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Article 2903 of comp.ai.philosophy:
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>From: ske@pkmab.se (Kristoffer Eriksson)
Newsgroups: comp.ai.philosophy,sci.philosophy.tech,sci.logic
Subject: Re: Penrose on Man vs. Machine
Message-ID: <6466@pkmab.se>
Date: 18 Jan 92 01:50:03 GMT
References: <1992Jan16.112129.7632@husc3.harvard.edu> <1992Jan16.123103.2429@arizona.edu> <1992Jan16.233232.7674@husc3.harvard.edu>
Distribution: world,local
Organization: Peridot Konsult i Mellansverige AB, Oerebro, Sweden
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In article <1992Jan16.233232.7674@husc3.harvard.edu> zeleny@zariski.harvard.edu (Mikhail Zeleny) writes:
>You misunderstand me, Bill: I said that I'll understand it, provided that
>there is anything to understand.  In other words, I trust that you will
>make it meaningful, rather than merely well-formed. 

What do you mean by a mathematical theory being "meaningful" ?

Actually, what do you mean by "understaning" a mathematical theory, at all?
I'm not sure why "understanding" would be a requirement for a sound mathe-
matical theory, except in the sense that one may either be familiar enough
with the theory's formal properties to be able to derive new formulas in it
(perhaps without looking it up in the book), or one may not.

In article <1992Jan9.110732.7279@husc3.harvard.edu> Mikhail Zeleny wrote:
>MZ:
>>> 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.
>
>DMC:
>>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.
>
>Our understanding of the results of first-order ZFC is related to our grasp
>of its standard model.  I invite you to meditate on the difference between
>the latter and its countable submodels before you emit any further
>exclamations.  While you are at it, feel free to characterize "sufficiently
>well" the predicate "... is finite" in a first-order language of your
>choice.  I eagerly await the results.

Regardles of which one of you two is right on this particular exchange, has
anyone of you considered the possibility to, not _express_ these higher-order
concepts directly in some first-order theory, but rather _code_ them in the
objects of this first-order theory, and then formulate the formal rules of
proof of the higher-order theory as expressions of this first-order theory?
Then, only by manipulating formulas in the first-order theory, you will be
able to derive results that represent, by way of a code, correctly derived
formulas of the higher-order theory.

I am of course thinking of the same techique as the one that was used by
Goedel to prove his incompletenes theorem, except that the theory he coded
was not of a higher order than the one he coded it in. (And note that I am
not simply invoking the incompletenes theorem, as is so usual in pseudo-
philosphical discussions, just using the same basic technique.)

The point is that even if the human mind embodies a first-order theory, it
would be able to reason with and about higher-order theories, if only it
were provided with the rules of those higher-order theories in a form that
it can use.

And as we all know, humans do have to learn or discover or invent the rules
of mathematical theories to reason with and about them; they are not
integral to the workings of our minds. And even more, especially to reason
_about_ some theory, we always have to make the rules of that theory
explicit to us, even if that theory is the theory of how our own minds
work. We can't reason _about_ a theory without having an explicit (although
mental) _represention_ (corresponding to the "code" above) of that theory,
even if that reasoning itself is proceeding by use of that very same theory.

One can also note that all proofs appearing in writing, manage to be captured
on the paper with a finite number of written characters, no matter how many
infinite sets of objects or other first-order-hostile objects they talk about
or how many infinite series of formal sentences they might use, indicating
that all reasoning is, on the uppermost level, finite. The rules that
produce these strings of characters that constitute written proofs, are
finite, and thus don't need a theory or mind of a higher order (just not
the same theory, often) to be grasped and manipulated. It should be obvious
that these rules could be told to a computer.

You might reply that you will immediately be able to invent a next-higher-
order theory, that will transcend the capabilities of the computer using
the previous rules, but then we're back where I started: let it code your
new theory under the previous theory, or preferably under the innermost
theory that operates the computer, corresponding to the theory that operates
our human minds. Preferably the innermost theory would be equipped with
facilities to ease the learning of new theories, and be capable of switching
between any number of different coded theories for different parts of the
same problem without loosing its wits, just like us humans.

-- 
Kristoffer Eriksson, Peridot Konsult AB, Hagagatan 6, S-703 40 Oerebro, Sweden
Phone: +46 19-13 03 60  !  e-mail: ske@pkmab.se
Fax:   +46 19-11 51 03  !  or ...!{uunet,mcsun}!mail.swip.net!kullmar!pkmab!ske


