my ki essay!!!!!

Name: Loh Ping Shun Subject Code: 9759 Centre/ Index No: 3016/ 3216

Euclidean Geometry: Certain and Objective?

Introduction

Euclidean geometry refers to the branch of mathematics that was first formalized by

Euclid of Alexandria in The Elements.

In this essay, the certainty of Euclidean geometry will be evaluated by the extent to

which it may be considered infallible (incapable of error), incorrigible (incapable of

being revised) or inductively certain. Its objectivity will be evaluated by the extent to

which its acceptance as knowledge is independent of individual minds. This paper

assesses the certainty and objectivity of Euclidean geometry first as a description of

physical reality (or space) and then as a purely formal study of relations. 1 I will argue

that Euclidean Geometry holds inductive certainty and objectivity as a purely formal

study of relations but that it only holds objectivity as a description of physical reality

and only when the measurements involved are small.2

1 This refers to Euclidean geometry when considered only as a study of the logical implications of the axioms without any relation to the physical world. 2 Small measurements usually refer to measurements on the order of the size of the earth. Non-Euclidean geometry applies to astronomic dimensions.

1


Evaluating the certainty and objectivity of Euclidean Geometry as a

description of physical reality

An early account of the nature of Euclidean geometry was given by Descartes. He

argued that Euclidean geometry was a certain and objective description of physical

reality because all the theorems are logically deduced from clear and distinct axioms

about physical reality. For example the first axiom is that it is possible to draw one

straight line from any two points. Such an axiom is simple to understand and strongly

self-evident and thus was considered to be a clear and distinct idea by Descartes.

Descartes argued that self-evident ideas or ideas that could be perceived clearly and

distinctly must be infallible and incorrigible because we cannot help but believe in

them and the presence of an all powerful and good God would not allow us to be

deceived into believing things we cannot help but believe in. Thus his argument is

premised on his belief that a supremely good and powerful God exists to ensure that

we are not deceived into believing what we cannot help but believe.

However Descartes’ argument for the existence of a supremely good God is circular

as he made use of the belief that he has a clear and distinct idea of a supremely

good God to prove that a supremely good God exists only to use this belief to then

justify complete and certain acceptance of the truth of ‘clear and distinct’ ideas. 3

Hence we cannot accept his conclusions regarding the certainty and objectivity of

Euclidean geometry as a description of physical reality because without a sufficient

proof for the existence of such a God, then we have no justification to accept ‘clear

and distinct ideas’ as truth.

3 Rene Descartes, Meditations on First Philosophy with selections from Objections and Replies, ed. by John Cottingham (Cambridge: Cambridge University Press, 1996),102-107.


Another philosopher who held that Euclidean geometry is certain and objective as a

description of physical reality was Immanuel Kant. He argued that this is so because

theorems and proofs follow with logical necessity from a few self-evident axioms the

certainty of which can be guaranteed as they are based on our unchangeable

intuition of space. 4According to Kant, our minds play an active role in structuring

sense data into perceptions through intuitions of the mind or what Kant calls the

forms of sensibility namely space and time. Space and time are not things in

themselves but merely intuitions or frameworks of the mind which we necessarily

impose on all our perceptions. Kant’s evidence for this is that though it is impossible

to perceive or grasp of an object outside of a spatial representation or temporal

representation we cannot ever perceive of space or time in themselves. In other

words, one cannot conceive of an object without perceiving it as occupying a region

of space or as persisting for a period of time. 5

Euclidean geometry according to Kant is merely a formal study of the nature of this

intuition of space which shapes our perceptions and his evidence for this is that we

cannot perceive objects (imaginary or not) without a spatial representation that is

Euclidean in nature.6 Prima facie, this seems to be true; it seems that we cannot

visualize the negations of any axioms or theorems in Euclidean geometry no matter

how hard we try. For example, to use the earlier example given, when we ask

ourselves to imagine if it is possible for more than 2 straight lines to be drawn

4 Immanuel Kant, A Critique of Pure Reason, trans. Werner S.Pluhar (Indianapolis: Hackett Publishing Company, 1996), 235-237.5 Matthew McCormick, “Immanuel Kant (1724-1804) Metaphysics,” The Internet Encyclopaedia of Philosophy, (2006), http://www.iep.utm.edu/k/kantmeta.htm.6 Loc. cit.

http://www.iep.utm.edu/k/kantmeta.htm


through two points, the answer seems to be a definite no. Hence since the axioms

necessarily represent that which we perceive through our intuition of space, then the

theorems which follow deductively from them must be an objective and certain

description of physical reality.

It has to be noted that Kant’s philosophy of geometry argues that Euclidean

geometry holds certainty in a much more limited sense as compared to Descartes’.

Descartes held Euclidean geometry to be both infallible and incorrigible while Kant

holds that Euclidean geometry is certain only to the extent that it is incorrigible but

not infallible. It is incorrigible because according to Kant our intuition of space is

unchangeable and hence produces knowledge that is unchangeable. It is fallible

(capable of error) because we can never perceive the real world as it is without this

intuition of space influencing and shaping our perceptions and hence can gain no

knowledge about the world outside this intuition.

Following the development of non-Euclidean geometry (alternative systems of

geometry developed based on one or more axioms that contradict the axioms of

Euclidean Geometry), formalists such as Gauss have called into question Kant’s

theory that Euclidean geometry is a study of our unchanging intuition of space.7

Formalists argued that the development of non-Euclidean geometry refuted Kant’s

theory since it showed that it was logically possible to conceive of alternatives to

Euclidean geometry. Hence they argued that empirical methods were needed to

determine the structure of space and hence the certainty of Euclidean geometry as a

7 Max Jammer, Concepts of Space, the History of Theories of Space in Physics (New York: Courier Dover Publications, 1993), 148.


description of reality was called into question since it was now dependant on

contingent empirical data for verification. However the mere development of non-

Euclidean geometry in itself is not sufficient to challenge a Kantian theory of

geometry for Kant never claimed that logical alternatives to Euclidean geometry were

impossible but that we could not possibly perceive of objects in a non-Euclidean

representation of space.8

A stronger challenge to Kant would come from Einstein’s theory of general relativity

which affirms that the true nature of spacetime is actually closer to Riemannian

geometry rather than Euclidean. This seems to lead us to reject Euclidean geometry

as a certain and objective description of physical reality. A possible initial Kantian

response may be that Euclidean geometry was meant to apply to space and not

spacetime and thus Einstein’s theory does not challenge its epistemic status at all

but one of the consequences of Einstein’s theory is that space cannot be adequately

described in the absence of time and thus such an objection would be untenable.

However, the ability of the theory of general relativity to refute Euclidean geometry

as a certain description of physical reality or space is questionable because the

structure of space as postulated by Einstein is dependent on the coordinative

definitions used by Einstein which can be changed to give completely different

results.9 Coordinative definitions correlate a particular geometrical entity to

observable physical entities and are necessary to allow for measurements to be

made of the structure or geometry of space or spacetime. However such definitions

8 Lawrence Bonjour, In Defence of Pure Reason, a Rationalist Account of a Priori Justification (Cambridge: Cambridge University Press, 1998), 220-221.9 Hans Reichenbach, The philosophy of Space and Time, trans. by Maria Reichenbach and John Freund (New York: Courier Dover Publications, 1958), 34-35.


are merely conventions and thus lack truth value.10 This is because geometrical

entities are not found in reality and we merely apply observable phenomena or

objects to geometrical models. For example, points or that which have no part do not

exist in reality not to mention the shortest distance between points. Hence the need

to choose coordinative definitions such as that ‘coordinating’ the path of a light ray to

straight lines (or the shortest distance between two points in space) in Einstein’s

theory of general relativity. Hence these definitions lack truth value because they are

a matter of human choice and cannot be true or false just like how a definition of

metre as a certain unit of length is neither true nor false, it just is.

Moreover the adoption of a different set of coordinative definitions could lead to the

conclusion that Euclidean geometry is the true structure of spacetime. This suggests

that since different theories based on different coordinative definitions postulating

different geometries of space are empirically equivalent, Einstein’s theory of relativity

does not pose a significant challenge to Kant’s theory of geometry.11 Even so Kant’s

position on geometry is still untenable as I shall show below.

This is because Kant’s argument that Euclidean geometry is a formal study of our

natural unchangeable intuition of space is premised on the claim that it is impossible

to perceive objects in non-Euclidean dimensions or visualize non-Euclidean

geometry. However Helmholtz has refuted this claim by showing that it is possible to

visualize non-Euclidean geometry entities and to perceive objects with a spatial

10 Ibid., 14-18.11 Lawrence Bonjour, op. cit., 221


representation that is non-Euclidean in nature.12 This implies that our perceptions of

physical reality are not necessarily constrained by a Euclidean intuition of space as

Kant claimed and thus refuting his view that Euclidean geometry is an incorrigible

description of reality.13

It seems then that Euclidean geometry as an infallible or at least incorrigible

description of physical reality is untenable since the Platonist and Kantian views

have both been shown to be flawed. At the same time, empirical evidence cannot

even show that Euclidean geometry is inductively certain as a description of physical

space because of the empirical equivalence of different measurements of the

structure of space depending on the coordinative definitions used. This is because

any measurements will then be in a sense, conventional since the coordinative

definitions on which they are based on are conventions and lack truth value.

Nevertheless it seems that Euclidean geometry may retain its objectivity as a

description of physical reality where measurements involved are small. This is

evident from the extensive use of Euclidean geometry in fields of physics such as

optics and architecture when measurements involved are small which clearly shows

that the acceptance of Euclidean geometry as a physical description of space and

hence the knowledge it provides of the external world is arguably independent of

individual minds and thereby objective.

12 Due to the constraints of this paper, discussion of his proof will not be provided but suffice to say it has been widely accepted by scientists and mathematicians such as Maxwell and Schlick. 13Gary Hatfield, “Spatial Perception and Geometry in Kant and Helmholtz,” Philosophy of Science Association, Volume Two: Symposia and Invited Papers (1984), http://www.jstor.org/stable/192527

http://www.jstor.org/stable/192527


However the objectivity of Euclidean geometry as a description of physical reality

has to be qualified in that it is objectively useful but not objectively ‘true’. This is

because scientists make use of it not because it corresponds to reality, but because

the margins of error caused by curvature of space (if we accept Einstein’s

measurements) are small enough to be negligible thus allowing the use of Euclidean

geometry as an accurate description of reality. To illustrate this point, consider the

use of Euclidean geometry as a description of physical reality where measurements

involved are small in comparison to the use of inexact values of π such as 22/7 for

calculations involving circles. We use these values of π not because they are ‘real’

but because the errors in calculation caused by the use of approximations are small

enough to be disregarded for our purposes. Similarly Euclidean geometry is used

where measurements are small as errors become small enough to be disregarded

and because working with Euclidean geometry would be simpler than Riemannian

geometry in calculations. 14Hence Euclidean geometry is objective as a description of

physical reality where measurements are small to the extent that it is close enough

to the truth to be useful.

14 Walter Prenowitz and Meyer Jordan, Basic Concepts of Geometry (New York : Ardsley House,1989), 116.


Evaluating the certainty and objectivity of Euclidean geometry as a formal

system of relations

In this section of the essay, I shall consider the certainty and objectivity of Euclidean

geometry as a formal study of relations or as the study of the logical implications of

axioms. Knowledge of Euclidean geometry in this sense includes both proofs and

theorems.

Prima facie, it seems that it is clear that Euclidean geometry is certain and objective

as a formal system of relations. This is because theorems are logically deduced from

the axioms and if we assume logical deduction to guarantee the transmission of truth

then it seems that Euclidean geometry as a whole must be certain and objective at

the very least as a formal system of relations. It also seems to be a valid assumption

that logical deduction assures certainty and objectivity in the transmission of truths

from premises to conclusions because it seems impossible to deny a logically

deduced conclusion and accept its premises without contradicting ourselves and it

seems that no new evidence can come to light that would refute the truth of the

conclusion relative to the premise.

However it could be argued that even as a formal system of relations, Euclidean

geometry does not necessarily provide certain knowledge because of the following

reasons.


Firstly truths deduced by reason are subject to human error.15Euclid’s Elements

contained numerous faulty proofs that were shown to be incomplete upon the

scrutiny of later mathematicians. Hilbert’s Foundations of Geometry, an

axiomatization of Euclidean geometry clearly shows the numerous implicit

assumptions (such as the axiom of betweenness) Euclid made use of in his proofs16;

clearly showing how Euclid’s reasoning was faulty. Furthermore, famous

mathematicians such as Descartes and Pascal had full faith in the geometrical

system clearly showing how difficult it was to spot the defects in Euclid’s Elements.

In light of this, perhaps it is not unreasonable to suggest that Hilbert’s Foundations of

Geometry may one day be shown to contain errors as well. Hence Euclidean

geometry even as the formal study of relations is corrigible and fallible.

Secondly, Euclidean geometry as a formal study of relations is ultimately based on

logical deduction yet reason, on which deduction is based on, offers no justification

why we should accept it as a justification for truth.17 We simply accept conclusions

derived through deduction or induction from true premises without justifying the truth

transmitting nature of reasoning. There is in fact no way of justifying reason as a

source of knowledge for any such justification would necessarily make use of reason

of some kind and hence would necessarily be circular. Hence Euclidean geometry as

a formal study of relations is fallible since we cannot know if reasoning is a certain

method of transferring truth from the axioms to the premises.

15 Lawrence Bonjour, op. cit., 110-112.16 Morris Kline, Mathematics: The Loss of Certainty (Oxford: Oxford University Press, 1982), 102-103.17 James Beebe, A Priori Skepticism, http://www.acsu.buffalo.edu/~jbeebe2/APSKEP.pdf

http://www.acsu.buffalo.edu/~jbeebe2/APSKEP.pdf


However, we may argue that even if proofs in Euclidean geometry have many a time

been shown to be in error, geometrical theorems proven by Euclid has withstood the

scrutiny of mathematicians till now and remain. None of the theorems in Euclid’s

Elements have been shown to be invalid though their proofs have been improved. 18

This is possibly because of the ability of empirical evidence to act as a check on the

validity of theorems. The fact that physical space approximates Euclidean space

(where measurements involved are small) means that any theorems wrongly

deduced from the axioms would most likely be quickly shown to be false by empirical

evidence such as drawings (which are used frequently in geometry) and thus swiftly

discarded. Hence due to the longevity of the acceptance of the truth of the theorems,

it is inductively certain that the theorems will continue to be accepted.

In fact while proofs have been improved on, this does not mean they lack certainty

completely, it may be argued that a degree of inductive certainty is nevertheless

present as mathematicians check on each other’s work each time proofs are

published. For example Moore proved that one of Hilbert’s axioms in the

Foundations of Geometry was redundant in 1902. This may explain why Euclid’s

Elements was able to withstand the scrutiny of many mathematicians before Hilbert

and suggests that the proofs in Hilbert’s improved axiomatization while perhaps not

beyond revision, does hold inductive certainty as well, albeit to a lesser degree than

the theorems.

18Clay Mathematics Institute, “Euclid’s Elements,” CMI Annual Report (2004), http://www.claymath.org/library/annual_report/ar2004/04report_featurearticle.pdf


Furthermore, we can certainly hold on to the claim that Euclidean geometry is at

least objective as a formal system of relations. Even if we cannot know for certainty if

any of the theorems or proofs is perfect, mathematicians generally accept the current

state of mathematical knowledge on Euclidean geometry as correct till proven wrong

and hence, the acceptance of theorems and proofs in Euclidean geometry in each

age and time can be said to be independent of individual minds and thus objective.

As a formal system of relations, Euclidean geometry does hold objectivity of a

different nature than as a description of physical reality since in the case of the latter,

Euclidean geometry is only objectively accepted as useful while in the former,

Euclidean geometry is objectively accepted as true as far as we know.


Conclusion

In sum, we have seen that the position that Euclidean geometry is a certain description of

reality is untenable because of its inherent weakness, that is, an assumption that Euclidean

geometry is rooted in our unchangeable intuition of space and not because as is popularly

believed, Einstein’s General Theory of Relativity refuted it. It lacks inductive certainty as

well due to the problem of coordinative definitions as discussed earlier. Nevertheless

Euclidean geometry is still objectively useful (though not objectively true) as a description of

physical reality when measurements involved are small.

Even when Euclidean geometry is considered as a formal study of relations, it only holds

inductive certainty due to the fallibility of human reasoning. However it is objectively true

within each time and age since mathematicians accept the truth of previously demonstrated

theorems and proofs unless proven wrong or imperfect.

(2997 words)


Bibliography

Books

Ayer, Alfred Jules. Language Truth and Logic. London : Penguin, 2001.

Ball, W.W. Rouse. A Short Account of the History of Mathematics. New York: Dover

Publications, 1893.

Berlinski, David. Infinite Ascent: A short history of mathematics. New York : Modern

Library, 2008.

Bonjour, Lawrence. In Defence of Pure Reason, a Rationalist Account of a Priori

Justification. Cambridge: Cambridge University Press 1998.

Carus, Paul. The foundations of mathematics: A Contribution to the philosophy of

Geometry. Chicago: the open court publishing company, 1902.

http://www.archive.org/details/foundmathemat00carurich

Descartes, Rene. Meditations on First Philosophy with selections from Objections

and Replies. Translated and edited by John Cottingham. Cambridge: Cambridge

University Press, 1996.

Devitt, Michael. “Reply to Bonjour” in Contemporary Debates in Epistemology, edited

by Ernest Sosa and Matthias Steup, 118-20. Cambridge, MA: Blackwell Publishers,

2005.

http://www.archive.org/details/foundmathemat00carurich


Ernest, Paul. Social Constructivism as a Philosophy of Mathematics. Albany: State

University of New York Press, 1998.

Einstein, Albert. Geometry and Experience. London: Methuen & Co. Ltd. 1922.

Euclid, Thomas Little Heath, Johann Ludwig Heiberg. The Thirteen Books of Euclid's

Elements. Translated by Thomas Little Heath. New York: Courier Dover

Publications, 1956

Helmholtz, Hermann von. “On the Origins and the Significant of the Axioms of

Geometry.” in Epistemological Writings. Edited by R.S.Cohen and Y.Elkana. Boston :

D. Reidel Publishing Company,1977.

Hilbert, David. The Foundations of Geometry. Illionis: The Open Court Publishing

Company, 1902. http://www.gutenberg.org/etext/17384

Hume, David and Tom L. Beauchamp. An Enquiry Concerning Human

Understanding. New York: Oxford University Press, 1999.

Hume, David. A Treatise of Human Nature. Harmonsworth: Penguin Classics, 1984.

Jammer, Max. Concepts of Space, New York: Courier Dover Publications, 1993

Kant, Immanuel, A Critique of Pure Reason. Translated by Werner S.Pluhar.

Indianapolis : Hackett Publishing Company, 1996.

http://rafflesinstitution.spydus.com.sg/cgi-bin/spydus.exe/ENQ/OPAC/BIBENQ/21719797?AUH_TYPE=B&AUH_NS=1&AUH=BEAUCHAMP%20TOM%20L


–––, Prolegomena to any future metaphysics. Edited by Günter Zöller. Translated by

Peter G. Lucas and Günter Zöller. Oxford: Oxford University Press, 2004.

Kline, Morris. Mathematics: The Loss of Certainty. Oxford: Oxford University Press,

1982.

Lagemaat, Richard Van De. Theory of Knowledge for the IB Diploma. Cambridge:

Cambridge University Press.

Lakatos, Imre. Proofs and Refutations: The Logic of Mathematical Discovery. Edited

by John Worrall and Elie Zahar. New York: Cambridge University Press 1999.

Magnani, Lorenzo. Philosophy and Geometry: Theoretical and Historical Issues.

Boston: Kluwer Academic Publishers, 2001.

Nerling, Graham. The shape of space. New York : Cambridge University Press ,

1994.

Plato. The Republic. Translated by Desmond Lee. London: Penguin 2007.

Prenowitz, Walter, and Meyer Jordan. Basic Concepts in Geometry. New York:

Ardsley House, 1989.


Reichenbach, Hans. The philosophy of Space and Time. Translated by Maria

Reichenbach and John Freund. New York: Courier Dover Publications, 1958.

Russell, Bertrand. The foundations of Geometry. Cambridge: Cambridge University

Press, 1897

Shapiro, Steward. Thinking about Mathematics, the Philosophy of Mathematics. New

York: Oxford University Press, 2000.

Sklar, Lawrene. Space, time and spacetime. Berkeley: University of California, 1974.

Torreti, Roberto. Philosophy of Geometry: From Riemann to Poincare. Boston: D.

Reidel Publishing Company, 1978.


Journals

Bealer, George “A theory of the A Priori” Pacific Philosophical Quarterly, 81 (2000).

http://pantheon.yale.edu/~gb275/TheoryofAPriori.pdf.

Blanchette, Patricia, "The Frege-Hilbert Controversy.", The Stanford Encyclopedia of

Philosophy, (Fall 2008 Edition), edited by Edward N.Zalta,

http://plato.stanford.edu/archives/fall2008/entries/frege-hilbert/.

Carson, Emily. “Kant on the method of mathematics.” Journal of the History of

Philosophy, Volume 37, Number 4 (October 1999),

http://muse.jhu.edu/login?uri=/journals/journal_of_the_history_of_philosophy/

v037/37.4carson.pdf.

Glymour, Clark. “The Epistemology of Geometry.” Noûs, Symposium on Space and

Time, Vol. 11, No. 3 (Sep 1977), http://www.jstor.org/stable/2214764.

Glymour, Clark and Frederick Eberhardt, "Hans Reichenbach." The Stanford

Encyclopaedia of Philosophy, (Fall 2008 Edition), edited by Edward N.Zalta,

http://plato.stanford.edu/archives/fall2008/entries/reichenbach/.

Grier, Michelle, "Kant's Critique of Metaphysics", The Stanford Encyclopaedia of

Philosophy, (Summer 2009 Edition), edited by Edward N.Zalta,

http://plato.stanford.edu/archives/sum2009/entries/kant-metaphysics/.

http://plato.stanford.edu/archives/sum2009/entries/kant-metaphysics/


http://plato.stanford.edu/archives/fall2008/entries/frege-hilbert/

http://pantheon.yale.edu/~gb275/TheoryofAPriori.pdf


Hatfield, Gary. “Spatial Perception and Geometry in Kant and Helmholtz.” Philosophy

of Science Association, Volume Two: Symposia and Invited Papers (1984),

http://www.jstor.org/stable/192527.

Murzi, Mauro. “Hans Reichenbach.”, The Internet Encyclopaedia of Philosophy

(2006), http://www.iep.utm.edu/r/reichenb.htm.

McCormick, Matthew. “Immanuel Kant (1724-1804) Metaphysics.” The Internet

Encyclopaedia of Philosophy, (2006) http://www.iep.utm.edu/k/kantmeta.htm.

Kamlah, Andreas. “Hans Reichehbach’s Relativity of Geometry.” Synthese, Hans

Reichenbach, Logical Empiricist, Part III, Vol. 34, No. 3 (Mar 1977),

http://www.jstor.org/stable/20115162.

Skirry, Justin. “Descartes , An overview.” The Internet Encyclopaedia of Philosophy,

(2008), http://www.iep.utm.edu/d/descarte.htm#SH6a.

Torretti, Roberto, "Nineteenth Century Geometry." The Stanford Encyclopaedia of

Philosophy, (Spring 2009 Edition), edited by Edward

N.Zalta, http://plato.stanford.edu/archives/spr2009/entries/geometry-19th/.

http://plato.stanford.edu/archives/spr2009/entries/geometry-19th/

http://www.iep.utm.edu/d/descarte.htm#SH6a


http://www.iep.utm.edu/k/kantmeta.htm

http://www.iep.utm.edu/r/reichenb.htm



Papers

Beebe, James “A Priori Skepticism.”


Websites

Clay Mathematics Institute, “Euclid’s Elements.” CMI annual report, 2004.

http://www.claymath.org/library/annual_report/ar2004/04report_featurearticle.pdf

“Euclidean Geometry.” http://math.youngzones.org/Non-Egeometry/index.html

Kiran S. Kedlaya. “Geometry Unbound”

http://math.mit.edu/~kedlaya/geometryunbound

Thomas McFarlan. “Kant and Mathematical knowledge”, Centre for Integral Science,

http://www.integralscience.org/sacredscience/SS_kant.html

http://math.mit.edu/~kedlaya/geometryunbound

http://math.youngzones.org/Non-Egeometry/index.html

http://www.claymath.org/library/annual_report/ar2004/04report_featurearticle.pdf


my ki essay!!!!!

Documents