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From: olaf@cwi.nl (Olaf Weber)
Subject: Re: Penrose and human mathematical capabilities
In-Reply-To: tim@maths.tcd.ie's message of 13 Jul 1995 03:40:25 +0100
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Date: Thu, 13 Jul 1995 09:59:30 GMT
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Xref: glinda.oz.cs.cmu.edu comp.ai.philosophy:30053 sci.logic:12287

In article <3u212p$5t@bell.maths.tcd.ie>, tim@maths.tcd.ie (Timothy Murphy) writes:
> olaf@cwi.nl (Olaf Weber) writes:

>> A proof that humans have no Goedelian limit must imply that they
>> transcend the limits of all universal TMs, and thus of all TMs.

> Am I alone in finding all this talk of Goedelian sentences and
> limits completely incomprehensible.

> Goedel's Incompleteness Theorems only apply to formal axiomatic
> systems (and only to certain formal systems at that).

Right.

> It does not make sense to speak of "the Goedel sentence of a TM",
> let alone of a human being.

In SotM (and TENM), Penrose spends a lot of time showing that there is
an isomorphism between formal systems and algorithms.  That is what
allows him to apply Gdel's theorem to Turing Machines (it translates
into a halting problem), and finally to conclude that

(G)	Human mathematicians are not using a knowably sound algorithm
	in order to ascertain mathematical truth. (page 76)

> To apply Goedel's theorems to human beings, you would have to show
> that human beings were (in some sense) formal systems.

Such a demonstration could go by showing that humans are (some sense)
computers running an algorithm.  For that algorithm you have a
corresponding formal system, for which Gdel's theorem may (or may
not) be applicable.

It is at this point (pages 130-1 and further) that his arguments
become really tricky and IMHO of doubtful quality.

> This would imply in particular that the human being could only take
> up an enumerable number of configurations, and could only change
> these configurations at discrete times.

So what?

-- Olaf Weber
