Sunday, 22. May 2005

Mathematics In Relativism Shock

Dennis Jerz linked to this gem of an LA Times article about mathematics going postmodern. Philip Davis, emeritus professor of mathematics at Brown University, shows that certain mathematical definitions drift over time, and proofs of mathematical "truth" (at least those on the level of difficulty where it takes computers to come up with one) are not absolute, but context-dependent, and socially constructed. For instance, hardcore mathematicians feel quite uneasy about proofs that can only be given using the help of computers, and cannot be verified "by hand".

This seems understandable, given the fact that, down on the hardware level, "digital" machines are undoubtably analog (the CPU's transitors are switched on and off by rising and ebbing electrical currents, and that motion is continous), and the famous "1"s and "0"s are really just an abstraction, which is prone to leak.

The notion that mathematics is not as resolvable a discipline as advertised is probably not so new; seems to me that the intuitionists argued along similar lines nearly a hundred years ago already. However, back then they were ridiculed by Hilbert and friends. Nobody is seen laughing about Davis now.

Impact on AI: if mathematicians doubt the correctness of mathematical proofs given by computers even now, then The Dawn of The Ultraintelligent Machines seems to be, erm, at least a good while off.

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