theory of science - mdh · theory of science information computationinformation, ... versa, th i i...

CDT403 Research Methodology in Natural Sciences and Engineering

Theory of ScienceINFORMATION COMPUTATIONINFORMATION, COMPUTATION,

KNOWLEDGE AND SCIENCE

Gordana Dodig-Crnkovic

School of Innovation, Design and Engineering

Mälardalen University 1

Science is (a well formed) knowledge structureScience is (a well formed) knowledge structure.

Knowledge is (a well formed) informationKnowledge is (a well formed) information structure.

Information is (a well formed) data structure.

All f th th lt f i t ti ithAll of them are the result of our interaction with the real world, with the universe.

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Science: Big PicturegScience and the UniverseThe Mytho-Poetic UniverseThe Medieval Divine Geocentric UniverseThe Clockwork (Mechanistic) UniverseThe Clockwork (Mechanistic) UniverseThe Computational Universe

Info-ComputationalismInformationComputationNatural computation – beyond Turing ModelNaturalist Understanding of CognitionInfo-Computational Knowlede Generation

Science, Knowledge, Truth and MeaningScienceScienceScientific MethodKnowledge and its productionJustification and TruthProof and Axiomatic systems Gödel theoremsProof and Axiomatic systems, Gödel theoremsMeaning

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Science: Big Picture Firstg

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Science and the Universe

htt // t b / t h? i43bN lKQN0http://www.youtube.com/watch?v=i43bNclKQN0MACROCOSMOS TO MICROCOSMOS.wmv

Similar:

http://www.youtube.com/watch?v=dAzJy_DECyQ&feature=relatedCosmic Super ZoomCosmic Super Zoom

http://www.youtube.com/watch?v=Vs5doooe2VY&feature=related From Milky Way to Quarks

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The Idea of UniverseThe Idea of Universe

The universe is an idea deeply rooted in our human culture, diff t i diff t l d d i diff t h Atdifferent in different places and during different epochs. At one time, it was a living organism (Tree of Life, Mother Earth), at yet another time, mechanical machinery - the Cartesian-Newtonian clockwork. Today’s metaphor for the universe is y pmore and more explicitly becoming a computer.

(Dodig Crnkovic 2006)

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The Idea of UniverseThe Idea of Universe

The universe is commonly defined as the totality of everything that exists, including all physical matter and energy, the planets, stars, galaxies, and the contents of intergalactic space, although this usage may differ with the context (see definitions, below). The term universe may be used in slightly different contextual senses, denoting such concepts as the cosmos the world or nature (Wikipedia Universe)cosmos, the world, or nature. (Wikipedia, Universe)

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Universe as RealityUniverse as Reality

The universe is defined as everything that exists According toThe universe is defined as everything that exists. According to this definition and our present understanding, the universe consists of three elements:

• space-timep• matter-energy • physical laws that govern the relationships between the two.

Those three elements correspond roughly to the ideas of Aristotle. In his book The Physics Aristotle divided everything that exists into three elements: matter (the stuff of which the

i i d ) f (th t f th t tt iuniverse is made), form (the arrangement of that matter in space) and change (how matter is created, destroyed or altered in its properties, and similarly, how form is altered). Physical laws were conceived as the rules governing thePhysical laws were conceived as the rules governing the properties of matter, form and their changes.

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Universe as RealityUniverse as Reality

Later philosophers such as Averroes and Spinoza discern twoLater philosophers such as Averroes and Spinoza discern two basic elements:

• the passive elements the fabric of the universe (naturathe passive elements, the fabric of the universe (natura naturata)

• the active principles governing the universe acting on the former elements (natura naturans)

Info-Computational Universe (Dodig Crnkovic 2006):• Information as structure• Computation as change

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Comment: Construction of KnowledgeComment: Construction of Knowledge

Our knowledge depends on our possibilities to interact with theOur knowledge depends on our possibilities to interact with the world – the nature and the humans as a part of natural world.

If we use scientific instruments such as microscopes telescopesIf we use scientific instruments, such as microscopes, telescopes or particle accelerators, our knowledge will be much more far reaching than if we only use our human bodily sensory organs.

We construct knowledge from pieces of information we get directly from the world or indirectly via other people (again either exchanging information personally or even moreeither exchanging information personally or even more indirectly from the information found in diverse kinds of documents.) (See Dodig-Crnkovic, Constructivist Research and Info-Computational Knowledge Generation http://www mrtc mdh se/ gdc/work/MBR09ConstructiveResearch pdf )Generation, http://www.mrtc.mdh.se/~gdc/work/MBR09ConstructiveResearch.pdf )

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The Mytho Poetic UniverseThe Mytho-Poetic Universe

In ancient Egypt the dome of the sky was represented by the dd N t Sh th i ht k d th th d Rgoddess Nut, She was the night sky, and the sun, the god Ra,

was born from her every morning. 11

The Medieval Geocentric Universe

From Aristotle Libri de caelo (1519). 12

The Clockwork (Mechanistic) Universe

The mechanicistic paradigm which systematically revealed physical structure in analogy with the artificial. The self-functioning automaton

- basis and canon of the form of the Universebasis and canon of the form of the Universe. Newton Philosophiae Naturalis Principia Matematica, 1687

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The Computational Universe

We are all living inside a gigantic t N t Th M t i thcomputer. No, not The Matrix: the

Universe.

Every process every change thatEvery process, every change that takes place in the Universe, may be considered as a kind of computation.

E Fredkin, S Wolfram, G Chaitin

The universe is on a fundamental level an info-computational phenomenon. GDC

http://www.nature.com/nsu/020527/020527-16.html 14

The Computational Universe

Konrad Zuse was the first to suggest (in 1967) that the physical b h i f th ti i i b i t d b i l lbehavior of the entire universe is being computed on a basic level, possibly on cellular automata, by the universe itself which he referred to as "Rechnender Raum" or Computing Space/Cosmos.

Computationalists: Zuse, Wiener, Fredkin, Wolfram, Chaitin, Lloyd, Seife, 't Hooft, Deutsch, Tegmark, Schmidhuber, Weizsäcker, Wheeler..Wheeler..

Pancomputationalism http://www.idt.mdh.se/personal/gdc/work/Pancomputationalism.mhtp p g p

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Info-ComputationalismInformation and computation are two interrelated and mutually defining phenomena – there is no computation without information (computation understood as information processing), and vice

th i i f ti ith t t ti ( ll i f tiversa, there is no information without computation (all information is a result of computational processes).

Being interconnected information is studied as a structure whileBeing interconnected, information is studied as a structure, while computation presents a process on an informational structure. In order to learn about foundations of information, we must also study computation.p

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Information

A special issue of the Journal of Logic, Language and Information (Volume 12 No 4 2003) dedicated to the different facets of informationdedicated to the different facets of information.

A Handbook on the Philosophy of Information (Van Benthem, Adriaans) is in preparation as one volume Handbook of the philosophy of science. http://www.illc.uva.nl/HPI/

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ComputationThe Computing Universe: Pancomputationalism

Computation is generally defined as information processing.(S B i M S R i Al ith S i M h(See Burgin, M., Super-Recursive Algorithms, Springer Monographs in Computer Science, 2005)

For different views see e.g.http://people.pwf.cam.ac.uk/mds26/cogsci/program.html Computation and Cognitive Science 7–8 July 2008, King's College Cambridge

The definition of computation is widely debated, and an entire issue of the journal Minds and Machines (1994, 4, 4) was devoted to the question “What is Computation?” Even: Theoretical Computer Science 317 (2004)

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Present Model of Computation: Turing MachinePresent Model of Computation: Turing Machine

............Tape

Read-Write headControl Unit

Tape

1. Reads a symbol2. Writes a symbol3. Moves Left or Right

http://plato.stanford.edu/entries/turing-machine/

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Computing Nature and Nat re Inspired Comp tationNature Inspired Computation

Natural computation includes computation that occurs in nature or is inspired by nature. Computing

Inspired by nature: •Evolutionary computation •Neural networks •Artificial immune systems

In 1623, Galileo in his book The Assayer - Il Saggiatore, claimed that the language of nature's book is mathematics and that the way to understand nature is

y•Swarm intelligence

Simulation and emulation of nature:F t l tmathematics and that the way to understand nature is

through mathematics. Generalizing ”mathematics” to ”computation” we may agree with Galileo – the great book of nature is an e-book!

•Fractal geometry •Artificial life

Computing with natural materials: •DNA computing p g•Quantum computing

Journals: Natural Computing and IEEE Transactions on Evolutionary Computation. 20

Turing Machines Limitations –Self-Generating Living Systems

Complex biological systems must be modeled as self-referential, self-organizing "component-systems" (George Kampis) which are selfsystems (George Kampis) which are self-generating and whose behavior, though computational in a general sense, goes far beyond Turing machine model.

“a component system is a computer which, when executing its operations (software) builds a new hardware.... [W]e have a computer that re-wires itself in a(software) builds a new hardware.... [W]e have a computer that re wires itself in a hardware-software interplay: the hardware defines the software and the software defines new hardware. Then the circle starts again.” (Kampis, p. 223 Self-Modifying Systems in Biology and Cognitive Science)

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Beyond Turing MachinesEver since Turing proposed his machine model which identifies computation with the execution of an algorithm, there have been questions about how widely the Turing Machine (TM) model isquestions about how widely the Turing Machine (TM) model is applicable.

With the advent of computer networks, which are the main paradigm p , p gof computing today, the model of a computer in isolation, represented by a Universal Turing Machine, has become insufficient.

The basic difference between an isolated computing box and a network of computational processes (nature itself understood as a computational mechanism) is the interactivity of computation. The most general computational paradigm today is interactive computingmost general computational paradigm today is interactive computing(Wegner, Goldin).

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Beyond Turing Machines

The challenge to deal with computability in the real world (suchas computing on continuous data, biological computing/organicas computing on continuous data, biological computing/organiccomputing, quantum computing, or generally natural computing) has brought new understanding of computation.

Natural computing has different criteria for success of a computation, halting problem is not a central issue, but insteadthe adequacy of the computational response in a network ofthe adequacy of the computational response in a network of interacting computational processes/devices. In many areas, wehave to computationally model emergence not being clearlyalgorithmic. (Barry Cooper)

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Correspondence Principle picture after Stuart A. Umpleby

http://www.gwu.edu/~umpleby/recent_papers/2004_what_i_learned_from_heinz_von_foerster_figures_by_umpleby.htm

TMNatural Computation

TM

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Info Computationalism Applied:Info-Computationalism Applied:Naturalizing Epistemology

(Understanding knowledge as a result of natural processes)

Naturalized epistemology (Feldman, Kornblith, Stich) is, in general, an idea that knowledge may be studied as a natural phenomenon --that the subject matter of epistemology is not our concept of knowledge, but the knowledge itself.

“The stimulation of his sensory receptors is all the evidence anybody h h d t lti t l i i i t hi i t f th ld Whhas had to go on, ultimately, in arriving at his picture of the world. Why not just see how this construction really proceeds? Why not settle for psychology? “("Epistemology Naturalized", Quine 1969; emphasis mine)mine)

I will re-phrase the question to be: Why not settle for computing?

Epistemology is the branch of philosophy that studies the nature, methods, limitations, and validity of knowledge and belief.

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Naturalist Understanding of Cognition

According to Maturana and Varela (1980) even the simplest organisms possess cognition and their meaning productionorganisms possess cognition and their meaning-production apparatus is contained in their metabolism. Of course, there are also non-metabolic interactions with the environment, such as locomotion, that also generates meaning for an organism bylocomotion, that also generates meaning for an organism by changing its environment and providing new input data.

Maturana’s and Varelas’ understanding that all living organismsMaturana s and Varelas understanding that all living organisms posess some cognition, in some degree. is most suitable as the basis for a computationalist account of the naturalized evolutionary epistemologyevolutionary epistemology.

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Info-Computational Account of Knowledge Generation

Natural computing as a new paradigm of computing goes beyond the Turing Machine model and applies

g

to all physical processes including those going on in our brains.

The next great change in computer science andThe next great change in computer science and information technology will come from mimicking the techniques by which biological organisms process information.

To do this computer scientists must draw on expertise in subjects not usually associated with their field, including organic chemistry, molecular , g g y,biology, bioengineering, and smart materials.

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Info-Computational Account of Knowledge Generation

At the physical level, living beings are open complex computational systems in a regime on the edge of chaos, p y g g ,characterized by maximal informational content. Complexity is found between orderly systems with high information compressibility and low information content and random systems with low compressibility and high information content. (Flake)

The essential feature of cognizing living organisms is their ability g g g g yto manage complexity, and to handle complicated environmental conditions with a variety of responses which are results of adaptation, variation, selection, learning, and/or reasoning. (Gell-Mann)

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Cognition as Restructuring of an Agent in Interaction with the Environment

As a result of evolution, increasingly complex living organisms arise that are able to survive and adapt to their environment It means theythat are able to survive and adapt to their environment. It means they are able to register inputs (data) from the environment, to structure those into information, and in more developed organisms into knowledge. The evolutionary advantage of using structured,knowledge. The evolutionary advantage of using structured, component-based approaches is improving response-time and efficiency of cognitive processes of an organism.

The Dual network model, suggested by Goertzel for modeling cognition in a living organism describes mind in terms of two superposed networks: a self-organizing associative memory network,superposed networks: a self organizing associative memory network, and a perceptual-motor process hierarchy, with the multi-level logic of a flexible command structure. 29

Cognition as Restructuring of an Agent in Interaction with the Environment

Naturalized knowledge generation acknowledges the body as our basic cognitive instrument. All cognition is embodied cognition, in bothbasic cognitive instrument. All cognition is embodied cognition, in both microorganisms and humans (Gärdenfors, Stuart). In more complex cognitive agents, knowledge is built upon not only reasoning about input information, but also on intentional choices, dependent on value

t t d d i d i tsystems stored and organized in agents memory.

It is not surprising that present day interest in knowledge generation places information and computation (communication) in focus asplaces information and computation (communication) in focus, as information and its processing are essential structural and dynamic elements which characterize structuring of input data (data →information → knowledge) by an interactive computational process going on in the agent during the adaptive interplay with the environment. 30

Natural Computing in Living Agents

- Agent-centered (information and computation is in the agent)- Agent is a cognizing biological organism or an intelligent machine or both- Interaction with the physical world and other agents is essential- Kind of physicalism with information as aKind of physicalism with information as a stuff of the universe- Agents are parts of different cognitive communities

S lf i ti- Self-organization- Circularity (recursiveness) is central for biological organisms

http://www.conscious-robots.com 31

What is computation? How does nature compute? Learning from Nature *compute? Learning from Nature

“It always bothers me that, according to the laws as we understand th t d it t k ti hi i fi it b fthem today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time …

So I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed and the laws will turn out to be simple like the chequerbe revealed, and the laws will turn out to be simple, like the chequer board with all its apparent complexities.”

Richard Feynman “The Character of Physical Law”

* 2008 Midwest NKS Conference, Fri Oct 31 - Sun Nov 2, 2008Indiana University — Bloomington, IN 32

Paradigm Shift

• Information/Computation• Discrete/Continuum• Natural interactive computing beyond Turing limit• Complex dynamic systems• EmergencyEmergency• Logic• Philosophy

H t i ( t t i )• Human-centric (agent-centric)• Circularity and self-reflection• Ethics returns to researchers agenda

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Info-Computational Paradigm ofInfo-Computational Paradigm of Knowledge

• Understanding of info-computational mechanisms and processesand their relationship to life and knowledge

• Argument for evolution of biological life cognition and intelligence• Argument for evolution of biological life, cognition and intelligence

• Development of new unconventional computational methods

• Learning from nature about optimizing solutions with limitedresources (Organic Computing)

• Providing a unified platform (framework) for specialist sciences to communicate and create holistic (multi-disciplinary/inter-co u cate a d c eate o st c ( u t d sc p a y/ tedisciplinary/transdisciplinary) views

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A mathematical analysis of the scientific method, the axiomatic method, and

Darwin's theory of evolution

G J Chaitin IBM Research

htt // i d / h iti / f j ht l

G. J. Chaitin, IBM Research

http://www.umcs.maine.edu/~chaitin/ufrj.html

http://www.cs.auckland.ac.nz/~chaitin/ufrj.html

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Chaitin’s work on Epistemology, Information Th d M t th ti i t t fTheory, and Metamathematics important for understanding of Formal Systems and their Relationship with Biology

http://www.umcs.maine.edu/~chaitin/ecap.pdf Epistemology

Relationship with Biology

http://www.umcs.maine.edu/ chaitin/ecap.pdf Epistemology as Information Theory: From Leibniz to Ω

http://www.umcs.maine.edu/~chaitin/mjm.pdf The Halting Probability O I d ibl C l it i P M th tiOmega: Irreducible Complexity in Pure Mathematics

http://www.umcs.maine.edu/~chaitin/unm.html Randomness in Arithmetic and the Decline & Fall of Reductionism in Pure Mathematicsand the Decline & Fall of Reductionism in Pure Mathematics

http://www.umcs.maine.edu/~chaitin/hu.html The Search for the Perfect Languageg g

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Despite the fact that there can be no TOE (Theory Of Everything) for pure mathematics as Hilbert hoped mathematicians remain enamored withhoped, mathematicians remain enamored with formal proof.

S th i l i f l f f th AMSSee the special issue on formal proof of the AMS Notices, December 2008 (From Chaitin’s lectures)http://www.ams.org/notices/200811/index.htmlp g

David Malone, Dangerous Knowledge, BBC TV, 90 minutes Google video vividly illustrates the search forminutes, Google video vividly illustrates the search for TOE in mathematicshttp://video.google.com/videoplay?docid=-5122859998068380459#

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http://www.umcs.maine.edu/~chaitin/jack.htmlMathematics, Biology and Metabiology

http://www.umcs.maine.edu/~chaitin/ev.htmlEvolution of Mutating Software

htt // i d / h iti / ht lhttp://www.umcs.maine.edu/~chaitin/mex.htmlSpeculations on biology, information and complexity

http://www scottaaronson com/writings/bignumbers htmlhttp://www.scottaaronson.com/writings/bignumbers.htmlWho Can Name the Bigger Number?

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Two Books On Universe as Quantum Information

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Science, Knowledge, Truth and Meaning

Critical thinkingWhat is science?What is scientific method?What is knowledge?Information and knowledgeTruth and meaninggLimits of formal systemsScience as learning processInfo-computational view of knowledge productionInfo computational view of knowledge productionComplexity

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Red Thread: Critical Thinking

“Reserve your right to think, y gfor even to think wrongly is better than not to think at all.”

Hypatia, natural philosopher and mathematician

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Haiku – Like Highlights

.遠山が目玉にうつるとんぼ哉

tôyama ga medamatôyama ga medama ni utsuru tombo kana

the distant mountainthe distant mountainreflected in his eyes...dragonfly

Kobayashi Issa (1763-1827)

(Haiku form: 5-7-5 syllables)

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What is Science?

EyeMaurits Cornelis Escher We can see Science from different perspectives…43

Definitions by Goal (Result) and Process (1)

science from Latin scientia, scire to know; 1: a department of systematized knowledge as an

object of study2: knowledge or a system of knowledge covering2: knowledge or a system of knowledge covering

general truths or the operation of general laws especially as obtained and tested through scientific

th dmethod

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Definitions by Goal (Result) and Process (2)

3: such knowledge or such a system of knowledge concerned with the physical world and its phenomena : natural science: natural science

4: a system or method reconciling practical ends with scientific laws <engineering is both a science and an g gart>

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Science: Definitions by Contrast

To do science is to search for repeated patterns, not simply to accumulate facts.

R b t H M A thRobert H. MacArthur

Religion is a culture of faith; science is a culture of doubtReligion is a culture of faith; science is a culture of doubt. Richard Feynman

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Empirical approach. Wh t S i th ?What Sciences are there?Dewey Decimal Classification®http://www.geocities.com/Athens/Troy/8866/15urls.html

000 - & Psychology200 - ReliComputers, Information & General Reference100 Phil h i100 - Philosophy gion300 - Social sciences400 - Language500 - Science500 - Science600 - Technology700 - Arts & Recreation800 - Literature800 Literature900 - History & Geography

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Dewey Decimal Classification®

500 – Science510 Mathematics520 Astronomyy530 Physics540 Chemistry550 Earth Sciences & Geology550 Earth Sciences & Geology560 Fossils & Prehistoric Life570 Biology & Life Sciences580 Plants (Botany)580 Plants (Botany)590 Animals (Zoology)

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Language Based SchemeClassical Sciences in their Cultural Context –

Logic

&

Mathematics

Culture

Natural Sciences(Physics,

Mathematics

1

Culture(Religion, Art, …)

5

yChemistry,Biology, …)

2

Social SciencesSocial Sciences(Economics, Sociology,

Anthropology, …)3

The Humanities(Phil h Hi(Philosophy, History,

Linguistics …)4 49

Understanding what science isUnderstanding what science is by understanding what scientists do

"Scientists are people of very dissimilar temperaments doing different things in very different ways. Among scientists are collectors classifiers and compulsiveAmong scientists are collectors, classifiers and compulsive tidiers-up; many are detectives by temperament and many are explorers; some are artists and others artisans. There are poet-scientists and philosopher-scientists and even a few mystics."

Peter Medawar, Pluto's Republic

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The Classical (Ideal) Model of Science

The Classical Model of Science is a system S of propositions and concepts satisfying the following conditions:

• All propositions and all concepts (or terms) of S concern a specific set of objects or are about a certain domain of being(s).

• There are in S a number of so-called fundamental concepts (or terms).

• All other concepts (or terms) occurring in S are composed of (or are definable from) these fundamental concepts (or terms).

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• There are in S a number of so-called fundamental propositions.

• All other propositions of S follow from or are grounded in (or are• All other propositions of S follow from or are grounded in (or are provable or demonstrable from) these fundamental propositions.

All propositions of S are true• All propositions of S are true.

• All propositions of S are universal and necessary in some sense or another.

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• All concepts or terms of S are adequately known. A non-fundamental concept is adequately known through its composition (or definition).

• The Classical Model of Science is a reconstruction a posteriori and sums up the historical philosopher’s ideal of scientificand sums up the historical philosopher s ideal of scientific explanation.

• The fundamental is that “All propositions and all concepts (or• The fundamental is that All propositions and all concepts (or terms) of S concern a specific set of objects or are about a certain domain of being(s).”

Betti A & De Jong W. R., Guest Editors, The Classical Model of Science I: A Millennia-Old Model of Scientific Rationality, Forthcoming in Synthese, Special Issue

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Science defined by its Method

Socratic Method Scientific Method

1. Wonder. Pose a question(of the “What is X ?” form).

1. Wonder. Pose a question.(Formulate a problem).(of the What is X ? form). (Formulate a problem).

2. Hypothesis. Suggest a plausible answer (a definition or definiens) from which some conceptually testable hypothetical propositions can be deduced.

2. Hypothesis. Suggest a plausible answer (a theory) from which some empirically testable hypothetical propositions can be deduced.

yp p p

3. Elenchus ; “testing,” “refutation,” or “cross-examination.” Perform a thought experiment by imagining a case which conforms to the definiens but clearly fails

3. Testing. Construct and perform an experiment, which makes it possible to observe whether the consequences specified in one or more of those hypothetical y

to exemplify the definiendum, or vice versa. Such cases, if successful, are called counterexamples. If a counterexample is generated, return to step 2 otherwise go to step 4

yppropositions actually follow when the conditions specified in the same proposition(s) pertain. If the test fails, return to step 2, otherwise go to step 4.

step 2, otherwise go to step 4.

4. Accept the hypothesis as provisionally true. Return to step 3 if you can conceive any other case which may show the answer to be defective

4. Accept the hypothesis as provisionally true. Return to step 3 if there are predictable consequences of the theory which have not been experimentally confirmedanswer to be defective. been experimentally confirmed.

5. Act accordingly. 5. Act accordingly. 54

The Scientific Method

EXISTING THEORIESAND OBSERVATIONS

1 Hypotesen måste

PREDICTIONS

3HYPOTHESIS

2

Hypothesis Hypothesis

must be

SELECTION AMONG

justeras

TESTS AND NEW

ypmust be

redefined

must be adjusted

The hypotetico deductive cycle

COMPETING THEORIES

6

TESTS AND NEW OBSERVATIONS

4

Consistency achievedThe hypotetico-deductive cycle

EXISTING THEORY CONFIRMED

(within a new context) or

NEW THEORY PUBLISHED

5The scientific-community cycle

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The Scientific MethodFormulating Research Questions and Hypotheses

Different approaches:

Intuition – (Educated) GuessAnalogySymmetryParadigmMetaphorMetaphorand many more ..

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The Scientific MethodCriteria to Evaluate Theories

When there are several rivaling hypotheses number of criteria can be used for choosing a best theory.

Following can be evaluated:

– Theoretical scope– Theoretical scope– Heuristic value (heuristic: rule-of-thumb or argument

derived from experience)Parsimony (simplicity Ockham’s razor)– Parsimony (simplicity, Ockham s razor)

– Esthetics– Etc.

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The Scientific MethodCriteria which Good Scientific Theory Shall Fulfill

Logically consistent– Logically consistent– Consistent with accepted facts – Testable– Consistent with related theories– Interpretable: explain and predict– Parsimonious– Pleasing to the mind (Esthetic, Beautiful)– Useful (Relevant/Applicable)

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Ockham’s Razor (Occam’s Razor)The Scientific Method

Ockham s Razor (Occam s Razor)(Law Of Economy, Or Law Of Parsimony, Less Is More!)

A philosophical statement developed by William of Ockham, (1285–1347/49) a scholastic that Pluralitas non est ponenda(1285–1347/49), a scholastic, that Pluralitas non est ponenda sine necessitate; “Plurality should not be assumed without necessity.”

The principle gives precedence to simplicity; of two competing theories, the simplest explanation of an entity is to be preferred.

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MAP OF SCIENCEScience as a result of Scientific Community

MAP OF SCIENCE

http://www.lanl.gov/news/albums/science/PLOSMapOfScience.jpg

Thi "M f S i " ill t tThis "Map of Science" illustrates the online behavior of scientists accessing different scientific journals, publications, aggregators, t C l t th i tifietc. Colors represent the scientific

discipline of each journal, based on disciplines

http://www.lanl.gov/news/index.php/fuseaction/nb.story/story_id/%2015965 60

Knowledgeg

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What is Knowledge?What is Knowledge?Plato´s Definition

Plato believed that we learn in this life by remembering knowledge originally acquired in a previous life, and that the soul already has knowledge, and we learn by recollecting what in fact the soul already knows.

[At present we know that we inherit some physical preconditions,[At present we know that we inherit some physical preconditions, structures and abilities already at birth. In a sense those structures of our brains and bodies may be seen as the result of evolution, so in a sense they encapsulate memories of the historical development of our bodies.]

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What is Knowledge?What is Knowledge?Plato´s Definition

Plato offers three analyses of knowledge, [dialogues Theaetetus 201 and Meno 98] all of which Socrates rejects.

Plato's third definition:" Knowledge is justified, true belief. "

The problem with this concerns the word “justified”. All interpretations of “justified” are deemed inadequate.

Edmund Gettier, in the paper called "Is Justified True Belief Knowledge?“ argues that knowledge is not the same as justified t b li f (G i P bl )true belief. (Gettier Problem)

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What is Knowledge?What is Knowledge?Descartes´ Definition

"Intuition is the undoubting conception of an unclouded and attentive mind, and springs from the light of reasons alone; it is more certain than deduction itself in that it is simpler."

“Deduction by which we understand all necessary inference from other facts that are known with certainty,“ leads to knowledgeother facts that are known with certainty, leads to knowledge when recommended method is being followed.

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What is Knowledge?What is Knowledge?Descartes´ Definition

"Intuitions provide the ultimate grounds for logical deductions. Ultimate first principles must be known through intuition while deduction logically derives conclusions from them.

These two methods [intuition and deduction] are the most certain routes to knowledge, and the mind should admit no others."routes to knowledge, and the mind should admit no others.

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What is Knowledge?

– Propositional knowledge: knowledge that such-and-such is the case.

– Non-propositional knowledge (tacit knowledge): the knowing how to do something.

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Sources of Knowledge

– A Priori Knowledge (built in, developed by evolution and inheritance) (resides the brain as memory)) ( y)

– Perception (“on-line input”, information acquisition)

– Reasoning (information processing)

– Testimony (network, communication)

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Blurring the Boundary BetweenBlurring the Boundary Between Perception and Memory

http://www.scientificamerican.com/article.cfm?id=perception-and-memory

http://www.sciencedaily.com68

Blue Brain vs Map of Science

http://online.wsj.com/article/SB124751881557234725.htmlIn Search for Intelligence, a Silicon Brain Twitches Sea c o te ge ce, a S co a tc es

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Computational BrainComputational BrainBrain Processing Information

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Cell Processing Information

http://www.youtube.com/watch?v=NJxobgkPEAo&feature=relatedFrom RNA to Protein Synthesis http://www.youtube.com/watch?v=3aVT2DTbtA8&feature=relatedReplication, Transcription, and Translation http://www.goldenswamp.com/page/2/

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Knowledge and Justification

Knowledge and Objectivity: Observations

Observations are always interpreted in the context of an a priori knowledge (Kuhn Popper)knowledge. (Kuhn, Popper)

“What a man sees depends both upon what he looks at and also upon what his previous visual conceptual experience has taughtupon what his previous visual-conceptual experience has taught him to see”.

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Knowledge and ObjectivityKnowledge and ObjectivityObservations

– All observation is potentially ”contaminated”, whether by our theories, our worldview or our past experiences.

– It does not mean that science cannot ”objectively” [inter-subjectivity] choose from among rival theories on the basis of empirical testing.empirical testing.

– Although science cannot provide one with hundred percent certainty yet it is the most if not the only objective mode ofcertainty, yet it is the most, if not the only, objective mode of pursuing knowledge.

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Perception and “Direct Observation”

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"Reality is merely an illusion, lb it i t t "albeit a very persistent one." -

Einstein

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Checker-shadow illusionhttp://web.mit.edu/persci/people/adelson/checkershadow_illusion.html

See even:See even:http://web.mit.edu/persci/gaz/gaz-teaching/index.htmlhttp://persci.mit.edu/people/adelson/publications/gazzan.dir/gazzan.htmLightness Perception and Lightness IllusionsLightness Perception and Lightness Illusions

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Direct Observation?!

A t i t f t hi h lit t i t t l t llAn atom interferometer, which splits an atom into separate wavelets, can allow the measurement of forces acting on the atom. Shown here is the laser system used to coherently divide, redirect, and recombine atomic wave packets (Yale University).

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Direct Observation?!

Electronic signatures produced by collisions of protons and antiprotons in th T t l tthe Tevatron accelerator at Fermilab provided evidence that the elusive subatomic particle known pas top quark has been found.

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Knowledge Justification

– Foundationalism (uses architectural metaphor to describe the structure of our belief systems. The superstructure of a belief system inherits its justification from a certain subset of beliefs –all rests on basic beliefs.)

– Coherentism

– Internalism (a person has “cognitive grasp”) and Externalism (external justification)(e te a just cat o )

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Truth (1)

– The correspondence theory

– The coherence theory

– The deflationary theory

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Truth (2)Truth (2)The Correspondence Theory

A common intuition is that when I say something true, my statement corresponds to the facts.

But: how do we recognize facts and what kind of relation is this correspondence?

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Truth (3)Truth (3)The Coherence Theory

Statements in the theory are believed to be true because being compatible with other statements.

The truth of a sentence just consists in its belonging to a system of coherent statements.

The most well-known adherents to such a theory was Spinoza (1632-77), Leibniz (1646-1716) and Hegel (1770-1831).

Ch t i ti ll th ll b li d th t t th b t th ld ldCharacteristically they all believed that truths about the world could be found by pure thinking, they were rationalists and idealists. Mathematics was the paradigm for a real science; it was thought that the axiomatic method in mathematics could be

d i ll iused in all sciences.

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Truth (4)Truth (4)The Deflationary Theory

The deflationary theory is belief that it is always logically superfluous to claim that a proposition is true, since this claim adds nothing further to a simple affirmation of the proposition itself.

"It is true that birds are warm-blooded " means the same thing asIt is true that birds are warm blooded means the same thing as "birds are warm-blooded ".

For the deflationist truth has no nature beyond what is captured inFor the deflationist, truth has no nature beyond what is captured in ordinary claims such as that ‘snow is white’ is true just in case snow is white.

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Truth (5)Truth (5)The Deflationary Theory

The Deflationary Theory is also called the redundancy th th di th th t th ththeory, the disappearance theory, the no-truth theory, the disquotational theory, and the minimalist theory .

63-70 see Lars-Göran Johansson63 70 see Lars Göran Johanssonhttp://www.filosofi.uu.se/utbildning/Externt/slu/slultexttruth.htmand Stanford Encyclopedia of Philosophyhttp://plato.stanford.edu/entries/truth-deflationary/

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Truth and Reality

Noumenon,"Ding an sich" is distinguished from Phenomenon "Erscheinung", an gobservable event or physical manifestation, and the two words serve as interrelated technical terms in Kant's philosophyterms in Kant s philosophy.

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Whole vs. Parts

• tusk → spear• tail → rope• trunk → snake• side → wall

l t• leg → tree

The flaw in all their reasoning is that speculating on theThe flaw in all their reasoning is that speculating on the WHOLE from too few FACTS can lead to VERY LARGE errors in judgment.

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Science and Truth

– Science as controversy (new science, frontiers)

– Science as consensus (old, historically settled)

S i k l d b t l t– Science as knowledge about complex systems

– Opens systems paraconsistent logicOpens systems, paraconsistent logic

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Proof

The word proof can mean:The word proof can mean: • originally, a test assessing the validity or quality of something.

Hence the saying, "The exception that proves the rule" -- the rule is tested to see whether it applies even in the case of the ( t) ti(apparent) exception.

• a rigorous, compelling argument, including: a logical argument or a mathematical proof– a logical argument or a mathematical proof

– a large accumulation of scientific evidence– (...)

(from Wikipedia) 93

Mathematical Proof

I th ti f i d t ti th t i t iIn mathematics, a proof is a demonstration that, given certain axioms, some statement of interest is necessarily true.

(from Wikipedia) 94

Mathematical Proof

Proofs employ logic but usually include some amount of natural p y g ylanguage which of course admits some ambiguity. In the context of proof theory, where purely formal proofs are considered, such not entirely formal demonstrations are called "social proofs".

The distinction has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term)called folk mathematics (in both senses of that term).

The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.language and logic in proofs, and mathematics as a language.

(from Wikipedia) 95

Mathematical Proof

Regardless of one's attitude to formalism, the result that is proved to be true is a theorem; in a completely formal proof it would be ; p y pthe final line, and the complete proof shows how it follows from the axioms alone. Once a theorem is proved, it can be used as the basis to prove further statements.

The so-called foundations of mathematics are those statements one cannot, or need not, prove. These were once the primary , , p p ystudy of philosophers of mathematics. Today focus is more on practice, i.e. acceptable techniques.

(from Wikipedia) 96

Pressupositions and LimitationsPressupositions and Limitations of Axiomatic Logical Systems

Axiomatic theory is built on a set of few axioms/postulates (ideas which are considered so elementary and manifestly b i th t th d t d t b f ldobvious that they do not need to be proven as any proof would

introduce more complex ideas). All the theorems (true statements) are derived logically from the

iaxioms.

When a system requires increasing number of axioms (as e.g. number theory does), doubts begin to arise. How many axioms are needed? How do we know that the axioms aren't mutually contradictory? Each new axiom can change the meaning of the

i tprevious system.

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GÖDEL: TRUTH AND PROVABILITY (1)

Kurt Gödel actually proved two extraordinary theorems. They have revolutionized mathematics, showing that mathematical truth is more than bare logic and computation.

Gödel has been called the most important logician since Aristotle. His two theorems changed logic and mathematics as well as the way we look at truth and proof.

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Gödels first theorem proved that any formal system strong enough to support number theory has at least one undecidable statement. Even if we know that the statement is true, the system cannot prove it. This means the system is incomplete. For this reason, Gödel's first proof is called "the incompleteness theorem".

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Gödel's second theorem is closely related to the first. It says that no one can prove, from inside any complex formal system, that it is self-consistent.

"Gödel showed that provability is a weaker notion than truth, no matter what axiomatic system is involved.

In other words, we simply cannot prove some things in mathematics (from a given set of premises) which we nonetheless can know are true “ (Hofstadter)nonetheless can know are true. (Hofstadter)

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TRUTH VS. PROVABILITY ÖACCORDING TO GÖDEL

After: Gödel, Escher, Bach - an Eternal Golden Braidby Douglas Hofstadter. 101

TRUTH VS PROVABILITYTRUTH VS. PROVABILITY ACCORDING TO GÖDEL

Gödel theorem is built upon Aristotelian logic. So it is true within the paradigm of Aristotelian logic.

However nowadays it is not the only logic existingHowever, nowadays it is not the only logic existing.

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LOGIC (1)

The precision, clarity and beauty of mathematics are the consequence of the fact that the logical basis of classical mathematics possesses the features of parsimony andmathematics possesses the features of parsimony and transparency.

Cl i l l i it i l t t th ff t fClassical logic owes its success in large part to the efforts of Aristotle and the philosophers who preceded him. In their endeavour to devise a concise theory of logic, and later mathematics they formulated so called "Laws of Thought"mathematics, they formulated so-called Laws of Thought .

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LOGIC (2)

One of these, the "Law of the Excluded Middle," states that every proposition must either be True or False.proposition must either be True or False.

When Parminedes proposed the first version of this law (around 400 B C ) there were strong and immediate objections400 B.C.) there were strong and immediate objections.

For example, Heraclitus proposed that things could be simultaneously True and not Truesimultaneously True and not True.

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NON STANDARD LOGICNON-STANDARD LOGICFUZZY LOGIC (1)

Plato laid the foundation for fuzzy logic, indicating that there was a third region (beyond True and False).

Some among more modern philosophers follow the same path, particularly Hegel.

But it was Lukasiewicz who first proposed a systematic alternative to the bi-valued logic of Aristotle.

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NON STANDARD LOGICNON-STANDARD LOGIC FUZZY LOGIC (2)

In the early 1900's, Lukasiewicz described a three-valued logic, along with the corresponding mathematics.

The third value "possible," assigned a numeric value between True and False.

Eventually, he proposed an entire notation and axiomatic system from which he hoped to derive modern mathematics.

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NON-STANDARD LOGICS

• Categorical logic• Combinatory logic• Conditional logic• Constructive logic

• Many-sorted logic• Many-valued logic• Modal logic• Non-monotonic logicg

• Cumulative logic• Deontic logic• Dynamic logic• Epistemic logic

g• Paraconsistent logic• Partial logic• Prohairetic logic• Quantum logicp g

• Erotetic logic• Free logic• Fuzzy logic• Higher-order logic

g• Relevant logic• Stoic logic• Substance logic• Substructural logicg g

• Infinitary logic• Intensional logic• Intuitionistic logic• Linear logic

g• Temporal (tense) logic• Other logics

g

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Meaning (1)

All meaning is determined by the method of analysis where the method of analysis sets the context and so the rules that aremethod of analysis sets the context and so the rules that are used to determine the “meaningful” from “meaningless”.

C J LoftingC. J. Lofting

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Meaning (2)

At the fundamental level meaning is reducible to distinguishingObjects (the what) from• Objects (the what) from

• Relationships (the where)

which are the result of process of• Differentiation or• Integrationteg at o

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Meaning (3)

Human brain is not tabula rasa (clean slate) on birth but rather contains

• behavioral patterns to particular elements of environment (gene-based)

• template used for distinguishing meaning based on the distinctions of “what” from “where”

• Meaning as use implies holistic rationality, and value systems (hence ethical views) are integrated in the aims of a rational agents.

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http://ndpr.nd.edu/review.cfm?id=12083 Notre Dame Philosophical Reviews 111

Philosophy of Science/Theory of SciencePhilosophy of Science/Theory of Science Assignments

– Assignment 2: Demarcation of Science vs. Pseudoscience (in groups of two) Di i f A i t 2 l– Discussion of Assignment 2 - compulsory

– Assignment 2-extra (For those who miss the discussion of the Assignment 2)

– Assignment 3: GOLEM: Three Cases of Theory Confirmation (in groups of two) Discussion of Assignment 3 compulsory– Discussion of Assignment 3 - compulsory

– Assignment 3-extra (For those who miss the discussion of the Assignment 3)

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Two Examples of Axiomatic Systems -Limitations and DevelopmentsLimitations and Developments

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Pressupositions and Limitations of Formal Logical Systemsof Formal Logical Systems

Axiomatic System of Euclid: Shaking up Geometryy g p y

Euclid built geometry on a set of few axioms/postulates (ideas which are considered so elementary and manifestly obvious that th d t d t b f ld i t dthey do not need to be proven as any proof would introduce more complex ideas).

When a system requires increasing number of axioms (as e.g. number theory does), doubts begin to arise. How many axioms are needed? How do we know that the axioms aren't mutually

t di t ?contradictory?

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Pressupositions and Limitations of Formal Logical Systemsof Formal Logical Systems

Axiomatic System of Euclid: Shaking up Geometry

Until the 19th century no one was too concerned about axiomatization.

y g p y

Geometry had stood as conceived by Euclid for 2100 years.

If Euclid's work had a weak point, it was his fifth axiom, the axiom about parallel lines. Euclid said that for a given straight line, one could draw only one other straight line parallel to it through a

i t h t id itpoint somewhere outside it.

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EUCLID'S AXIOMS (1)

1. Every two points lie on exactly one line.

2. Any line segment with given endpoints may be continued in either directioneither direction.

3. It is possible to construct a circle with any point as its center and with a radius of any length. (This implies that there is neither an upper nor lower limit to distance. In-other-words, any distance, no mater how large can always be increased, and any distance, no mater how small can always be divided.)

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EUCLID'S AXIOMS (2)

4. If two lines cross such that a pair of adjacent angles are congruent, then each of these angles are also congruent to any other angle formed in the same way. (Says that all right angles are equal to one another.)

5. (Parallel Axiom): Given a line l and a point not on l, there is one and only one line which contains the point and is parallel to land only one line which contains the point, and is parallel to l.

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NON-EUCLIDEAN GEOMETRIES (1)

Mid-1800s: mathematicians began to experiment with different definitions for parallel lines.

Lobachevsky, Bolyai, Riemann: new non-Euclidean geometries by assuming that there could be several parallel lines through the outside point or there could be no parallel lines.

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NON-EUCLIDEAN GEOMETRIES (2)

Two ways to negate the Euclidean Parallel Axiom:

– 5-S (Spherical Geometry Parallel Axiom): Given a line l and a– 5-S (Spherical Geometry Parallel Axiom): Given a line l and a point not on l, no lines exist that contain the point, and are parallel to l.

– 5-H (Hyperbolic Geometry Parallel Axiom): Given a line l and a point not on l, there are at least two distinct lines which contains the point and are parallel to lthe point, and are parallel to l.

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Reproducing the Euclidean World in a model of the Elliptical Non-Euclidean World.

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Spherical/Elliptical Geometry

In spherical geometry lines of latitude are not great circles (except for the equator), and lines of longitude are. Elliptical Geometry takes the q ), g p yspherical plan and removes one of two points directly opposite each other. The end result is that in spherical geometry, lines always intersect in exactly two points, whereas in elliptical geometry, lines always intersect in one point.

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Properties of Elliptical/Spherical Geometry

In Spherical Geometry, all lines intersect in 2 points. In elliptical geometry lines intersect in 1 pointgeometry, lines intersect in 1 point.

In addition, the angles of a triangle always add up to be greater than 180 degrees. In elliptical/spherical geometry, all of Euclid's postulates still do hold with the exception of the fifth postulatepostulates still do hold, with the exception of the fifth postulate.

This type of geometry is especially useful in describing the Earth's surface.

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Hyperbolic Cubes

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DEFINITION: Parallel lines are infinite lines in the same plane that do not intersectParallel lines are infinite lines in the same plane that do not intersect.

Hyperbolic Universe Flat Universe Spherical Universe

Einstein incorporated Riemann's ideas into relativity theory to d ib th t fdescribe the curvature of space.

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MORE PROBLEMS WITHMORE PROBLEMS WITH AXIOMATIZATION…

Not only had Riemann created a system of geometry which put commonsense notions on its head, but the philosopher-commonsense notions on its head, but the philosophermathematician Bertrand Russell had found a serious paradox for set theory!

He has shown that Frege’s attempt to reduce mathematics to logical reasoning starting with sets as basics leads to contradictions.contradictions.

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HILBERT’S PROGRAM

Hilbert’s hope was that mathematics would be reducible to finding proofs (by manipulating the strings of symbols) from a fixed system of axioms, axioms that everyone could agree were true.

Can all of mathematics be made algorithmic, or will there always be new problems that outstrip any given algorithm, and so requirenew problems that outstrip any given algorithm, and so require creative mind to solve?

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AXIOMATIC SYSTEM OF PRINCIPIA:AXIOMATIC SYSTEM OF PRINCIPIA: PARADOX IN SET THEORY

Mathematicians hoped that Hilbert's plan would work because axioms and definitions are based on logical commonsense intuitions such as e g the idea of setintuitions, such as e.g. the idea of set.

A set is any collection of items chosen for some characteristic common for all its elementscommon for all its elements.

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RUSSELL'S PARADOX (1)

There are two kinds of sets:

– Normal sets, which do not contain themselves, and – Non-normal sets, which are sets that do contain themselves.

The set of all apples is not an apple. Therefore it is a normal set. The set of all thinkable things is itself thinkable, so it is a non-normal set.

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Let 'N' stand for the set of all normal sets.

Is N a normal set?

If it is a normal set, then by the definition of a normal set it cannot be a member of itself. That means that N is a non-normal set, one of those few sets which actually are members of themselves.

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But on the other hand N is the set of all normal sets; if weBut on the other hand…N is the set of all normal sets; if we describe it as a non-normal set, it cannot be a member of itself, because its members are, by definition, normal.

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Russell resolved the paradox by redefining the meaning of 'set' to exclude peculiar (self-referencing) sets, such as "the set of all normal sets“.

Together with Whitehead in Principia Mathematica he founded mathematics on that new set definition.

They hoped to get self-consistent and logically coherent system …

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… However, even before the project was complete, Russell's expectations were dashed!

The man who showed that Russell's aim was impossible was Kurt Gödel, in a paper titled "On Formally Undecidable Propositions of Principia Mathematica and Related Systems."

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Critique of Usual Naïve Image of Scientific Methodof Scientific Method

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Critique of Usual Naïve ImageCritique of Usual Naïve Image of Scientific Method (1)

The narrow inductivist conception of scientific inquiry

1 All facts are observed and recorded1. All facts are observed and recorded.2. All observed facts are analyzed, compared and classified,

without hypotheses or postulates other than those necessarily involved in the logic of thoughtinvolved in the logic of thought.

3. Generalizations inductively drawn as to the relations, classificatory or causal, between the facts.

4 Further research employs inferences from previously4. Further research employs inferences from previously established generalizations.

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This narrow idea of scientific inquiry is groundless for several reasons:

1. A scientific investigation could never get off the ground, for a collection of all facts would take infinite time, as there are infinite number of facts.

The only possible way to do data collection is to take only relevant facts. But in order to decide what is relevant and what is not, we have to have a theory or at least a hypothesis about what is it we are observingare observing.

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A hypothesis (preliminary theory) is needed to give the directionto a scientific investigation!

2. A set of empirical facts can be analyzed and classified in many different ways. Without hypothesis, analysis and classification are blind.

3. Induction is sometimes imagined as a method that leads, by mechanical application of rules from observed facts to generalmechanical application of rules, from observed facts to general principles. Unfortunately, such rules do not exist!

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Why is it not possible to derive hypothesisWhy is it not possible to derive hypothesis (theory) directly from the data? (1)

– For example, theories about atoms contain terms like “atom”, “electron”, “proton”, etc; yet what one actually measures are spectra (wave lengths), traces in bubble chambers, calorimetricspectra (wave lengths), traces in bubble chambers, calorimetric data, etc.

– So the theory is formulated on a completely different (and more– So the theory is formulated on a completely different (and more abstract) level than the observable data!

The transition from data to theory requests creative imagination!– The transition from data to theory requests creative imagination!

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Why is it not possible to derive hypothesisWhy is it not possible to derive hypothesis (theory) directly from the data?* (2)

– Scientific hypothesis is formulated based on “educated guesses” at the connections between the phenomena under study, at regularities and patterns that might underlie their occurrence. Scientific guesses are completely different from any process of systematic inference.

– The discovery of important mathematical theorems, like the discovery of important theories in empirical science, requires inventive ingenuity.

*Here is instructive to study Automated discovery methods in order to see how much theory must be used in order to extract meaning from the “raw data” 138

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