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**CDT403 Research Methodology in Natural Sciences and Engineering**

Theory of Science SCIENCE VERSUS PSEUDOSCIENCE AS CRITICAL THINKING VERSUS WISHFUL THINKING Gordana Dodig-Crnkovic School of Innovation, Design and Engineering Mälardalen University

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THEORY OF SCIENCE Lecture 1 INFORMATION, COMPUTATION, KNOWLEDGE AND SCIENCE Lecture 2 SCIENCE AND CRITICAL THINKING. PSEUDOSCIENCE AND WISHFUL THINKING - DEMARCATION Lecture 3 SCIENCE, RESEARCH, TECHNOLOGY, SOCIETAL ASPECTS. PROGRESS. HISTORY OF SCIENTIFIC THEORY. POSTMODERNISM AND CROSSDISCIPLINES Lecture 4 PROFESSIONAL & RESEARCH ETHICS

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RECAPITULATION OF THE FIRST LECTURE: HISTORICAL DEVELOPMENT OF THINKING ABOUT THE WORLD/NATURE/UNIVERSE

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MYTHOPOETIC THINKING Mythopoetic (myth + poetry) truth is revealed through myths, stories and rituals. Myths are stories about persons, where persons may be gods, heroes, or ordinary people. Myth allows for a multiplicity of explanations, where the explanations are not logically exclusive (can contradict each other) and are often humorous, exciting and colorful. Mythic traditions are conservative. Innovation is slow, and radical departures from tradition rare. The Egyptian king Akhenaton and Queen Nefertiti making offerings to the Aton.

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MYTHOPOETIC THINKING Myths are self-justifying. The inspiration of the gods was enough to ensure their validity, and there was no other explanation for the creativity of poets, oracles, and prophets than inspiration by the gods. Thus, myths are not argumentative.

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**GREEK NATURAL PHILOSOPHY – PROTOSCIENCE**

Anaximander of Miletus (c c. 546 bc) a pupil of Thales, and as other members of the Ionian School, was an early scientist. He constructed the first geometrical model of the universe, and made maps of the earth and the skies. He said that the arche ('beginning and basis') of existing things is an apeiron ('limitless') nature of some kind, from which come the heavens and the kosmos ('world order'). Anaximander advocated the idea of biological evolution with human beings, like other animals, evolved from fish. Anaximander was the author of the first written work of philosophy in ancient Greece, On Nature, which has been lost. His philosophy was a natural dialectics.

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**SCIENCE: THE MECHANICAL 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 Universe – the clockwork. Newton Principia, 1687

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**SCIENCES: THE UNIVERSE AS A COMPUTER**

We are all living inside a gigantic computer. No, not The Matrix: the Universe. Every process, every change that takes place in the Universe, may be considered as a kind of computation. K Zuse, E Fredkin, S Wolfram, S Lloyd, G Chaitin and many more Konrad Zuse, Computing Space

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**Three Major Paradigm Shifts**

Mytho-poetic, God-Centric Universe Info-Computational Human-Centric Universe (Classical) Mechanic Universe Dodig-Crnkovic G and Müller V, A Dialogue Concerning Two World Systems: Info-Computational vs. Mechanistic. In: INFORMATION AND COMPUTATION , World Scientific Publishing Co. Series in Information Studies. Editors: G Dodig-Crnkovic and M Burgin,

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**MAKING SENSE – CONSTRUCTION OF MEANING**

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**EMBODIED COGNITION – KNOWLEDGE AS A PHYSICAL PHENOMENON (PROCESS/STRUCTURE)**

From: Ago Ergo Cogito - "I act, therefore I think“ – Certesian divide bridged. No separation mind-body

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**BOTTOM-UP (BODY->MIND) VS**

BOTTOM-UP (BODY->MIND) VS. TOP DOWN (MIND->BODY) VIEW OF SENSE-MAKING

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**BOTTOM-UP VIEW, COGNITIVE AGENCY-BASED**

Meaning (1) All 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 the result of process of Differentiation and Integration or identification of differences and similarities, recognition of patterns.

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Meaning (3) Human brain is not tabula rasa (clean slate) on birth but rather contains (gene-based, evolutionary acquired) morphological structures used for meaning production based on the distinctions of same/different, what/how, where/when etc. behavioral patterns, etc.

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**TOP-DOWN VIEW, LANGUAGE-CENTERED**

SEMIOTICS (1) Semiotics, the science of signs, looks at how humans search for and construct meaning. Semiotics: reality is a system of signs! (with an underlying system which establishes mutual relationships among those and defines identity and difference, i.e. enables the description of the dynamics.)

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**Three Levels of Semiotics (Theory of Signs)**

syntactics semantics pragmatics Three Levels of Semiotics (Theory of Signs)

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SEMIOTICS (2A) syntactics semantics pragmatics

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**SEMIOTICS (3) Reality is a construction.**

Information or meaning is not 'contained' in the (physical) world and 'transmitted' to us - we actively create meanings (“make sense”!) through a complex interplay of perceptions, and agency based on hard-wired behaviors and coding-decoding conventions. The study of signs is the study of the construction and maintenance of reality.

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SEMIOTICS (4) 'A sign... is something which stands to somebody for something in some respect or capacity'. Sign takes a form of words, symbols, images, sounds, gestures, objects, etc. Anything can be a sign as long as someone interprets it as 'signifying' something - referring to or standing for something.

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**SEMIOTICS (5) CAT (signified) (signifier) The sign consists of**

(signified) (signifier) CAT The sign consists of signifier (a pointer) signified (that what pointer points to)

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**This is Not a Pipe . . . by Rene Magritte. . . . Surrealism**

SEMIOTICS (6) This is Not a Pipe by Rene Magritte Surrealism

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SEMIOTICS (7) Reality is divided up into arbitrary categories by every language. [However this arbitrariness is essentially limited by our physical predispositions as human beings. Our cognitive capacities are defined to a high extent by our physical constitution.] The conceptual world with which each of us is familiar with, could have been divided up in a very different way. The full meaning of a sign does not appear until it is placed in its context, and the context may serve an extremely subtle function.

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COMMUNICATION Communication is imparting of information, interaction through signs/messages. Information is the meaning that a human gives to signs by applying the known conventions used in their representation. Sign is any physical event used in communication. Language is a vocabulary and the way of using it.

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**HIERARCHICAL STRUCTURE OF LANGUAGE**

Object-language Meta-language In dictionaries of SCIENCE there is no definition of science! The definition of SCIENCE can be found in PHILOSOPHY dictionaries.

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**AMBIGUITIES OF LANGUAGE (1) Lexical ambiguity**

Lexical ambiguity, where a word have more than one meaning: meaning (sense, connotation, denotation, import, gist; significance, importance, implication, value, consequence, worth) sense (intelligence, brains, intellect, wisdom, sagacity, logic, good judgment; feeling) connotation (nuance, suggestion, implication, undertone, association, subtext, overtone) denotation (sense, connotation, import, gist) …

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**AMBIGUITIES OF LANGUAGE (2)**

Syntactic ambiguity like in “small dogs and cats” (are cats small?). Semantic ambiguity comes often as a consequence of syntactic ambiguity. “Coast road” can be a road that follows the coast, or a road that leads to the coast.

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**AMBIGUITIES OF LANGUAGE (3)**

Referential ambiguity is a sort of semantic ambiguity (“it” can refer to anything). Pragmatic ambiguity (If the speaker says “I’ll meet you next Friday”, thinking that they are talking about 17th, and the hearer think that they are talking about 24th.) Vagueness is an important feature of natural languages. “It is warm outside” says something about temperature, but what does it mean? A warm winter day in Sweden is not the same thing as warm summer day in Kenya.

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**AMBIGUITIES OF LANGUAGE (4)**

Ambiguity of language results in its flexibility, that makes it possible for us to cover the whole infinite diversity of the world we live in with a limited means of vocabulary and a set of rules that language is made of. On the other hand, flexibility makes the use of language complex. Nevertheless, the languages, both formal and natural, are the main tools we have on our disposal in science and research.

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**USE OF LANGUAGE IN SCIENCE. LOGIC AND CRITICAL THINKING. PSEUDOSCIENCE**

LOGICAL ARGUMENT DEDUCTION INDUCTION REPETITIONS, PATTERNS, IDENTITY CAUSALITY AND DETERMINISM FALLACIES PSEUDOSCIENCE

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REASONING Use of reason, especially to form conclusions, inferences, or judgments. Evidence or arguments used in thinking or argumentation. The process of drawing conclusions from facts, evidence, etc.

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LOGICAL ARGUMENT An argument is a statement logically inferred from premises. Two sorts of arguments: Deductive general particular Inductive particular general

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**LOGICAL ARGUMENT Constituents of a logical argument: premises**

inference and conclusion

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JUDGMENT It is important to notice that all reasoning basically depends on judgment (the ability to perceive and distinguish relationships; the capacity to form an opinion by distinguishing and evaluating) “Now, the question, What is a judgment? is no small question, because the notion of judgment is just about the first of all the notions of logic, the one that has to be explained before all the others, before even the notions of proposition and truth, for instance.” Per Martin-Löf On the Meanings of the Logical Constants and the Justifications of the Logical Laws; Nordic Journal of Philosophical Logic, 1(1):11 60, 1996.

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INDUCTION Empirical Induction Mathematical Induction

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**EMPIRICAL INDUCTION The generic form of an inductive argument:**

Every A we have observed is a B. Therefore, every A is a B.

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**An Example of Inductive Inference**

Every instance of water (at sea level) that we have observed has boiled at 100 C. Therefore, all water (at sea level) boils at 100 C. Inductive argument will never offer 100% certainty! A typical problem with inductive argument is that it is formulated generally, while the observations are made under some particular, specific conditions. ( In our example we could add ”in an open vessel” as well. )

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**Inductive Inference has Limited Validity**

An inductive argument have no way to logically (with certainty, with necessity) prove that: the phenomenon studied do exist in general domain that it continues to behave according to the same pattern According to Popper, inductive argument only supports working theories based on the collected evidence.

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Counter-example Perhaps the most well known counter-example was the discovery of black swans in Australia. Prior to the point, it was assumed that all swans were white. With the discovery of the counter-example, the induction concerning the color of swans had to be re-modeled.

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**MATHEMATICAL INDUCTION**

The aim of the empirical induction is to establish the law. In the mathematical induction we have the law already formulated. We must prove that it holds generally. The basis for mathematical induction is the property of the well-ordering for the natural numbers.

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**THE PRINCIPLE OF MATHEMATICAL INDUCTION**

Suppose P(n) is a statement involving an integer n. Than to prove that P(n) is true for every n n0 it is sufficient to show these two things: 1. P(n0) is true (the basis step) 2. For any k n0, if P(k) is true, then P(k+1) is true. (the induction step)

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**THE TWO PARTS OF INDUCTIVE PROOF**

the basis step the induction step. In the induction step, we assume that statement is true in the case n = k, and we call this assumption the induction hypothesis.

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**THE STRONG PRINCIPLE OF MATHEMATICAL INDUCTION**

Suppose P(n) is a statement involving an integer n. In order to prove that P(n) is true for every n n0 it is sufficient to show these two things: 1. P(n0) is true. 2. For any k n0, if P(n) is true for every n satisfying n0 n k, then P(k+1) is true.

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**INDUCTION VS DEDUCTION, TWO SIDES OF THE SAME COIN**

Deduction and induction occur as a part of the common hypothetico-deductive method, which can be simplified in the following scheme: Ask a question and formulate a hypothesis (educated guess) - induction Derive predictions from the hypothesis - deduction Test the hypothesis and its predictions - induction.

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**INDUCTION VS DEDUCTION, TWO SIDES OF THE SAME COIN (1)**

Deduction, if applied correctly, leads to true conclusions. But deduction itself is based on the fact that we know something for sure (premises must be true). For example we know the general law which can be used to deduce some particular case, such as “All humans are mortal. Socrates is human. Therefore is Socrates mortal.” How do we know that all humans are mortal? How have we arrived to the general rule that our deduction is based on? Again, there is no other method at hand but (empirical) induction.

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**INDUCTION VS DEDUCTION, TWO SIDES OF THE SAME COIN (2)**

Even the process of induction implies use of deductive rules. On our way from specific (particular) up to universal (general) we use deductive reasoning. We collect the observations or experimental results and extract the common patterns or rules and regularities by deduction. For example, in order to infer by induction the fact that all planets orbit the Sun, we have to analyze astronomical data using deductive reasoning.

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**INDUCTION & DEDUCTION: Traditional View**

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**Deduction-Induction Roller Coaster (A Loop)**

general deduction induction particular

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**INDUCTION & DEDUCTION GENERAL PARTICULAR Problem domain INDUCTION**

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INDUCTION & DEDUCTION “There is actually no such thing as a distinct process of induction” said Stanly Jevons; “all inductive reasoning is but the inverse application of deductive reasoning” – and this was what Whewell meant when he said that induction and deduction went upstairs and downstairs on the same staircase.” …(“Popper, of course, is abandoning induction altogether”). Peter Medawar, Pluto’s Republic, p 177.

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INDUCTION & DEDUCTION In short: deduction and induction are - like two sides of a piece of paper - the inseparable parts of our recursive thinking process.

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**FALLACIES - ERRORS IN REASONING**

What about incorrectly built arguments? Let us make the following distinction: A formal fallacy is a wrong formal construction of an argument. An informal fallacy is a wrong inference or reasoning.

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**FORMAL FALLACIES (1) An example: “Affirming the consequent"**

"All fish swim. Kevin swims. Therefore Kevin is a fish", which appears to be a valid argument. It appears to be a modus ponens, but it is not! If H is true, then so is I. (As the evidence shows), I is true. H is true This form of reasoning, known as the fallacy of "affirming the consequent" is deductively invalid: its conclusion may be false even if premises are true.

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**FORMAL FALLACIES (2) Incorrect deduction from auxiliary hypotheses**

If H and A1, A2, …., An is true, then so is I. But (As the evidence shows), I is not true. H and A1, A2, …., An are all false (Comment: One can be certain that H is false, only if one is certain that all of A1, A2, …., An are all true.)

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INFORMAL FALLACIES (1) An informal fallacy is a mistake in reasoning related to the content of an argument. Appeal to Authority Ad Hominem (personal attack) False Cause (synchronicity; unrelated facts that appear at the same time coupled) Leading Questions

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**INFORMAL FALLACIES (2) Appeal to Emotion**

Straw Man (attacking the different problem) Equivocation (not the common meaning of the word) Composition (parts = whole) Division (whole = parts) See more on:

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**SOME NOT ENTIRELY UNCOMMON “PROOF TECHNIQUES”**

Proof by vigorous hand waving Works well in a classroom or seminar setting. Proof by cumbersome notation Best done with access to at least four alphabets and special symbols. Proof by exhaustion Proof around until nobody knows if the proof is over or not… Read the rest on

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**UNDERSTANDING PHENOMENA IN NATURE: CAUSALITY AND DETERMINISM**

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**CAUSALITY AND DETERMINISM**

Causality establishes that one thing causes another. Practical question (object-level): what was the cause (of an event)? Philosophical question (meta-level): what is the meaning of the concept of a cause?

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CAUSALITY Early natural philosophers, concentrated on conceptual issues and questions (why?). Later natural philosophers and scientists concentrated on more concrete issues and questions (how?). The change in emphasis from conceptual to concrete coincides with the rise of empiricism.

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**ARISTOTLE’S CAUSALITY: The Four Causes**

The material cause - constituents, substratum or materials. This reduces the explanation of causes to the parts (factors, elements, constituents, ingredients). The formal cause - form, pattern, essence, whole, synthesis or archetype. The account of causes in terms of fundamental principles or general laws - the influence of the form (essence). The efficient cause - 'what makes what is made and what causes change of what is changed - agency, nonliving or living, acting as the sources of change. The final cause or telos is the purpose or end that something is supposed to serve. Omitted from present day causal explanations.

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CAUSALITY David Hume was probably the first philosopher to postulate a entirely empirical definition of causality. Of course, both the definition of "cause" and the "way of knowing" whether X and Y are causally linked have changed significantly over time. Some natural philosophers deny the existence of "cause" and some natural philosophers who accept its existence, argue that it can never be known by empirical methods. Modern scientists, on the other hand, define causality in limited contexts (e.g., in a controlled experiment).

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DETERMINISM Determinism is the philosophical doctrine which regards everything that happens as solely and uniquely determined by what preceded it. From the information given by a complete description of the world at time t, a determinist believes that the state of the world at time t + 1 can be deduced; or, alternatively, a determinist believes that every event is an instance of the operation of the laws of Nature.

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**Critique of Usual Naïve Image of Scientific Method**

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

The naive inductivist idea of scientific inquiry sees scientific process as consisting of the following steps: All facts are observed and recorded. All observed facts are analyzed, compared and classified, without hypotheses or postulates other than those necessarily involved in the logic of thought. Generalizations inductively made about the relations, structural or causal, between the facts. Further research consists of inferences (deductions) from previously established generalizations.

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**Critique of Usual Naïve Image of Scientific Method (2)**

This narrow idea of scientific investigation 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 observing.

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**Critique of Usual Naïve Image of Scientific Method (3)**

A hypothesis (preliminary theory) is needed to give the direction to 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 general principles. Unfortunately, such rules do not exist!

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**Why is it not (yet)* possible to derive theory directly (automatically) 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, calorimetric data, etc. 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. * However, we cannot exclude the possibility of intelligent automated process of discovery!

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**Why is it not (yet) possible to derive theory directly (automatically) 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 it 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”

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**Knowledge and Objectivity: Observations**

KNOWLEDGE AND JUSTIFICATION Knowledge and Objectivity: Observations Observations 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”.

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**KNOWLEDGE AND OBJECTIVITY Observations**

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. Although science cannot provide one with hundred percent certainty, yet it is the most, if not the only, objective (inter-subjective) mode of pursuing knowledge.

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**Perception and “Direct Observation”**

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**Perception and “Direct Observation”**

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**Perception and “Direct Observation”**

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**Perception and “Direct Observation”**

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

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**Perception and “Direct Observation”**

Checker-shadow illusion See even: Lightness Perception and Lightness Illusions

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Direct Observation?! An 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 the Tevatron accelerator at Fermilab provided evidence that the elusive subatomic particle known as 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)

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**TRUTH (1) The correspondence theory The coherence theory**

The deflationary theory

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**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) 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 ( ), Leibniz ( ) and Hegel ( ). Characteristically 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 used in all sciences.

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**TRUTH (4) The Deflationary Theory**

The deflationary theory is belief that it is always logically unnecessary 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 case snow is white.

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**TRUTH (5) The Deflationary Theory**

The Deflationary Theory is also called the redundancy theory, the disappearance theory, the no-truth theory, the disquotational theory, and the minimalist theory. See: Stanford Encyclopedia of Philosophy

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**If you want to learn more here are reach sources of further reading…**

A Mathematical Analysis of The Scientific Method, The Axiomatic Method, and Darwin's Theory Of Evolution, G. J. Chaitin

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Chaitin’s work on Epistemology, Information Theory, and Metamathematics important for understanding of Formal Systems and their Relationship with Biology: Epistemology as Information Theory: From Leibniz to Ω The Halting Probability Omega: Irreducible Complexity in Pure Mathematics Randomness in Arithmetic and the Decline & Fall of Reductionism in Pure Mathematics The Search for the Perfect Language

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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 with formal proof. See the special issue on formal proof of the AMS Notices, December (From Chaitin’s lectures) David Malone, Dangerous Knowledge, BBC TV, 90 minutes, Google video vividly illustrates the search for TOE in mathematics

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

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**Whole vs. Parts tusk spear tail rope trunk snake side wall**

leg tree The 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 With respect to the truth content, there are different views of science: Science as controversy (new science, frontiers) Science as consensus (old, historically settled) Science as knowledge about complex systems Open systems with paraconsistent logic

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**PROOF The word proof can mean:**

a test assessing the validity or quality of something. a rigorous, compelling argument, including: a logical argument or a mathematical proof a large accumulation of scientific evidence and alike In mathematics, a proof is a demonstration that, given certain axioms, some statement of interest is necessarily true.

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**PROOF OF PYTHAGORAS THEOREM**

Einsteins Proof Water proof Geometrical proof

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**Pressupositions and Limitations of Axiomatic Logical Systems**

Axiomatic theory is built on a set of few axioms/postulates (ideas which are considered so elementary and obvious 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 those axioms. Thus axiomatic theories are closed logical systems. For open axiomatics, see Unconventional Algorithms: Complementarity of Axiomatics and Construction, Dodig Crnkovic G. and Burgin M. 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 previous system.

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**GÖDEL: TRUTH AND PROVABILITY (1)**

Kurt Gödel 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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**GÖDEL: TRUTH AND PROVABILITY (2)**

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: TRUTH AND PROVABILITY (3)**

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)

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**TRUTH VS. PROVABILITY ACCORDING TO GÖDEL**

After: Gödel, Escher, Bach - an Eternal Golden Braid by Douglas Hofstadter.

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**TRUTH 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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CRITICAL THINKING (1) Critical thinking is rationally deciding what to believe or do. To rationally decide something is to evaluate claims to see whether they make sense, whether they are coherent, and whether they are well-founded on evidence, through inquiry and the use of criteria developed for this purpose. Critical Thinking

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**CRITICAL THINKING (2) How Do We Think Critically? A. Question**

First, we ask a question about the issue that we are wondering about. For example “What is going on?" B. Answer (hypothesis) Next, we propose an answer or hypothesis for the question raised. A hypothesis is a "tentative theory provisionally adopted to explain certain facts." We suggest a possible hypothesis, or answer, to the question posed. For example “A phenomenon occurs under certain conditions."

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**CRITICAL THINKING (3) C. Test**

Testing the hypothesis is the next step. With testing, we draw out the implications of the hypothesis by deducing its consequences (deduction). We then think of a case which contradicts the claims and implications of the hypothesis (inference). For example, “If phenomenon really exists it will systematically occur under certain conditions.“ Criteria for truth Criteria are used for testing the truth of a hypothesis such as self-consistency, consistency with existing knowledge, empirical confirmation, etc.

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PSEUDOSCIENCE (1) A pseudoscience is set of ideas and activities resembling science but based on fallacious assumptions and supported by fallacious arguments. Martin Gardner: Fads and Fallacies in the Name of Science

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PSEUDOSCIENCE (2) Motivations for promotion of pseudoscience range from simple lack of knowledge or no skills in the scientific method, to deliberate deception for winning a power, financial gain or other benefits. Some people consider some or all forms of pseudoscience to be harmless entertainment or sort of counseling. Others, such as Richard Dawkins, and James Randi consider all forms of pseudoscience to be harmful, whether or not they result in immediate harm to their followers. Richard Dawkins, Professor of the Public Understanding of Science at Oxford University web page.

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PSEUDOSCIENCE (3) Typically, pseudoscience fails to meet the criteria met by science generally (including the scientific method), and can be identified by one or more of the following rules of thumb: asserting claims without supporting experimental evidence; asserting claims which contradict experimentally established results; failing to provide an experimental possibility of reproducible results; or violating Occam's Razor (the principle of choosing the simplest explanation when multiple viable explanations are possible.

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**PSEUDOSCIENCE (4) Astrology Dowsing Creationism ETs & UFOs**

Supernatural Parapsychology/Paranormal New Age Divination (fortune telling) Graphology Numerology Velikovsky's, von Däniken's, and Sitchen's theories Pseudohistory Homeopathy Healing Alternative Medicine Cryptozoology Lysenkoism Psychokinesis Occult & occultism

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PSEUDOSCIENCE (5) James Randi Exposes Telekinesis (1:48) Randi Tests An Aura Reader (3:24) James Randi on Astrology (1:36) James Randi Tests a Dowser (5:46) James Randi and Richard Dawkins (6:23) Dawkins debunks dowsing (4:57) Richard Dawkins on alternative medicine and the nature of science (1:14) Astrology Numerology and You-4 (3:26) Carl Sagan on Astrology (8:35) Ben Goldacre on Homeopathy (3:04)

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**PSEUDOSCIENCE AS WISHFUL THINKING**

No science can predict human future with certainty –pseudosciences fulfill human wish to know their future. No science can cure all diseases – but pseudosciences fulfill human wish to have cure for every disease. No science can what pseudosciences claim to be able to! While sciences support critical thinking, pseudosciences apply wishful thinking.

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**PSEUDOSCIENCE (6) http://skepdic.com/ The Skeptic's Dictionary,**

Skeptical Inquirer The Swedish Skeptic movement (in Swedish) Scientific Evidence Supporting Evolution Scientific American, July 2002: 15 Answers to Creationist Nonsense Human Genome, Nature 409, (2001)

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**THE PROBLEM OF DEMARCATION (1)**

After more than a century of active dialogue, the question of what marks the boundary of science remains formally unsettled. As a consequence the issue of what constitutes pseudoscience continues to be controversial. Nonetheless, reasonable consensus exists on a number of issues.

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**THE PROBLEM OF DEMARCATION (2)**

Criteria for demarcation have traditionally been coupled to philosophy of science. Logical positivism, for example, held that only statements about empirical observations are meaningful, effectively asserting that statements which are not derived in this manner (including all metaphysical statements) are meaningless.

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**THE PROBLEM OF DEMARCATION (3)**

Karl Popper attacked logical positivism and introduced his own criterion for demarcation, falsifiability. Thomas Kuhn and Imre Lakatos proposed criteria that distinguished between progressive and degenerative research programs.

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**THE PROBLEM OF DEMARCATION (4)**

Read a book by astrophysicist Carl Sagan against pseudoscience: The Demon-Haunted World (13:56) (2:49:35) The book explains the scientific method and encourage people to learn critical thinking. It explains methods to help distinguish between science, and pseudoscience by means of critical thinking.

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**THEORY OF SCIENCE ASSIGNMENTS**

Assignment 2: Demarcation of Science vs. Pseudoscience (in groups of two) 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 Assignment 3-extra (For those who miss the discussion of the Assignment 3) Deadlines

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**Assignment 2: Demarcation: Pseudoscience vs**

Assignment 2: Demarcation: Pseudoscience vs. Science (done in groups of two) [1] Hansson, Sven Ove, "Science and Pseudo-Science", The Stanford Encyclopedia of Philosophy (Fall 2008 Edition), Edward N. Zalta (ed.), [2] [3] The Astrotest A tough match for astrologers (Rob Nanninga), Further reading Popper on Demarcation (Stanford Encyclopedia): Astrology Fact Sheet (North Texas Skeptics),

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**Assignment 2: Demarcation: Pseudoscience vs**

Assignment 2: Demarcation: Pseudoscience vs. Science (done in groups of two) Use the template (Answer form) Leave the template unchanged, write down your answer after each question. Think critically! You are expected to work in groups of two. Your text should not be shorter than two A4 pages written in usual text format. Prepare for the discussion in the class! Please write the file name in the following format: name1_name2_a2.doc

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**APPENDIX For additional reading**

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ISLANDS OF KNOWLEDGE “You see, you have all of mathematical truth, this ocean of mathematical truth. And this ocean has islands. An island here, algebraic truths. An island there, arithmetic truths. An island here, the calculus. And these are different fields of mathematics where all the ideas are interconnected in ways that mathematicians love; they fall into nice, interconnected patterns. But what I've discovered is all this sea around the islands.” Gregory Chaitin, an interview, September 2003

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**Two Examples of Axiomatic Systems -Limitations and Developments**

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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). 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?

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Pressupositions and Limitations of Formal Logical Systems Axiomatic System of Euclid: Shaking up Geometry Until the 19th century no one was too concerned about axiomatization. Geometry has 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 point somewhere outside it.

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**EUCLID'S AXIOMS (1) Every two points lie on exactly one line.**

Any line segment with given endpoints may be continued in either direction. 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) 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.) (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. 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 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 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 spherical 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 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'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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**MORE PROBLEMS WITH AXIOMATIZATION…**

Not 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! He 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 (manipulating the strings of symbols) from a fixed system of 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 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 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 set.

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**RUSSELL'S PARADOX (2) 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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RUSSELL'S PARADOX (3) 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'S PARADOX (4) 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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RUSSELL'S PARADOX (5) … 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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LOGIC

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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 and transparency. Classical 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".

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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. 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 LOGIC FUZZY 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 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 Cumulative logic Deontic logic Dynamic logic Epistemic logic Erotetic logic Free logic Fuzzy logic Higher-order logic Infinitary logic Intensional logic Intuitionistic logic Linear logic Many-sorted logic Many-valued logic Modal logic Non-monotonic logic Paraconsistent logic Partial logic Prohairetic logic Quantum logic Relevant logic Stoic logic Substance logic Substructural logic Temporal (tense) logic Other logics

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**MATHEMATICAL INDUCTION**

The aim of the empirical induction is to establish the law. In the mathematical induction we have the law already formulated. We must prove that it holds generally. The basis for mathematical induction is the property of the well-ordering for the natural numbers.

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**THE PRINCIPLE OF MATHEMATICAL INDUCTION**

Suppose P(n) is a statement involving an integer n. Than to prove that P(n) is true for every n n0 it is sufficient to show these two things: 1. P(n0) is true. 2. For any k n0, if P(k) is true, then P(k+1) is true.

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**THE TWO PARTS OF INDUCTIVE PROOF**

the basis step the induction step. In the induction step, we assume that statement is true in the case n = k, and we call this assumption the induction hypothesis.

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**THE STRONG PRINCIPLE OF MATHEMATICAL INDUCTION (1)**

Suppose P(n) is a statement involving an integer n. In order to prove that P(n) is true for every n n0 it is sufficient to show these two things: 1. P(n0) is true. 2. For any k n0, if P(n) is true for every n satisfying n0 n k, then P(k+1) is true.

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**THE STRONG PRINCIPLE OF MATHEMATICAL INDUCTION (2)**

A proof by induction using this strong principle follows the same steps as the one using the common induction principle. The only difference is in the form of induction hypothesis. Here the induction hypothesis is that k is some integer k n0 and that all the statements P(n0), P(n0 +1), …, P(k) are true.

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**Example. Proof by Strong Induction**

P(n): n is either prime or product of two or more primes, for n 2. Basic step. P(2) is true because 2 is prime. Induction hypothesis. k 2, and for every n satisfying 2 n k, n is either prime or a product of two or more primes.

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**Statement to be shown in induction step:**

If k+1 is prime, the statement P(k+1) is true. Otherwise, by definition of prime, k+1 = r·s, for some positive integers r and s, neither of which is 1 or k+1. It follows that 2 r k and 2 s k. By the induction hypothesis, both r and s are either prime or product of two or more primes. Therefore, k+1 is the product of two or more primes, and P(k+1) is true.

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The strong principle of induction is also referred to as the principle of complete induction, or course-of-values induction. It is as intuitively plausible as the ordinary induction principle; in fact, the two are equivalent. As to whether they are true, the answer may seem a little surprising. Neither can be proved using standard properties of natural numbers. Neither can be disproved either!

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**This means essentially that to be able to use the induction principle, we must adopt it as an axiom.**

A well-known set of axioms for the natural numbers, the Peano axioms, includes one similar to the induction principle.

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**PEANO'S AXIOMS 1. N is a set and 1 is an element of N.**

1. N is a set and 1 is an element of N. 2. Each element x of N has a unique successor in N denoted x'. 3. 1 is not the successor of any element of N. 4. If x' = y' then x = y. 5. (Axiom of Induction) If M is a subset of N satisfying both: 1 is in M x in M implies x' in M then M = N.

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CAUSALITY What does the scientist mean when (s)he says that event b was caused by event a? Other expressions are: bring about, bring forth produce create… …and similar metaphors of human activity. Strictly speaking it is not a thing but a process that causes an event.

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CAUSALITY Analysis of causality, an example (Carnap): Search for the cause of a collision between two cars on a highway. According to the traffic police, the cause of the accident was too high speed. According to a road-building engineer, the accident was caused by the slippery highway (poor, low-quality surface) According to the psychologist, the man was in a disturbed state of mind which caused the crash.

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**CAUSALITY But what was The cause of the accident?**

An automobile construction engineer may find a defect in a structure of a car. A repair-garage man may point out that brake-lining of a car was worn-out. A doctor may say that the driver had bad sight. Etc… Each person, looking at the total picture from certain point of view, will find a specific condition such that it is possible to say: if that condition had not existed, the accident might not have happened. But what was The cause of the accident?

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CAUSALITY It is quite obvious that there is no such thing as The cause! No one could know all the facts and relevant laws. (Relevant laws include not only laws of physics and technology, but also psychological, physiological laws, etc.) But if someone had known, (s)he could have predicted the collision!

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CAUSALITY The event called the cause, is a necessary part of a more complex web of circumstances. John Mackie, gives the following definition: A cause is an INSUFFICIENT BUT NECESSARY part of a complex of conditions which together are UNNECESSARY BUT SUFFICIENT for the effect. This definition has become famous and is usually referred to as the INUS-definition: a cause is an INUS-condition.

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CAUSALITY The reason why we are so interested in causes is primarily that we want either to prevent the effect or else to promote it. In both cases we ask for the cause in order to obtain knowledge about what to do. Hence, in some cases we simply call that condition which is easiest to manipulate as the cause.

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CAUSALITY Summarizing: Our concept of a cause has one objective and subjective component. The objective content of the concept of a cause is expressed by its being an INUS condition. The subjective part is that our choice of one factor as the cause among the necessary parts in the complex is a matter of interest.

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CAUSE AND CORRELATION Instead of saying that the same cause always is followed by the same effect it is said that the occurrence of a particular cause increases the probability for the associated effect, i.e., that the cause sometimes but not always are followed by the effect. Hence cause and effect are statistically correlated.

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**CAUSE AND CORRELATION X and Y are correlated if and only if:**

P(X/Y) > P(X) and P(Y/X) > P(Y) [The events X and Y are positively correlated if the conditional probability for X, if Y has happened, is higher than the unconditioned probability, and vice versa.]

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**CAUSE AND CORRELATION Reichenbach's principle:**

If events of type A and type B are positively correlated, then one of the following possibilities must obtain: A is a cause of B, or ii) B is a cause of A, or iii) A and B have a common cause.

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**CAUSE AND CORRELATION The idea behind Reichenbach’s principle is:**

Every real correlation must have an explanation in terms of causes. It just can’t happen that as a matter of mere coincidence a correlation obtains.

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CAUSE AND CORRELATION We and other animals notice what goes on around us. This helps us by suggesting what we might expect and even how to prevent it, and thus fosters survival. However, the expedient works only imperfectly. There are surprises, and they are unsettling. How can we tell when we are right? We are faced with the problem of error. W.V. Quine, 'From Stimulus To Science', Harvard University Press, Cambridge, MA, 1995.

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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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**The Classical (Ideal) Model of Science**

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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**The Classical (Ideal) Model of Science**

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 scientific explanation. 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, KNOWLEDGE, TRUTH, MEANING**

SCIENCE, KNOWLEDGE, TRUTH, MEANING. FORMAL LOGICAL SYSTEMS AND THEIR LIMITATIONS The science is not about the search for truth (“absolute truth”) but the search for meaning in the form of explanations/models/ simulations that work: “No such (scientific) model, however comprehensive, coherent or well entrenched it might be, can lay an automatic claim to objective truth, even though contextually it may provide a reliable and successful explanatory tool for making sense of what is going on around us.” Edo Pivčević, The Reason Why: A Theory of Philosophical Explanation, KruZak, 2007 Knowledge networks in communities of practice - Language

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**(Philosophy, History, Linguistics …)**

Classical Sciences in their Cultural Context – A Language Based Scheme Culture (Religion, Art, …) 5 Natural Sciences (Physics, Chemistry, Biology, …) 2 Social Sciences (Economics, Sociology, Anthropology, …) 3 The Humanities (Philosophy, History, Linguistics …) 4 Logic & Mathematics 1

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CRITICAL THINKING (1) Critical thinking is rationally deciding what to believe or do. To rationally decide something is to evaluate claims to see whether they make sense, whether they are coherent, and whether they are well-founded on evidence, through inquiry and the use of criteria developed for this purpose. Critical Thinking

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**CRITICAL THINKING (2) How Do We Think Critically? A. Question**

First, we ask a question about the issue that we are wondering about. For example, "Is there right and wrong?" B. Answer (hypothesis) Next, we propose an answer or hypothesis for the question raised. A hypothesis is a "tentative theory provisionally adopted to explain certain facts." We suggest a possible hypothesis, or answer, to the question posed. For example, "No, there is no right and wrong."

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**CRITICAL THINKING (3) C. Test**

Testing the hypothesis is the next step. With testing, we draw out the implications of the hypothesis by deducing its consequences (deduction). We then think of a case which contradicts the claims and implications of the hypothesis (inference). For example, "So if there is no right or wrong, then everything has equal moral value (deduction); so would the actions of Hitler be of equal moral value to the actions of Mother Theresa (inference)? as Value nihilism ethics claims"

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**CRITICAL THINKING (4) 1. Criteria for truth**

Criteria are used for testing the truth of a hypothesis. The criteria may be used singly or in combination. a. Consistent with a precondition Is the hypothesis consistent with a precondition necessary for its own assertion? For example, is the assertion "there is no right or wrong" made possible only by assuming a concept of right or wrong - namely, that it is right that there is no right or wrong and that it is wrong that there is right or wrong?

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**CRITICAL THINKING (5) b. Consistent with itself (self-consistent)**

Is the hypothesis consistent with itself? For example, is the assertion that "there is no right or wrong" itself an assertion of right or wrong? c. Consistent with language Is the hypothesis consistent with the usage and meaning of ordinary language? For example, do we use the words "right" or "wrong" in our language and do the words refer to concepts and meanings which we consider "right" and "wrong"?

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**CRITICAL THINKING (6) d. Consistent with experience**

Is the hypothesis consistent with experience? For example, do people really live as if there is no right or wrong? e. Consistent with the consequences Is the hypothesis consistent with its own consequences, can it actually bear the burden of being lived? For example, what would the consequences be if everyone lived as if there was no right or wrong?

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Knowledge & Faith Dr. Carl J. Wenning Department of Physics Illinois State University.

Knowledge & Faith Dr. Carl J. Wenning Department of Physics Illinois State University.

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