paul budnik - diversity and the limits of reason

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Diversity and the Limits of Reason Paul Budnik Mountain Math Software www.mtnmath.com [email protected] G¨ odel’s Incompl eteness Theorems imply that diversity must be a complement to reason for unbounded creativ- it y . Understandi ng this is crucial in this age of gl obal- ization. Introduction Logic discovered its own limitations in G¨ odel’s Incompleteness Theorems. Before G¨ odel’s proofs, mathematicians sought a logical mechanistic process that could, with enough time and resources, dec ide any mat hematical question. Gödel proved this was impossible. Any consistent sufficiently powerful set of mathematical assumptions or axioms can be expanded in simple obvious ways and complex ones. The standard example is to add the axiom that the system itse lf is consi sten t. G¨ odel proved this cannot be derived within any consistent system strong enough to define an ideal computer. An ideal computer can run forever error free and has access to unlimited storage. Any single path approach to expanding mathematics will encounter a Gödelian limit. 1 Progress can be made forever, but all the results, over an infinite time, are subsumed in a single axiom that will never be disco ve red. Only a divergent process that explores an ever increasing number of alternatives, providing increasing resources to each viable path, can escape a Gödelian limit. There is no way to select between alternative expan- sions of mathematics beyond eliminating those that are inconsistent or proved wrong for more complex reasons. For example, a con tradiction exists only if two consequences are inconsistent, but there may still be an infinite sequence of consequences that are mutually inconsistent. The proof that all mathematical truth can be explored in this way is trivial. If resources are availabl e, all alternatives can be consid ered , each with ever greater effort. There is evidence that this is relevant to the evolution of the mathematically capable human mind, to the future expansion of mathematics and to cultural creativity in general. 1 Roger Penros e has argue d that quantu m effect s in the brain allo w mathemat icia ns to trans cend the limitation of Gödel’s proof [7]. This is not necessary to explain the mathematically capable human mind. All that requires is the enormou s div ersi ty of biol ogica l evo lution as discu ssed below. There is no significant evidence to support Penrose’s idea. 1

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Page 1: Paul Budnik - Diversity and the Limits of Reason

8/8/2019 Paul Budnik - Diversity and the Limits of Reason

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mailto:[email protected]

http://www.mtnmath.com/

http://www.mtnmath.com/

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http://video.google.com/videoplay?docid=-8677521434864225474

http://www.mtnmath.com/willbe.html

http://www.mtnmath.com/willbe.html

http://math.stanford.edu/~feferman/papers/newaxioms.pdf

http://www.wwnorton.com/catalog/spring99/gunsgerms.htm

http://www.mtnmath.com/pom.html

http://www.mtnmath.com/willbe.html

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