From newshub.ccs.yorku.ca!ists!helios.physics.utoronto.ca!news-server.csri.toronto.edu!rpi!usc!cs.utexas.edu!uunet!psinntp!scylla!daryl Thu Jan  9 10:34:30 EST 1992
Article 2593 of comp.ai.philosophy:
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>From: daryl@oracorp.com
Subject: Re: Penrose on Man vs. Machine
Message-ID: <1992Jan9.131829.15232@oracorp.com>
Organization: ORA Corporation
Date: Thu, 9 Jan 1992 13:18:29 GMT

Mikhail Zeleny writes:

> 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.

All of these concepts *have* been represented in first-order language!
They all can be expressed within the first-order language of
set-theory, ZFC. The concepts cannot be characterized completely in a
first-order language, but they can be characterized sufficiently well
to accommodate human reasoning about them. If you think otherwise,
point out a result in any of these fields that is not in fact a
theorem of ZFC.

> 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.

There is no evidence that humans can go beyond formalized reasoning, so
it is not clear what you are demanding be explained.

Daryl McCullough
ORA Corp.
301A Harris B. Dates Dr.
Ithaca, NY 14850-1313





