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CDT403 Research Methodology in Natural Sciences and Engineering

Theory of ScienceSCIENCE, KNOWLEDGE, TRUTH, MEANING

Gordana Dodig-Crnkovic

School of Innovation, Design and Engineering

Mälardalen University

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Theory of ScienceTheory of Science

Lecture 1 SCIENCE AND KNOWLEDGE: TRUTH MEANINGLecture 1 SCIENCE AND KNOWLEDGE: TRUTH, MEANING. FORMAL LOGICAL SYSTEMS PRESUPPOSITIONS

Lecture 2 LANGUAGE AND COMMUNICATION. CRITICAL THINKING. PSEUDOSCIENCE - DEMARCATION

Lecture 3 SCIENCE AND RESEARCH: TECHNOLOGY, SOCIETAL ASPECTS PROGRESS HISTORY OF SCIENTIFICSOCIETAL ASPECTS. PROGRESS. HISTORY OF SCIENTIFIC THEORY. POSTMODERNISM AND CROSSDISCIPLINES

Lecture 4 GOLEM LECTURE. ANALYSIS OF SCIENTIFIC CONFIRMATION THEORY OF RELATIVITY COLD FUSIONCONFIRMATION: THEORY OF RELATIVITY, COLD FUSION, GRAVITATIONAL WAVES

Lecture 5 COMPUTING HISTORY OF IDEAS

Lecture 6 PROFESSIONAL & RESEARCH ETHICS

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Science, Knowledge, Truth and Meaning

CRITICAL THINKING

WHAT IS SCIENCE?

WHAT IS SCIENTIFIC METHOD?WHAT IS SCIENTIFIC METHOD?

WHAT IS KNOWLEDGE?

KNOWLEDGE, TRUTH AND MEANING

FORMAL SYSTEMS AND THE CLASSICAL MODEL OF SCIENCE

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Course Material

– Course web page http://www idt mdh se/kurser/ct3340/http://www.idt.mdh.se/kurser/ct3340/

– Theory of Science CompendiumTheory of Science Compendium G Dodig-Crnkovic

– The Golem: What You Should Know about Science Harry M. Collins & Trevor Pinch

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Red Thread: Critical Thinking

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

Hypatia, natural philosopher and mathematician

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Haiku – Like Highlights

Toyama gat ki i ttsuki ni utsurutonbo kana

A distant mountainshimmers in the eye of a dragonflyeye of a dragonfly..

Kobayashi Issa (1763-1827)

(Haiku form: 5-7-5 syllables)

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What Is Science?

7EyeMaurits Cornelis Escher

1. Definitions by Goal (Result) and Process (1)

i f L ti i ti i t kscience from Latin scientia, scire to know; 1: a department of systematized knowledge as an

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

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

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1. Definitions by Goal (Result) and Process (2)

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

4: a system or method reconciling practical ends with4: a system or method reconciling practical ends with scientific laws <engineering is both a science and an art>

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2. Science: Definitions by Contrast

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

Robert H. MacArthur

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

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

000 - & Psychology200 - ReliComputers, Information & General Reference100 - Philosophy gion300 - Social sciences400 - Language500 - Science600 - Technology600 Technology700 - Arts & Recreation800 - Literature900 - History & Geography900 - History & Geography

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3. Dewey Decimal Classification®

500 – Science510 Mathematics510 Mathematics520 Astronomy530 Physics540 Ch i t540 Chemistry550 Earth Sciences & Geology560 Fossils & Prehistoric Life570 Biology & Life Sciences580 Plants (Botany)590 Animals (Zoology)590 Animals (Zoology)

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4. Language Based SchemeClassical Sciences in their Cultural Context –

Logic

&

Classical Sciences in their Cultural Context

N l S i

&

Mathematics

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Culture(Religion, Art, …)

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Natural Sciences(Physics,

Chemistry,Biology, …)

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Social Sciences(Economics, Sociology,

A h l )Anthropology, …)3

The Humanities(Philosophy, History,

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( p y, y,Linguistics …)

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5 U d t di h t i i5. Understanding what science is by understanding what scientists do

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

Peter Medawar Pluto's RepublicPeter Medawar, Pluto s Republic

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6 C iti f U l N ï I6. Critique of Usual Naïve Image of Scientific Method (1)

The narrow inductivist conception of scientific inquiry

1. 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 thought.

3 Generalizations inductively drawn as to the relations3. Generalizations inductively drawn as to the relations, classificatory or causal, between the facts.

4. Further research employs inferences from previously bli h d li iestablished generalizations.

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C iti f U l N ï ICritique of Usual Naïve Image of Scientific Method (2)

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 infinitecollection 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 ffacts. 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 observing.

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C iti f U l N ï ICritique of Usual Naïve Image of Scientific Method (3)

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

2. A set of empirical facts can be analyzed and classified in many diff t With t h th i l i d l ifi tidifferent 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 general principles Unfortunately such rules do not exist!principles. Unfortunately, such rules do not exist!

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Wh i it t ibl t d i h th iWhy 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 areelectron , proton , etc; yet what one actually measures are spectra (wave lengths), traces in bubble chambers, calorimetric data, etc.

– So the theory is formulated on a completely different (and more abstract) level than the observable data!abst act) e e t a t e obse ab e data

– 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, atat 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 inferencesystematic inference.

– The discovery of important mathematical theorems, like the fdiscovery of important theories in empirical science, requires

inventive ingenuity.

19*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”

Socratic Method Scientific Method

1 Wonder Pose a question 1 Wonder Pose a question1. Wonder. Pose a question(of the “What is X ?” form).

1. Wonder. Pose a question.(Formulate a problem).

2. Hypothesis. Suggest a plausible answer (a definition or definiens) from

2. Hypothesis. Suggest a plausible answer (a theory) from which some empirically testable

which some conceptually testable hypothetical propositions can be deduced.

hypothetical propositions can be deduced.

3. Elenchus ; “testing,” “refutation,” or “cross-examination ” Perform a thought

3. Testing. Construct and perform an experiment which makes it possible tocross-examination. Perform a thought

experiment by imagining a case which conforms to the definiens but clearly fails to exemplify the definiendum, or vice versa Such cases if successful are

an experiment, which makes it possible to observe whether the consequences specified in one or more of those hypothetical propositions actually follow when the conditions specified in the sameversa. Such cases, if successful, are

called counterexamples. If a counterexample is generated, return to step 2, otherwise go to step 4.

conditions specified in the same proposition(s) pertain. If the test fails, return to 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 confirmed.

205. Act accordingly. 5. Act accordingly.

The Scientific Method

EXISTING THEORIESAND OBSERVATIONS

1 Hypotesen

PREDICTIONS

3HYPOTHESIS

21 Hypotesen

måste justeras

Hypothesis must be

redefined

Hypothesis must be adjusted

SELECTION AMONG COMPETING THEORIES

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TESTS AND NEW OBSERVATIONS

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The hypotetico-deductive cycle

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Consistency achieved

EXISTING THEORY CONFIRMED

(within a new context) or

NEW THEORY PUBLISHED

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5The scientific-community cycle

Formulating Research Questions and Hypotheses

Different approachesDifferent approaches:

Intuition – (Educated) GuessIntuition (Educated) GuessAnalogySymmetryParadigmMetaphorand many moreand many more ..

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Criteria to Evaluate Theories

When there are several rivaling hypotheses number of criteria can b d f h i b t thbe used for choosing a best theory.

Following can be evaluated:

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

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

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Criteria which Good Scientific Theory Shall Fulfill

– Logically consistent– Consistent with accepted facts – Testable– Consistent with related theories

Interpretable: explain and predict– Interpretable: explain and predict– Parsimonious– Pleasing to the mind (Esthetic, Beautiful)g ( , )– Useful (Relevant/Applicable)

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

A philosophical statement developed by William of OckhamA philosophical statement developed by William of Ockham, (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 competingThe principle gives precedence to simplicity; of two competing theories, the simplest explanation of an entity is to be preferred.

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Wh t I K l d ?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 alreadyoriginally acquired in a previous life, and that the soul already has knowledge, and we learn by recollecting what in fact the soul already knows.

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

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Wh t I K l d ?What Is Knowledge?Plato´s Definition

Plato's third definition:" Knowledge is justified true belief "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 BeliefEdmund Gettier, in the paper called "Is Justified True Belief Knowledge?“ argues that knowledge is not the same as justified 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 isattentive 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 knowledge when recommended method is being followed.

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Wh t I K l d ?What Is Knowledge?Descartes´ Definition

"Intuitions provide the ultimate grounds for logical deductions. Ultimate first principles must be known through intuition whileUltimate 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."

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What Is Knowledge?

– Propositional knowledge: knowledge that such-and-such is the case.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 andA Priori Knowledge (built in, developed by evolution and inheritance)

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

Reasoning (information processing)– Reasoning (information processing)

– Testimony (network, communication)y ( , )

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K l d d Obj ti itKnowledge and Objectivity:Observations

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

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

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K l d d Obj ti itKnowledge and ObjectivityObservations

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

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

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Perception and “Direct Observation”

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Perception and “Direct Observation”Perception and Direct Observation

"Reality is merely an illusion, albeit a very persistent one." -

Einstein

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

See even:http://persci.mit.edu/people/adelson/publications/gazzan.dir/gazzan.htmp p p p p g g

Lightness Perception and Lightness Illusions

http://www.ihu.his.se/~christin/Vetenskapsteori/Vetenskapsteorikurser

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Direct Observation?!

An atom interferometer, which splits an atom into separate wavelets, can allow th t f f ti th t Sh h i th lthe 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 p yprotons and antiprotons in the Tevatron accelerator at Fermilab provided evidence that the elusive subatomic particle known as top quark has been ffound.

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Knowledge Justification

– Foundationalism (uses architectural metaphor to describe the structure of our belief systems. The superstructure of a beliefstructure 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)

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Truth (1)

The correspondence theory– The correspondence theory

– The coherence theoryThe coherence theory

– The deflationary theory

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T th (2)Truth (2)The Correspondence Theory

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

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

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T th (3)Truth (3)The Coherence Theory

Statements in the theory are believed to be true because being tibl ith th t t tcompatible with other statements.

The truth of a sentence just consists in its belonging to a system of h t t t tcoherent statements.

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

Characteristically they all believed that truths about the world could b f d b thi ki th ti li t d id li tbe 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 used in all sciences

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used in all sciences.

T th (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 claimsuperfluous 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 as "birds are warm-blooded ".

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

Th D fl ti Th i l ll d th d dThe Deflationary Theory is also called the redundancy theory, the disappearance theory, the no-truth theory, the disquotational theory and the minimalist theorythe disquotational theory, and the minimalist theory .

63-70 see Lars-Göran Johanssonhttp://www filosofi uu se/utbildning/Externt/slu/slultexttruth htmhttp://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 fromdistinguished from Phenomenon "Erscheinung", an observable event or physical manifestation, and the two wordsmanifestation, and the two words serve as interrelated technical terms in Kant's philosophy.

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Whole vs. Parts

• tusk → speart il• tail → rope

• trunk → snakeid ll• side → wall

• leg → tree

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

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The Classical Model of Science

The Classical Model of Scienceis a system S of propositions and concepts satisfying the following conditions: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• 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 provable or demonstrable from) these fundamental propositions.

• 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 itsfundamental 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 scientific explanation.

• The fundamental is that “All propositions and all concepts (or terms) of S concern a specific set of objects or are about aterms) 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

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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

Science and Truth

– Science as controversy (new science, frontiers)

– Science as consensus (old, historically settled)

– Science as knowledge about complex systems

– Opens systems, paraconsistent logic

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Scientific Truth (1)

– Physics professor is walking across campus, runs into Math professor Physics professor has beeninto Math professor. Physics professor has been doing an experiment, and has worked out an empirical equation that seems to explain his data, p q p ,and asks the Math professor to look at it.

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Scientific Truth (2)

– A week later, they meet again, and the Math professor says the equation is invalid By then theprofessor says the equation is invalid. By then, the Physics professor has used his equation to predict the results of further experiments, and he is getting p , g gexcellent results, so he asks the Math professor to look again.

– Another week goes by, and they meet once more. The Math professor tells the Physics professor the equation does work, ”but only in the trivial case where the numbers are real and positive "

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where the numbers are real and positive.

Proof

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

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

• a rigorous, compelling argument, including: – a logical argument or a mathematical proof– a large accumulation of scientific evidencea large accumulation of scientific evidence– (...)

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Mathematical ProofMathematical Proof

In mathematics, a proof is a demonstration that, given certain axioms, some statement of interest is necessarily true.

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Mathematical ProofMathematical Proof

Proofs employ logic but usually include some amount of natural language which of course admits some ambiguity. In the context of proof theory, where purely formal proofs are considered, such

t ti l f l d t ti ll d " i l f "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).

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

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Mathematical Proof

Regardless of one's attitude to formalism the result that is provedRegardless 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 the final line, and the complete proof shows how it follows from the axioms alone Once a theorem is proved it can be used asthe 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 study of philosophers of mathematics. Today focus is more onstudy of philosophers of mathematics. Today focus is more on practice, i.e. acceptable techniques.

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Pressupositions and Limitations of Formal Logical Systems

Axiomatic System of Euclid: Shaking up Geometry

Euclid built geometry on a set of few axioms/postulates (ideas which are considered so elementary and manifestly obvious that they do not need to be proven as any proof would introduce more complex ideas). p )

When a system requires increasing number of axioms (as e.g. b th d ) d bt b i t i H inumber theory does), doubts begin to arise. How many axioms

are needed? How do we know that the axioms aren't mutually contradictory?

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Pressupositions and Limitations of Formal Logical Systems

Until the 19th century no one was too concerned about

Axiomatic System of Euclid: Shaking up Geometry

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

Geometry had stood as conceived by Euclid for 2100 yearsGeometry had stood as conceived by Euclid for 2100 years.

If Euclid's work had a weak point, it was his fifth axiom, the axiom b t ll l li E lid id th t f i t i ht liabout parallel lines. Euclid said that for a given straight line, one

could draw only one other straight line parallel to it through a point 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 direction.

3 It is possible to construct a circle with any point as its center and3. 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 anycongruent, 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 l.

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NON-EUCLIDEAN GEOMETRIES (1)

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

Lobachevsky, Bolyai, Riemann: new non-Euclidean geometries by assuming that there could be several parallel lines through theassuming 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 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– 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 l.

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Reproducing the Euclidean WorldReproducing 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 spherical plan and removes one of two points directly opposite eachspherical 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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intersect in one point.

Properties of Elliptical/Spherical Geometry

In Spherical Geometry all lines intersect in 2 points In ellipticalIn Spherical Geometry, all lines intersect in 2 points. In elliptical geometry, 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 postulate.

This type of geometry is especially useful in describing the Earth'sThis 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 intersect.

Hyperbolic Universe Flat Universe Spherical Universe

Einstein incorporated Riemann's ideas into relativity theory to describe the curvature of space

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describe the curvature of space.

MORE PROBLEMS WITHMORE PROBLEMS WITH AXIOMATIZATION…

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

He has shown that Frege’s attempt to reduce mathematics toHe has shown that Frege s attempt to reduce mathematics to logical reasoning starting with sets as basics leads to 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 fixedproofs (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 require creative mind to solve?

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

Mathematicians hoped that Hilbert's plan would work because i d d fi iti b d l i laxioms and definitions are based on logical commonsense

intuitions, such as e.g. the idea of set.

A set is any collection of items chosen for some characteristic common 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 setset 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 itIf 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 themselvesthemselves.

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But 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 allexclude peculiar (self referencing) sets, such as the set of all normal sets“.

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

They hoped to get self consistent and logically coherent systemThey 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!expectations were dashed!

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

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GÖDEL: TRUTH AND PROVABILITY (1)

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

Gödel has been called the most important logician since AristotleGödel 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 undecidableto 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 incompletenessFor 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 isone can prove, from inside any complex formal system, that it is self-consistent.

"Gödel showed that provability is a weaker notion than truth noGödel 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 inIn other words, we simply cannot prove some things in mathematics (from a given set of premises) which we nonetheless can know are true. “ (Hofstadter)

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TRUTH VS. PROVABILITY ACCORDING TO GÖDEL

87After: Gödel, Escher, Bach - an Eternal Golden Braidby Douglas Hofstadter.

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 existing.

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LOGIC (1)

The precision, clarity and beauty of mathematics are the f th f t th t th l i l b i f l i lconsequence of the fact that the logical basis of classical

mathematics possesses the features of parsimony and transparency.

Classical logic owes its success in large part to the efforts of Aristotle and the philosophers who preceded him In theirAristotle 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".

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LOGIC (2)

One of these the "Law of the Excluded Middle " states that everyOne of these, the Law of the Excluded Middle, states that every 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 objections.

For example, Heraclitus proposed that things could be simultaneously 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).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 alternativeBut 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.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 e tua y, e p oposed a e t e otat o a d a o at c systefrom which he hoped to derive modern mathematics.

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NON-STANDARD LOGICS

• Categorical logic• Combinatory logic• Conditional logic

• Many-sorted logic• Many-valued logic• Modal logic• Conditional logic

• Constructive logic• Cumulative logic• Deontic logic

• Modal logic• Non-monotonic logic• Paraconsistent logic• Partial logicg

• Dynamic logic• Epistemic logic• Erotetic logic• Free logic

g• Prohairetic logic• Quantum logic• Relevant logic• Stoic logic• Free logic

• Fuzzy logic• Higher-order logic• Infinitary logic

• Stoic logic• Substance logic• Substructural logic• Temporal (tense) logicy g

• Intensional logic• Intuitionistic logic• Linear logic

( ) g• Other logics

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NON-STANDARD LOGICS

http://www.earlham.edu/~peters/courses/logsys/nonstbib.htmhtt // th d bilt d / h t /l i /

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http://www.math.vanderbilt.edu/~schectex/logics/

The Limits of Reason - G J Chaitin

The limits of reasonScientific American 294, No. 3 (March 2006), pp. 74-81.

Epistemology as information theory: from Leibniz to ΩCollapse 1 (2006), pp. 27-51. Reprinted in Teoria algoritmica della complessità, 2006. p g p ,

Meta Math!first paperback edition Vintage, 2006. tage, 006

Speculations on biology, information and complexityBulletin of the European Association for Theoretical Computer Science 91 (February 2007) pp 231-237Science 91 (February 2007), pp. 231 237.

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Meaning (1)

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

C. J. Lofting

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Meaning (2)

At the fundamental level meaning is reducible to distinguishingAt the fundamental level meaning is reducible to distinguishing• Objects (the what) from• Relationships (the where)Relationships (the where)

which are the result of process of• Differentiation or• Integration

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Meaning (3)

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

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

f• 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 99

Assignments

– Assignment 2: Analyze Pseudoscience vs. Science (in groups of two)(in groups of two)

– Discussion of Assignment 2 - compulsoryAssignment 2 extra (For those who miss the– Assignment 2-extra (For those who miss the discussion of the Assignment 2)Assignment 3: GOLEM: Analyze Three Cases of– Assignment 3: GOLEM: Analyze Three Cases of Theory Confirmation (in groups of two)

– Discussion of Assignment 3 - compulsoryDiscussion of Assignment 3 compulsory– Assignment 3-extra (For those who miss the

discussion of the Assignment 3)100

g )

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