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Andrew Olson Center for Inquiry Indiana

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In the 19 th century, scientists felt it desirable to develop a logical foundation for all of science. Something similar to Euclid’s five postulates for plane geometry Leibnitz broke with tradition, set by Aristotle, that all reasoning could be reduced to syllogisms. Georg Boole published his “Boolean Algebra” in 1847, a significant advance

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C. S. Pierce (1885) and Ernst Schroeder (1890) extended Booles’ work. Guiseppe Peano proposed axioms to formalize arithmetic in 1889, but it did not include inference rules. Gottlob Frege (1848-1925) developed concept of formal system in his 1893 treatise, using a difficult notation. He also axiomatized arithmetic with one axiom. Bertrand Russell simplified Frege’s system with Pierce’s notation. He found a paradox in it, showing it was inconsistent (1902).

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Russell’s Paradox Set, S, of all sets not members of themselves If S is one of its sets, then it is not a member of itself. If S is not a member, it is a member of itself by construction! Epimenides from Crete said “All statements of Cretans are lies.” Cantor’s paradox The power set, S 2, of a set, S, is the set of all subsets of S. It’s cardinality > that of S. The universal set, U, is set of all sets. Cardinality of U 2 > cardinality of U. This contradicts the definition of U.

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David Hilbert (1862-1943), famous mathematician Proposed Program to establish foundation for science. Ten specific problems whose solutions would lead to this foundation. Leading three: The “Continuum Hypothesis” The consistency of the axioms of arithmetic Cantor’s 1883 conjecture: The real numbers can be “well-ordered”.

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Ernst Zermelo proved a generalization of Cantor’s conjecture in 1904. Also, an axiomatized set theory in 1908. Not until 1963 was Hilbert’s first problem, “the Continuum Hypothesis” was conquered. Paul Cohen showed that it is not “solvable”, i.e., provable, with axioms of set theory. He used set theory methods Kurt Gödel developed in 1938. Bertrand Russell and Alfred North Whitehead wrote (1910-1913) treatise, to give a logical foundation for all the mathematics known then.

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They used theory of “types” to avoid self-reference in statements like “a set of sets”. For example, consider the sets, s 0 1 ={1}, s 0 2 ={2}, s 0 3 ={3}, s 0 4 ={4}, s 0 5 ={4, 5}, s 0 6= {6},…, as well as sets of such sets: {{3}}, {{1}, {4, 5}},… Let sets of the form s 1 1 ={s 0 1, s 0 2 }, s 1 2 ={s 0 2, s 0 3 }, s 1 3 ={ s 0 3, s 0 4, s 0 5 }, s 1 4 ={s 0 6 },… have a type of one higher order. Let the set s t+1 i have type one higher than s t i. Then, s 2 1 ={s 1 1, s 1 3 }, s 2 2= { s 1 1, s 1 3, s 0 1 }, … have type one higher than s 1 i, i = 1, 2, 3,… The populations of elements these are made from increases as t increases.

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Three approaches to Hilbert’s Program: Logicists: Russell and Frege viewed logic as universal system in which all mathematics should be derived. Validity of mathematical statement depends on its logical meaning. This was apparently a Platonic view. Formalists: Hilbert, et al, said mathematics justified only by the consistency of formal systems with finite symbol manipulation following syntatic rules. A logical positivist view Constructivists : Restrict mathematics to finite, operations and “potentially” infinite objects. Replaced idea of “arbitrarily small” by explicit object on which operations could be performed.

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By 1928, it was shown that consistency of was reducible to proving system of arithmetic (i.e., for the natural numbers, 1, 2, 3, 4,…) was consistent. Thus, Hilbert’s Program required an algorithm: Algorithm: Procedure with distinct steps that generates final value after finite number of operations.

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Kurt Gödel: born April 28, 1906 in Born, Moravia, Hapsburg Empire. Father, Rudolf, trained technically, became partner in Born textile factory. They became well to do. Mother, Marianne, well educated, very cultured. Older brother, Rudolf, studied to be a doctor. Kurt was inquisitive and precociousness. He was also reticent and doted on his mother. This was compounded by rheumatic fever age 6. At 8, he discovered this could damage his heart. This made him health anxiety that persisted (unjustifiably).

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His brother, Rudolf, stated: “My brother had a very individual and fixed opinion about everything and could hardly be convinced otherwise. Unfortunately he believed all his life that he was always right not only in mathematics but also in medicine, so he was a very difficult patient for doctors. After severe bleeding from a duodenal ulcer... for the rest of his life he kept to an extremely strict ( over strict? ) diet which caused him slowly to lose weight.”

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Kurt enrolled in the University of Vienna in the fall of 1924, There he studied mathematics, physics and philosophy with He wanted to be a physicist. Kurt fell in love with the views of Plato during Heinrich Gomperz’ lectures. He declared Gomperz was the one person who most influenced his life. From 1925 on, he was an implacable Platonist. Gödel rated Philip Furtwängler’s lectures (number theory) among the most stimulating he had. This may be cause of his change to mathematics in 1926.

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Greek Sophists : Travelling philosophers, one of whom was Progatoras. He said, “Man is the measure of all things.” This referred to the moral sphere, not the world in general. The Sophists felt there were no objective rights or wrongs, only differences of opinion. Plato argued against this, claiming objective truth in all aspects of life. He based this objectivity of abstract reality. This is reachable by reason. Life, he argued, should be the practice of impassioned reason. This philosophy was very influential until Renaissance.

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Gödel accepted number theory as the strongest evidence of conceptual realism. At the age of 22 in 1928, this goal lead him to pursue mathematical logic. From 1926 on, he frequently engaging in discussions with the faculty at Vienna about important topics of mathematics, physics and philosophy. People were impressed by Gödel’s intellectual gifts during these discussions.

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Vienna Circle was major influence in Vienna from 1924-1933. Hans Hahn, mathematician of broad interests, founded it. A group of eminent faculty members of the University met weekly to discuss major ideas in logic. Admission was by invitation only, but Hans Hahn invited Gödel to join in 1926. Karl Popper was not invited. Ludwig Wittgenstein refused to join, although he was invited, and carried on conversations with it The Circle was greatly influenced by reading his works. The Circle’s members were logical positivists. Niels Bohr, Max Born and Werner Heisenberg were not members despite being positivists.

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Positivists’ goal: remove pure philosophy from the sciences. They followed David Hume (1711-1776), who claimed that a priori reasoning provides no meaning about world – it concerns only concepts’ relationships. Only empirical means can show propositions about fact or existence to be “true” or “false”. They accept mathematical logic as valid a priori knowledge of synthetic form, but having no content about the world (semantic knowledge).

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Platonists: Included both Einstein and Gödel Viewed mathematical “truth” as descriptive of the world, but not empirical. Mathematical “truths” are independent of humans. A descriptive proposition is “true” or “false”, not just because of its meaning, but also because it depends on the pertinent facts in the world. Gödel denied that mathematics is the nothing but the syntax of language. He claimed his results disproved this.

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Gödel was quite reserved in the Circle’s meetings, perhaps for his deep disagreement with its positivistic ideas. It stimulated the development of own approach. For example, his work on his doctoral dissertation under Hans Hahn. This proved a type of completeness of “first order logic.” The consistency had already been proved. His theorem says the inference rules of predicate calculus suffice to derive all consequences of the axioms in a first order logic. This is not “completeness” with negation, his topic of his later incompleteness papers.

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He presented his dissertation in a session for short papers on the 2nd day of a symposium in Königsberg, Germany, September 5-7, 1930. Main speakers presented the first day. Results of the previous days were discussed on the last day. Members of the Vienna Circle and other logical positivists dominated the symposium. Not a major player, Gödel spoke little. Only near the end of the third day did he make a terse, offhand comment that he could prove incompleteness of a formal system

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Gödel’s comment went in one ear and out the other of the participants. It is not even in the symposium’s Minutes. Only John von Neumann, from Princeton Univ., noticed. He queried Gödel about it and became convinced of the claim’s validity. He began corresponding with him and discussed it with his colleagues. At the age of 24, Gödel completed the formal proof of his first general “incompleteness theorem” by October, 1930, publishing it, along with the second theorem, in 1931. He wanted to qualify for the position of Privatdozent. Following this, he was quite prolific, publishing many papers and developing in international reputation.

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Gödel‘s first theorem says, specifically: Theorem 1 :Any formal system, S, (with the negation operator) that is consistent and includes number theory contains an unprovable statement, i.e., it is incomplete. The second theorem states: Theorem 2: The consistency of a system, S, satisfying the conditions of Theorem 1, cannot be proved within the system

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His father’s death in 1929 was a blow to Marianne. At that point, she joined the brothers in Vienna so they could care for her. After a long courtship Kurt, married Adele nee Porkert on September 20, 1938. His family was strongly opposed to this because she was divorced, several years his senior, a Catholic, of a lower class, and, worst of all, a dancer.

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He and Adele moved to the Institute of Advanced Study, at Princeton University to escape the draft and Nazi persecution. They arrived in San Francisco on March 4, in 1940 after an arduous trip across Russia Although he was not Jewish, he looked every bit the intellectual and had had many Jewish colleagues, including Hans Hahn. There he made friends with von Neumann and Oskar Morgenstern, He had difficulty forming sincere relations because of his austere personality.

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Albert Einstein was the one true friend with whom he could communicate. They shared many interests, despite the great differences in their personalities and interests. While he was a member of the Institute for Advanced Studies, he was awarded several prestigious honors. After Einstein died, Gödel withdrew into himself. His paranoia worsened. He had had breakdowns in Austria before. He relied on Adele to protect him from being “poisoned”. When she had to enter the hospital for 6 months, he refused to eat. He finally died on January 14, 1978, due to mental illness leading to starvation.

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A formal system of logic, S, is a collection consisting of an alphabet (symbols, such as +, *, AND, ¬, OR, =>), a set of rules forming proper statements, a set of rules for drawing inferences, and a set of statements { s }, each properly composed from the alphabet. Certain of the proper statements are designated axioms, also called “theorems by definition”. A theorem is a proper statement in S that is derivable with a finite sequence of applications of the inference rules to theorems in S. Such a sequence is called a proof.

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A statement in S is provable in S means it is a theorem in S. The alphabet of S contains the negation operator, ¬. S is consistent means that S has no statement, s, for which both s and ¬s are theorems in S. A statement, s, in S is decidable in S means that either s is a theorem or ¬s is a theorem, or both. A proper statement, s, in S is undecidable in S means that neither s nor ¬s are theorems in S.

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Statements ¬s decidable s s Theorems ¬s’ undecidable s’s’ Consistent, Complete System Consistent, Incomplete System Figure 1 Figure 2

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Statements decidable s ¬s s inconsistent s’ s’ ¬s’

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all inconsistent s s Statements=Theorems ¬s Inconsistent, Complete System Figure 4

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Truth or Consequences Unless you get the following use of the terms “true” and “false” right, you will not make the correct interpretation of mathematicians’ results. The only reference to something being “true” is to a proposition, not a theorem. A proposition is a relationship among statements (represented by symbols) in S, so its being “true” means it holds for some particular instances in S of those symbols. A statement has the truth value “true” for some set {t} of terms, t, having valid values in the vocabulary of S means it holds for all such values. It has the truth value “false” for any other such values if it does not hold for them.

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To say the proposition “x - 6 = y + 2” is “true”, where is S a system involving integer arithmetic, means it applies precisely to all integer pairs (x, y) for which x-6=y+2. This is set of integer pairs on line with slope 1 passing through point (0, -8) shown in Figure 5. The proposition is “false” for any other integer pair, such as (0, 0). It has no truth value for any pair like (¾, -7¼), despite the fact that it satisfies the equation (in rational number arithmetic); similarly for (π, π-8) (in real number arithmetic).

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y (16, 8) 0 x (8, 0) (0, -8) (-8, -16) Figure 5 “Goldbach’s conjecture is true.” This means every natural number, n, satisfies either 1<=n<3, n is odd, or n=p+q for some pair (p, q) of primes. No one knows whether on not Goldbach’s conjecture is true, but its condition for being true can be stated.

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In summary: Cannot conclude from claim “s is true in S” that s is provable, i.e., the consequence of a finite chain of theorems in S according to the rules of inference in S. There may, or may not, be such a chain despite the claim.

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Gödel Numbers Proof of Theorem 1 follows the ideas of Russell and Whitehead. Gödel showed how to represent any formal logical system, S, that includes number theory by a hierarchical system of numbers. Assign digits to elementary logical symbols; e.g., ¬, ALL, IMPLIES, THERE EXISTS, =, 0, s, and punctuation symbols, ), (, and. s(…) means immediate successor of its argument, e.g., s(2) = 3

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Alphabet also contains three kinds of variables: numerical (e.g., x, where x could have the value 6 or 11), sentential (representing a logical expression formed from the numerical and elementary symbols, and predicate (express a property of numbers or numerical expressions, like even, prime, or more than) Syntax rules used to form any logical statement in system; e.g., the statement THERE EXISTS x (x = s(x )) Gödel assigned each elementary symbol a specific integer (its Gödel number), such as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 Numerical variables, like x, are represented by their Gödel numbers. These are, in sequence, prime numbers > number of elementary symbols.

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Coded like this, the symbols in example logical statement above are: 4 11 9 11 5 7 9 11 10 8 8 Condense this string to single, unique integer, its Gödel number. Because this example has eleven symbols, calculate its Gödel number by forming product of the first eleven primes, each raised to the corresponding number in this string’s list: 2 4 *3 11 *5 9 *7 11 *11 5 *13 7 *17 9 *19 11 *23 10 *29 8 *31 8 whose value ison order of 270.094*10 95 Inference chains: x IMPLIES x generates the sequence 11 3 11 10 with Gödel number 2 11 *3 3 *5 11 *7 10 = 7,626,831,723*10 11

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This Gödel coding procedure is an algorithm to compute Gödel number of an expression. It is reversible, giving algorithm to determine the expression corresponding to a given Gödel number. Every integer has unique factorization into primes. If two expressions have the same Gödel integer, they have the same factorization into primes. The translation of an expression into its Gödel number is via factorization into primes. If two Gödel numbers are equal, they have the same prime factorization. This is the Gödel coding factorization. So, the two expressions must be the same. Gödel integers encode only valid statements. They do not necessarily encode theorems.

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Let F be function producing propositions in S with single argument, e.g., F(y), where y is valid expression in S that reduces to a positive integer, e.g., y could be n compositions of the operator, s. Example: F(s(s(0))) would reduce to an expression in S. Let GÖ be a function on the propositions in S that produces the Gödel number of its argument proposition. That is, GÖ(p) = Gödel number of p. We have seen how to compute GÖ. We will cite an important result: Fixpoint Theorem :For any F, there is a Gödel integer, n, for which n = GÖ(F(n p )). Here, the argument, n p of F, is interpreted as n compositions of the operator, s, as illustrated above.

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Let PR be a function on the positive integers so that PR(n) has value: “true” if n is the Gödel number of a provable statement, t, in S, and “false” for all other integers. Given any positive integer, n, there is an algorithm to calculate the value of PR(n). Can write the claim, “p is a theorem in S”, in the arithmetical form: PR(GÖ(p)), for any p, sensible or not. Gödel’s proof of Theorem 1 needed about thirty pages because he did not have benefit of many newer results, like the Fixpoint Theorem.

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Theorem 1: Any formal system, S, (with the negation operator) that is consistent and includes number theory contains an undecidable statement, i.e., it is incomplete. Proof: The proposition, ¬PR(n) has the value “true” precisely when n is not Gödel number of a theorem in S. Let proposition q p = ¬PR(n), so that: q p has value “true” precisely when n is not Gödel number of theorem in S. Apply the Fixpoint Theorem to case: F(n p ) = q p, where the argument, n p, of F represents the n-iterated composition, s(s(s(…s(0),,,))). Then there is a Gödel integer, g, for which g = GÖ(F(g p )) = GÖ(q p ). That is, g is the Gödel integer of q p, giving the result: R: q p has the value “true” precisely when g is not Gödel number of a theorem in S. Q.E.D.

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Therefore, q p has value “true” precisely when it is not provable. Suppose q p is provable in S. Then, g is the Gödel number of a theorem in S, i.e., q p, This implies q p is not “true”, i.e., ¬q p is “true”. The proof showing q p is a theorem in S shows ¬q p is “true” because the relation R is expressible in S. That is, ¬q p is “true” is a theorem in S. Thus, both q p and ¬q p are provable in S. This implies S is inconsistent, contrary to hypothesis. Therefore, q p cannot be provable, but it has the value “true”. Q.E.D.

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Theorem 2: The consistency of a system, S, satisfying the conditions of Theorem 1, cannot be proved within the system. Proof: Let C represent the proposition “S is consistent”. Suppose the truth of “C” follows from the logic in S, i.e., “C” is “true” is a theorem. Then, q p is “true” is a consequence in S of this, i.e., q p is “true” is a theorem in S. R, in the body of Theorem 2, implies q p is “true” is equivalent to “q p is not provable in S”. But, we have just proved in S that q p is “true”. The alternative is that “C” cannot be a theorem in S, i.e., it is not provable in the logic of S. Q.E.D.

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The Good Case 1g: Interpreted properly, Gödel’s theorems provide insight into proper way to view problems related to completeness and consistency. This required a deep understanding of issues involved and illustrated a way to deal with many of the paradoxes that supposedly underlay these concepts. Unfortunately, it requires careful reasoning and a grasp of these insights to apply his results properly.

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The Ugly Case 1u: Suppose that S is some formal, consistent system and contains arithmetic. There is some statement, q p, not provable in the logic of S, in particular, not an axiom. Adding q p to the set of axioms of S, gives another system, S, in which q p is a theorem. But Theorem 1 says S is not complete either, Can continue extending S forever. Case 2u: Theorem 2 says that “consistency of S”, i.e. “C”, is not provable in S. Adding statement, “C”, to axioms of S, gives extended system, S Theorem 2 says cannot prove consistency in S either. That is, we cannot construct a system in this way that we can prove is consistent.

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Case 3u: Hilbert wanted to prove all formal theories consistent based on arguments using only finite reasoning, and no infinite sets or methods. Gödel’s theorems do show that this is a vain hope, as Cases 1 and 2 illustrate. Case 4u: In Requirements Engineering, engineer gathers information about a physical system to build and poses these as formal statements describing constraints the system must satisfy, what it must do, how it must perform. (see notes for more details)

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The Bad Case 1b: “By Gödel’s theorem, a system is either incomplete or inconsistent. Thus, logically speaking, it is impossible for us to fully “prove” any proposition.” It is a misstatement of his results, and expresses a gross misunderstanding of them.

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Case 2b: Nagel and Newman claim that Gödel proves: “that it is impossible to establish the internal logical consistency of a very large class of deductive systems – elementary arithmetic, for example, unless one adopts principles of reasoning so complex that their internal consistency is as open to doubt as that of the systems themselves.” Between the words “to” and “unless”, this is a reasonable statement. Beyond that, the authors seem to be are trying cover their inability to understand the implications of the theorems.

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Case 3b: A common misconception of Gödel’s results often has the form: “According to Gödel’s incompleteness theorem, understanding our own minds is impossible, yet we have persisted in seeking this knowledge through the ages!” This is, were we able to construct a system that represents our mind, it would not say whether or our mind is even self consistent. This extrapolates to “understanding” its modes of thinking, and from there, to comprehending one’s self. Sometimes, this argument is reversed to conclude that the human mind exceeds the capability of any “machine”, and so, cannot be a machine. This is a great leap away from of Gödel’s Theorems. The real issue here is the capability being discussed is that of the “mind not knowing what the mind knows.” This is clearly a paradoxical statement arising from its self-reference. Gödel’s theorems do not directly involve self-reference; rather, they involve equating the “truth” of a proposition with its provability. As we saw, “truth” and provable are two distinct concepts.

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Case 5b: Michael Guillen claims “the only possible way of avowing an unprovable truth, mathematical or otherwise, is to accept it as an article of faith.” Daniel Graves, following his line of reasoning from Case 4b, continues this idea with, “In other word, scientists are as subject to belief as non-scientists.” This is a gross misunderstanding of what scientists do when they postulate a property or pose a hypothesis. In the first example, a postulate is considered just a property that may have not any relation to the ‘real world’ at all. The criterion for its inclusion in the system is whether or not it results in an interesting mathematical system. In many cases, however, “interesting” is based on how well the system models a ‘real world’ system. In this case, choosing the postulates is somewhat empirical. In the second example, a hypothesis is chosen to match an expected reality in the physical system under study. The resulting system is subjected to explicit tests to see verify its fit with reality. If poor, the hypothesis is modified so the system fits better. No “belief” is involved at all.

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Case 6b: Najamuddin Mohammed argues from Theorem 1 that, given an unprovable statement, s, in S, a one can choose to include s in a new system, S, while another can choose ¬s in a new system, S. This generates a chain of systems he claims leads to many self-consistent, contradictory systems of reasoning. He demands to know which is right and wrong. It seems that answer depends upon psychological states of mind of those involved. “Correctness, in this state of affairs has no meaning in these cases,” he says; “all this can lead to agnostic and atheistic stances.” This reasoning underlies the position of “postmodernism,”. One of its theses is that social norms are relative, depending solely on the culture around them A norm in one culture is just as valid as its analogue in another. Mohammed is railing against this view, demanding an absolutist philosophy. Neither party is correct in believing that Gödel’s theorems support this view. Mathematicians, and scientists in general, do not wander off generating systems at random. The ones they create must obey strict criteria, along the lines discussed in Case 5b. The surviving mathematical tools have been studied and improved by numerous people. Over time, these are winnowed to just those that have proved most useful in providing a window into the real world. This is not to say that there will be only one system that describes a mathematical structure well. Geometry is an example where there are three systems, each describing a geometry that is useful in an appropriate, real context.

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Nature and Construction of Knowledge in Mathematics

Nature and Construction of Knowledge in Mathematics

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