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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: <1992Jan8.114628.7224@husc3.harvard.edu>
Date: 8 Jan 92 16:46:27 GMT
References: <1992Jan7.031553.24886@oracorp.com> <1992Jan7.105117.7193@husc3.harvard.edu> <1585@ariel.ucs.unimelb.edu.au>
Organization: Dept. of Math, Harvard Univ.
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Nntp-Posting-Host: zariski.harvard.edu

In article <1585@ariel.ucs.unimelb.edu.au> 
jcollier@ariel.ucs.unimelb.edu.au (John Donald Collier) writes:

>In <1992Jan7.105117.7193@husc3.harvard.edu> 
>zeleny@zariski.harvard.edu (Mikhail Zeleny) 
>writes in reply to Daryl McCullough:

MZ:
>>And, as I have pointed out, your reasoning can only convince those who
>>share your formalist understanding of mathematics.  Since the notions of a
>>program halting on a given input, or a theory being consistent are
>>fundamentally second-order, i.e. non-recursive, our ability to understand
>>them is sufficient evidence of our ability to perform non-algorithmic
>>tasks.  Indeed, it is arguably true that all understanding is fundamentally
>>non-algorithmic; however, in view of our past disagreements, I shan't
>>repeat an argument to that effect, limiting myself to the claim that it is
>>intuitively obvious to me that I am capable of understanding.

JDC:
>What has seemed intuitively obvious in the past has often turned out
>on closer examination to be completely false. 

In the absense of examples, this statement reduces to pure eristic.  If you
believe that introspection can never result in certainty, then say so.

JDC:
>It is quite unclear that we understand the second-order concepts
>involved in any way that avoids the arguments that you are trying to
>reply to. How can you tell that your understanding is not
>representable in first order language? Certainly anything you can do
>with it is so representable. So what's the point in arguing for
>something more?  It seems pretty empty to me.

Sorry, John, but it's your objection that seems empty and ill-informed to
me.  Try representing in a first-order language set-theoretic concepts like
*countable set* and *finite set*, or topological concepts like *open set*
and *continuous function*, or analytic concepts like *set of measure 0*, or
probabilistic concepts like *random variable*.  It seems to me that our
success in discovering and manipulating such concepts amounts to prima
facie evidence of our ability to grasp non-recursive abstract entities.
Now, if you wish to explain this evidence away, it is incumbent upon you to
demonstrate that our ability can indeed be represented in an accounted for
by a first-order language used formalistically.  In other words, we are
back to Searle in a "Mathematical Room".  Be my guest, but bear in mind the
historical fate of logical positivists...

>-- 
>John Collier 			Email: jcollier@ariel.ucs.unimelb.edu.au
>HPS -- U. of Melbourne		  	Fax:   61+3 344 7959
>Parkville, Victoria, AUSTRALIA 3052


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