Glutamate Hypothesis



GODEL'S THEOREM/PROOF. The Czech Republic-born American logician and mathematician Kurt Godel (1906-1978) formulated his "incompleteness theorem" in 1931, which states that in any formal system that employs arithmetic, it is possible to develop statements that are true but that cannot be proved within the system. Another way to express Godel's theorem is: All consistent axiomatic formulations of number theory include undecidable propositions. This "theorem of incompleteness" means that all mathematics is based on a set of axioms; some mathematical truths cannot be derived from these axioms, and the set of axioms, therefore, is incomplete. In its barest form, Godel's formulation involves the translation of an ancient paradox (i.e., Epi-menides' paradox, or the liar paradox) in philosophy into mathematical terms. Epi-menides (c. 6th century, B.C.) was a Cretan prophet who made one immortal statement: "All Cretans are liars" (cf., Titus 1:12 in the

New Testament of the Bible). Other versions of this statement are, simply, "I am lying;" or "This statement is false." This latter version of Epimenides paradox violates the usually assumed dichotomy of statements into categories of "true" and "false," because if you tentatively think the statement is "true," then it immediately backfires on you and makes you think it is "false." On the other hand, once you have decided the statement is "false," a similar backfiring returns you to the idea that it must be "true." Godel's theorem had a significant effect on logicians, mathematicians, and philosophers interested in the foundations of mathematics because it showed that no fixed system - no matter how complicated - could represent the complexity of the whole numbers (0, 1, 2, 3, etc.). Godel's theorem/proof, also, has had a bearing on psychology, especially in the area of artificial intelligence (AI). For instance, computers must be programmed for AI, but there is only a finite number of possible programs. Humans, on the other hand, are capable of an unlimited number and variety of behaviors. Therefore, any set of existing computer programs would be incomplete and, consequently, this fact indicates that it is impossible to construct a machine (AI) that behaves like a human being. Godel's theorem/proof has been used to argue against both the "strong AI" position (that "conscious awareness/thought" may be explained in terms of computational principles) and the "weak AI" position (that "conscious awareness/ thought" may be simulated by computational procedures). In this context, then, Godel's theorem/proof argues in favor of the viewpoint that a fundamental difference exists between "human intelligence" and "artificial intelligence." See also ARTIFICIAL INTELLIGENCE; CELLULAR AUTOMATON MODEL. REFERENCES

Godel, K. (1931). Uber formal unentscheid-bare satze der Principia Mathe-matica und verwandter systeme I. Monatshefte fur Mathematik und Physik, 38, 173-198. Godel, K. (1962). On formally undecidable propositions. New York: Basic Books.

Hofstadter, D. R. (1979). Godel, Escher, Bach: An eternal golden braid. New York: Basic Books.

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