From: holmes@opal.idbsu.edu (Randall Holmes)
Subject: Re: Russell's Paradox (was re: Of apples and their skins)
Organization: Boise State University
Date: Tue, 13 Oct 1992 18:20:09 GMT

In article <1992Oct10.105913.9955@samba.oit.unc.edu> Robert.Vienneau@bbs.oit.unc.edu (Robert Vienneau) writes:
>[Discussion by Randall Holmes and Mikhail Zeleny of Russell's Paradox
>and set theory deleted.]
>
>You both obviously have a much better grasp of the technical issues than
>I do. Perhaps you can help me with what has always been my problem with
>set theory.
>
>I, and I daresay many other not very advanced mathematicians, have an
>intuitive notion of sets and numbers. As far as I can tell, when one
>tries to rigorously express these notions in a consistent manner, one
>needs to introduce some sort of aesthetically displeasing complications
>that are no part of my intuition. For example, Russell's theory of
>types, with its extensive hierarchy and distinctions. (Even with
>Ramsey's simplifications, which I do not understand, I do not think the
>theory can be said to agree with any simple intuition).

The simple theory of types, in which one has a type 0 of individuals,
type 1 of sets of individuals, type 2 of sets of sets of individuals,
has always seemed pretty intuitive to me.  After a while, what with
typical ambiguity, it gets to seem a bit overpopulated, I agree (that
is, one gets a different number 3 in each type, etc.)

 Or one has a
>distinction between propositions that can define a set and those that
>cannot. I leave out such craziness as Godel's theorems and the
>Skolem-Lowenstein (sp?) theorem. I do know that I need the Axiom of
>Choice for much useful applied math. Yet the A.C. has consequences that
>seem silly. I cannot decide if the A.C. accords with my naive intuition
>or not. 

There is no reason to expect things to stay "intuitive" outside the
bounds of ordinary experience.  Godel's theorem and the Lowenheim
Skolem theorem both actually "make sense" when properly understood
(and neither shakes the foundations of reason for someone not prepared
to look for cracks in the foundations of reason); but they were
certainly a surprise when they were first found.

The Axiom of Choice has peculiar consequences because infinite sets
and processes can lead to counter-intuitive results.  Since our naive
intuition is founded on experience with finite sets and finite (or
perhaps potentially countable) processes (this could be argued up to
the power of the continuum in some cases), it should be hardly
surprising that it runs into trouble.  The intuition can be retrained,
however.


>
>So what's the upshot? That mathematicians (and all others) must abandon
>their naive intuition about the most basic entities they use
everywhere?

Only when and where it proves to be incorrect.  Remember that the
results you find alarming were found by people working with their
trained intuition...


>That we do not have any consensus about what we are are talking about
>when we are doing math? Does mathematics need a foundation? Am I right
>to think the current situation is comparable to Quantum Mechanics in
>physics: "if you do not think the theory is crazy, you have not been
>paying attention" (Niels Bohr, probably I'm misquoting)?

I think a lot of people think this way; but I think that there is
really very little cause for alarm (though there might be a few more
surprises in store!).  A lot of this (with regard to QM as well) has
to do with the spirit of the times; philosophical attacks are being
mounted on the validity of reason even as reason is achieving its
greatest triumphs.

>
>In short, I find most engineers and mathematicians (!) are sleepwalking.
>They think there some's basis for their manipulations, and they think
>they know what they're doing. On the other hand, specialists in logic
>and set theory have been producing a century's worth of unreadable technical
>monographs showing that the practitioners have no understandable basis
>for their practice.

This is just not true.  You may have trouble reading the monographs
(we might have trouble reading what you read or write, too) but they
do not contain any proofs that your practice is not understandable.
They do contain proofs that some things you might like to do cannot be
done (we can't solve the halting problem or build an all-purpose
automatic theorem prover), but I fail to see why we should have
expected to be able to do these things.



Is something drastically wrong here? Am I right
>to feel uncomfortable? Is the current situation unsubstainable?

No (not with mathematics at least; maybe with society at large).
Maybe yes (find out more and you may feel less uncomfortable).  It
depends; there is a failure of communication here which should be
corrected, which may be partly due to malice in some quarters.


>
>                                  Robert Vienneau
>                                  rob@kaman.com
>--
>   The opinions expressed are not necessarily those of the University of
>     North Carolina at Chapel Hill, the Campus Office for Information
>        Technology, or the Experimental Bulletin Board Service.
>           internet:  bbs.oit.unc.edu or 152.2.22.80


-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu
Newsgroups: sci.philosophy.meta