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Article 2871 of comp.ai.philosophy:
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>From: zeleny@zariski.harvard.edu (Mikhail Zeleny)
Newsgroups: comp.ai.philosophy,sci.philosophy.tech,sci.logic
Subject: Re: Penrose on Man vs. Machine
Message-ID: <1992Jan18.131648.7770@husc3.harvard.edu>
Date: 18 Jan 92 18:16:46 GMT
References: <1992Jan15.140324.27354@oracorp.com>
Organization: Dept. of Math, Harvard Univ.
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Nntp-Posting-Host: zariski.harvard.edu

In article <1992Jan15.140324.27354@oracorp.com> 
daryl@oracorp.com writes:

>Mikhail Zeleny writes:

MZ:
>> In view of non-categoricity of first-order PA, any mathematical
>> definition that involves quantification over the integers, whether in
>> object language or in meta-language (e.g. by appealing to the
>> conventional notion of a proof as a *finite* sequence of propositions)
>> is ipso facto second-order.  Is this so hard to understand?

DMC:
>It is hard to understand why you want to distinguish first-order and
>second-order definitions if every definition is second-order. Anyway,
>I think your use of "second-order" is nonstandard; as I said, the
>usual definition has to with whether the definition involves explicit
>quantification over sets.

If reference is to be interpreted as fixed at least up to model isomorphism
by a concept of the referent, then all reference to arithmetical objects is
accomplished through the mediation of second-order concepts.  Which is to
say, assuming that we understand the natural numbers up to isomorphism, we
must have the means of grasping second-order concepts.  (Incidentally, my
use of `second-order' is quite standard in semantics; reflect that, in
mathematics, theories are classified as first-, second-, third-,...
\omega-order, and so on, in virtue of their syntax alone.)  On the other
hand, if you reject the assumption that we understand the natural numbers
up to isomorphism, it seems to me that you lose your epistemic entitlement
to the use of classical mathematics involving Markov's abstraction of
potential realizability (the guarantor of potential infinity required e.g.
by the Peano axioms), including classical intuitionism.  So you would be
limited to something like Yessenin-Volpin's conception of feasible numbers.
So either change your math, or change your mind.

MZ:
>> Thanks for the elementary recursion theory lesson; now that you managed to
>> get the didactic compulsion out of your system (I hope), please observe the
>> term `intensional' in the above request. Now kindly make an effort to
>> discuss the same issue as I am addressing, viz. the question of
>> proof-theoretic strength of the language needed to give an exhaustive,
>> categorical definition of the above notions.

DMC:
>Mikhail, the fact on which you seem to be basing all this stuff about
>the necessity of second-order logic is the following: To uniquely
>characterize the natural numbers, one must augment the Peano axioms
>with the second-order statement:
>
>     For all sets S of natural numbers, if S contains 0 and is closed
>     under the successor operation (if x is in S, then x+1 is in S),
>     then S contains all of the natural numbers.
>
>This statement is inherently second-order because there is no way in
>first-order logic that one can say "For all sets S". However, in
>first-order logic, you can certainly make sure that the above holds
>for every *first-order definable* set S. My belief is that that is the
>best that we can do in characterizing the naturals, or at least I don't
>know of any evidence to the contrary.

The evidence to the contrary is provided by introspection: I am fully
confident that I am capable of characterizing the integers up to
isomorphism, so restricting the induction axiom to first-order definable
sets is clearly insufficient.  However, it seems that our intuitions differ
on this issue.  On the other hand, I don't know of any empirical evidence
to the effect that we can characterize non-feasible numbers, so perhaps by
your own criteria you are stuck with ultra-intuitionism after all.

>Daryl McCullough
>ORA Corp.
>Ithaca, NY


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