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From: minsky@media.mit.edu (Marvin Minsky)
Subject: Re: Penrose & Banach-Tarski/Axiom of Choice
Message-ID: <1994Oct23.154319.804@news.media.mit.edu>
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References: <385i1s$69h@toves.cs.city.ac.uk> <burt.782758488@aupair.cs.athabascau.ca>
Date: Sun, 23 Oct 1994 15:43:19 GMT
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In article <burt.782758488@aupair.cs.athabascau.ca> burt@aupair.cs.athabascau.ca (Burt Voorhees) writes:
>jampel@cs.city.ac.uk (Michael Jampel) writes:
>
>>Does any of this make sense?
>
>>My question: is the Axiom of Choice TRUE. Or, equivalently, is the
>>Banach-Tarski paradox true. What would a human A say? He might say
>>``No''. What would human B say? He might say ``Yes''. Therefore two
>>different humans might have different sets of rationally held beliefs
>>(both at the semantic level i.e. there is no `proof' of AC).
>>Therefore, Penrose says that one of them is not intelligent. Which shows
>>how silly he is.
>
>>Is this argument correct? If not, why not?
>
>No, it tries to draw a general
>conclusion from a particular
>example.
>
>Penrose doesn't say that humans,
>or at least intelligent humans,
>can always recognize things which
>are true but unprovable, only that
>sometimes this can occur, while it can
>never happen for a computer.  He doesn't
>even claim that this "intuitive"
>recognition is always correct.

Well, that's really silly because it's self-inconsistent -- because a
computer can "guess" *all* hypotheses, by enumerating them.  And of
course it can do much better than this by using heuristics.

What is intuition, anyway? I suspect that, for mathematicians, it
usually means using heuristics that you're not aware of using.


