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Article 2850 of comp.ai.philosophy:
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>From: kubo@zariski.harvard.edu (Tal Kubo)
Newsgroups: comp.ai.philosophy
Subject: Re: Penrose on Man vs. Machine
Summary: principle of insufficient reason doesn't cut the mustard
Message-ID: <1992Jan17.163200.7722@husc3.harvard.edu>
Date: 17 Jan 92 21:31:57 GMT
References: <1992Jan16.142652.7552@oracorp.com>
Sender: Tal Kubo
Followup-To: <1992Jan16.142652.7552@oracorp.com>
Organization: Dept. of Math, Harvard Univ.
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In article <1992Jan16.142652.7552@oracorp.com> daryl@oracorp.com writes:
>> (...)
>> 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.

I did allow for heuristics.  In my posting, I suggested some
plausible strategies to focus the effort on the proof of important
theorems, in aid of which further heuristics might be enlisted.  Let me
even propose a simple heuristic for isolating important lemmas and
definitions: lemmas/definitions = segments of proofs/theorem statements
used disproportionately often relative to their length. What I dispute
is not the value of heuristics in solving problems, but that any intuition
can be embodied in the form of program code.  The intuitions exist only in
the minds of the conscious agents who construct the algorithms. 

Formalization is possible only when the scope of a theory has been
determined.  It is entirely possible that much of our externally
communicated mathematical reasoning is computable, once formulated in
in appropriate framework.  Thus I don't take impossibility proofs such
as halting problems as the main evidence against formalizability; it might
even be that our present reasoning in specific areas, such as abstract
halting problems,  is itself computable.  Not so for our intuitions!
We can set up theorem provers for Euclidean geometry because our 
intuition has surpassed it and we can conceive its full extent. To
faithfully represent any ideas in a formal system requires a semantic 
grasp sufficient to understand their limitations. 

My argument in no way relies on the state of the art. I grant that
theorem provers today can duplicate many of the theorems known in
ancient times.  Perhaps future systems will be able to derive all proofs
extant in the literature today.  So what? If mathematics were to grind to
a halt today, it would be trivial for computers to equal the human
theorem production: just write down all known proofs. 

Notice that I have not questioned some of your rather radical premisses,
e.g. mathematics coextensive with first-order ZFC, satz-beweis the only
substantive ('observable') content of mathematics.  What I have argued
is that computers will never beat humans even at that limited game. 
I take the observed ability of some humans to perceive difficult
mathematical truths, even lacking the appropriate means of proof, as
evidence of conceptual content in mathematics beyond symbol-pushing, and
thus beyond automation.  Maybe some future incarnation of cognitive 
science will explain it, but I refuse to dismiss it as 'heuristics' 
without further evidence.

>
>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.
>

Heavier-than-wood flotation would have been a better example.  A
man-made bird has been a plausible idea since the time of the
Trojan Horse.

Computer technology will develop further? Ain't seen nothing, you say?
Optimism alone is not convincing.  I insist on an equal optimism
concerning human capabilities.  An unending tower of abstractions,
of which today's mathematics is just the beginning, will up the ante at
least as fast as technology can catch up with it.  On what grounds do you
believe that thought is computable, other than lack of refutation?  Are
you similarly hopeful about faster-than-light travel?  I won't quibble
about time travel, as daylight savings time has solved this problem in 
the spirit of AI's approach to the mind problem.  Maybe next year's
tachyonic cars will save Detroit... 

>Daryl McCullough
>ORA Corp.

Tal Kubo   kubo@zariski.harvard.edu


