understanding of everything else must also be
incomplete. Corpus Hermeticum II, §§12-13 in Copenhaver, B.P. (trans. and ed.) Hermetica Cambridge
University Press 1992, p.11 463 “The principles of nature and of grace based upon reason” 1714 Loemker p.636, §1
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Leibniz’s concept of the pre-established harmony ties the incorporeal part of the universe
where souls “reside” with the corporeal. A single soul monad in the incorporeal domain can
control many monads in the corporeal domain at once. The example of the human body was
just what I thought was an easy way of showing how a single soul monad can neatly be tied to
a multiplicity of monads in the corporeal domain – which I am calling “body monads” for
convenience – through the pre-established harmony.
The soul monad is simple in that it is indivisible, has independent perceptions and exists in
incorporeality. Any given soul monad is tied to a body monad through the pre-established
harmony.464 A single soul monad in the incorporeal domain can control many monads in the
corporeal domain at once. We will use the example of the human body to demonstrate how a
single soul monad can be tied to a multiplicity of monads in the corporeal domain through the
pre-established harmony. Since Leibniz writes that every soul monad is tied to a body monad
through the pre-established harmony, it must be said that, for all intents and purposes, a soul
monad only consciously acts on compound monads. For example, a human mind which
Leibniz largely conflates with the soul can move its little finger or their whole body. However,
even the little finger comprises trillions of independent body monads. The soul does not have
any business with each of those trillions of individual body monads but only with the whole
that we call “the finger”. We cannot rule out the possibility that a soul monad has a pre-
established harmonic individually with each of the trillions of independent body monads that
make up the little finger compound monad.
Thus, we make Leibniz’s framework more precise by adding that the soul monad is tied to a
definable boundary compound body monad through the pre-established harmony. For a
person, that boundary compound body monad is commonly called the “human body”. This is
the largest compound monad with which the soul has a pre-established harmony.
Alternatively, we could say that the human body is the largest compound monad over which
the human soul (which is itself a monad) has direct “control” though that control is really just
a manifestation of the pre-established harmony. The soul monad is also tied to a multiplicity
of smaller compound body monads that comprise the boundary compound monad. For
example, the left leg forms a compound monad that is part of the boundary monad. Though
Leibniz did not extend the idea much further, we can say that that the boundary of control can
be extended indefinitely by the building of machines and other, wider forms of influence of
humans over the physical universe. However, when humans influence vast areas through
projects such as dam-building, railway construction, farming and forestation, it is not only
human souls that are at work. Rather, the decisions of human souls are influencing processes
that will function in a new or altered way – at least for a time – even without ongoing human
intervention.
Another extension we can make is that soul monads usually work as compounds too. In the
case of people, most people take their perception and appetitions from others or from their
perception of the group’s perception and appetitions. Still, even an individual soul monad’s
perception of the group is its own. Each of two monads in a mob who are each blindly
following their own perception of the mob has a radically different perception of the mob. The
464 Monadology, p.262 §78
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difference between those two may not appear to be very great judging from the
actions/phenomena that arise from those monads’ “membership” of the mob.
Figure 5: How a soul monad typically communicates with another
Figure 6: More general schematic of soul monads and body monads
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We now return to elucidating on Leibniz’s conception of monads. As Figure 5 indicates, there
is no direct communication between the two body monads except via ideals manifested in the
laws of physics. Figure 5 also helps conceptualise how a soul monad can be tied to a set of
compound monads whose simple monad constituents are all members of some fixed set.
Passive vs active, or body vs soul
The theory of pre-established harmony does not imply that the body is a ship without captain
or crew which somehow reaches its destination regardless (as Bayle complained).465 This is
because God mediates between the two to make it work. The mediation is in conformity with
“pre-established harmony” which is like the law of gravity. God would never decide to act
against this law any more than he would suspend gravity for a day, as there would not be a
sufficient reason to do so and it would diminish the harmony and grandeur of the universe.
It is significant that Leibniz notes that everything is a monad and that monads are substance
which would cease to exist without ongoing perception and activity. In other writings, Leibniz
agreed with Aristotle that there are whorls in every part of matter, which never cease in their
motion. These two positions are consistent.466
If an imperfect monad can only be acted upon, i.e. can only suffer,467 how can we distinguish
an extremely imperfect and thus passive created monad from an asteroid which is a physical
body and thus a compound monad? The difference is two-fold. First, the monad albeit
imperfect is eternal whereas the asteroid is not. Second, the monad has some perfection
however minute whereas the asteroid (qua asteroid, i.e. rock) is entirely passive albeit that it
comprises relatively passive monads with unceasing internal activity.
There are two extreme kinds of monad: body and soul. In fact, monads are on a continuum
between these extremes. The former kind operates in the domain of efficient causes or
physics, which is nothing but metaphysical principles and mathematical rules – i.e. ideals.
Thus, body monads are passively subject to those “laws of nature”. The latter kind is driven
by perceptions and appetite, which do not always correspond since the passage “from mind to
heart is so long.”468 Thus, soul monads can make decisions. These are known as final causes.
Body monads are controlled by the laws of physics and by the decisions of corresponding
soul monads. The clearer and more distinct the perceptions of a soul monad, the more power
it has to affect other monads. (On the ground of ability to affect other monads, God has
465 1705, Loemker pp.586-7 466 Also consistent is Hermes Trismegistus. “There is nothing that is not a product of the cosmic fecundity. In
moving, it makes all things live, and it is at once the location and craftsman of life.” Corpus Hermeticum IX
§6 Copenhaver, p.28. Also see Corpus Hermeticum IX §8 in Copenhaver, p.29. In Corpus Hermeticum V §5,
Trismegistus writes that motion is indispensable to – indeed, part of – the fecundity of the cosmos and “This
is the order of the cosmos, and this is the cosmos of order” in Copenhaver p.19. 467 Monadology, p.245 §49 468 “Hence it comes that our soul has so many means of resisting the truth which it knows, and that the passage
from mind to heart is so long. Especially is this so when the understanding to a great extent proceeds only by
faint thoughts, which have only slight power to affect, as I have explained elsewhere. Thus the connexion
between judgement and will is not so necessary as one might think.” Theodicy, p.311 §314
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infinite power.) Leibniz’s theory of intention subsists in the power and tendency of monads to
affect other monads which will not be investigated further in this thesis.469
The theory of pre-established harmony is a harmony between efficient and final causes.
Accordingly, Leibniz says that neither efficient nor final causes are adequate without the
other.470 This is Leibniz’s resolution to the problem of dualism and his answer to Descartes’
occasionalism.471
Nothing can be found in a simple substance or created monad but perceptions and their
changes.472 However, the ability to perceive does not imply consciousness.473 So not all
created monads are conscious.
Actions of monads
Latta writes that a created thing can act outwardly to the degree of its perfection, and can but
be acted upon to the degree of its imperfections.474 No monad can act outside of itself.475
However, it would appear for all intents and purposes that it can act:
(a) On the body via pre-established harmony, albeit that that harmony was put there or
implemented by God.
(b) On other monads via the intermediation of God albeit this is always in conformity
with ideals.
At a precise technical level, however, Latta is correct.
A created monad, due to confused understanding (i.e. “perceptions”), can cause its
corresponding body to ingest beer so causing further confusion and leading to a nobbling of
the mind/soul monad’s ability to control the body. That is, the pre-established harmony
reduces the refinement in the physical motion of the body in line with the confusion in the
perception of the soul monad which the alcohol somehow caused. In short, pre-established
harmony works both ways. The alcohol also nobbles the ability to reason and remember.
Platonically, referring to Meno, reasoning and remembering are the same thing noting that the
memory activated in Meno is of thoughts that have always been with the soul, i.e. since the
beginning of time, albeit in indistinct form. Thus, the physical action of the chemical of
alcohol on the brain is affecting the perceptions of the created monad or soul.476
469 C.f. Hill, J.C.R. Leibniz’s metaphysics of intentionality PhD 2008, National University of Singapore 470 Loemker, p.588 middle of first paragraph 471 Robinson, H. “Dualism” 2003 revised 2011, in Stanford Encyclopedia of Philosophy. Accessed at
http://plato.stanford.edu/entries/dualism/ 21 Feb 2012 472 Monadology, p.228 §17 473 Loemker, p.588 7th last line 474 Monadology, p.245 §49 475 Monadology, p.245 footnote 79 476 Without referring to monads, Friedrich Schiller addressed the impact of the body on the soul and vice-versa in
“On the connection between the animal and spiritual nature in Man”, Letter V of Schiller’s Philosophical
Letters Tapio Riikonen, T. and David Widger, D. (eds) accessed at http://www.gutenberg.org/files/6799/6799-
h/6799-h.htm on 1 Jan 2011
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Immaterial realm is in harmony with material realm
It is not just pairs of monads, simple and/or compound, that are in harmony with one another.
The entire realm of final causes (souls) is in harmony with that of efficient causes (bodies).477
This does not mean that they move in lockstep and are copies of one another. However, it
must mean that they are derivable each from the other because harmony implies that the
“connection of ideas” is indubitable when properly worked out.478
Life
Souls are themselves vital principles with perception and appetite. The appetite means that
they must also have intention provided they are not passive.479 If they have memory, then they
are intelligent to varying degrees.480
Substantial forms include rational souls, but not in an organic body such as a stone. Further,
there is no part of matter that does not include an infinity of organic animated bodies. Thus,
even the stone does. However, there is matter which nevertheless is not animated in spite of
what is in it. So a stone is not animated though it has in it an infinity of animated bodies. A
pond is not animated even though it has fish in it.
Q&A
In order to clarify some of the concepts raised, we segue to a question and answer format with
these five bullet points:
What are such non-animated bodies? Compound monads
Are they merely ideas? Yes. Are they actual but non-permanent and without memory. No
Since a thing without motion – though its parts have motion – does not exist, then such
non-animated bodies do not exist. Correct
Does this include all compound monads? Yes
If they are ideas, then are they not provably and permanently true? Yes, but they are
implied by the entirety of the rest of the universe. When the rest of the universe changes,
including their constituent monads, they are no longer implied and thus are not necessary.
A vital principle is a substantial form. Only animated matter, animated bodies and conscious
monads are substantial forms. Thus, a vital principle is “life” and not all monads have life.
Vital principles are not eternal for a living thing may die.481
477 Monadology, p.263 §79 478 Loemker, p.638 §5 479 C.f. Hill, J.C.R. Leibniz’s metaphysics of intentionality PhD 2008, National University of Singapore 480 1705, Loemker p.586 and Monadology pp.244-5 §48 481 To Hermes Trismegistus, death is merely dissolution. Corpus Hermeticum VIII §4 in Copenhaver pp.25-26,
Corpus Hermeticum XII §16 in Copenhaver pp.46-47
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Sentiment is perception accompanied by memory, i.e. there is an echo of the perception for
some time afterwards. A living being with sentiments is an animal, its monad is a soul. When
the soul has reason, it is a spirit.
Since everything comprises monads and monads are with perception and thus animated, there
is “life” everywhere, even in the coldest asteroid.482 For Leibniz, every cubic millimetre of the
universe is packed full of substance; that is, the universe is a plenum.483 For Leibniz, God
would not create a universe in which there was any empty space or in which two things were
alike because either would amount to redundancy, violating the principle of sufficient reason.
Since the asteroid qua an asteroid is inanimate, it is only an idea – i.e. it does not really exist.
This is even though the compound monad must conform to metaphysical principles and
mathematical rules, as must its behaviour and that of its constituent created monads. This is
consistent with the consequence of Leibniz’s theory that only God and simple substances
exist. Leibniz affirms this when he writes, “Sensible things, however, and composite things in
general, or the substantiated things, so to speak, are in flux and become rather than exist.”484
For example, the asteroid is undergoing a constant exchange of simple monads with its
environment. As far as biological organisms on earth are concerned, such exchange with the
environment and ingested nutrients is well-established and confirmed by experiment. Leibniz
is saying that this is true of all composite monads.
The Hermetic monad
In a discourse of Hermes to Tat, Hermes explained that God sent down to the physical
universe a great mixing bowl which was “Mind”, inviting all human hearts to immerse
themselves in it. Those who did, and who do, participate in knowledge and become
“perfect people because they received mind”.485 The mixing bowl or “Mind” is the
monad of Hermes. Hermes says that the monad is in all things as a root and beginning,
presumably because all things have mind to some degree however small. Nicolaus of
Cusa’s conception of mind is similar to that of Hermes, for tells a story similar to
Hermes’ mixing bowl whence all souls may sup of Mind and so acquire their own
mind.486 482 Restated in Leibniz’s 1714 essay at Loemker p.636 §1 last two lines, and p.637 §4 first two lines. 483 For Hermes Trismegistus, there is nothing which is empty. Even hollow containers are “full of air and spirit”.
Hermes finds it almost blasphemous to imply that any of the things that are, is empty. Rather, all that is is full
of substance. Corpus Hermeticum II §§10-11 in Copenhaver, pp.10-11 484 Loemker, p.592 (Letter to Hansch, July 25, 1707) 485 Corpus Hermeticum IV §4 in Copenhaver, pp.15-16 486 For, to Cusa, when treating of an individual mind, he says that the mind is the first and the most simple image
of the divine enfolding. The mind comprises the power of understanding, reasoning, imaging and sensing as
its elements, meaning these various powers can be combined in multifarious ways. When the Philosopher
asks the Layman where the mind gets these powers, the Layman answers, “From unity,” which would seem
to correspond to Hermes’ mixing bowl which was sent down by God. Miller, C.L. (trans. and intro.) Nicolaus
of Cusa Idiota de Mente Abaris Books, New York 1979, pp.51, 83. Credit goes to Miller for tying together
p.51 and p.83 and putting them neatly side-by-side in his introduction. The connection we have drawn and
not taken from Miller is that between Cusa’s conception of mind as coming “from unity” and the mixing
bowl of Hermes.
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Things without any mind, in Leibnizian terms, would be pure body and for Leibniz this is
extremely rare and does not get much elucidation from Leibniz. For Leibniz, each thing that is
real – except God – is a monad whether or not it has developed an intellectual faculty worthy
of the name “mind” (small “m”).
Since souls are what the universe is made of, and since humans have incorporeal souls
(which, popularly, is the only kind of soul), let us examine Leibniz’s theory of death.
What survives death?
Since souls are eternal, what happens after that which we call “death”? Leibniz addressed
Pythagoras’ conception of transmigration of souls487 and wrote that his own theory implies
even more. It is not only the soul that is indestructible.488 Not only does the animal’s soul
subsist after death but the entire animal does. So there is no metempsychosis but there is
metamorphosis.489 This is clarified when Leibniz explains that an animal might change its
“organic slough” as its “machine may often perish in part”.490
The personality of beasts is not preserved on death, though the personality of humans is
preserved. While beasts’ souls are merely indestructible, human souls are immortal. In the
case of humans, “not only the soul but also the personality subsists. In saying that the soul of
man is immortal one implies the subsistence of what makes the identity of the person,
something which retains its moral qualities, conserving the consciousness, or the reflective
inward feeling, of what it is”. However, “this conservation of personality does not occur in the
souls of beasts: that is why I prefer to say that they are imperishable rather than to call them
immortal.” (emphasis in original)491
Human souls differ from souls of beasts
Leibniz is emphatic that animals have souls too, in opposition to the Cartesians. However, he
is also emphatic that the intelligence of humans is such that humans must and should rule over
the beasts.492 Descartes argued that beasts are mere machines, and Bayle took Descartes’ side
against Leibniz.493 Presumably, no soul monad is a mere machine, though soul monads –
especially those which correspond to passive matter – may often approximate mere
mechanism. Indeed, from the superficial perspective of textbook physics, this is what makes
the behaviour of inanimate matter amenable to human study and human control.
487 Loemker p.589 488 Monadology, p.262 §77 489 1714, Loemker p.638 §6 last paragraph and p.637 §4 490 Monadology, p.262 §77 491 Theodicy, §89 p.171 492 Langley, A.G. (ed.) New Essays on Human Understanding, 1949, p.97 493 Theodicy, pp.40-41
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Leibniz muses, “It is certain that God sets greater store by a man than a lion; nevertheless it
can hardly be said with certainty that God prefers a single man in all respects to the whole of
lion-kind.”494 Later, he says the maxim “that all is made solely for man” is “old and somewhat
discredited”.495 Far from making Leibniz an environmentalist or animal-lover, we can really
only conclude that Leibniz is opposed to gratuitous mistreatment of animals. Leibniz says that
punishing and even killing animals would be justified if it effectively prevented beasts from
causing disorder or other harm in human society.496,497
Animals have rational souls, as do humans. However, human souls are “raised to the rank of
reason and to the prerogative of minds”. Minds are “images of the Deity” whereas ordinary
souls are not.498,499
Monads are the same as one another
All that exists must be interrelated, and therefore everything must express the same nature but
in a different way. All minds have intercourse with each other and express the same nature.500
Consistent with this is that every monad contains within it complete information on every
other monad and, thus, complete information on the entire universe. However, such
perceptions are confused and indistinct.501 Because all monads are on this equal footing,
Leibniz says that all of history, both sacred and profane, is confirmed.502 (We also discern
from this that no monad – much less a soul monad – is a mere machine but can potentially
manifest complex behaviour. The internal state of such entities would be orders of magnitude
greater in richness than their overt behaviour may appear to be, though not prohibitive of
being understood.)
494 Theodicy, §118 p.188 495 Theodicy §118, p.189 496 Theodicy §70, p.160 497 Hermes wrote that all creatures should multiply, though of course only intelligent creatures would be able to
comprehend the diktat: “[G]od immediately spoke a holy speech: ‘Increase and increasing and multiply in
multitude, all you creatures and craftworks.’” Corpus Hermeticum I §18 in Copenhaver, p.4. Of course, in
Genesis I:28, God directed God to, “Be fruitful and multiply.” 498 Monadology, pp.264-265 §§82-83 499 Hermes wrote, “Nothing is more godlike than mind, nothing more active nor more capable of uniting humans
to the gods and gods to humans. … Blessed is the soul completely full of mind, wretched the soul completely
empty of it.” (Corpus Hermeticum X §23 in Copenhaver, p.35) This and the mixing bowl or monadic “Mind”
of Hermes (Corpus Hermeticum IV §4 in Copenhaver, pp.15-16) suggest that Hermes regards soul’s embrace
of mind as a choice for humans in the degree to which a human mind imbibes of Mind. Hermes does not
suggest that animal souls can choose to embrace Mind more, in order to be elevated to the form or state of
humans. However, Hermes does say that the greater the degree to which a human soul embraces Mind, the
closer they reach to God (though Hermes uses a lower case “g” and sometimes refers to “gods”). Whether a
human soul embraces body – and its pleasures – or Mind is a choice that it is within the power of humans to
make. This is consistent with Leibniz’s view of a continuum of activity and passivity between human minds.
The more active have to a greater degree embraced Mind, in the Hermetic sense. 500 Loemker, p.365 14th last to 10th last lines of 2nd-last paragraph 501 Loemker p.590 last 3 lines of last paragraph, and Monadology p.248 §56 502 “On the method of distinguishing real from imaginary phenomena” Loemker, p. 365
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Since simple monads’ perceptions are “indeterminate”, i.e. non-ideal, the choice of the best
possible universe is a choice leading to superior guidance of our and other simple monads’
internal perceptions and so affect their “appetitions” and thence their decisions. Indeed, even
galactic events are impinging on the perceptions of humanity, and so are affecting our
decisions and behaviour.503,504
Conclusion
Leibniz formulated a coherent system which embraces material reality, immaterial reality,
God, ideas and thought. There are many exciting directions to take from here, not least the
one which Leibniz prescribed: creating a Universal Characteristic.
The Leibnizian conception of the physical universe as outlined in this chapter shows Leibniz
to be Nominalist. However, Leibnizian Nominalism is qualified by the necessity of the Best,
since only the best may ever manifest in the physical universe. Things can be the Best only
when they correspond to ideas, though we do not know whether correspondence with ideas is
sufficient – it probably is. Thus, we can increase our understanding of the physical universe
by studying ideas because, of all possible thoughts and concepts, only ideas can be
instantiated in the physical universe. Since ideas have structure and clarity, they can often be
given mathematical form and are subject to calculation. This is why mathematics is
indispensable to physics, and why mathematics sometimes precedes physics.
Perhaps the most important idea of all is that of the Best, since it subsumes all other ideas.
Arguably, Leibniz’s “the Best” is what Plato referred to in The Republic as “the good”. An
idea is only instantiated at a particular point in any series of events if it is consistent with the
Best. It is the need to understand “the Best” which often places metaphysics, aesthetics and
art, and theology prior to mathematics and physics. The first three situate, contextualise and
bound how close human thought can approach “the Best” in their endeavours through
mathematics and physics.
503 Loemker, p.640 §13, 2nd sentence, 2nd paragraph 504 A possible example of galactic events directly impinging on humanity is given by the paper on the effect of
our solar systems passage through a spiral arm of the Milky Way on earthly climate. See Shaviv, N.J.
“Cosmic Ray Diffusion from the Galactic Spiral Arms, Iron Meteorites, and a possible climatic connection?”
Physical Review Letters 2002 Vol. 89 Issue 5
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Chapter 8: The ongoing role of thought and the ongoing creation of the Best
Introduction
The Best is made more complicated – or richer, or more harmonious and beautiful – by the
proactive and wilful role played by thought in the running or “unfolding” of the physical
universe. These are the thoughts of souls – in particular, human souls and other souls equally
capable as human souls though we presently do not have direct evidence of any of the latter –
which are in symbiosis with the appetitions of, or the choices made by, souls. Choices are also
made by God in the interaction between souls and the physical universe. Physics cannot be
complete without subsuming the role of these thoughts, appetitions and choices. A calculus or
some kind of mathematics is needed to help our human minds cope with it in a structured
way. These thoughts and choices are indispensable to the “onward and upward” unfolding of
the universe in the Best possible way. Without these thoughts, appetitions and choices, the
universe would be inferior to what it is, it would certainly not be the Best, and it – for reasons
not yet understood by this writer – might not even be possible.
There is an apparent contradiction in the fact that God makes choices from moment-to-
moment in the unfolding of the universe, and the fact that Leibniz held that a universe that
requires intervention by God must be imperfect. This contradiction vanishes when we
understand that the universe only appears to be running or operating from moment-to-
moment. It is actually being re-created anew from moment-to-moment per the perceptions or
thoughts of soul monads which, to varying degrees, are changing in tandem with the
unfolding of the physical universe. The relationship involved in that tandem-ness requires
further study, and it essentially is nothing but Leibniz’s concept of the pre-established
harmony.
In this chapter, these matters are addressed: the choices God makes, the ongoing creation of
the universe, the role of the thoughts of souls, the role of free will and the necessity of
including intellectual intent in physics.
Choices God makes, continuously
While the physical universe functions without divine intervention, a contradiction in Leibniz
is that the Monadology has God making choices continuously.505 The “choices” are “made” as
a result of the universe and the intentionality built into its design. Thus, the choices are not
directly of God but are of the universe. The universe functions almost as an intelligent entity
505 The same apparent contradiction is in Hermes when he says, “sensation and understanding enter from outside
… but the cosmos got them once and for all when it came to be, and, having got them, it keeps them by god’s
agency.” (Corpus Hermeticum IX §8 in Copenhaver p.29) How can it be said that the cosmos has them once
and for all when god’s agency is needed to keep them? Further, as we know from Hermes, once something is,
it can never not be. We can only suppose that the contradiction is to be resolved as with Leibniz. The
“agency” by which these things continue to exist is only indirectly of God; directly, it is of the cosmos which
was fathered by God, to use Hermes’ words.
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or system that makes choices continuously. We now explain Leibniz’s theory using his
terminology.
First kind of choice: God mediates between monads
No monad can affect another of its own accord. This is natural since monads are all self-
sufficient. However, monads act on one another through the mediation of God. We remind the
reader that we are forced to read Leibniz as meaning “through the intentionality of the
universe which God created”. Such mediation is only ever in accord with ideals.506 Thus,
there is not really a “choice” for God, because he always finds and chooses the best
possibility though he has the power not to so choose. Whence, we understand why Leibniz
says that (the functioning of?) the entire universe relies on “God” for its existence each and
every moment.
Grace cannot be understood by reason but only by revelation. We leave this to one side since
the natural workings of the physical universe are untouched by grace, as they are governed by
metaphysical principles and mathematical rules. For God to arbitrarily modify these would be
an imperfection in the order of things.507 Thus, the reliance on God’s mediation is not through
grace as such, for God would not choose to act in any way other than the best and that, in
turn, is in accord with ideals.508
Perhaps the pre-established harmony is a subclass of the interaction between monads in
conformity with ideals through the mediation of God. Since it adheres to ideals, it is in that
sense pre-established. However, it is not pre-established in the sense that the perceptions of
the soul are indistinct and so the choices unpredictable or, at least, non-ideal. Or, to God, they
are largely predictable because he knows the internal perceptions, and their changes, of simple
substances or created monads.509
Nicolas Malebranche was a philosopher who was a contemporary of Leibniz. Malebranche
lived in Paris from 1638 to 1715. Malebranche concluded that “there is only one true cause
because there is only one true God; …the nature or power of each thing is nothing but the will
of God; … all natural causes are not true causes but only occasional causes”.510 This is now
known as occasionalism. Leibniz’s thesis that God mediates between monads seems to verge
on occasionalism, it has very different conclusions. While occasionalism denies any causation
aside from God, monads by contrast determine their own internal states. Thence, interactions
between monads are affected. That is, the internal states of monads are independently driving
unfolding order of the universe. Leibniz’s doctrine appears similar to occasionalism, but it
leads to a conclusion contrary to occasionalism.
506 Monadology, p.246 §51 507 1702, Loemker p.556 final paragraph 508 Loemker p.639 §10 509 Monadology pp.228-9 §17 510 Robinet, A. (ed.) Malebranche, N. Oeuvres complètes de Malebranche Vrin Paris 1958–84, Vol. II, p.312.
Quote in Lennon, T.M. and Olscamp, P.J. (trans.) Malebranche, N. The Search for Truth and Elucidations of
the Search for Truth Cambridge University Press 1997
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Second kind of choice: God chooses the best possible from what is possible
There is an infinite number of possible universes511 all in accord with ideals or they would not
be possible. God chooses the best of all that are possible.512 Ideas are as they are, and ideas
cannot be mutually contradictory or some of them would not be ideas. Therefore, the
difference between possible universes can only be our and all monads’ perceptions. Another
way of considering this is to “distinguish metaphysical necessity from moral necessity”.513
Furthermore, God does not act purely “according to mathematical laws, following an absolute
necessity”.514
In short, “nature leads to grace and grace perfects nature by using it.” Here, “nature” means
metaphysical principles and mathematical rules represented by the laws of physics.515 This
use of the word “grace” does not mean “grace” in the usual sense of miraculous divine
intervention, for a propulsion towards the best possible outcome is subject to an ideal or rule.
Can a calculus of such “best outcomes” be produced? However, we pre-empt ourselves for we
need a calculus of metaphysical principles first. Indeed, we are a long way even from an
overarching calculus of mathematical rules. However, it might make sense to start on all three
projects simultaneously as they might assist each other.
God must be making a choice continuously based on sufficient reasons for the best of all
possibilities which God perceives in his mind all at once. This is not Zeusian arbitrariness in
playing with the universe and humans. If we had the all-encompassing perceptive ability of
God, we would know what he will choose.516 Leibniz suspends judgement on the extent of the
role of grace in God’s making the choices,517 though we do know that Leibniz denied the
necessity of grace to this universe being the best possible one.
Were it not for this selection undertaken by God, the principle of plenitude would imply that
the domain of ideas bears a one-to-one correspondence with the physical universe. However,
this cannot be true because there is an infinite number of possible universes which conform to
metaphysical principles and mathematical rules, when in fact only one universe exists.518
511 Monadology, p.247 §53 512 Also, “The dominion of his will relates only to the exercise of his power, he gives effect outside himself only
to that which he wills, and he leaves all the rest in the state of mere possibility.” Theodicy, p.243 §183. Also
see Loemker p.640 §12. 513 Theodicy, p.313 §310 514 Theodicy, p.399 §8 515 Loemker p.640 2nd half §15 which is related to last 4 lines of Loemker p.639 §9 516 Monadology p.248 §55 517 Loemker p.590, end of first paragraph 518 This indicates that Leibniz has much more fine-grained criteria than Hermes Trismegistus who says, to
paraphrase, what is will always be and what is not never will. (Corpus Hermeticum II §13 in Copenhaver
p.11) Or has Hermes assumed the best possible world doctrine in referring to that which has it in its nature to
be? Even if he has, Leibniz’s view is more refined because Leibniz considers the unfolding of events so that
what may be now is different from what may be after, say, the earth has completed its current orbit around the
sun. Hermes regards being as a timeless or eternal matter, whereas Leibniz allows for change and trajectories
of events.
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Unanswered question re first kind of choice
When does God need to mediate or communicate between soul monads, albeit he always does
so in conformity with ideals? Alternatively, when do two soul monads ever communicate or
impinge upon each other except via the laws of physics via corresponding body monads?
Leibniz does not provide an example. It is unlikely that God ever does. However, Leibniz
probably does not wish to contradict the Bible by excluding the possibility of miracles. Most
likely, Leibniz believes that there is a scientific explanation for what apparently are
miracles.519
Consideration re first kind of choice: God’s choices vs intelligent monads’ choices
Ideals matter in the outcome of interactions between monads. They also happen to produce
maximum harmony and beauty, and the best possible outcome. They govern choices by God
“on behalf of” passive substance (e.g. planets) in order that our perception may grasp this
harmony and be improved by it. God makes such choices because they please him.520
So, in any case, God is extremely active (infinitely so) in producing “the most beautiful
combination of justice and goodness which could be wished”.521 It is through this continuous
selection process that God creates a repayment of good from evil plus interest.
Clear and distinct perceptions in a monad are different from appetition. Leibniz says little
about what determines appetite. Presumably, clarity on certain ideals (such as justice and
goodness) will generally create concomitant appetite, whereas clarity on other ideals (such as
plenitude) without clarity on, say, justice and goodness might lead to evil appetite. However,
this is not necessarily the case. The passage “from mind to heart” is long.522 Such monads
might have power for good or evil in the universe, respectively, depending on the combination
of ideals and conceptions perceived and the degree of clarity with which they are understood
by the monad. Monads capable of any of these were made by God in his image. Leibniz also
calls such monads “spirits”.523 “The mind is not a part but an image of divinity, a being which
represents the universe, a citizen of the divine kingdom.”524
Leibniz draws parallels and distinctions between God’s knowledge and a created monad’s
perceptions, and between God’s will and a created monad’s appetite.525 The latter qualities (of
519 Leibniz did believe that it is scientifically possible for water to turn into wine, and for the body of Christ to be
in several places simultaneously. He wrote metaphysical explanations as to how these phenomena are
possible, which we will not address in this thesis. 520 Loemker p.575, 16th line from the bottom 521 1705, Loemker p.590 522 “Hence it comes that our soul has so many means of resisting the truth which it knows, and that the passage
from mind to heart is so long. Especially is this so when the understanding to a great extent proceeds only by
faint thoughts, which have only slight power to affect, as I have explained elsewhere. Thus the connexion
between judgement and will is not so necessary as one might think.” Theodicy, p.311 §314 523 1705, Loemker p.590; Loemker p.637 §4 first paragraph; Loemker p.640 §14; Loemker p.640 §15 first half 524 Loemker, p.595 (Letter to Hansch, July 25, 1707) 525 Monadology, p.245 §48
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the monad) are only imitations of the former qualities (those of God), with a quality
corresponding to the degree of perfection of the monad.
Question re second kind of choice
What is an example of where there might be a choice, both options in conformity with the
laws of physics, one tending to the better and the other to the worse – where God of course
chooses the former?
One is reminded of the postmodern “chance event” films where the falling of a cigarette butt
in a particular square centimetre of floor triggers a course of events which changes a person’s
day or life. But even in this case there is no place for a choice to be made by God, for the
coordinates of the location where the cigarette butt falls is determined by the laws of physics
and since people have free will God cannot intervene in their perceptions, decisions or
reactions.
The overriding contradiction is that Leibniz said in several places that a universe which
required intervention or involvement by God at all, much less continuously, would not be a
perfect universe and so not the kind of universe that God would have created. We cannot think
that Leibniz did not realise this, and so do not find it a stretch to say that, in this context, by
“God” Leibniz meant “the universe that God created”. Indeed, since God would in any case
choose the best, why would he not create a universe that does so “on autopilot”? Autopilot
programmed or built in to the universe by God is many (perhaps infinitely-many) orders of
magnitude beyond ad lib “thinking on your feet” human creativity.
Answer to the question re second kind of choice: what God does from moment-to-moment
In short, God has to maintain the correspondence between souls or active substance, that is
incorporeal, and body which is passive substance and corporeal.
Leibniz objects to God being required to act at every moment, and thus explains that his
theory of pre-established harmony is superior to Descartes’ theory of occasional causes. Pierre
Bayle agrees. Yet Leibniz’s theory of ongoing choice of the best possible universe526 also
requires ongoing intervention by God though we have explained above how we prefer to
understand this. The difference is that pre-established harmony seeks only to explain the
movement of body under the action of soul/mind. God should not have to get involved in
mere body.
Leibniz’s explanation of the ongoing choosing of the best possible universe must leave the
laws of physics to work as they do, while allowing God to make the “choice” that remains of
the still-infinite possibilities even with the universe constrained by the laws of physics. The
only intervention needed is to maintain the pre-established harmony. That is, “to change the
natural course of the thoughts of the soul to adapt them to the impressions of the body” and
526 Monadology, §53
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vice-versa.527 God could choose to tamper with the pre-established harmony, and presumably
not impact the functioning of the laws of physics at all, but chooses not to because it would
derogate from the universe functioning in the best possible way.
This is an enormously significant point that cuts across theology, morality, psychology and
physics. If a human (or animal or bacterial) soul had a thought or intention that was not given
effect to in the corporeal realm, then not a scientist in the world would notice. Yet it would
make a very real difference to phenomena in the corporeal world. What, indeed, should that
correspondence be between a thought and an action? Is it possible to define the
correspondence as a mathematical relation? Leibniz argues that God links thoughts with
actions in the best possible way.
Creating the universe anew in each moment
Arguably, Leibniz remains true to his dictum that a universe that requires God’s ongoing
intervention would be imperfect, while allowing that God does have an ongoing influence
over creation.528 The ongoing intervention is not of God but of soul monads. soul monads
have thoughts which collectively form a space or a field which is incorporeal. Each soul
monad’s thoughts impact the physical universe in the best possible way, indeed, causing
change in or of the physical universe. A different set or field of thoughts of soul monads
corresponds not only to a different universe because soul monads and their thoughts are part
of the universe, but also results in a different physical universe.
An entirely different universe would result if such an entirely new set of thoughts were
possible and if the best possible manifestation of those thoughts were an entirely different
universe. Of course, such an entirely different universe would need to be in conformity with
ideas. However, soul monads’ thoughts do not change very often and often move in lockstep
with one another – at least, those of human soul monads do – and thus we see correspondingly
little change in the physical universe.
There is ongoing change in soul monads’ thoughts creating a new space or “thought field”
causing change in the physical universe. We say “change” because it manifests as an
incremental “delta”. However, for Leibniz to be correct it cannot be mere change because
“change” implies a running or functioning or ongoing operation, which cannot require God’s
intervention and therefore cannot have a pre-established harmony since God has a hand in the
pre-established harmony. Rather, the universe is being re-created every moment according to
the thoughts of soul monads which act as a diktat on the physical universe in the best possible
way via the pre-established harmony.
527 Loemker, p.587, 6th last line of 1st paragraph 528 Leibniz, G. W. Second letter to Samuel Clarke 1715-16 paragraph 8, “I do not say the material world is a
machine or watch that goes without God’s interposition, and I have sufficiently insisted that the creation
wants to be continually influenced by its creator. But I maintain it to be a watch that goes without wanting to
be mended by him; otherwise we must say that God bethinks himself again. No, God has foreseen
everything. He has provided a remedy for everything beforehand. There is in his works a harmony, a beauty,
already pre-established.” in Loemker, p.679
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Soul monads individually are eternal and indestructible while their perceptions change as a
result of experience of each other and of the physical universe. Again, these perception-
change or thought-changing experiences are preceded by the action of God between body
(passive substance or physical matter, which includes manifestations of the thoughts of other
souls which is the only way we can experience or know the thoughts of other souls) and the
soul. While each monad (active and passive alike) is itself indestructible and eternal, the
nature of the universe and the way monads interact with one another is transitory and
dynamic. It is these changing relationships along with the dynamicism in the internal
perceptions of soul monads that characterise each new incarnation of the universe as a whole.
The way such “characterising” works is a function of the pre-established harmony. The pre-
established harmony needs to be investigated as a serious branch of metaphysics. Ultimately,
when our understanding is sufficiently clear, the pre-established harmony would become a
branch of physics. An analogy is that while H2O molecules do not change, the way in which
they relate to one another as well as their internal (energy) state can differ greatly. This gives
us the difference between ice, water and steam. Thus, individual monads can change greatly
in their interrelationships and internal state creating greatly different universes. The
differences of concern in this analysis is in the incorporeal (active or soul) part of the universe
as well as in the corporeal/physical (passive or body) part of the universe which is controlled
by the incorporeal part, as well as in the entire universe regarded holistically.
An open question is what soul monads are there aside from those of simple living organisms,
animals and humans? It may be that other soul monads are very close to being passive (i.e.
body monads) but because in their dullness they work in such large numbers and with such
uniformity (far moreso than the soul monads of, say, humans do) they have enormous
influence in the physical universe via the pre-established harmony. Such would be the source
of the “controlling” field of creation of stars, galaxies, their motion and evolution, and the
behaviour of cosmic radiation. All cosmic phenomena apparently involving passive (i.e. non-
living, inanimate) matter would fall in this category. Consistent with this is that, according to
Leibniz, there is no matter that is entirely inanimate. The identity and nature of these soul
monads – which correspond to non-living (in the conventional biological sense) body monads
– requires further investigation.
Is it “Hello Humphrey Appleby” or, Does God control everything while appearing not to?
If God controls in the best possible way the relations between monads, what actual role is
there for free will? It seems that God via the Best actually controls everything. The central
factor that God does not control is the internal perceptions and appetitions of monads. It is
only when these are clear and distinct that the relations between monads and thus
considerations of the Best come into play.
Our free will, and that of all soul monads to the extent that they are active/soul, consists in
our ability to choose what we focus on, our physical experiences (to the extent that we can),
our appetitions, our use of rationality in making interpretations and our intellectual input. Of
course, the extent to which any individual can make those choices is limited too, but that very
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limitation is itself a function of the pre-established harmony because the scope of those
choices is a function of interactions between our soul monad and other monads. Perhaps this
is why Leibniz advocated top-down policy towards structure society in such a way as to
maximise the positive intellectual and cultural potential of the common mass of people to
enable them to make superior choices for ongoing betterment of themselves and of humanity
as a whole.
Mystical theology
Leibniz’s essay “On the true theologia mystica”529 is notable for what it does not say about
God. However, it is very clear about what is not real. “Most knowledge and invention
[Tichten] belongs to the shadow way.”530 He says that, “Corporeal things are but shadows
which flow away, glimpses, shapes, truly dreams.”531 Leibniz later restates this in a letter
written in Hanover on 30 June 1704.532 He goes on, “Essential truth is in the spirit alone. But
inexperienced men take the spiritual for a dream and what is tangible for the truth.”533
Leibniz is neither Gnostic nor a mystic. In another place, he wrote that witchcraft and sorcery
are based on delusion or fraud. He wrote of what sounds like a description of transcendental
meditation, “There are some who imagine a world of light in their brains.” However, “this is
not the light but only a heating of their blood.”534 He says that those who claim to experience
mystical phenomena are only in an aberrant psychological state. We should try to “conserve
them in this beautiful frame of mind, just as one preserves a curiosity or a cabinet-piece.”535
Leibniz cannot even be said to be a Cabbalist, for he explains why the idea of the “universal
soul” is untenable.536
Leibniz is amending and extending the actual Theologia Mystica537 by Dionysius the
Areopagite (“Dionysius”). Leibniz’s title suggests that his intention is to supplant it. Due to
the status and institutional theological significance of Dionysius, it is likely that Leibniz is
being diplomatic and agrees with very little of it. It is too “mystical” for Leibniz. Dionysius is
emphatic about what we cannot know, and the uselessness of mind. It reads almost like a
manual in transcendental meditation in that it instructs the seeker to abandon mind, body and
everything that he has.538 However, in “On the true theologia mystica” Leibniz criticises “the 529 c.1690, Loemker pp.366-8 530 c.1690, Loemker pp.367 531 p. 368, emphasis added 532 p.536 first paragraph 533 Ibid., p.368 534 Ibid., p.367 535 Letter to Electress Sophia in 1691, footnote 4, Loemker p.369 536 “Reflections on the doctrine of a single universal spirit” 1702, Loemker, p.555 537 Dionysius the Areopagite, Mystical Theology http://www.esoteric.msu.edu/VolumeII/MysticalTheology.html
accessed 26 October 2010. Another translation is at
http://www.ccel.org/ccel/pearse/morefathers/files/areopagite_06_mystic_theology.htm accessed 28 June 2011
which is Parker, J. (trans.) Dionysius the Areopagite, Works London: James Parker and Co. 1897, transcribed
by Roger Pearse, Ipswich, UK, 2004. 538 “For by the resistless and absolute ecstasy in all purity, from thyself and all, thou wilt be carried on high, to
the superessential ray of the Divine darkness, when thou hast cast away all, and become free from all.”
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denial of self” saying that the only denial of self we should consider is hatred of our non-
being which is the source of sin. Sin does not come from God; rather, “original sin has arisen
in some creatures from their nonbeing and hence out of nothingness”.
For Leibniz, the incorporeal is subject to rational inquiry just as much as the physical universe
is amenable to experiment. However, Dionysius seems to deny all kinds of knowledge and
inquiry when he writes, “We pray to enter within the super-bright gloom, and through not
seeing and not knowing, to see and to know that not to see nor to know is itself the above
sight and knowledge.”539 Thus, Dionysius advocates removal of mind, which would certainly
have played a role in rousing Leibniz to write a response to counter Dionysius. On the
contrary, not only does Leibniz say that a human is something, but he also explains the nature
of self-knowledge: “The only self-knowledge is to distinguish well between our self-being
and our nonbeing.” Furthermore, “one must make use of sensual things and must view the
shadow pictures only as an aid or a tool and not rest in them.”540
For Leibniz, it makes sense to inquire into the nature of God and, indeed, it is necessary to do
so to progress in science. However, Dionysius has made the concept of God entirely mystical.
Leibniz also writes that, “Essential truth is in the spirit alone. But inexperienced men take the
spiritual for a dream and what is tangible for the truth.”541 Dionysius does not agree that truth
is in the spirit but has detached the concept of truth from rationality. Indeed, it may be that
Dionysius is denying the concept of truth altogether though he does not explicitly say so; he is
certainly instructing the “adept” to abandon the pursuit of truth.
While, for Leibniz, “Corporeal things are but shadows,”542 he repeatedly says that what is
physically tangible or “body” must be used by humans as a tool to further advance our soul
which is to progress in understanding and intellectual capability. However, “God belongs to
me more intimately than my body”. Analysing body is effective in helping us to come up with
useful ways of doing things, which can help us in day-to-day life.543 In doing so, we can
follow Leibniz’s art of reasoning approach, which is not far from Newton’s prescription.
Likewise, in his concept of metaphysical substance or materia prima, Leibniz is using his
reductionist principle of his pragmatic “art of reasoning”.544 Leibniz finds it useful to
distinguish primary from derivative when in discussing forces545 and in discussing
phenomena.546 Using this method of reduction, Leibniz explains why the Cartesian concept of
extension is not fundamental. He explains that extension or distance and time are merely
relative ideas, in some ways anticipating Einstein and Einstein’s forebears in relativity
theory.547
Section I, Caput I “What is the divine gloom?” of Mystic Theology accessed at
http://www.ccel.org/ccel/pearse/morefathers/files/areopagite_06_mystic_theology.htm 28 June 2011 539 Ibid. 540 Loemker, p.368 541 Ibid. 542 Ibid. 543 Loemker, p.283 544 Wiener, pp.78-80 545 Loemker, p.537, 4th paragraph 546 Loemker, p.536, end of 1st paragraph 547 Letter to Burcher de Volder professor at the University of Leyden, 30 June 1704, Loemker, p.536
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Mathematical determinism
Yet 14 years later Leibniz refers to “phenomena” and “actual events”.548 This seems to be
shorthand for constituents of the physical universe as far as we can ever vouch for it. More
specifically, it seems to be the impact of ideals on substance. More accurately, it is the
adherence of the universe to the best possible because God has designed it that way. The “best
possible” happens to be in conformity with ideals. Why this is the best possible is a question
of huge importance, which is beyond the scope of this thesis.
Leibniz restates conformity of the laws of physics with ideals by saying that it cannot be
otherwise for nothing ever happens which violates any of the most exact rules of
mathematics.549 Paradoxically Leibniz does not think that mathematical determinism suffices
for an all-encompassing corpus of physics or of the unfolding of the physical universe. This is
due to:
(a) possibly, the self-affecting dynamical nature of the universe;
(b) the imperfect character of perception and the non-ideal decisions by monads that flow
from the appetite resulting from those imperfect perceptions;
(c) the creative ability of some monads which surpass other monads in their clarity and
distinctness of perception, and purity of their appetition; and
(d) the fact that God does not act purely according to mathematical rules.550
If we restrict consideration in (a) to passive substance, then its behaviour is likely to be
calculable provided we knew the exact location of every atom in the universe, and full
knowledge of every possible metaphysical and mathematical rule. Of course, this will never
be possible for humans, but we can comprehend that it might theoretically be possible.
An example of (b) is the result of geopolitical manipulations leading to humans blowing each
other up in a senseless war. An example of (c) is humans improving upon nature by, for
example, redirecting a river to bring life to a portion of desert. These are by definition outside
what can be addressed by ideals, and metaphysics and mathematics are restricted to ideals.551
In other words, perception and all that depends on it is inexplicable on mechanical grounds.552
Perceptions are no small consideration even though they are outside the corpus of
deterministic textbook physics, which only even attempts to address efficient causes.553
548 Loemker, p.536, end of 1st paragraph 549 Loemker, p.583 550 God is not like “the blind nature of the mass of material things, which acts according to mathematical laws,
following an absolute necessity, as the atoms do in the system of Epicurus.” Theodicy, p.399 §8. Yet the fact
that he acts for the best has a certain comprehensible and reasoned harmony to it, if not “absolute necessity”. 551 On the other hand, the example in (b), for Leibniz, might lead to something better, just as Westphalian
sovereignty followed the 1648 Treaty of Westphalia which concluded the Thirty Years War. The example in
(c) is humans acting like the divine in improving the universe and enhancing its plenitude. 552 Monadology, pp.228-9 §17 553 Loemker, pp.639-640 §11
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Indeed, only perceptions can be found in a simple substance554 which is a created monad.555
This class of influences on the universe is called final causes.556 All internal activities of
created monads are a result of perceptions which are also a kind of thought or mental image
or notion. Each created monad is independent of everything as if there were nothing in the
universe but itself and God. This is notwithstanding the fact that every monad perceives every
other monad, since perception is not the same thing as direct influence. Apparent “influence”
of a monad on another (e.g. of a person holding another in their thrall) is actually the latter
monad voluntarily pursuing the perceptions of the former or having their perceptions
manipulated by the former; in common parlance, the latter has ceded its will or personal
sovereignty to the former.
Another way of considering (b) and (c) are that God’s raw material is us, along with monads
below and above us. He needs to wait especially for us and monads above us to improve, and
gives us every conceivable opportunity to do so without violating our free will. He would not
force us because that would not be the best way. The way this unfolds leading to our realising
our co-creative ability is outside metaphysics and mathematics, but is a function of the
“thought process” of a monad.
How can this thought process be improved to help along God’s intention for the universe? Of
course, the Universal Characteristic as Leibniz conceived it would aid thought. The Universal
Characteristic can now be regarded as a means whereby some of the most powerful
fundamental constituents of reality – active monads or minds – can empower themselves to
fulfil their function even more effectively. First, the Universal Characteristic must encompass
the discovery and use of metaphysical principles and mathematical rules. To be complete it
must be able to assist or systematise calculations with both (b) and (c), and accommodate
problems with respect to those. Somehow, (d) needs to be taken into account too. To
understand (d), we need to understand the question under the above subheading “Unanswered
question re second kind of choice”.
Is everything planned in advance?
That creation should unfold in the best possible way is pre-ordained because this is built into
the design of the universe. At the same time, there cannot anyway be sudden and bizarre
changes due to the principle of sufficient reason. Human free choice adds a certain amount of
variability, within a particular range. If we had perfect knowledge, then we could forecast how
things are going to work, subject to (or modulo) the free choice of human souls and all the
other soul monads in the universe. With any given amount of knowledge, the best choice in a
given scenario is deterministic. Thus, if we knew all that was important to a person and what
their inclinations were, then we would know in advance what they were going to decide. As
mentioned in the above section “Creating the universe anew in each moment”, soul monads
do not change their thoughts very significantly very often though over time there can be huge
554 Monadology, p.228 §17 555 Monadology, p.229 §18 556 Loemker, pp.639-640 §11
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changes. Of course, there are decisions that can be made randomly or without reason.
However, these have little impact on the overall course of history.557 The large changes on
history are caused by changes in the “thought field” of large numbers of monads, which of
course are sparked by insights had by particularly active soul monads such as that of Christ.
In the sense that the architectonic design allows us to foresee what will happen provided we
have perfect knowledge of all facts and all possible ideals, yes.558 But note that this is much
harder than it might sound. We would need to know the internal perceptions of all created
monads, and know which outcome will produce the best possible conditions for
improving/clarifying their perceptions. We will also need to know the impact certain
experiences will have on these perceptions, and their perceptions of their own and of other
monads’ perceptions, the best possible choice from that point, and so on. In short, we would
need to be God.
A priorism
Leibniz makes an art and a science of reading God’s mind. This is because it is rational to do
so since God is the Creator, and because it is possible to do so as God’s mind is rational.
“Reason is also choice” says Milton in Paradise Lost.559 Aquinas also makes an art and a
science of reading God’s mind.560
Leibniz says that the best science is done by attempting to read God’s mind like Cusa and
Einstein did.561 Experimental results are useful to confirm or deny what we think that God
would have done, or to check the accuracy of our reasoning, much as we would check our
accuracy in carrying out a long and complicated piece of arithmetic.
Science is a result only of our ability to grasp demonstrable truths such as in logic, number
and geometry for they “make the connection of ideas indubitable and their conclusions
infallible”.562 “The mathematical sciences, moreover, which deal with eternal truths rooted in
the divine mind, prepare us for the knowledge of substances.”563
Is a priorism better than empirics?
557 Loemker, p.640 §13 first paragraph, and Loemker p.641 §16 first sentence 558 Loemker, p.640 §13, 1st paragraph 559 Milton, J. Paradise Lost Book III, line 108 in Ricks, C. (ed.) Paradise Lost and Paradise Regained The
Signet Classic Poetry Series, The New American Library, New York 1968 560 See Thomas Aquinas’ Summa Theologica accesssed at http://www.newadvent.org/summa/ 26 May 2011 561 For Cusa, see De Docta Ignorantia and Idiota de Mente, and for Einstein see
http://web.ceu.hu/yehuda_einstein_and_god.pdf accessed 30 Oct 2010 562 Loemker p.638 §5 563 Loemker, p.592 (Letter to Hansch, July 25, 1707)
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Yes, when it is available to a human mind.564 However, it is rare that any human’s
“intellectual perceptions” (i.e. the internals of any created monad belonging to a human) are
clear and distinct in anything. All created monads are nearly always confused and indistinct in
their perceptions with respect to all things. Even in the angels and blessed “there is always
some confused perception mingled with distinct knowledge”.565
How could innate truths regarding topics other than physics – i.e. outside of metaphysics and
mathematics – ever be found through empirics?566
Leibniz’s “empiricism” is simply that sensory observation prompts thoughts and checks
reasoning, including the soundness of reasoning. “The external senses, properly speaking, do
not deceive us. It is our inner sense which often makes us go too fast.”567 Empiricism says
that the senses give us knowledge and that thoughts are only needed to organise the
knowledge that the senses give us. This is at odds with Leibniz’s position as explained in New
Essays.568
For humans, ideas and empirics are interdependent
Similarly, entirely detached from sense perception, ideas are not possible. At least, we never
have ideas that are so abstract that they are entirely disconnected from sense perception.569,570
Is this effectively saying that we can never hold a pure idea in our mind, but only something
mixed with empirics? Perhaps not. Leibniz says in New Essays that we already know a great
deal. It is sense perception or general experience that draws it out of us or which triggers
thoughts from our memory. In this, Leibniz follows the doctrine of reminiscence from Plato’s
Meno.571 As we will explain below, if we have one Idea within us, then we probably have all
ideas within us, with varying degrees of distinctness. This is not surprising because “each
distinct perception of the soul includes an infinity of confused perceptions which envelop the
564 Recall the above footnote regarding Plato’s view on reason versus sense perception. In The Republic (602 d),
Plato says, “A stick will look bent if you put it in the water, straight when you take it out, and deceptive
differences of shading can make the same surface seem to the eye concave or convex; and our minds are
clearly liable to all sorts of confusions of this kind.” 565 Theodicy §310, p.314 566 Bennett, J. (trans.), Leibniz, G.W. New Essays on Human Understanding , pp.97-99 and Chapter III, 1st ed.
Feb 2005, amended April 2008. Accessed at www.earlymoderntexts.com/jfb/leibne.pdf 10 May 2011 567 Theodicy §65, p.109 568 Bennett, J. (trans.), Leibniz, G.W. New Essays on Human Understanding Book IV Chapter XII §13
“Philalethes: Although I recommend experimentation, I don’t lack respect for probable hypotheses; they can
lead us to new discoveries and are at least great helps to the memory. But our mind is very apt to go too fast,
and to be content with flimsy conjectures rather than taking the time and trouble needed to test them against a
multitude of phenomena.” 1st ed. Feb 2005, amended April 2008, pp.212-213 Accessed at
www.earlymoderntexts.com/jfb/leibne.pdf 10 May 2011 569 Ibid., p.19 570 Hermes writes, “Both sensation and understanding flow together into humans, intertwined with one another,
as it were. For without sensation it is impossible to to understand, and without understanding it is impossible
to have sensation.” Corpus Hermeticum IX §2 in Copenhaver p.27 Even Plato’s Meno, written centuries after
the Corpus Hermeticum, bears this out because the uneducated servant boy is given prompts in the form of
appropriate questions and a diagram to aid him in “remembering” geometrical truths. 571 Langley, G.A. (trans.), Leibniz, G.W. New Essays on Human Understanding Open Court, La Salle, Illinois
1949, p.105
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entire universe.”572 Similarly, “Each soul knows the infinite, knows everything, but
confusedly.”573
If one rejects the idea of remembering things that we already knew, then the need for
experience to trigger thoughts which lead to ideas might mean that the domain of ideas is
actually bound up with the impressions on our senses that, by definition, can only occur in the
physical universe. This leads to the “sufficient reason” for the existence of the physical
universe.
That no human thought is ever entirely abstract might encompass the fact that humans
“experience” or “conceive” ideas in a cultural context. Perhaps it is merely the perceived
context, whereas the idea itself is universal and objective. Perfect objectivity is not possible
for humans, and different cultures facilitate the conception of the universe in different
ways.574
Free will
Gose in his discussion of right reason refers to the removal of choice by addictions to
pastimes such as drinking and smoking.575 This is how Gose explains that the corollary of
Milton’s concept of “right reason” is that only the wise can be free. “With his accent on
liberty and freedom, Milton is quick to affirm that liberty is not license. … Milton asserts that
with the ‘dignity and freedom of individual man’ comes a responsibility for individual
discipline.” The addict cannot exercise that discipline.
Leibniz would agree that only the wise are truly free. However, this is because they have a
greater appreciation of what will lead to true happiness, which is pursuit of God’s mind. Thus,
we would say that Milton meant that “To exercise Reason is also choice”576 and the wise tend
to choose to.
Gose makes a couple of other useful remarks. He says that “The classicists as well as other
non-Christian thinkers” do not deny that Reason is a source of (i.e. a source of knowledge
of?) God. But, Gose says, “it is limited by human imperfection.” This echoes Nicolaus of
Cusa. Next, Gose says, “Milton says that the unwritten law of God: is no other than that law
of nature given originally to Adam, and of which a certain remnant, or imperfect illumination,
572 Loemker, p.640 §13 1st paragraph 573 Loemker, p.640 §13 2nd paragraph 574 There is evidence that the way that the “plastic brain” is structured is partly a result of culture, causing
physically different wiring of synapses between cultures. Thus the question goes beyond sense perception but
to what the term objective means and what it means to think; these concepts might have different meanings to
radically different brains. This does not mean that ideals differ between minds, but perceptions of ideals
could differ with the resultant conclusion that what a particular mind perceives as idea I is in fact idea J or is
K which is not an idea at all. Refer to Doidge, N. The Brain That Changes Itself: Stories of Personal Triumph
from the Frontiers of Brain Science Penguin 2007 575 Gose, M. Right Reason: Milton’s Ethical Standard
http://globalvillage.pepperdine.edu/GoseWriter/rreason.html accessed 4 October 2010 576 Milton, J. Paradise Lost Book III, line 108 in Ricks, C. (ed.) Paradise Lost and Paradise Regained The
Signet Classic Poetry Series, The New American Library, New York 1968
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still dwells in the hearts of all mankind; which, in the regenerate, under the influence of the
holy spirit, is daily tending towards a renewal of its primitive brightness.” Here we see the
Platonic-cum-Leibnizian memory of what the soul already knows, but by a different method.
Gose says, “The classicists as well as other non-Christian thinkers … are in need of divine
revelation and love to complete their search for truth.” One could be forgiven for reading
Cusa in this way, and perhaps this is what Cusa meant. It is certain that Cusa believed we
could never equal God. However, contrary to Cusa,577 Leibniz is clear that divine revelation in
the sense of a mystical enlightening or a rapturous experience is no way to achieve
knowledge. We might sometimes feel that that is how we are discovering – or recalling –
something, but in fact it is a process of Reason, and not necessarily a deductive one, that gets
us there. On the other hand, if we are remembering, then perhaps there is a place for
revelation though Leibniz never allowed for it.
To Leibniz, there is no such thing as freedom in the sense of liberalism or of “licence”.578
Leibniz often used the pejorative barbarism. However, he also noted that men from all
cultures have a sense of justice even if they do not apply it to all things and times or to the
extent that Christians do or would if they followed Christ’s teachings all the time. In New
Essays he gave the example of the barbarisms of the native Indians of America and of some
native tribes of Peru, but noted also that in some matters those peoples adhere to the same
standards of justice as Christians of Europe.
What is an evil person?
An “evil person” is a soul monad with perceptions that are contradictory to ideas. One may
ask, “Contradictory to which ideas?” The answer is, “To all ideas.” Remember that “idea” has
a specific meaning for Leibniz. They are not merely thoughts or concepts. An idea is correct
by definition, and to be “correct” it must be consistent with everything that exists and has ever
existed. An evil person is further away from ideas than a good one.
This definition encompasses moral wrong, and deliberately emphasizes that moral truths are
as much subject to rational proof as a mathematical principle may appear to be. Perceptions
contradictory to ideas may have a degree of clarity and distinctness greater than many “good”
soul monads have of concepts that bear some resemblance to ideas. To anticipate how Leibniz
deals with the problem of: It is the clarity with which the evil soul perceives what is
contradictory to ideas that forces other soul monads to clarity and drive their own conceptions
closer to clear and distinct perceptions of ideas.
577 Nicolaus of Cusa wrote that he experienced an epiphany during a sea voyage, in which God opened up his
understanding. 578 Echoed by Alexander Hamilton http://www.catholiceducation.org/articles/history/us/ah0015.html accessed 4
Oct 2010 “Freedom was a great treasure, Hamilton agreed with the ideologues, propagandists, social
engineers, and manipulators of all ages, but it must be organized against abuse. Unrestrained man’s freedom
degenerated into license and anarchy.”
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“Evil” soul monads are active just as are “good” soul monads, and so are superior in to and
able commandeer more passive monads and passive matter just as well as the good souls are,
if the more good souls allow them to. The definition of an “evil” soul monad is that it works
in contradiction to ideas or relatively so in comparison to “good” soul monads. Thus, nothing
that evil soul monads do can last, thought they have can cause destruction indefinitely as long
as they are allowed to before that uplift themselves, or are uplifted by more correct
perceptions, i.e. by perceptions that are closer to ideas. Since only ideas can exist, evil works
in conformity with non-being or nothingness, which is precisely what Leibniz wrote in “On
the true theologia mystica”.579 That is, sin is not from God but from nothingness or the non-
being of certain creatures.
In order to wreak havoc, evil soul monads make use of discoveries that have been made in
conformity with ideas, which permit machines of control and weapons of destruction all of
which arise from engineering and science. Leibniz wrote that a bad European is much worse
than a bad tribal native because the bad European has at his disposal all the power arising
from the positive scientific culture of European civilisation.
The problem of evil
One of the most frequent criticisms of Leibniz’s system of metaphysics is his “best of all
possible worlds” doctrine which is actually a consequence of the principle of sufficient
reason. If this is the best of all possible worlds, then why does evil exist? Davies raises this
objection to the “Leibnizian optimism” but leaves it aside as an “ethical issue”.580 Yet it is
clear that question of justice, goodness and morals generally are as much a part of ideas and
the workings of “the mind of God” as are metaphysical principles and mathematical rules.
Davies then considers “maximum variety” as a substitute for “the best”.581 This might better
have been considered under Leibniz’s doctrine of plenitude.582 Nonetheless, Davies pursues a
promising line of enquiry when he considers beauty as a guide to truth.583
Leibniz explained that the truest and most enduring happiness comes from pursuing the
understanding of the whole which ultimately means the mind of God.584 However, this pursuit
must be free in order to be fruitful. Thus, evil is expected in this pursuit but those minds
which know better are likewise free to best the proponents of evil conclusions reached
through misguided reasoning, and to struggle against those forces and attain supremacy over
them.
Erasmus is close to Milton on free will, e.g. “exhortations, commands, choice, reward and
punishment, all present in scriptures with respect to salvation, would be meaningless in the
579 ca.1690, in Loemker pp.367-370 at p.368 580 Davies, P. The Mind of God Penguin Books Camberwell Victoria 1992, p.173 581 Ibid. 582 Kepler refers to plenitude in passing when he says that “mere space without body is a contradiction” and
“Being has precedence over not being”. M.C., Chapter XI, p.129 Duncan, trans., Abaris Books 583 Davies, P. The Mind of God Penguin Books Camberwell Victoria 1992, p.175 584 We return to this below under the heading “Interdependence of ideas”.
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absence of free will.”585 This is precisely what Milton has God say to his Son in Paradise
Lost.586 Leibniz’s explanation of free will befits his rationalism and best of all possible worlds
doctrine, whose corollary is that people tend to make the best decision they can at any given
juncture, and the outcome however good or bad leads us to a wiser or otherwise better state in
the end. Leibniz even says that a murder will ultimately lead to something better than had the
murder not taken place. However, he also takes pains to emphasise that this does not mean
that those who do wrong should not be punished – they should. However, he says, ultimately
sin punishes itself anyway. The past has occurred for the best but we must not acquiesce in the
present for the future as the quietists advocate; we should always do our best.587
Also see the effect of evil and what God permits.588 God does not accidentally allow evil to
occur; he does provide the tools or circumstances which can be used for evil undoubtedly
knowing that this “risk” exists.589 God permits evil because he can bring a greater good out of
it. The whole of creation actually derives a net gain from evil.590 Moreover, Leibniz seems to
be saying that certain kinds of good would not be possible at all were it not for evil. Leibniz
does not see the relationship between good and evil as a balance sheet, where we allow evil to
debit us one unit so we can be credited two units worth of good. Rather, evil can trigger
entirely qualitatively new categories of good and catalyse possibilities for good that would not
otherwise have been considered or embraced.
We can hearken back to Milton’s Paradise Lost, where God says to his Son that Satan should
not be stopped from carrying out his plan. Humans need to be left free to choose whether they
shall yield to temptation or not. Given the reality of free choice, the inevitable result is that
some humans will yield to temptation while some will not and innocents will necessarily
suffer. This evil is not merely accidental or part of an optimisation but is an intended and, in
view of the whole, a good result which works for the betterment of the innocents as well as
those who yielded. Perhaps innocents are not innocent, for all must take a responsibility for
the whole.
Leibniz writes that he cannot “approve the opinion of certain moderns who maintain boldly
that what God has done is not supremely perfect but that he could have done much better.”
Rather, he holds that, “God does nothing for which he does not deserve to be praised.”591 So
does God deserve to be praised for the starving children and the murderers and rapists?
Leibniz addresses this when he asks, “Why does such a Judas, who is merely possible in the
idea of God, actually exist?” about which he says, “no answer can be expected here on earth,
585 From http://www.siue.edu/~evailat/psr-Lut-Er.html accessed 4 Oct 2010 586 Milton, J. Paradise Lost Book III, lines 100 to 130 in Ricks, C. (ed.) Paradise Lost and Paradise Regained
The Signet Classic Poetry Series, The New American Library, New York 1968 587 “We must rather act in accordance with the presumptive will of God, so far as we are able to know it, trying
with all our might to contribute to the general welfare…” with emphasis in the original. (Loemker, p.305 §4) 588 “Discourse on metaphysics” 1686 in Loemker, p.307 589 Leibniz writes that it is, “true that God co-operates in evil in the actual performance of introducing these
forms into matter” Theodicy, p.353 §381. Separately, Leibniz explains, “how it is to be understood that God's
will takes effect, and concurs with sin, without compromising his wisdom and his goodness.” p.399 §8 590 “On the true theologia mystica” c.1690, Loemker p.368 591 “Discourse on metaphysics”, Loemker, p.304
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except the general one that since God has found it good that he should exist in spite of the sin
which God foresaw, this evil must be compensated for with interest in the universe and that
God will draw a greater good from it and that it will turn out finally that this sequence of
events, including the existence of this sinner, is the most perfect among all other possible
kinds.”592 Evil plays the role of revealing the good and, over time, we step back in order to
leap forward better.593
Leibniz’s conception of evil is more as natural imperfections that are an inevitable part of the
process of upwards evolution or moral upshift, ultimately leading to every kind of
improvement, enrichment and increased power. Ultimately, when realised in human
endeavour, such upshifts includes kinds that are perceptible in the physical universe through
projects such as those discussed in Chapter 3. So for Leibniz the defining characteristic of evil
is not that which is judged as such by God, but it is more closely linked with that which has a
role built into the design of the universe, just as that which is good has such a role.
Scientific comprehensibility of God’s plan
While we are identifying Leibniz as a Neoplatonist – as he says “I tend more towards Plato,
Mr Locke to Aristotle” but on some things we differ from both of these great ancient writers –
we must note that Leibniz is more rationalist than anyone before him.594
If there are two things we can take away from New Essays, the first is that all is
comprehensible. The universe was made in such a way that if a mind knows all or is admitted
into the secrets of things, then there will be reasonable explanations for things and all will be
understood. Second, the discovery process is a remembering of what the soul already knows.
Hill argues that “Leibniz’s views of intentionality are closely bound up with key elements of
his metaphysics and epistemology – especially his understanding of relations, his commitment
to the explanatory power of theism and the role of the divine ideas”.595
The objective way in which God works embodied in the architectonic universe is probably
Leibniz’s most significant contribution, and we would argue that this is rationalism. What
Kepler discovered and described for the solar system, Leibniz explained for the universe in
toto. In fact, Leibniz continued Kepler’s programme of, in Franklin’s words, constructing a
theory of everything.596
592 “Discourse on metaphysics” §30, 1686, Loemker 1969 p.322 593 “Reply to Bayle’s Dictionary article Rorarius”, 1702, Loemker 1969 p.582 594 Leibniz’s Preface to his New Essays Concerning Human Understanding, p.2 595 Hill, J. C. R. Leibniz’s metaphysics of intentionality PhD Thesis, National University of Singapore 2008, p.3,
viewed 27 August 2010
<https://scholarbank.nus.edu.sg/bitstream/handle/10635/16595/HillJCR.pdf?sequence=1> 596 Franklin, J. The Science of Conjecture: Evidence and probability before Pascal J. H. Press Baltimore and
London 2002, pp.150-1
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In New Essays it is very clear that God would not create the universe in any way other than
the best. Thus, the universe could not be otherwise than as it is. Therefore, we study the
universe as it is (i.e. do science) by reference to that, not to God per se. However, we can
come up with hypotheses by considering (the) perfection which is God’s nature – and all the
(other) characteristics that God necessarily must have. Writers like Wenck would say this is
“constraining God”. God’s nature, however, is to act for the best. There is a mathematical
kind of certainty to how God would act, but our reason is so clouded and our knowledge so
miniscule that we are infinitely far from knowing or understanding “the best”. The concept of
“the best” is an objective one and is just as discoverable as any of the ideas. We are still using
the strict definition of “idea”.
While ideas are perceived in the mind of God, it is likely that everything we think is an idea,
in fact, is not. Indeed, each idea is bound up with the entire universe and cannot exist or be
understood except as part of the entire universe. Only God can hold the entire universe in
mind in a single thought. Do ideas actually exist? As discussed above, according to Leibniz,
ideas so do not exist even though they are perceived by the mind of God or held in God’s
mind. Ideas are realized or manifested in the physical universe. Indeed, ideas are all that is
manifested in the physical universe: this is one of the definitions of an idea.
Intention and the Scholastic Leibniz project
The closest we get to a Platonic world of ideas in Leibniz is the “intelligible world in the
divine mind, which I also usually call the region of ideas”.597 We know that they have the
effect of defining the best. However, it is only a “world” in the sense that God has a lot of
ideas in his mind at once. A capable human mind could also have in it a world of ideas.
Hill argues that the scholastics believed in the existence of such a domain of ideas:598
the scholastic philosophers developed a “package” of metaphysical claims which
underlay a common approach to intentionality. These claims were that relations have
reality outside the mind; that God’s ideas function as exemplars of their objects; and
that the mind abstracts “intelligible species” from the objects of perception. These
views allowed them to defend the intuition that a thought of X is linked to X in some
way by the relations of both similarity and causation. … Leibniz held views that were
structurally similar to the [scholastic] “package”, which allowed him to hold a similar
approach to intentionality.
According to Hill,599 the scholastic theory of intentionality was supported by three primary
metaphysical doctrines:
1. “Ideas” function as exemplars, and are located primarily in the divine understanding.
2. When the active intellect contemplates something, it actually takes on the same form
or species as that thing.
597 Loemker, p.592 (Letter to Hansch, July 25, 1707) 598 Hill, J.C.R. Leibniz’s metaphysics of intentionality PhD National University of Singapore 2008, p.iii 599 Hill, J.C.R. Leibniz’s metaphysics of intentionality PhD National University of Singapore 2008, pp.8-9
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3. Relations, or at least some relations, exist outside the mind and cannot be reduced to
monadic (or atomic?) predicates.
Each of these contradicts Leibnizian ideas in various ways. Each of these contradicts
Leibnizian ideas in various ways. Taking each of the points in turn:
1. For Leibniz, an idea is a concept that is consistent with the entire universe and so is an
expression of the entire universe. An idea is not located in the divine understanding or
anywhere. An idea has no location. It is true that only the divine understanding can
ever grasp an idea because only the divine understanding can grasp the entire universe
in a single thought. Human understanding can approximate ideas but never fully.
2. The intellect, active or otherwise, never takes on the form as a thing that it
contemplates. The intellect is an active monad or a thing in its own right. An active
monad – meaning, a mind – can have impressions of things upon it which are nothing
but perceptions.
3. A relation is itself a concept that might approximate reality to the extent that its
predictions are confirmed by goings-on in the physical universe. A concept only exists
as a mental perception. If a relation did exist outside the mind, like anything that exists
outside the mind, it would be part of the physical universe. Once we accept that
something is part of the physical universe, it is no longer a mere concept and therefore
cannot be a relation. The concept of relation is merely a thought tool to help us
advance our concepts closer to the likeness of ideas.
Hill’s project of demonstrating Leibniz’s use of or, at least, agreement with scholastic ideas is
not isolated in the literature. In his 1686 “Discourse on metaphysics” Leibniz wrote, “the
opinions of the Scholastic philosophers and theologians are much sounder than has been
imagined, provided that they are used appropriately and in their proper place. I am even
convinced that if some exact and thoughtful mind were to take pains to clarify and assimilate
their thoughts after the manner of the analytic method of geometricians, he would find a great
treasure of very important and strictly demonstrative truths.”600 Farrer says that Leibniz was a
scholastic but was bent on reforming if not rewriting scholasticism.601
Leibniz notes with approval that Scholastics at one time expressed that God is the light of the
soul. Thus, Leibniz is actively looking for areas to agree with the Scholastics.602
As far as intention goes, it seems that the ultimate or umbrella intention of all is God’s. We
only have ideas of things in our soul because of God’s continuous action on it. Further, as
already discussed, ideas are all that we have that can cause phenomena (in fact, to which
600 Loemker, p.309 601 Farrer, A. (commentator), Huggard, E.M. (trans.) Theodicy, p.12 602 “God is the sun and light of souls … and this opinion has not been invented only today. In addition to the
Holy Scriptures and the Fathers, who were always more Platonists than Aristotelians, I recall having
observed long ago that at the time of the Scholastics, several believed that God is the light of the soul … ‘the
active intellect of the rational soul’. The Averroists gave this a bad turn of meaning, but others … understood
it in a way worthy of God and capable of elevating the soul to a knowledge of its true good.” (“Discourse on
metaphysics” 1686 §28, Loemker, p.321)
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phenomena correspond because phenomena are the best possible, to be explained further
below under the final heading in this chapter before the conclusion) as notions are too
indistinct.603
When we say ideas “cause” phenomena we mean, in fact, to which phenomena correspond
because phenomena are the best possible. This is explained above under the headings:
Ideas and “the Best” are outside all that exists
The apparent ruling of actual things by ideas
and below under the heading:
Foundations of a programme for discovery.
The apparent physical world as far as we can tell has been provided as a way to help our
indistinct and confused intellect come up with ideas or remember ideas that it has always had
within it, but which in some cases have never been brought to the conscious fore of the mind.
As we know from our experience, real ideas are usually bound up with moral purpose and a
kind of detached, i.e. not heatedly emotional, passion.
Regarding relations, Hill says that Leibniz’s “apparently contradictory statements about
relations can best be understood by distinguishing between two kinds of relations in his
thought. He thinks that relational properties have extra-mental reality, but ‘inter-substantial’
relations do not, and are mere abstracta. In this, Leibniz is very similar to his scholastic
forebears.” It would appear plain that relational properties exist beyond the mind due to the
reality of the things of which they are properties. On the other hand, “inter-substantial”
relations do not have a reality beyond the mind because they are not real since all substances,
i.e. all monads, are independent (despite their being the same and being in constant inter-
communication).
Hill writes:
Leibniz speaks about God’s ideas a great deal – not just those of human beings – and
assigns them a part-cause in creation, with a deliberate appeal to Augustine. These
ideas are conceived in a fundamentally relational way. Leibniz rejects Malebranche’s
identification of our ideas with God’s, but he does think there is some important
connection between them; for us to have an idea is to exist in some relation to God’s
ideas.
However, that relation does not seem to be any different from the relation between two souls
having the same idea. For example, if Jack and John both prove that the angle sum of a
triangle is 180° then there is a relation between their minds when both have the image of that
idea in their mind.
Hill writes:
I argue that an interpretation of Leibniz as a nominalist about ideas, which has been
defended by some recent commentators, is not accurate. 603 Leibniz, G. W. “Discourse on metaphysics” 1686 §28, Loemker, p.321
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Leibniz himself seems to be in support, “As long as we have only a nominal definition, we
cannot be sure of the consequences drawn from it, for if it concealed some contradiction or
impossibility, we could draw conflicting conclusions. This is why truths do not depend on
names and are not arbitrary, as some modern philosophers have thought.”604 However, recall
that to not be a nominalist is to believe either universals exist or abstract objects exist in a
Platonic sense.605 Leibniz believes neither of these and therefore is not a nominalist. Believing
we are forced inexorably to conclusions that are correct or true is different from believing that
truths exist as real objects. Rather, as explained when Leibnizian ideas were introduced, what
is an idea is bound up and ultimately defined by the totality of the best of all possible
universes.
Continuing with Hill:606
According to this [nominalist] interpretation, Leibniz believes that to have an idea of
something is simply to have a disposition to think about it. I argue, however, that this
is only Leibniz’s understanding of ideas when they are not being thought of; when
they are being thought of, they are objects of thought.
It is unclear what an “object of thought” is. However, Leibniz does not take an “idea to be an
immediate object of thought or for some permanent form”.607 Leibniz says that our mind can
manifest whatever it desires to itself, “As a matter of fact, our soul always does have within it
the disposition to represent to itself any nature or form whatever, when an occasion arises for
thinking about it.” This makes thought akin to a “phantasm”.
Hill writes:608
Moreover, the divine ideas certainly cannot be understood dispositionally at all. In
fact, the divine ideas are identical with their objects, a claim which is fundamental to
Leibniz’s overarching argument in the Theodicy. Here again, there are strong
similarities between Leibniz’s position and that of the scholastics.
…
Leibniz distinguishes between “a” concept of something and “the” concept of the same
thing; the latter is identical with God’s idea of that thing, which is also identical with
the thing itself.
We would disagree that in Leibniz’s analysis God’s idea or concept of a thing is any more real
than a human mind’s idea or concept. An idea is not a real thing but a mental perception of
real or potentially real things. Leibniz distinguishes mental images, even God’s mental images
of ideas, from real things. Thus, divine ideas cannot be identical in the sense of being the
same as their objects, for a mental image cannot be real. Leibniz distinguishes a mental
image, even God’s mental images of ideas, and real things.
604 Leibniz, G. W. “Discourse on metaphysics” 1686, Loemker p.319 605 Rodriguez-Pereyra, G., “Nominalism in Metaphysics” Stanford Encylopedia of Philosophy Zalta, E.N. (ed.)
Article first published 2008 and substantially revised 2011 accessed at
http://plato.stanford.edu/entries/nominalism-metaphysics/ 28 July 2012 606 Hill, J.C.R. Leibniz’s metaphysics of intentionality PhD National University of Singapore 2008, p.10 607 Loemker p.320 §26, lines 6-9 608 Hill, J.C.R. Leibniz’s metaphysics of intentionality PhD National University of Singapore 2008, p.11
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Leibniz is strict about what it means to have an idea. “One can boast of having an idea of a
thing only when one is assured of its [the thing’s] possibility.”609 Leibniz also writes, “it is
obvious we have no idea of a concept when it is impossible.” Further, “in the case of merely
suppositive knowledge, even if we may have an idea, we do not grasp that idea, for such a
concept is known only in the same way as are those concepts which involve a hidden
impossibility; even if it is possible, we cannot learn of its possibility by this way of knowing
[i.e. by supposing it is so].”610
Overall, Hill is looking at the pieces of Leibniz’s philosophy and letting us know where it
seems to be scholastic. The overall intention of Leibniz differs from the overall intention of
most scholastics. Leibniz as a self-described “more Platonic than Aristotelian” thinker
undoubtedly adopted some ideas similar to the scholastics and in some cases fully agreed with
the scholastics.
Hill discusses the kinds of ideas that Aquinas says that God can have - i.e. T-ideas (ideas as
types) versus TE-ideas (ideas as types and as exemplars).611 Hill’s discussion around Aquinas
misses the fact that God acts for the best because it is in God’s nature to do so. God is not
impelled by nature but by his own nature, albeit that his nature is to act for the best; thus, we
see ideas in what becomes the laws of physics (i.e. nature) only due to God’s nature.
God’s thoughts are made real by his will
Upon thinking of something, for God to will it is for it to become real. Thus, intention for God
is a very different matter from intention for humans. At the same, while action not will alone
is needed for humans, action is a result of will. As far as the soul is concerned, will is action
for the soul. Pre-established harmony causes the will of the soul, or a particular kind of will of
the soul, to invoke changes in the compound body monad which effects physical changes in
the universe.
God only chooses to will into reality thoughts that are in conformity with ideas. Could
humans act to bring into reality plans that are not in conformity with ideas? Of course, though
action by humans is much slower and more painstaking that will is for God.
We must leave open the possibility that there are ideas accessible by God that require faculties
that God has which humans do not and never will have. For example, God’s mind can
comprehend the entire universe and all of its possibilities in a single thought.612 This may well
609 Leibniz, G. W. “Discourse on metaphysics” 1686, §23, Loemker p.318 610 Ibid., §25 p.319 611 Hill, J.C.R. Leibniz’s metaphysics of intentionality PhD National University of Singapore 2008, pp.32-33 612 Friedrich Schiller reiterated this concept of a single act of thought: “Reason insists, in accordance with its
necessary laws, upon absolute totality of perception, and without letting itself be rebuffed by the necessary
limitation of the power of imagination, the mind requires from it a complete summation of all the parts of a
given quantum in one simultaneous mental image.” Johnson, S. trans. Of the Aesthetic Estimation of
Magnitude (1793) Accessed at http://www.schillerinstitute.org/transl/Schiller_essays/magnitude.html 1 Jan
2011
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open up an entire class of ideas which are of no practical use to humans without the capability
of instantaneous pan-universal comprehension. However, humans could presumably still
grasp and reason about the process, structure and harmony in the ideas used; it is just that we
could never make practical use of those ideas in the way that God does. We might still be able
to make practical use of those ideas for smaller matters. However, it might be possible that
there are some ideas that only make sense when the entire universe and all possibilities are
under consideration.
Thoughts that soul monads have
Ideas are the foundation of the unfolding of the universe even though ideas are not of the
universe or in the universe, nor of the immaterial realm of souls (which too is part of reality)
nor even part of God. Thoughts too are merely images or perceptions. Nevertheless, both
ideas and thoughts follow an order, can be structured and demand study. Indeed, they
motivate the Universal Characteristic.
How can we speak of “them” if they do not exist? Language and speech are nothing but
vehicles for representing thought, and a thought may be about “any nature or form whatever”,
thing, notion or idea. In any case, all we are doing is progressively bringing out of our
memory in clearer and more distinct form ideas that have always been with us. They would
have to have been, since our soul – like all monads – perceives all other monads and therefore
has knowledge of the entire universe. However, the perception is unclear and indistinct.
A mind/soul can represent to itself a thing that is not real. However, the universe was (in the
design phase) constructed in God’s mind. The perfect universe (of the many so conceived by
God’s mind) was willed into existence. That universe is in conformity with ideas, which only
means that the universe is consistent with itself and is good, i.e. is a reflection of God himself.
So by understanding the universe we understand God. The universe, in turn, manifests every
possible good. Ideas are simply mental constructs for understanding the universe and hence
the good (God) as well as vice-versa, i.e. ideas are mental constructs for understanding the
good (God) and hence the universe.613
There is a paradox in that what is most evanescent offers the greatest power, perhaps the only
power, to affect the physical universe. Three concepts are considered: ideas, which do not
exist; souls, which are immaterial; and perceptions within souls which are merely mental
impressions. Ideas are not even real and so are “less” material than the immateriality of souls
and are also less material than the images which comprise the perceptions within souls. Yet it
is from ideas that souls derive their power. In particular, it is by the distinctness with which
ideas are perceived that their power is measured. This is only because a thought can only
correspond to physical reality to the extent that it represents an idea. Riemann formulated a
theory of thought that appears to be a maturation and consolidation of the mechanism and
613 Hermes Trismegistus identifies God with and as good, whereas Leibniz identifies God with that than which
there is no greater. Nicolaus of Cusa identifies God with “Absolute Maximality” or that than which there is
no greater, meaning the infinite. De Docta Ignorantia Book I Chapter 2 §5 in Hopkins, J. (trans.) Nicolaus of
Cusa De Docta Ignorantia The Arthur J. Banning Press, Minneapolis 3rd ed. 1988, p.6
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role of Leibniz’s “perceptions”. Riemann used the term Geistesmassen which can be
translated as thought masses, spirit masses or mind masses.614 The term “mass” could
mislead. Riemann’s emphasis is on the fact that Geistesmassen are the locus of activity of
intelligent souls. Mixing in Leibniz’s terminology, when a thought mass is sufficiently clear
and distinct, it can give rise to action by the corresponding body monad. The closer the
thought mass is to representing an idea, the more effective the consequences of its guidance of
the corresponding body monad in the physical world will be. For example, if a scientific
concept is incorrect, then it is not an idea and physical structures or machines will not work if
their design is based on that concept. If a scientific concept is correct, even if only within a
particular domain, i.e. it is closer to being an idea, then it can be used for new engineering
possibilities.
When monad X distinctly and unconfusedly perceives the indistinct perceptions in monad Y,
are the perceptions of monad X ideas? Yes, because such distinct perception would
understand the long chains of causes and implications of the indistinct perceptions of Y. Such
perceptions by X give it indirect power over Y because it understands Y.
Humankind as a subject of physical science
Due to the relationship of God to Man, since God is the starting point for a priori scientific
thought, it is necessary to understand Man in the a priori fashion too. Indeed, just as it is only
possible to theorize about God in the a priori fashion and possibly have some conclusions
thereby reached confirmed or denied by experiment or mathematics, the same is true of Man.
Even an Atheist, to whom the concepts of divinity of Man and God’s relationship to Man are
meaningless if not offensive, the fact that Man has a conscious creative power not found
elsewhere in nature begs special attention.
The difference between our ideas and our brand of creation versus that of nature (i.e. what
God chooses to allow, aside from humans) is critical to our understanding the character of
intellection beyond nature but below God’s. God made nature, but could do much more;
instead, God wants humans to extend nature. This can only be to allow humans the
opportunity to improve themselves, since God could have done and could do anything that
humans might ever do. The upwards development of humans is part of the process of
perfecting the universe. A teacher gives its pupils opportunities to create, and does not jump in
and take over out of impatience. The teacher’s task is for each student to successfully create
autonomously.
At the same time, there is nothing in nature which could do what Man can do. Only a self-
conscious entity such as Man can undertake creation of the kind that Man can. Thus, Man is
God’s natural instrument for that very kind of creation. Necessary to the self-consciousness
that must go hand-in-hand with the divine creativity that Man has is political conflict,
614 Riemann, B. Gesammelte mathematische Werke, wissenschaftlicher Nachlass und Nachträge “Collected
mathematical works, scientific deductions and supplements” , Springer-Verlag, Berlin Heidelberg New York
1990. In particular, see I. Zur Psychologie und Metaphysik “I. On psychology and metaphysics” pp.541-2 of
Fragmente philosophischen Inhalts “Fragments philosophical content” for a concise introduction.
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oligarchism, etc. Because we are self-conscious, we will naturally have trouble understanding
our own nature or character, just as it took thousands of years to discover various natural
physical laws. On the other hand, Man is not God’s instrument. Man is God’s project. God
wants Man to succeed in autonomously developing or evolving the physical universe upwards
but, again, for the sake of Man not the universe per se.
“Ideas” (in the strict monadological sense) about humankind
Ideas, which are often presented as laws, about ourselves and our creative capability cannot
be studied empirically in the way that some physical phenomena can be studied using
physical experiment. As yet, we have no systematic empirical way of discovering such laws.
Thus, we have been limited to the a priori or, at least, purely intellectual method.
Sensory perception and concepts that are able to be visualised clearly in a way that is akin to a
sensory perception but held in the mind are the locus or raw materials of the Empiricist mode
of thought and investigation. These loci are not a very effective way of discovering such laws
because there is little in the external world that is indicative of the potential, nature or role of
Man. First of all, humanity largely lives in the realm of final causes which, by definition, does
not permit the use of empirics. Second, manifestations of humankind in the realm of efficient
causes generally require the discovery to be made first and then be implemented into human
society and human law and government. It may then take decades or centuries for the results
to manifest themselves which can then be pointed to as proof of whatever the presumption
about human nature had been. Thus, only the a priori method is of significant value and this is
based purely on thought.
There have been many political experiments in history. There are libraries of books on the
results of those experiments. Arguably, many political systems resulted in disaster because the
a priori work had not been done or had flaws: such flaws would be regarded as moral flaws.
As mentioned above under the heading “The problem of evil”, moral ideas are as much a part
of the domain of discoverable truths as any “physical scientific” question of, say, gravity or
air pressure. The point is the moral truths are physical scientific, and the nature of Man is a
physical scientific question because Man is a force for change in the universe. Man is a
particularly interesting question of this kind because Man intentionally effects physical
change in a way that is bounded but is not restricted, as such, by the physical laws that govern
inanimate matter or the laws that govern living biological matter.
Historically, very few people think. Thus, progress is slow. Ideas about human nature impact
human institutions which means that oligarchical or control structures are threatened. This
discourages mentation on such ideas, further slowing progress. People generally take their
signals from empirics and can thus be corralled relatively easily, discouraging or preventing
mentation on ideas about our self-conscious selves and even more strongly suspending their
promulgation, implementation or even serious consideration.
Thus, progress in the realm of final causes – which is the immaterial or soul domain – is far
slower than our progressive understanding of “efficient causes”. Yet the realm of final causes
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is more powerful or, at least, enormously more influential in Man’s impact on the universe.
Yet, it is certain that such ideas can be reasoned about. Plato’s The Republic is a starting point
for a demonstration of how. Leibniz explains that it is so in New Essays and provides further
guidance as to the “how” throughout his writings.615
Foundations of a programme for discovery
We now have established sufficient background to be able to resolve and draw together a
number of apparently conflicting threads in Leibniz’s and Nicolaus of Cusa’s conceptions of
the role of ideas and possibilities for scientific discovery.
Leibniz is unambiguous that ideas are not real, in that they have no metaphysical/ontological
existence and certainly no physical existence. However, ideas are a standard by which
perfection is measured (at least, sometimes) and, therefore, define components of - and
totalities within - the physical universe. To Leibniz, there is nothing “more perfect” about a
geometrical square than a roughly cut square paver in the physical world, because the
physical world is (a) real and (b) the best of all possible worlds. The roughness of the paver is
necessary, is a part of its beauty and that beauty includes its role in the totality of the universe,
and is a characteristic of the unfolding upshifting of the universe - and the people in it - to
betterment.
It seems that there is an independent concept of “the Best”. Or is it simply God-defined? If
“Ideas” are given to us by the best possible universe, they are given to us by “the Best” which
is itself an idea. Because we do not know what ideas are - or which concepts are ideas - unless
we see them in all the complexity of the universe, we need to define them (and the Best) by
the universe. God does not care about a perfect square’s ideality. He cares about how to get
the best possible universe, and a perfect square may or may not be part of that.
This may be why Riemann suggested that physics (as a study of the actual, physical universe)
comes first and mathematics second. It often happens that we find ideas in this Best universe.
Thus, it is useful for us to investigate ideas (or what we in our limited understanding think of
or perceive as ideas) so we have advanced ones “on the shelf” for when our studies in physics
need them. After all, the principle of plenitude says that everything that is possible - is.616
Ultimately, we do not know whether what we think is an idea is an idea until it has been tested
in the universe or by checking against the universe (i.e. by experiment). Theoretically, we
could know without “testing” but that would require us to - like God - hold the entire universe
(and all of its possible unfoldings) in our mind in a single thought. Similarly, as it is hard to
the degree just described which concepts are ideas, it is just as hard (perhaps by orders of
magnitude) to know what “the Best” is. Again, the complexity is such that it may be that it
can only be known by holding the entire universe in the mind at once. No doubt, however,
there are principles for the discovery of which we need to start forming hypotheses, as
615 Langley, A.G. (ed.) Leibniz, G.W. New Essays Concerning Human Understanding Open Court Publishing,
Illinois 1949, p.95ff 616 As does Hermes Trismegistus. Corpus Hermeticum II §13 in Copenhaver, B.P. Hermetica Cambridge
University Press 1992, p.11
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Leibniz has done. They ought to be simply. Leibniz believed that the hypotheses can be
formed or worked out using his Universal Characteristic. Thus, it is possible that it is not
necessary to have to be able to hold the entire universe and all its possible unfoldings in our
mind at once in a single thought in order to be able to determine what the Ideas are or even
what the Best is.
We are still left with the question, (a) does the Best Universe define the Ideas (which includes
the Best), or (b) is it vice-versa? Because, from De Docta Ignorantia, we will never equal
God, assuming (a) is more correct. However, we necessarily chip away with (b). The
Universal Characteristic would provide a series of leaps in the pursuit of (b). The Universal
Characteristic itself would necessarily have limitations or a “ceiling” per De Docta
Ignorantia, and so would each success incarnation or version of the Universal Characteristic.
We see in all this how tightly woven science associated with experience of the universe as it
is, (a), is with a priori thought and the methods of Reason which include development of the
Universal Characteristic, (b).
Conclusion
As a result of their perceptions, appetitions and capacity for rational thought, human and other
souls make decisions that impact the physical universe. God plays an intermediating role, as
shown in Figure 7. Thus, the consideration of passive matter alone cannot give a complete –
and probably not even close to correct – understanding of the universe. The domain of physics
needs to be expanded to include the incorporeal, since the incorporeal control the corporeal,
as well as to include the connection between the two since that is the only way it is possible to
understand how the incorporeal controls the corporeal. More work is needed to understand the
pre-established harmony, which is in the middle part of Figure 7, and on how thoughts (i.e.
the internal activities of soul monads) can attain more closely to ideas which is the left node
in Figure 7.
Two open questions present themselves. The first relates to the pre-established harmony.
Is it possible to define the correspondence between the soul or thoughts and the corresponding
body monads as a mathematical relation? Leibniz argues that God links thoughts with actions
in the best possible way. This leads us to the second. Can we discover a general definition for
“the Best” perhaps for improvement over time? Since the universe corresponds with “the
Best” possible, which is not always the same as what we think of as ideals or mathematicals,
can we formulate an overarching criterion, or criteria, for the Best without having to hold the
universe and all of its possible unfoldings in our mind in a single thought like God can? We
know from Kepler that the reality of the universe is harmonious, beautiful and brilliant, but
those do not serve as working criteria nor were they intended to.
The ongoing re-creation of the universe as directed by the thoughts of soul monads
necessitates a design which places thinking soul monads at the top of the universe. The
Creator has given soul monads the prerogative to create if they are able to be active (i.e. to
think rationally, in the strict Leibnizian sense) and choose to be. Indeed, even if soul monads
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choose dullardism, they are by the nature of the role allotted them still the ones to which
God’s attention turns in the unfolding of creation via the pre-established harmony. Thus,
Leibniz vindicates the Renaissance view of Man.
Figure 7(a): pre-established harmony between a body and soul pair of monad counterparts
Figure 7(b): pre-established harmony at the collective level of all reality between all that is
incorporeal (all active matter) and all that is corporeal (all passive or physical matter)
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Chapter 9: A priorism in science
Introduction
Leibniz sought to re-orient and enhance the burgeoning program of experimental science.
Leibniz wanted a primary emphasis on structured thought with experiment playing a
supporting role to the reasoning process. Leibniz saw that in his lifetime, there was a risk that
experiment would become primary with reasoning relegated to a secondary and supporting
role. Indeed, some writers such as Sarpi and Newton as discussed above sought to exclude
reason, and substitute experimental evidence or the testimony of the senses for the capabilities
of the mind.
Wiener notes an oscillation in Leibniz’s writings “between his a priori system of irreducible
real definitions and the experimental aspect of his program”.617 Leibniz certainly gave
attention to experimentation, even writing that he is in the habit of writing out a catalogue of
experiments to be done when he examines some matter of physics. He will ensure that the list
includes all experiments needed to find the cause of what is in question “through
demonstration and not through Hypothesis”.618 This shows that Leibniz was not an a priori
fanatic, but – as in all other things – a pragmatist.
Nonetheless, a discussion of the importance of a priorism in science would be incomplete
without giving examples of its use. Are there examples to which we can point and say that a
priori thought was critical to a discovery or a process of discovery whose results had enduring
value? We will discuss three examples: Leibniz’s work in creating the calculus, the discovery
of non-Euclidean geometry and Kepler’s work in Mysterium Cosmographicum.
A priori thought is often equated with seeking to understand the mind of the Creator with
observed physical phenomena playing the positive role of acting as a challenge influencing
the direction of this quest by introducing new questions or challenging existing answers.619
Therefore, the chapter ends with a brief section on some contemporary references to the mind
of God which are not in agreement with the Leibnizian a priori programme.
The obvious challenge is: name one discovery that was made by observation alone.
Discoveries are rarely made by pure observation and, when they are, much more hard work
that requires thought as well as observation is required to give meaning to the initial
interesting observation. Contradictions might be found between observations. Problems in
accuracy and consistency are found with observations. Critical information is found through
observation, but observation does not give new understanding. Typically, observations while
sometimes answering questions usually lead to more questions. Eratosthenes might have
hypothesized the shape of the earth from an observation. But the same observation had been
617 Wiener, p.xxvi 618 Letter to Berthey 1677, Ibid., p.xxiv 619 This is to borrow the words of Meli but to use them in a different context. Meli, D.B. Equivalence and
Priority: Newton versus Leibniz Clarendon Press, Oxford 1993, p.19
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made by others many times before without Eratosthenes’ hypothesis springing to their mind.
Eratosthenes might then have used observations to determine the radius of the earth, and thus
answer a question, but in his mind he had already formed a view as a result of observations.
Differences in the lengths of shadows at the same time of day exposed a contradiction which
Eratosthenes attempted to resolve by forming an hypothesis. Once he had done so, he was
able to ask a question which could then be answered by further observation, which was really
interpretation of experience through the lense of his understanding/hypothesis.
The calculus’ development from Archimedes to Barrow to Leibniz involved little or no
practical experimentation. The same can be said of the development of Euclidean and non-
Euclidean geometry. However, the discoverers were affected by thinking that resulted from
experience. For example, Archimedes and Leibniz were engaged in experimentation and
thinking related to physics, which must have had some impact on their treatment of the
infinitesimal.
Clearly, Kepler’s development of planetary motion and gravity was partly guided and partly
set off in new directions by empirical observation, as was Copernicus’ work. Their
development of ideas relating to properties of the universe may not have been made had it not
been for the development of precision in astronomical observations in their time. To have
discerned those ideas from principles pertaining to how God works (“that the universe could
not be other/better than it is”) would have been too much of a leap for Copernicus’ and
Kepler’s time, and perhaps even for today. This does not mean that Kepler’s discoveries
would, in principle, have been impossible without the telescope. Leibniz’s point is that the
indistinctness and confusedness of our reasoning leads to erroneous theory, and generally
prevents us from learning about the universe without at least some empirical information
versus by way of pure thought. Leibniz agrees that more precise observations can reveal
errors in theory. Kepler and Copernicus’s striving to correct the disagreements between the
Ptolemaic theory and increasingly precise astronomical observations is nothing but an
example of new empirical information revealing the “confusedness and indistinctness” of our
mental conceptions and reasoning.
Indeed, pure thought even with human imperfection and our general dependence on
observation maintains a power that observation cannot have. Franklin’s paper, on two
perspectives of Leibnizian optimism explains how we can, and often do, work out
mathematically – i.e. through pure thought – what is not possible and thus what things we
could never observe because they could never occur.620 The paper notes that proving what
must be is a long way from proving what cannot be. Nonetheless, Franklin argues that pure
thought can be a very powerful thing and can provide results that no experiment ever could: a
truth that holds always and everywhere.
In the three domains of discovery considered here, a priori thinking dominated and was
probably decisive. Most domains of discovery bear fruit over several generations of thinkers
who build on one another, so a complete argument that a discovery resulted from a priori
620 Franklin, J. “Two caricatures II: Leibniz’s best world” International Journal for the Philosophy of Religion 52
(2002), 45-56, accessed at http://web.maths.unsw.edu.au/~jim/caric2.pdf
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thinking requires consideration of those thinkers over multiple generations and centuries. We
do not have the space for a full study of that kind.
Example 1: The infinitesimal and the calculus
Leibniz’s development of the calculus involved much more than the infinitesimal though the
infinitesimal is a core concept. However, the great ancillary work did not involve empirical
observation though it did involve mental images and diagrams of curves to help clarify ideas.
Bagni says that the development of the calculus often assumed “an aprioristic platonic
epistemological perspective”.621
This perspective was not limited to curves in the abstract, nor did it even begin with abstract
curves. In 1675, Leibniz generalized the calculation of centre of gravity of a figure, and the
calculation of the moment of a figure about any given line.622 Leibniz acknowledges that
Cavalieri pioneered the use of the infinitesimal idea for moments.623 Child points out that
Cavalieri used the phrase “incrementum difforme gravitas” to “connote a gradual increase
that follows a definite law” (emphasis in Child).624 In 1679 in a Letter to Tschirnaus, Leibniz
wrote, “Huygens, who thought me a better geometer than I was, gave me to read the letters of
Pascal, published under the name of Dettonville; and from these I gathered the method of
indivisibles and centres of gravity, that is to say the well-known methods of Cavalieri and
Guldinus.” 625 In July 1676, when investigating the “inverse method of tangents” (quadrature)
Leibniz wrote, “There is indeed another method that is more general and a priori, namely, by
the intersection of two tangents, which should always intersect between the two points at
which they touch the curve, as near one another as you can imagine”.626
621 Bagni, G.T. 94–103 “Exhaustion argument and limit concept in the History of Mathematics: educational
reflections” In Furinghetti, F. Kaiser, S. & Vretblad, A. (Eds.), Proceedings of HPM–2004, History and
Pedagogy of Mathematics, Uppsala July 12–17, 2004, at p.4 622 Leibniz, G. W. “Analytical quadrature by means of centers of gravity” in Child, J. M. (ed. and trans.) The
early mathematical manuscripts of Leibniz; tr. from the Latin texts by Carl Immanuel Gerhardt with critical
and historical notes by J. M. Child The Open Court Publishing Company 1920, republished by the University
of Michigan Library, p.66 623 “In 1629, Cavalieri, a Jesuati – an adherent to the Rule of St. Augustine – was appointed to the chair in
mathematics at the University of Bologna, a post he occupied until his death, largely through the
recommendation of Galileo, who proclaimed him the foremost Italian mathematician of the day. His
Geometria indivisibilibus contains the first systematic exposition, as it pertains to the principles of
summation, of what we now know as the calculus. He accomplished this by employing the concept of
‘indivisibles,’ or ‘infinitesimals,’ which served the same purpose as ‘the method of exhaustions’ employed by
Archimedes and other Greek mathematicians. In principle these approaches were the same but the system of
notation for indivisibles was much more concise and convenient.” Accessed at
http://www.brown.edu/Facilities/University_Library/exhibits/math/textfr.html#bon.html 11 May 2011. The
original text referred to is Geometria indivisibilibus continuorum nova quadam ratione promota ...
Typographia de Duciis, Bologna 1635 624 Child, J.M. (ed. and trans.) The early mathematical manuscripts of Leibniz; tr. from the Latin texts published
by Carl Immanuel Gerhardt with critical and historical notes Open Court Publishing Company, Chicago and
London, 1920 reprinted by University of Michigan Library, pp.208-9 n.25 625 Ibid., p.215. Also see further quotes at p.215 n.36. 626 Ibid., p.119
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Leibniz’s principle of continuity was based on his conception of how God thinks, via the
principle of sufficient reason which is a consequence of the best of all possible worlds
principle. Of course, the infinitesimal is a corollary of continuity. We do not know whether the
mathematical conception of continuity or the metaphysical principle of continuity came first,
but it is difficult to believe that one did not influence the other or perhaps the influence
worked in both directions. Yet, Cusa understood that, “As the mind considers it, the
continuum is divided into what is always further divisible and the number mounts to infinity.
But in actually dividing a line, one comes to a part that is actually indivisible. It is this I call
the atomic unit.” Yet Cusa knew that the atomic unit is “nothing” for he quoted Boethius
saying that, “If you add one point to another, you effect no more than if you join nothing to
nothing” and then wrote “if you connect the ends of two lines, you do make a longer line, but
the point of connection constitutes no length at all”. The emphasis here is on how “the mind
considers it”.627 It is on this intellectual artifice that the infinitesimal is based. Katz and Sherry
argue that the artifice of the infinitesimal is not fatal to the well-foundedness of the calculus.
Indeed, they say that the artifice is testament to the forwardness of Leibniz’s thinking, by
saying, “Leibniz’s was a remarkably modern insight that mathematical expressions need not
have a referent, empirical or otherwise, in order to be meaningful.”628
The presumption that continuity has atomic components – which are metaphysical only – with
which humans can calculate is the stride that Leibniz takes with his calculus. Indeed, the ratio
Δy/Δx approaches a ratio of atomic lengths which is called dy/dx. Had Cusa not brought the
idea of the infinitesimal out into the open, and discussed and analysed it, future generations of
thinkers might not have had the confidence to use the “atomic unit” as Leibniz did. Without
Leibniz’s conception of the infinitesimal, there would have been no Leibnizian calculus.
Further, Cusa’s analogy of the quadrature of the circle is so central to Leibniz’s work that
Grosholtz refers to the “infinite-sided polygon” perspective of the circle (for which Cusa is
famous) as “Leibniz’s infinite-sided polygon”.629
Leibniz’s a priori progress into the infinitely small allowed him to develop a method of
general applicability. Of course, thinkers before Leibniz had ventured into the infinitely small.
Child tells us that Leibniz’s sole introduction was the infinitely small division between
ordinates in the method of Cavalieri c.1672.630 From there, Leibniz considered the infinitely
small triangle which he called the Characteristic Triangle. (Note the synonymous terminology
with the sought-after Universal Characteristic which Leibniz believed could be found as the
basis for a general method to solve any problem.) Leibniz knew that this triangle was
“indefinite” and so it was not possible to perform calculations on it directly. However, “yet he
perceived that it was always possible to find definite triangles similar to it.”631 Incidentally,
the use of of dy/dx predominantly in reference to the Characteristic Triangle is one reason
627 Miller, C.L. (trans. and intro.) Nicolaus of Cusa Idiota de Mente Abaris Books, New York 1979, pp.71-73 628 Katz, M. and Sherry, D. “Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, and
Their Foes from Berkeley to Russell and Beyond” Erkenntnis Springer 2012 Vol. 77 629 Grosholtz, E. “Was Leibniz a Mathematical Revolutionary?” pp.126, 133 630 Child, J.M. (ed. and trans.) The early mathematical manuscripts of Leibniz; tr. from the Latin texts published
by Carl Immanuel Gerhardt with critical and historical notes Open Court Publishing Company, Chicago and
London, 1920 reprinted by University of Michigan Library, p.38 631 Ibid., p.39
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why Richard Brown considers Leibniz’s calculus as fundamentally different from Newton’s
fluxions which assists Leibniz’s cause in the calculus priority debate.632
We have just described a priori thought that is removed from observation by two degrees of
abstraction. First of all, the definite triangles on which Leibniz performs calculations are
themselves creations of the mind. Second, he uses those calculations to make inferences about
the indefinite triangle about which he can presume to know very little directly.
The calculus is not necessarily complete. New understanding of the infinitesimal domain
might be reached in the future causing the Leibnizian calculus to give way to a better and
more powerful calculus. For example, there may be a time in the future when we find certain
qualities of the indefinite triangle which need not be regarded as true or which are provably
false in the infinitesimal domain. That is, just because an indefinite triangle is similar to a
definite one, why must they necessarily share certain qualities? The edifice on which the
calculus is built may collapse to give way to a more general calculus.
Leibniz reached a certain point in his development of the calculus when the calculations were
entirely algebraic. This in itself was an achievement. However, the differentiability or
integrability of a curve then depended on the form of its algebraic equation. As is standard for
mathematicians today but was not in the 17th Century, Leibniz considered equations of
“infinite prolixity”633 and systematically showed that such equations did not pose a problem
for his calculus.
Leibniz proceeded to integrate the powers of x and stopped at x3 for, as Weissenborn said, “his
soul is in the throes of creation”.634 He proceeds to derive the method of the integration of a
square, such as f(x)2 and thence a particular case of integration by parts xf(x).635 Shortly
thereafter, Leibniz notes that integration has the effect of raising the dimension of the thing
integrated regardless of how large the dimension might be.636 For example, integrating a line
over an interval produces a 2-dimensional area, integrating an area through some revolution
produces a 3-dimensional volume, and so on.
632 Brown, R. C. Leibniz unpublished, Chapter 11 “Epilogue”, p.177. For Leibniz, dy/dx “is simply a device to
calculate the subtangent t from the equation dy/dx = y/t.” However, “Had Leibniz been influenced by
Newton” we would expect Leibniz to have treated dy/dx as an independent object, function or rate of
change.” We have avoided the priority debate and it remains beyond the scope of this thesis. However,
Leibniz knew that curves can be used to represent all manner of quantity such as distance, velocity, rate of
change itself, or any quantity one seeks to measure. So dy/dx might have represented many quantities and
Leibniz was certainly aware that it was a rate of change. By finding the subtangent, Leibniz knew that he was
opening up the facility for calculation with any quantity one happened to be analyzing in the form of the
curve to which one sought the subtangent. He was aware of the generality of mathematics and he knew that
the tangent represented an instantaneous rate of change. However, it is true that Leibniz’s focus on dy/dx was
due to its role with respect to the Characteristic Triangle which was so important to the development of his
method. 633 Child, J.M. (ed. and trans.) The early mathematical manuscripts of Leibniz; tr. from the Latin texts published
by Carl Immanuel Gerhardt with critical and historical notes Open Court Publishing Company, Chicago and
London, 1920 reprinted by University of Michigan Library, p.77 634 Ibid., p.75 635 Ibid., p.80 636 Ibid., p.80
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In 1675, Leibniz completes working which he believes will enable him to find the quadrature
(integral) of any conic. Child notes that the effort comes to nought and, even if it had not, he
would have obtained a large quadratic whose roots would have been too complicated to
use.637 Nonetheless, in the process, Leibniz exhibits a number of qualities of a priori
reasoning.638
Arithmetical progression
Leibniz considers how y changes when x changes in arithmetical progression. This is an
“artifice” whereby he draws out details on the expression for the curve for the following
purpose of eliminating terms.
Eliminating terms
Given that Leibniz has established himself in purely algebraic manipulation, he seeks
“artifices” which may simplify the working. Where there are more equations than there are
unknowns, solution is problematic. However, by taking a variable to be in arithmetical
progression, he is able to create additional equations and thus eliminate an undesired
unknown.
Moments
By employing an argument that is partly geometrical and partly algebraic, Leibniz proceeds
with an alternate route under the heading of “moments”. That is, he is able to produce two
expressions for the same thing with one slightly different due to its being resolved into two
parts with the one being infinitely small compared with the other.639
We can understand why Cauchy complained that Leibniz’s infinitesimal lacks precision.640
This is because Leibniz uses “artifices” to trick his formulae into simplification so as to “lay
bare” their secrets.
In addressing claims that indefinable real numbers cannot exist because they are not
denumerable and cannot be written down in decimal format, Franklin and Newstead argued
that the fact that we cannot see something directly does not mean that it is not there. Leibniz
has an unusual variation on this problem with his own critics. He took the view that if we
have forced something out into the open that we did not know to be there, we can still use it
even if we do not understand why it is there.
Further, the more we appreciate the “precision” which the limit conception (or δ/ε definition)
of the infinitesimal brings, the more it seems to be at the cost of distracting or misdirecting us
from the more exciting – and, probably, ultimately more fruitful – hunt for what Leibniz was
637 Ibid., pp.97-103 638 Ibid., pp.93-103 639 Ibid., p.98 and in footnote 34 on the same page, Child notes the advance in Leibniz’s thinking exhibited by
Leibniz’s use of the idea that one quantity is “infinitely small compared with” another allowing the former to
be discarded. 640 This is referred to by Boyer many times uncritically. See Boyer, C. The History of the Calculus and its
Conceptual Development Dover Publications: New York 1949
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actually taming. There seems to be a wild beast (or orchid) in the domain of ideas of which
Leibniz had only begun to get a smell.
Leibniz defended and encouraged the use of “artifice” to permit calculation with the
continuous. For Leibniz, it was not necessary that his conception of the infinitesimal actually
exists because it is a useful tool. Why would he so water it down, by disconnecting it from
how he regarded the universe to actually be? We suggest three possible reasons:
(a) He did not want society to be denied the benefit of his calculus as a pragmatic aid to
human development. Further, he did not think that the Republic of Letters would reach
agreement on his metaphysical arguments on continuity any time soon.
(b) He knew that he was expressing a property of metaphysical reality in precise terms,
and was not so arrogant as to think that such a profound idea could not be expressed
with greater fidelity than he had expressed it. In the meantime, he hoped his calculus
to nonetheless be able to enable a great contribution to human development.
(c) He intended his infinitesimal calculus to be a tool of engineers and other “men of
action” who had no interest in metaphysics or theology.
Beyond Leibniz, it is by the a priori method that Riemann qualifies the validity of Leibniz’s
infinitesimal triangles. Over II §2 and III §1 Riemann makes it clear that infinitesimal
triangles are only applicable in a space of curvature zero and in which the angle sum of a
triangle is two right angles.641
Example 2: The Fifth Postulate and Non-Euclidean geometry
The arguments surrounding Euclid’s Fifth Postulate and tacit assumptions made by Euclid are
rich with a priori reasoning with far-reaching ramifications for geometry and for human
conceptions of the structure of the physical universe.
Euclid’s “postulates” were in the original Greek merely “requests”.642 Torretti says that Euclid
did not regard his “requests” as self-evident. Nonetheless, Keyser wrote that Euclid’s Fifth
Postulate (as it is called) was “perhaps the most famous single utterance in the history of
science”.643 Euclid’s The Elements is often regarded as a triumph of the ideal, or a testament
to what the mind can create through pure logic beginning with some basic postulates. Thus,
Euclidean geometry might suffice as a persuasive example of the power of a priori thought. It
is perhaps not surprising that it should be found wanting by a priori methods. Nonetheless,
641 Spivak, M. A Comprehensive Introduction to Differential Geometry Vol. 2 Publish or Perish, Inc., Houston,
1999 p.160 642 Torretti, R. “Nineteenth Century Geometry” in Stanford Encyclopedia of Philosophy 1999 revised 2010.
Accessed at http://plato.stanford.edu/entries/geometry-19th/ on 18 March 2011. Indeed, the word Euclid uses
is “Αίτήματα” (pronounced aiteemata) from Mourmouras, D.E. (ed.) Euclid The Elements accessed at
http://www.physics.ntua.gr/~mourmouras/euclid/book1/elements1.html 19 March 2011 which Google
translates as “requests”. The Greek Lexicon translates “aίτημα” as a request or demand. (An Intermediate
Greek-English Lexicon: founded upon the Seventh Edition of Liddell and Scott’s Greek-English Lexicon,
Oxford University Press) 643 Keyser, C. J. Mathematical philosophy : a study of fate and freedom E. P. Dutton & Co., New York 1922
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because Euclidean geometry has been so important, we regard the devising of non-Euclidean
geometry as a legitimate example of a priori mathematical and – as we shall see when we get
to Riemann – scientific thought. The consequences of non-Euclidean geometry are more
significant for physics than its own discovery, because non-Euclidean geometry allows
physics to be done with mathematics.
The Fifth Postulate was queried for many centuries before its true nature was found. For
example, in the 13th Century, Nasir al-Din (Eddin) al-Tusi, astronomer to Hulagu Khan
(brother of Kublai Khan and grandson of Genghis Khan) thought that it could be proven by
the first four postulates and therefore did not have to be included as a postulate.644,645 Boyer
and Merzbach write that al-Tusi was the last of the Arabaic precursors to non-Euclidean
geometry, that John Wallis published al-Tusi’s work in the 17th Century which was the starting
point for the work of the Jesuit Girolamo Saccheri in the early 18th Century.
Saccheri assumed that the two summit angles of a quadrilateral are less than 90 degrees and
tried to find a contradiction. Due to his preconception of that the two summit angles had to be
right angles, he “found” a contradiction with his initial assumption that the upper angles are
less than 90 degrees. This was published by Saccheri in a booklet entitled “Euclid Cleared of
Every Flaw”. Since Saccheri’s did not convince mathematicians, it enlivened interest in the
non-necessity of the fifth postulate even though this was not the conclusion reached by
Saccheri himself.646
A breakthrough came with Lobachevsky and Bolyai simultaneously and independently.
Lobachevsky motivated his Pangeometry by the fact that the consequences of the Fifth
Postulate “which, although they appear simple, nevertheless, appear arbitrary, and
consequently, inadmissible.”647 He begins by defining a line by the locus of intersection of
equal circles centred at two fixed points, and a plane by the locus of intersection of equal
spheres centred at two fixed points.648
Lobachevsky makes it clear that he regards his new non-Euclidean geometry, “Pangeometry”,
as an a priori construction which is “a complete geometric doctrine”.649 It is of interest
because it provides “methods that are useful for the computation of various geometric
quantities.” He says that the idea that the angle sum of a triangle is constant is “not a
necessary consequence of our notions of space” and then says “only experience can confirm
the truth [or otherwise] of this assumption”. So from something that is inadmissible because it
appears arbitrary, Lobachevsky has made a prediction that if we examine real rectilinear
triangles – especially large ones – we will find that their angle sum varies.
644 Boyer, C.B., Merzbach, U.C. A History of Mathematics John Wiley and Sons 2011, Chapter 11 “The Islamic
Hegemony” 645 Mathematics Illuminated Geometries Beyond Euclid published by Annenberg Learner §8.3 Non-Euclidean
Geometry accessed at http://www.learner.org/courses/mathilluminated/units/8/textbook/03.php 4 June 2011 646 Ibid. 647 Lobachevsky, N.I., Papadopoulos, A. (trans.) Pangeometry, European Mathematical Society Zurich, 2010 p.3 648 Ibid., p.4 Papadopoulos notes that the definition of plane is akin to Leibniz’s: that surface which divides space
into two congruent parts. 649 Ibid., p.75
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Lobachevsky and Bolyai independently developed non-Euclidean geometry at almost the
same time. Riemann, working in the post-Fifth Postulate environment, distinguished
“unboundedness and infinite extent”650 which removes the necessity for infinitude of the line
and creates circumstances wherein the Fifth Postulate fails.651 Wolfe pointed out that Euclid
unconsciously assumed the infinitude of the line, which is a weak point of the Fifth Postulate,
in his proof of Proposition I.16.652 This is a case of the “‘hidden lemma’ or unacknowledged
assumption” referred to by Richard Brown, the uncovering of which is central to the
dialectical development of mathematics.653 Joyce points out that the first 15 propositions of
Book I of The Elements hold in elliptic geometry but not Proposition 16, saying “When a
‘straight line’ is extended, its ends eventually meet so that, topologically, it becomes a circle.
This is very different from Euclidean geometry since here the ends of a line never meet when
extended.”654 These considerations are all from thought alone.
It may be argued that the realm of geometry is by its nature the realm of thought. However,
geometry and physics are intertwined. Let us consider someone relatively recent, namely,
Riemann. Riemann found the basis of geometry as inseparable from physics when he said that
if space is a discrete manifold, then its metric relations can be discerned from within our own
understanding of that discrete manifoldness. If it is continuous, then we must understand the
binding forces which act upon it in order to understand its metric relations. These questions
are asked a priori and they provide the setting for the research programme to be undertaken in
understanding the universe.655
There may be critical experiments that can be designed to determine whether space is a
discrete or a continuous manifold. Such design would necessitate a priori investigation.
Indeed, the work already done with Lie groups and algebraic geometry in general can only be
regarded as a priori. Once we have determined whether space is a discrete or a continuous
manifold or some third possibility, then a priori speculations will need to begin anew. In the
end, we see that according to thinkers such as Riemann, physics not only can but must be
carried out just as a priori as can geometry. Physics since Riemann seems to have borne this
out, with most progress occurring in the mathematical domain punctuated by experimental
results, rather than vice-versa.
650 Riemann, B., Clifford, W.K. (trans.) “On the Hypotheses which lie at the Bases of Geometry” Nature, 8
(1873), pp.14-17, 36-37. Accessed at http://www.mat.ub.es/EMIS/classics/Riemann/index.html on 15 March
2011 and in Spivak, M. A Comrehensive Introduction to Differential Geometry Vol. 2 Publish or Perish, Inc.,
Houston, 1999 p.160 651 Wolfe, H.E. Non-Euclidean Geometry, Holt, Rinehart and Winston 1945, p.8 652 Ibid., p.6 653 Brown, R. C. “The Deconstruction of Mathematics” in Brown, R. C. Are Science and Mathematics Socially
Constructed? A Mathematician Encounters Postmodern Interpretations of Science World Scientific
Publishing Co. Singapore 2009, p.165 654 Joyce, D.E. Euclid’s Elements accessed at http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI16.html
on 27 Apr 2011 655 Riemann, B., Clifford, W.K. (trans.) “On the Hypotheses which lie at the Bases of Geometry” Nature, 8
(1873), pp.14-17, 36-37. Accessed at http://www.mat.ub.es/EMIS/classics/Riemann/index.html on 15 March
2011 and in Spivak, M. A Comrehensive Introduction to Differential Geometry Vol. 2 Publish or Perish, Inc.,
Houston, 1999 p.160
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Riemann identifies a domain of growing interest in which only a priori speculation can lead
to progress when he concludes in the fourth-last paragraph that, “it seems that the empirical
notions on which the metrical determinations of space are founded, the notion of a solid body
and of a ray of light, cease to be valid for the infinitely small.” Thus, he says, we are free to
suppose the metric relations of space in the infinitely small do not conform to the hypotheses
of geometry. Moreover, he says, we should suppose this as soon as it permits a simpler
explanation of phenomena, which is nothing but a criterion for the desirability of an a priori
speculation.656
Typical of the conclusion of a priori work in physics from Riemann’s view is the work of
Archytas from the 4th Century BCE.657,658 The physical result of doubling the cube arose from
manipulating curves in a way that only the mind’s eye could conceive and which is not
subject to empirical observation (unless it has already been constructed deliberately).
Wolfe points out that intuition is not always reliable.659 Thus, Pasch and others introduced the
Postulates of Order which may seem to be obvious,660 but which do not always hold, and
which Euclid assumed in proving Proposition I.21. It would seem that such tacit assumptions
of Euclid would be among the difficulties that Riemann referred to when he said “the
empirical notions on which the metrical determinations of space are founded … cease to be
valid for the infinitely small.” Leibniz would agree that some geometry that holds for the
infinitely small, for there must be structure at all levels, but such a geometry is not something
that would correspond to intuitive conceptions resulting from our day-to-day experience.
We know what Leibniz says about continuity: the principle of sufficient reason demands
continuity always in the universe, or no discontinuities. Dedekind provides what is generally
regarded as a satisfactory definition of continuity, but questions whether it holds even in
regard to an arbitrary straight line. He says that he cannot prove it and nor can it be proved.661
Dedekind’s postulate of continuity is that if all points of a straight line fall into two classes,
such that every point of the first class lies to the left of every point in the second class, then
there is one and only one point which produces this division of the line into two classes (or
this severing of the line into two portions).662 For example, one can imagine singularities in
656 Riemann, B., Clifford, W.K. (trans.) “On the Hypotheses which lie at the Bases of Geometry” Nature, 8
(1873) pp.14-17, 36-37. Accessed at http://www.mat.ub.es/EMIS/classics/Riemann/index.html on 15 March
2011 and in Spivak, M. A Comrehensive Introduction to Differential Geometry Vol. 2 Publish or Perish, Inc.,
Houston, 1999 p.160 657 Huffman, C. “Archytas” first published 26 Jun 2003 with substantial revision 25 Jul 2007, accessed at
http://plato.stanford.edu/entries/archytas/ on 27 Apr 2011 658 Rivest, F. and Zafirov, S. “Duplication of the cube” undated, accessed at
http://www.cs.mcgill.ca/~cs507/projects/1998/zafiroff/ on 27 Apr 2011 where it is said of Archytas’ solution,
“it is not a construction in a plane but a bold construction in three dimensions, determining a certain point as
the intersection of three surfaces of revolution, (1) a right cone, (2) a cylinder, (3) a tore or anchor-ring with
inner diameter nil. The intersection of the two later surfaces gives (says Archytas) a certain curve (which is in
fact a curve of double curvature), and the point required is found as the point in which the cone meets this
curve.” 659 Wolfe, H.E. Non-Euclidean Geometry, Holt, Rinehart and Winston 1945, p.9 660 Ibid., p.13 661 Ibid., p.10 662 Ibid.
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the domain of the infinitesimal in which there is no concept of left and right, or in which one
actually has a single point which has nothing to its left and nothing to its right. This might not
be as fatal to geometry as it might seem, as Wolfe says that a large portion of Euclidean and
non-Euclidean geometry can be constructed without the principle of continuity.663 This
vindicates Riemann’s uncertainty as to whether space comprises a discrete or continuous
manifold.
Leibniz’s belief that the universe must be continuous due to the principle of sufficient reason
was based on a priori reasoning regarding the nature of God. Suppose it was found that in the
realm of the infinitesimal – which is precisely where most discussions about continuity take
place – Dedekind’s continuity criterion fails. Such an outcome would be rich with paradox,
and could itself only be reached via a priori reasoning.
In discussing the foundations of mathematics it could be expected that a priori thought is
prominent. There is often an unavoidable segue into metaphysics, philosophy and even
theology to reach conclusions more sound from those which were held previously. Yet the
tools and presumptions of physicists, engineers and even policymakers emerge from these
foundations. Thus, the processes followed in exploring and concluding in the foundations of
science have vast real-world consequences over centuries of human history. Therefore, the
role and method of a priori thought is best explored, understood and refined rather than
ignored or cast aside in favour of putative pure empirical observation.
Example 3: Elliptical orbits in a heliocentric solar system
We can say that Kepler’s wiping the slate clean of epicycles upon epicycles (i) was – arguably
– justified by “common sense” (which is easy to say in retrospect) but (ii) was argued by
Kepler as more consistent with the simplicity and beauty of how he expects the Creator to
work. With Kepler’s theological background and clear orientation towards “understanding
how God thinks” as being the purpose of science, we would say that (ii) was more likely the
critical component which persuaded Kepler that there was something fundamentally wrong
with the epicycles approach and to “start again”. We doubt whether Kepler gave “common
sense” any importance at all.
This is diluted, certainly, by the fact that for Kepler to add even more epicycles to refine the
epicycles theory to fit the observations a posteriori was “getting a little bit ridiculous”.
However, we should hesitate to ascribe such vulgar late 20th Century tests on Kepler.
Kepler had to discard the concept that the perfectness of circles makes them a necessary
component of any theory of how the heavens move. We add that Kepler was not just
influenced by, but was an intellectual disciple of, Nicolaus of Cusa. Thus, he had to have been
aware that no human presumption of truth (such as that planets must move in circles) was
necessarily permanent. Rather, more is to be gained by exploring the domain of ignorance,
since that domain is inevitable and infinite. The question to ask would have been, “What if
663 Ibid., p.12
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we’re wrong about circles being necessary?” It is only by doing so that we may possibly come
up with a better theory. Cusa’s theory of docta ignorantia is entirely couched in the eternal
journey of the human soul towards the understanding that God has and towards an
understanding of God.
Kepler begins Chapter XI of Mysterium Cosmographicum, entitled “On the arrangement of
the solid, and the origin of the Zodiac”, with a paragraph both characteristically Platonic and
Pythagorean.664 He writes, “I have deduced the natural properties of the planets from
immaterial things and mathematical figures” and “I dare to investigate the origins of the
circles [of planetary orbits] from frankly imaginary cross sections.” Kepler then justifies the
view that God is a God that wills based on absolute reason, and nothing exists except by his
will. We will call this the Divine Architectonic Principle (“DAP”). For the scientist, the effect
of DAP is that we must limit our enquiry to the bounds of the inscrutable powers of the
Founding Wisdom. In long hand, this means that a scientist should work within the truth of
the fact that God gave effect to the mathematicals by using them as the foundation for the
architecture of the universe. Thereby, Kepler’s intention in Chapter XI is draw “likely
explanations” for the way universe is from “the quantities” meaning numbers and which, by
extension, means the Pythagorean corpus.
We must pair this with Kepler’s earlier footnote that God, as well as a tongue, has a finger. We
must not attribute meaning to God’s words that contradict the works of God’s finger as seen in
the physical universe.
Notwithstanding Kepler’s footnote 1 that the chapter could be omitted for it carries no weight,
in footnote 2, written apparently 25 years after he wrote Mysterium Cosmographicum, Kepler
says that working within the DAP has repaid him with interest over the last 25 years even
though in this instance (i.e., M.C. Chapter XI) it did not lead to a happy result. In particular,
he is talking about the chapter’s aim in demonstrating the arrangement of the zodiac and
numbers. He does not resile from his modus operandi and the intellectual criteria for a useful
argument that follow from DAP.
For the scientist, the Mathematicals are the cause of natural things and the reason for this is
that the Creator had the Mathematicals as archetypes with him from eternity in their simplest
divine state of abstraction even abstracted from numbers (when we consider numbers in their
material or empirical/intuitive aspect as quantities of things). It may be that it is with this
“shadow domain” that Leibniz was grappling, as outlined under the previous heading
“Infinitesimal”.
Kepler notes that Aristotle carped at the theory that the mathematicals are the cause of things.
Given that Aristotle denied the existence of the archetypes, it is not surprising that he denied
the existence of a Creator and decided that the universe was eternal.665 Kepler admits that the
archetypes would have possessed no force had God not had regard to them in the act of
664 Kepler, J. Duncan, A.M. (trans.) Mysterium Cosmographicum Abaris Books, Janus Series, Opal Publishing
1981 pp.123-125 665 But since ideas do not exist, then to Leibniz the archetypes do not exist, but they do subsist.
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Creation. This hearkens forward to Leibniz’s Theodicy (§184), where Leibniz writes, “It is
true that an atheist may be a geometrician: but if there were no God, geometry would have no
object. And without God, not only would there be nothing existent, but there would be
nothing possible.”666
It seems that Kepler is saying that we find the expression of God in the universe through (and,
perhaps, only through) the subsistence of the archetypes in the design/architecture of the
universe. This reminds of Leibniz’s architectonic conception of the universe.667
Ultimately, Kepler concludes that it must be that “splendid and plainly necessary causes” for
the eccentricities of the orbits of the planets nearer the sun:668
1. Can be derived from the harmonies as if from an archetype, and
2. Rest on the fact that God is the architect of the universe since the archetypes are
ineffectual by themselves.
Kepler sticks to this line even though, as he says, in this chapter the application of the
principle did not have a happy result. As Leibniz would say, our perceptions are confused and
indistinct so the final arbiter on all of our a priori reasoning, however cogent it may be, is the
physical universe, i.e. the works of God’s finger.
Alternative views of the Creator’s mind
Given that the conception of the Creator’s mind was so important to Leibniz and Kepler
among other scientists in formulating their a priori conceptions, it is apt to mention some
contemporary views of the Creator’s mind.
Bruce Lipton comes close to saying that a study of the universe is a study of a mind; we add
that the only mind that that could be is that of the Creator.669 He writes that, “Physicists are
being forced to admit that the universe is a ‘mental’ construction.” Leibniz is clear that the
universe is not a mental construction in the sense of being an illusion, and that the universe
exists independently of any single mind including the mind of God.
Richard Conn Henry writes that the universe has a mental nature. However, he seems to be
verging on Kantian subjectivism. Elsewhere, he quotes Sir James Jeans, “The stream of
knowledge is heading toward a nonmechanical reality; the universe begins to look more like a
666 Huggard, E.M. (trans.), Leibniz, G. W. Theodicy: Essays on the Goodness of God, the Freedom of Man and
the Origin of Evil Released 24 Nov 2005 [EBook #17147] from C.J. Gerhardt's Edition of the Collected
Philosophical Works, 1875-90 La Salle, Illinois 61301 667 Leibniz, G.W. Tentamen anagogicum c.1696, in Loemker, pp.477-9 668 Duncan, A.M. (trans.), Kepler, J., Mysterium Cosmographicum Abaris Books, Janus Series, Opal Publishing
1981 p.125, footnote 2, fifth-last line 669 Lipton, B. “Embracing The Immaterial Universe Toward a New Noetic Science” Shift: At the Frontiers of
Consciousness No. 9, Dec 2005-Feb 2006, pp. 8-12 the quarterly publication of the Institute of Noetic
Sciences (IONS); website: www.noetic.org accessed at http://www.brucelipton.com/biology-of-
belief/embracing-the-immaterial-universe
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great thought than like a great machine. Mind no longer appears to be an accidental intruder
into the realm of matter … we ought rather hail it as the creator and governor of the realm of
matter.” However such snippets and references to quantum physics as Henry indulges in
remind of the quasi-scientific essays of the now-discredited postmodern milieu.670
While Henry has only quoted Jeans in brief, and so we cannot be sure of Jeans’ full meaning,
we would not agree that the universe is merely a thought. Leibniz made it clear that the
universe is real and is separate from God. It is true that God can create with merely an act of
will which is arguably tantamount to a mental act, but in doing so God creates something that
is separate and independent of his mind.
Superficially Henry appears to agree with our contentions regarding Leibniz’s conception of a
priorism and the power of a priorism, because by understanding God’s mind before or
without observation we can understand physics. Nonetheless, for the reasons explained, we
distance ourselves from Henry’s position.
Conclusion
We have shown that a priori thought played at least a key enabling role in three examples,
namely, in the development of the calculus, non-Euclidean geometry and the Keplerian theory
of the solar system. By becoming more aware of the a priori foundations of other scientific
domains and of science today in general, the study of a priori foundations may receive a fillip
which will allow for greater benefit to be derived from experimental efforts. Equally
importantly, a new array of theories may arise from contradictory or unexplained
experimental data already collected.
670 Henry, R.C. “The Mental Universe” Nature 436: 29, 2005
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Chapter 10: Conclusion – Leibniz’s humanist and Neoplatonic agenda
Introduction
In this short final chapter, we recap on Leibniz’s historical role and describe Leibniz’s agenda
referred to in the thesis title. Leibniz continued an ancient “agenda” certainly of the
Neoplatonists but also pre-dating the Neoplatonists to the Egypt of Hermes Trismegistus and
perhaps before that. In Chapter 3, it was explained how Leibniz helped usher in modernity.
Chapter 4 through 9 explored how Leibniz helped define, direct and fight for the kind of
thinking – and the kind of thinking about thinking – that the onset modernity relied upon. For
modernity to shift upwards rather than collapse, it would be well to maintain Leibniz’s
methods and orientation, and undertake some of Leibniz’s projects which have been embraced
in a serious way by only a few, such as the Universal Characteristic.
Leibniz and humanism
To compare Leibniz’s view of Man with that of other thinkers, such as Kant, is beyond the
scope of this thesis. However, an account of Leibniz’s metaphysics and the underpinnings of
his approach to science requires mention of his conception of Man’s nature and Man’s role in
the universe. In this, Leibniz was in the humanist tradition of Dante and distinguished from
the quasi-mystical, albeit at the same time somewhat Christian humanist, attitude of Leibniz’s
British contemporaries such as Boyle. Leibniz might not have said like Dante that humanity is
the jewel of Creation or the highest part of Creation, but he did say that humanity and humans
can act like the Divine in the miniature. This is not far from saying that humanity has a special
place in Creation, for Leibniz did not say that any other kind of creature can act like the
divine in miniature.
It must be asked what it means that Man is able to reason in the way that he can. Kepler put
this down to the uniqueness of Man in all Creation and the special role that God prescribed
for Man by giving Man the power of Reason. If nothing else, it can be concluded that there is
something special about Man. From here, it is easy to enter the Dantean Humanist
Renaissance. Leibniz was the kind of humanist that Pico della Mirandola and Giordano Bruno
were, who regarded Man as magus who can arrogate and wield great power and effect vast
change in the universe like a race of gods. As mentioned in this thesis, this is aligned with the
Hermetic conception of humankind.
Human progress
The intention to benefit humankind is implicit in Leibniz’s descriptions of, and optimistic
hopes for, human knowledge and in his idea of progress. For Leibniz, progress in itself and
progress for humankind were synonymous. Leibniz wrote and promoted designs for social
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organisation to promote improvements in culture and knowledge among the general
population, believing that the best of human attainments were intellectually and morally
accessible to all people.
Leibniz was at a historical juncture which he helped create. He was a classicist and a
medievalist himself. He was a conciliatory eclectic thanks to Thomasius as Leibniz’s teacher
and also thanks to the efforts of Thomasius to ensure a lasting peace after the Thirty Years
War.
Leibniz’s optimism is inherent in his best of all possible worlds doctrine. This doctrine is the
logically necessary and otherwise natural outcome of Leibniz’s metaphysics and its interplay
with his theology. In many ways, Leibniz’s metaphysics is contained in or is at least derivable
from his theology.
Broadening the field of debate for Platonism
Leibniz was a universalist and “Renaissance man” who helped create modernity with all of
the power of modern rigor. No longer was it Cusa arguing against the Aristotelians on
metaphysical grounds. Rather, it was a Renaissance-type humanist arguing against the neo-
Aristotelians, Averroists and Empiricists on many grounds other than metaphysical. Leibniz
combined the power of the Platonic dialogue with the scientific success of Kepler and
Galileo, and with his own and his associates’ achievements in mathematics, invention,
machinery, metaphysics and physics.
Considering the big questions first
In Chapter 9, it was shown that working through the steps of discovery for fundamental
questions of theology and metaphysics can affect where we end up in the natural sciences.
Conversely, as Riemann indicated and as Leibniz allowed, empirical discoveries can indicate
where our a priori reasoning went wrong. To address the basic questions in metaphysics while
conducting work in the natural sciences was always the approach of the Neoplatonists. That
background work forms the largest context for natural science. Our answers to the “big” a
priori questions can be tested with our work in the natural sciences. That background work,
the “big” a priori questions, must include the nature of humankind and the role of humankind.
This is because in a universe resulting from the Creator, minds with the capability to address
such questions first of all cannot be accidental and have a purpose different from that of, say,
the nearest passing meteor.671
Human capability
The nature of the universe cannot be understood without including the role of humans as
physical force and as a uniquely dynamic law of nature. Humanity’s capability and potential
671 Also see Penrose, R. The Emperor’s New Mind Penguin 1991
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to exercise its power as collective magus or as a wilful physical force must be taken into
account. Related endeavours such as investigations in mathematics are aided or hindered by
the scaffolding provided by the culture as much as anything else, which includes the
language, art, literature, music as well as philosophy, mathematics (of whatever kind) and
science of the age, which creates conditions for particular kinds of thought to flourish. How
does a culture develop conditions conducive to particular kinds of thought and creativity? We
can say that a culture’s preoccupation with a particular class or kind of Platonic ideas, absent
interruptions caused by war, deliberate sabotage or other factors, may lead to a raft of
discoveries in that domain over successive generations. Such discoveries can feed into
economic, military and cultural breakthroughs creating prosperity for the civilisation or
culture concerned. The political framework and conception of humanity that creates social
fertile conditions for any such flourishing of thought was of great interest to Leibniz. He
promoted related ideas through his circle of influence.
Human knowledge
Leibniz to our knowledge did not use Cusa’s term docta ignorantia “learned ignorance”. He
was optimistic about what was already known and not written down or organised. He was
more optimistic still, and even excited, about what humans would come to know in the future.
He sometimes seemed a little daunted by the extent of what humanity does not know. Even
though Leibniz was aware of the rigor and power of mathematical tools, he did not believe
that everything could be explained or subsumed by mathematical methods.
In responding to Descartes’ Principia Philosophie Part I point-by-point in 1692, Leibniz
wrote about the horizons of human knowledge:672
On Article 26. Even though we are finite, we can yet know many things about the
infinite: for example, about asymptotic lines, or lines which approach each other
continuously when infinitely produced but never meet; about spaces which are infinite
in length but not greater in area than a given finite space; and about the sums of
infinite series. Otherwise we should also know nothing with certainty about God.
However, it is one thing to know something about a matter and another to comprehend
the matter, that is, to have within our power all that is hidden in the matter.
On what humans can know and what humans should investigate:
On Article 28. As for the end God has proposed to himself, I am fully convinced both
that they can be known and that it is of the highest value to investigate them; and that
to disdain this inquiry is not without danger or suspicion. In general, whenever we see
anything that is particularly useful, we may safely assert that one, among others, of the
ends which God has proposed to himself in creating this thing is precisely to render
these services, since he both knew and planned this use of it. I have elsewhere pointed
out, and shown by examples, that certain concealed physical truths of great importance
672 Loemker, L. E., trans. and ed., Gottfriend Wilhelm Leibniz: Philosophical Papers and Letters 2nd ed. D.
Reidel Publishing Company, Dordrecht Hollard 1969, p.387; first ed. 1956 University of Chicago Press,
Chicago
Page 195
can be discovered by considering final causes, which not have been discovered as
easily by efficient causes.
Considering final causes is tantamount to pursuing an a priori program. Article 28 also states
that no phenomenon in the universe is accidental, and the deliberate causes of things are
knowable by humans.
A new metaphysics as a blueprint for the good in human society
We referred to Loemker in Chapter 3 who said that Leibniz had hoped his metaphysics
“would be adopted and made a blueprint, so to speak, by men of good will (honestas) for the
restoration of European order”.673 Counterposed to Leibniz was an “alternative interpretation
of human nature, human thought, and the good”. Loemker places this under the subheading
“The New Way of Ideas: Locke” and that new way is based on “another theory of ideas – that
of nominalism and positivism”.674
Mercer’s thesis on Leibniz makes much of the concept of “conciliatory eclecticism”.675 To be
sure, much of Leibniz’s work and thought can be regarded in this way. Raynaud begins his
commentary by saying, “In comparison to Descartes, or Hobbes, or Locke, Leibniz presents
himself as a moderate or a conciliator: he means to rescue a portion of the heritage of Plato
and Aristotle, he rejects Hobbes’s radical nominalism as well as his legal positivism, and he
breaks with Cartesian mechanism in order to make a significant place for natural
teleology.”676
Flexibility in scientific method, and the ultimate journey
In Chapter 6 under the heading “Science may use indemonstrables”, Raynaud’s conclusions
were given regarding Leibniz’s pragmatism. For Leibniz, methodological doubt was not a
reason to forego deny useful benefits derived from particular lines of enquiry.677 Further, the
requirement to answer a priori questions before conducting was also not absolute though it
can lend great power. “For,” Raynaud says, “if science may employ indemonstrables, that also
means that the initial absence of such principles does not impede one from progressing on the
path of reason, without having to ‘cast into doubt’ common beliefs.”678 For Leibniz, there are
many useful and legitimate paths. For Raynaud, Leibniz “establishes a new continuity
673 Loemker, L. E. Struggle for Synthesis: The Seventeenth Century Background of Leibniz’s Synthesis of Order
and Freedom Harvard University Press: Cambridge USA 1972, p.127 674 Ibid., pp.127-8 675 Mercer, C. Leibniz’s Metaphysics: Its Origins and Development Cambridge University Press: Cambridge,
2001, pp. 80-110 676 Raynaud, P. “Leibniz, Reason and Evil” in McCarthy, J. C., ed. and trans. Modern Enlightenment and the
Rule of Reason CUA Press Washington D.C. 1998, p.150 677 Ibid., p.152 678 Ibid.
Page 196
between science and the active life, between knowledge and practical judgement, and between
reason and faith.” 679
Benefit to humankind is but a self-evidently good side-effect of the seeking after the kingdom
of God. We will end with Leibniz’s words in which he reiterates Kepler’s advocacy of seeking
scientific insight to improve ourselves but, unlike Kepler, he also recognises the pragmatic
benefits to the day-to-day lives of humans made possible by scientific endeavour:680,681
I have shown on several occasions that the final analysis of the laws of nature leads us
to the most sublime principles of order and perfection, which indicate that the universe
is the effect of a universal intelligent power.682 As the ancients already held, this truth
is the chief fruit of our investigations; without mentioning Pythagoras and Plato,
whose primary aim was such an analysis, even Aristotle sought to demonstrate a prime
mover in his works, particularly in his Metaphysics. It is true that these ancient
thinkers were not informed about the laws of nature as are we since they lacked many
of the methods which we have and of which we ought to take advantage. The
knowledge of nature gives birth to the arts, it gives us many means of conserving life,
and it even provides us with conveniences; but the satisfaction of spirit which comes
from wisdom and virtue, in addition to being the greatest ornament of life, raises us to
what is eternal, whereas this life, in contrast, is most brief. As a result, whatever serves
to establish maxims which locate happiness in virtue and show that everything follows
the principle of perfection is infinitely more useful to man, and even to the state, than
all that serves the arts. Discoveries useful to life, moreover, are very often merely the
corollaries of more important insights; it is true here too that those who seek the
kingdom of God find the rest on their way.
679 Ibid. 680 Plauché, G. A. writes in his essay Ancient vs. Modern Political Thought 9 April 2011, “The premodern
political philosophers whose thought achieved dominance – Plato, Aristotle, Cicero, Augustine, Aquinas, and
others – were primarily concerned with the search for right order. They generally accepted essentialism,
teleology, eudaimonism, and natural law-type virtue or deontic ethics. Modern political philosophers tend to
be more concerned with the search for peace and order, consequentialist or deontic ethical systems concerned
primarily with social order, and are more likely to be rationalists or empiricists and base their theories on
reductionist foundations.” In this sense, Kepler’s philosophy was more pre-modern while Leibniz’s was
modern. Accessed at http://gaplauche.com/blog/2011/04/09/ancient-vs-modern-political-thought/ 12 May
2011 681 Leibniz, G. W. c1696 “Tentamen Anagogicum: An anagogical essay in the investigation of causes” Loemker
1969, p.477 682 These are precisely the words of Cicero in De natura rerum. Also see Boyle V, 515ff. quoted in Burtt, E.A.
The Metaphysical Foundations of Modern Physical Science Routledge London 1932 2nd ed. reprinted 1950
p.189, “the consideration of the vastness, beauty, and regular motion of the heavenly bodies; the excellent
structure of animals and plants; besides a multitude of other phenomena of nature, and the subserving of most
of these to man; may justly induce him as a rational creature, to conclude, that this vast, beautiful, orderly,
and (in a word) many ways admirable system of things, that we call the world, was framed by an Author
supremely powerful, wise and good, can scarce be denied by an intelligent and unprejudiced considerer.”
Page 197
Leibniz’s policy for nations
With Leibniz, the humanist conception of Man as magus and as the jewel of Creation was
crystallised in pragmatic policy and, in particular, in national government policy. Leibniz was
born in 1647 the year before the Treaty of Westphalia. This treaty enshrined the doctrine of
national sovereignty in the law of nations, and established the modern nation state system,
which today is known as the Westphalian system.
Leibniz was partly a product of the optimism inevitably brought by the end of the Thirty Years
War (1618-1648) as well as of the cultural optimism of his milieu. For example, Bach lived
and worked in Leipzig at the same time as Leibniz though it is not known whether they knew
each other. The possibilities of the new era were being conceived during Leibniz’s lifetime
and Leibniz helped conceive them. Leibniz and Denis Papin were early recruits into the
Academy of Sciences in Paris under its inaugural leader Christian Huygens. The Academy
was charged by Louis XIV with developing new power sources for France. Thus, it was a
scientific academy with a national and industrial purpose. From the start, Leibniz was forced
to grapple with the idea of power and of increasing Man’s physical power. He accepted the
challenge. Leibniz came to understand partly by exposure to Huygen’s gunpowder
experiments that controlled explosive force had a higher order of potential than the passive
force emphasized by the ancients. Perhaps inspired by the mission of the French academy,
Leibniz later advised Tsar Peter the Great to establish the Russian Academy of Sciences. Tsar
Peter agreed and invited Leibniz to be its first president, though Leibniz declined that offer
partly due to his existing responsibilities and commitments, and – possibly – partly because
Russia was regarded as a dangerous place. In any case, Leibniz’s circle of influence included
many nations other than France and Russia.
In the Westphalian era, the idea of Man as magus was coming into its own, and was being
enshrined in forward-thinking policy by European and Eurasian leaders partly under Leibniz’s
influence. The concept of Man as magus emerged from Neoplatonism. It was joined at the hip
with the optimistic conception of a plentiful and vast physical universe, in which Man has a
decisive and central role. Arguably, it was for promoting such plenitude and optimism against
the established fixedness of prevailing policy that Giordano Bruno was burned at the stake.
Far from being mystical, vague or otherwise divorced from reality, the Neoplatonic
orientation was a living force which demanded that humanity adopt the scientific stance and
take its rightful place as commander of itself and of the universe. Leibniz’s legacy survives
today in the policies of scientific-industrial nation state. A month before the closing words of
this thesis were written, the NASA rover Curiosity landed on Mars. Leibniz would be pleased,
but much remains to be done.
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Appendix 1: To the greater glory of God
The entirety of what follows in this Appendix is a quote from an anonymous circular which
we have reason to believe was written by Leibniz. It is referred to in Chapter 3 of this thesis.
Ad majorem Dei gloriam [“to the greater glory of God”]683
Profound meditation is urged upon researchers concerning the propulsive power of
gunpowder; and their perceptive intelligence is challenged to realize the possibility of
diverting the vast power of gunpowder to healthier applications than those hitherto
known. Only irreverent thinking can deny such an incontrovertible truth as that in the
sight of God only wholesome applications and uses appertain to all that is created and
can be manufactured therefrom; for everything exists for the benefit of mankind, if
only they had a serious desire for it. Nevertheless it is only too well known that the
corrupted and misled spirit of man is concerned in countless cases not with salutary
application, but allows itself to become passionately concerned with applying all its
acumen to discovering how to misuse things, which according to the intention of the
Creator should be useful to mankind, so that they should have every cause to praise
him on their account.
Such misuse had induced some men to describe the discoverer of gunpowder simply
as a sorcerer in monk’s garb instructed by satan; for owing to its violent power it
seems impossible to achieve aught but explosions, loss of life and destruction.
Similarly doubtless in the immemorial past, people thought of the propulsive power of
flowing water and of wind, before they had been made serviceable for useful purposes
for mankind by wise and industrious mechanical craftsmen who used first simple
wheels and later toothed wheels. The above mentioned view of the satanic origin of
gunpowder should therefore be set aside and replaced by the following:
1. The inventor of gunpowder, whoever he was, was a capable chemist.
2. The skilful achievements of chemistry are hated neither by God nor by Nature, nor
are they directed against God’s will; since they can instantly convert active poisons
into blessed and healing potions.
3. It is possible by means of some form of control to force the aforesaid propulsive
power of gunpowder, however sudden and violent it may now be, into ordered
channels, so that it should be adaptable depending on the arrangements made for
driving an ordinary mill or for performing other work; and this aim may be obtained if
earnest prayer for divine support is combined with enthusiastic pyro-mechanical
labours, and if mind and hand are ceaselessly busied with this work; and above all if
the aforementioned fundamental demonstration [of a pious use of the gunpowder and
the praise of the Almighty Creator are kept in mind, - rather than its direct tangible
683 Klemm from Ad majorem Dei gloriam published in Nouvelles de la Republique des Lettres Amsterdam 1695,
pp. 218-220
Page 199
use, which divine Providence will in its wisdom confer on the present century or on
future ones.
More than two and a half years have elapsed since a number of researchers were
publicly charged with the said problem concerning gunpowder, namely that it could
and should be used for other purposes than hitherto. And the researchers should
thereby be urged to the bold attempt gradually to give up their enthusiastic researches
which were placing the misapplied use of the violent power of gunpowder . . . in a
very favourable light. And at the same time they were to devote a part of their efforts,
for the glory of the Creator, to discovering a new use for gunpowder, which had been
latent in it from the beginning but had gone unheeded, because all who have hitherto
occupied themselves with its application have stood under the spell of that terrible
misconception that gunpowder was useful only for an idle and wasteful display of its
flashes and flying sparks, or else to wound, to kill, to explode, to burst, to ruin, in short
to unhinge the whole world. It is but too easy to suppose that our researchers, under
the influence of this prejudice, have given either no thought or not sufficiently serious
thought to a useful application of gunpowder, especially as no prospect of an
important or obvious application attracted them in this direction. Only one man has
been found who, with a view to a possible advantageous application, has freed himself
from the above mentioned preconceived opinion … and has honoured the attempt with
his attention, in a French letter of the 24 May 1686.
‘I have received the problem communicated by you concerning a new
application of gunpowder. In my opinion it may certainly be hoped to attain
this end. Seven or eight years ago I showed to Monsieur Colbert an engine
which I had had built for this very purpose, and which was illustrated in the
Proceedings of our Academy.
‘It worked as follows; a tiny quantity of gunpowder, about a thimbleful, was
able to raise some 1,600 lb. five feet high; and not with such violence as is
usual, but with moderate and steady power. Four or five servants, whom
Monsieur Colbert ordered to pull the rope attached to the engine, were quite
easily lifted up into the air. Nevertheless there was a certain difficulty in
constantly reproducing this Power.’
The writer of this letter communicated two quite unusual and really incomparable
discoveries:
1. One or two drachmas of gunpowder, a thimbleful, will raise a weight of 1,600 lb.
five feet; and furthermore,
2. This occurs without the usual violence, but with moderate and steady power.
The first of these discoveries arouses admiration, but is in conformity with principles
already accepted. The effectiveness of the powder could naturally be increased either
by addition of more powder or by improvement of the piston. Nevertheless the second
discovery would appear to extend beyond these known principles, and must be the
Page 200
more highly valued since it approaches the miraculous. Doubtless therefore all those
who are plagued by curiosity to see this simple and really useful experiment could take
the trouble to make such a machine or another which is suitable to impel any weight
chosen as desired. They should allow themselves to be helped therein by men who are
familiar not only with the use and misuse of gunpowder but also with the art of
mechanics. Especially they will need for this, the magnificent work of the late
Monsieur Bondel The Art of Shooting with iron balls filled with powder, a work that
would perhaps be more correctly entitled “The Art of Thoroughly Understanding the
Nature and Characteristics of Natural and Violent Motion”. In it will be found many
demonstrations directed toward the aim here mentioned. And there can no longer be
any doubt that the fact that a tiny quantity of powder can raise 1,600 lb. so high, can,
some day, be put to general use as soon as an inventor turns his attention to solving the
many difficulties, especially those which obstruct the repetition of regular action. It
remains only to add that at this point besides the description and exhibition, a drawing
of the machine itself could very easily have been shown, by means of which its
manner of working could have been shown; did not the ease with which this can be
manifested and understood seem to make this quite superfluous, especially since its
effectiveness has already been more than sufficiently demonstrated. Moreover, gifted
investigators have had sufficient reason to believe positively that the making of such
experiments, all too rare, each of which needs special consideration, may ultimately
lead to a fruitful contribution useful to everyone.
Meantime, the decision rests with God alone. He will according to his merciful
judgment at the right time make it evident that all creation is appointed for the welfare
and service of mankind. It is therefore the duty of man not only to believe this truth,
but to work with all his power that he may use and enjoy everything with
acknowledgement and gratitude. Praised therefore be the most holy name of Him
through whose goodness the first stage of this apparent impossibility (namely the
useful application of gunpowder) has been overcome; Praised, say I, be his name to all
eternity! Amen.
Page 201
Page 202
Appendix 2: A proof from basics
The entirety of what follows in this Appendix is a quote from a letter by Leibniz.
Tentamen anagogicum [“an essay proceeding from the basics to prove something firm”]684
The inquiry into final causes in physics is precisely the application of the method
which I think ought to be used, and those who have sought to banish it from their
philosophy have not adequately considered its usefulness. For I do not wish to do them
the injury of thinking that they have evil designs in doing this. Others followed them,
however, who have abused their position, and who, not content with excluding final
causes from physics but restoring them elsewhere, have tried to destroy them entirely
and to show that the Creator of the universe is most powerful, indeed, but without any
intelligence. There have been still others who have not admitted any universal cause,
like the ancients who recognized nothing in the universe but a concourse of
corpuscles. This seems plausible to those minds in whom the imaginative faculty
predominates, because they believe that they need to use only mathematical principles,
without having any need either for metaphysical principles, which they treat as
illusory, or for principles of the good, which they reduce to human morals; as if
perfection and the good were only a particular result of our thinking and not to be
found in universal nature.
I recognize that it is rather easy to fall into this error, especially when one’s thinking
stops at what imagination alone can supply, namely, at magnitudes and figures and
their modifications. But when one pushes forward his inquiry after reasons, it is found
that the laws of motion cannot be explained through purely geometric principles or by
imagination alone. This is also why some very able philosophers of our day have held
that the laws of motion are purely arbitrary. They are right in this if they take arbitrary
to mean coming from choice and not from geometric necessity, but it is wrong to
extend this concept to mean that laws are entirely indifferent, since it can be shown
that they originate in the wisdom of their Author or in the principle of greatest
perfection, which has led to their choice.
This consideration gives us the true middle term that is needed for satisfying truth as
well as piety. We know that while there have been, on the one hand, able philosophers
who recognized nothing except what is material in the universe, there are, on the other
hand, learned and zealous theologians who, shocked at the corpuscular philosophy of
and not content with checking its misuse, have felt obliged to maintain that there are
phenomena in nature which cannot be explained by mechanical principles; as for
example, light, weight, and elastic force. But since they do not reason with exactness
in this matter, and it is easy for the corpuscular philosophers to reply to them, they
injure religion in trying to render it a service, for they merely confirm those in their
error who recognize only material principles. The true middle term for satisfying both
684 c.1696, Loemker 1969, pp.477-9
Page 203
truth and piety is this: all natural phenomena could be explained mechanically if we
understood them well enough, but the principles of mechanics themselves cannot be
explained geometrically, since they depend on more sublime principles which show
the wisdom of the Author in the order and perfection of his work.
The most beautiful thing about this view seems to me to be that the principle of
perfection is not limited to the general but descends also to the particulars of things
and of phenomena and that in this respect it closely resembles the method of optimal
forms, that is to say, of forms which provide a maximum or minimum, as the case may
be - a method which I have introduced into geometry in addition to the ancient method
of maximal and minimal quantities. For in these forms or figures the optimum is found
not only in the whole but also in each part, and it would not even suffice in the whole
without this. For example, if in the case of the curve of shortest descent between two
given points, we choose any two points on this curve at will, the part of the line
intercepted between them is also necessarily the line of shortest descent with regard to
them. It is in this way that the smallest parts of the universe are ruled in accordance
with the order of greatest perfection; otherwise the whole would not be so ruled. It is
for this reason that I usually say that there are, so to speak, two kingdoms even in
corporeal nature, which interpenetrate without confusing or interfering with each other
- the realm of power, according to which everything can be explained mechanically by
efficient causes when we have sufficiently penetrated into its interior, and the realm of
wisdom, according to which everything can be explained architectonically, so to
speak, or by final causes when we understand its ways sufficiently. In this sense one
can say with Lucretius not only that animals see because they have eyes but also that
eyes have been given them in order to see, though I know that some people, in order
the better to pass as free thinkers, admit only the former. Those who enter into the
details of natural machines, however, must have need of a strong bias to resist the
attractions of their beauty. Even Galen, after learning something about the function of
the parts of animals, was so stirred with admiration that he held that to explain them
was essentially to sing hymns to the honor of divinity. I have often wished that an able
physicist would undertake to prepare a special work whose title - or whose aim at least
- would be The Hymn of Galen.
Page 204
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