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Article 2776 of comp.ai.philosophy:
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>From: daryl@oracorp.com
Newsgroups: comp.ai.philosophy
Subject: Re: Penrose on Man vs. Machine
Message-ID: <1992Jan16.142652.7552@oracorp.com>
Date: 16 Jan 92 14:26:52 GMT
Organization: ORA Corporation
Lines: 45

Tal Kubo writes:

> Start with a very powerful computer, with infinite memory, and a
> well-defined formal system which suffices to express statements of,
> say, first-order ZFC. The computer now executes the following infinite
> program: construct a comprehensive list of the true statements, and in
> the process, continually attempt to shorten the proofs of the already
> proven statements.  Of course, after an infinite runtime, the machine
> will have discovered all mathematical truths and their optimal proofs
> -- "The Book" as Paul Erdos calls it.
> 
> 
> ...Now, let us consider the temporal evolution of our computer
> program.  Let program start correspond to 4000 BC (Babylonian mathematics).
> After some point, the program will have produced a list of proofs which
> subsumes all of the mathematics a formalist would consider as known in
> Andre Weil's time. This corresponds to 1945 AD.  
> 
> My question is, will our computer program, using whatever heuristics you
> like, prove the Weil conjectures before 1975?  Before 2000? Before 10,000?
> And how long will it take it to find a proof of length comparable to
> Deligne's?  Under 100,000 years?

I agree with the statement that simply attempting to prove or disprove
every statment of ZFC is unlikely to produce a proof of Weil's
conjecture in any reasonable amount of time. It would waste too much
time trying to prove completely useless theorems. However, if you
allow for heuristics, that changes things enormously. To be able to
construct mathematical proofs in a reasonable amount of time, one
(whether human or machine) needs intuition about what lemmas and
definitions are useful and interesting. Once again, the fact that I
don't know how to program such heuristics doesn't prove anything to
me; there is no evidence that these heuristics are non-computable.

Computer science is only a few decades old, so it is no more valid to
say that the nonexistence of good automatic theorem provers is
empirical evidence for the impossibility of good automatic theorem
provers than it would have been in 1890 to say that heavier-than-air
flight was empirically impossible. The fact that we don't know how to
do something is not evidence that it is impossible, only evidence of
our limitations.

Daryl McCullough
ORA Corp.
Ithaca, NY


