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Article 2388 of comp.ai.philosophy:
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>From: zeleny@zariski.harvard.edu (Mikhail Zeleny)
Newsgroups: sci.philosophy.tech,sci.logic,sci.math,comp.ai.philosophy
Subject: Re: Penrose on Man vs. Machine
Summary: mathematics includes empirical truths, pending their formalization
Keywords: the limits of human understanding: no such thing
Message-ID: <1991Dec23.135321.6894@husc3.harvard.edu>
Date: 23 Dec 91 18:53:19 GMT
References: <1991Dec23.165606.5935@oracorp.com>
Organization: Dept. of Math, Harvard Univ.
Lines: 148
Nntp-Posting-Host: zariski.harvard.edu

In article <1991Dec23.165606.5935@oracorp.com> 
daryl@oracorp.com writes:

DMC:
>>>Therefore, in these cases, Penrose' arguments amount to the following:
>>>
>>>    (1) Assuming that we can tell which Turing machines H are partially
>>>        correct for solving the halting problem, then our reasoning
>>>        is nonalgorithmic.
>>>
>>>    (2) Assuming that we can tell which theories are sound, then our
>>>        reasoning is nonalgorithmic.
>>>
>>>In other words, assuming that we can do things that no machine can do,
>>>then we can do things that no machine can do.

MZ:
>>Quite so.  As I stated earlier, all analytic arguments can be
>>construed as a form of begging the question by an opponent who doesn't
>>accept the intuitive validity of their premises.

DMC:
>Agreed, the point of such an argument is to reduce the truth of some
>questionable statement to the truth of some statements that we find
>more intuitively valid. However, I think that Penrose' premises, that
>we can solve all instances of the halting problem, and that we can
>tell whether an arbitrary collection of axioms is consistent, is much
>*less* plausible than his conclusion, that our minds are not
>algorithmic.

Our intuitions differ.  This, in effect, is what your criticism of Penrose
amounts to.  Fine; this is how all philosophy, from Plato to Kripke, has
been done.  Just try not to delude yoursenf into thinking that you have
refuted his argument.

DMC:
>It seems to me that there is no empirical evidence suggesting the
>truth of the premises. As far as I know, every rigorous proof in
>mathematics (at least in the limited domains of number theory and
>analysis) has proved a theorem of ZFC, so there is no example of a
>mathematical result that is beyond purely formal methods.

There is, however, ample *empirical* evidence of consistency of ZFC; surely
this result must be considered as part of mathematics.  On the role of
inductive reasoning in estabilishing mathematical truth, see the well-known
books by Polya.  I suggest that eventually this inductive study will
engender a stronger theory proving the consistency of ZFC.

MZ:
>> I hope that you would understand that your claim about the purely
>> formal nature of reflexion principles as illustrated by Penrose was
>> made in error.

DMC:
>Let me be more precise. While I might agree that the meaning of a
>reflection principle may be beyond what is formalizable in first-order
>logic (just as the meaning of arithmetic is beyond such
>formalization), there is a way to formalize a fragment of that meaning
>in set theory. My claim is that there is no mathematical result that
>depends on a reflection principle that isn't also a consequence of the
>fragment of the reflection principle that ZFC is capable of.
>Therefore, there is no evidence that reflection principles enable us
>to go beyond what is provable in ZFC.

However, the reflection principle for ZFC is ipso facto unprovable within
ZFC.  My point is that reflecting on a theory must of necessity transcend
its syntax.  This is how I understand Penrose's claim that reflection
principles provide the very antithesis of formalist reasoning (p.110).

DMC:
>Another point that makes Penrose' claim about the power of reflection
>principles dubious: Penrose claims that, given an a syntactic theory,
>by reflecting on the *meaning* of the symbols, we can arrive at truths
>that are not derivable in that theory. This implies that our minds
>are, in a sense, more powerful than any syntactic theory whose meaning
>we can understand. However, to go from this claim to the claim that
>our minds are more powerful than *any* syntactic theory, it's
>necessary to assume that we can figure out a meaning for an
>*arbitrary* syntactic theory, not simply variants of first-order
>arithmetic and set-theory!

I assume that syntax is dependent on semantics: take care of the sense, and
the sounds will take care of themselves.  All that we have to assume is
that we can figure out a meaning of any syntactic theory we can devise.
That seems reasonable to me.

DMC:
>I find that assumption quite unbelievable: it would imply that given
>an arbitrary consistent r.e. set of axioms, we can figure out a
>meaning for the symbols that makes all the axioms true, and that
>allows us to use reflection to get a new, true but unprovable
>statement. Penrose only considered theories like PA that come with a
>ready-made interpretation (because they were invented by humans, with
>a particular human-understandable meaning in mind).

Sorry, from where I stand it appears that PA was *discovered* by humans.
My proviso stands: I see no reason to postulate an inherent limitation of
mathematical inquiry.

DMC:
>If you don't buy the fact that we can make sense out of an arbitrary
>consistent collection of axioms, then Penrose' argument leads not to
>his desired conclusion, but to the conclusion that:
>
>     If I were given a first-order axiomatization of my own mind, I
>     wouldn't be able to make sense of it.

As far as I can tell, a subjunctive counterfactual with an impossible
antecedent must be trivially true.  In any case, I claim that no
axiomatization devised by human means can stymie human understanding.

DMC:
>You might be interested in a brief conversation that I had with
>Penrose following his lecture at Cornell on The Emperor's New Mind.
>
>Daryl: Professor Penrose, it seems to me that there is no mathematical
>       result that human mathematicians can rigorously derive that is
>       not a consequence of ZFC, a purely formal system.
>
>Penrose: We know that ZFC is consistent, which is something that ZFC cannot
>         prove.
>
>Daryl: Wait a minute! What's your argument that ZFC is consistent?
>
>Penrose: If it's not consistent, why would anybody bother using it?

Penrose's last claim makes sense as a statement of empirical, rather than
analytic truth.  Perhaps he misunderstood your qualification of rigorous
derivation; perhaps he just has a different idea of rigor.

>-------------------------------------------------------------------
>
>
>Daryl McCullough
>ORA Corp.
>Ithaca, NY

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