From: holmes@opal.idbsu.edu (Randall Holmes)
Subject: Re: Report on Philosophies of Physicists
Organization: Boise State University Math Dept.
Date: Fri, 11 Sep 1992 15:21:00 GMT

In article <1992Sep10.205022.15408@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>In article <1992Sep10.132003.15495@sei.cmu.edu> firth@sei.cmu.edu (Robert Firth) writes:
>>In article <1992Sep10.034627.3965@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>>
>>>Today we all firmly believe that mathematics is a human activity.
>>
>>Include me out.  I'm a genuine, hard core, ostrakon carrying Platonist,
>>and I believe the entities discovered by mathematicians have a real
>>existence.

It's nice to know that there is one more of us.

>
>Yeah, they're pretty compelling, I must admit.  Just like the program
>that ghost-writes for half the contributors to this group.  Bet you
>believe all those people are real too.  A very convincing program, its
>author deserves a Turing award.
>
>The problem with Platonism is that the entities discovered by
>mathematicians depend for their existence on which axioms one assumes.
>One might think that axioms are beyond question, if only by definition
>of "axiom".  However it is not obvious to anyone, even today (in fact
>especially today), what the right axioms should be.

The axioms are determined by empirical investigation of what agrees
with our "intuition" of the actual entities.  How else would we know
that the Godel sentence is true?  (which shows that our concept of the
natural numbers determines the Peano axioms, NOT vice versa!)

>
>The axiomatic picture changed dramatically in the 1870's with Cantor's
>development of set theory, which at the time met with great
>controversy.  That it won out in the end was not the result of
>Platonists going out and doing the necessary experiments to confirm
>Cantor's theory the way physicists would, but rather the result of the
>mathematical community finding that Cantor's theory provided an
>attractive and workable foundation for mathematics.

But that is exactly what it means to "go out and do the necessary
experiments" in the context of mathematics.

  There is no a
>priori reason why Cantor's set theory need be the only workable
>foundation.  Nor is there any reason why alternative equally workable
>foundations will all create the exact same class of mathematical
>entities and theorems.

There are known alternative foundations (I'll spare you the
disquisition on Quine's "New Foundations" which those who know me
expect at this point).  They do not necessarily describe (not
"create") the same entities, nor do they prove the same theorems, but
there tends to be a way to translate parts of one theory into the
other and vice versa, which gives us a clear idea that they are
talking about the same things.  Think of different sets of axioms as
different "frames of reference" covering the same underlying reality.
There is "relativity" for the frames of reference, but the underlying
relaity is absolute.

>
>The possibility has by no means been ruled out that a different
>foundations for mathematics exists which allows us to interpret our
>environment and predict its future more or less as usefully and
>reliably as with our present mathematics, but which changes our outlook
>on the universe.  Some in fact claim this has already been done, most
>notably the category theorists, starting with Bill Lawvere in the
>1960's, and continuing with as recent an article as Colin McLarty's in
>J.Symbolic Logic 56:1243 (Dec. 1991) on categorical foundations.  While
>I'm currently neutral myself on the right choice of foundations, I do
>think the foundational category theorists are on to something
>important.

As I said before, this has already been done, several times.  ZFC
remains most attractive to most mathematicians.

>
>In the first half of this century those who worried about foundations
>seemed to have converged on ZFC, the axioms of Zermelo-Fraenkel set
>theory together with the Axiom of Choice, as definitive of mathematical
>reality.  But then Goedel and Cohen showed that Choice was independent
>of ZF (and the continuum hypothesis CH independent of ZFC), and later
>various reasonable axioms were found, some independent of Choice and
>some, such as the Axiom of Determinacy for certain games, actually
>contradicting Choice.  These showed that the notion of set was not the
>clear-cut notion that a Platonist might think.

No, it just shows that we do not know certain things.

  While this outcome
>would not surprise many of Cantor's contemporaries, it comes as a shock
>to later generations brought up to believe Cantor unquestioningly.
>Those who saw the swamp before the landfill went in know it's just
>landfill and vulnerable to earthquakes, those who didn't are skeptical
>when you tell them the ground they're standing on is not perfectly
>solid!
>
>A reasonable position today is to say that "everyone" accepts Z, tame
>Zermelo set theory (though that too could change in the future).

This is certainly not true.  There are intuitionists.  There are
mathematicians who will express doubts of the consistency of Peano
arithmetic in print (Edward Nelson in _Predicative Arithmetic_).
There are, of course, the few (one?) of us that works in NFU + axioms
of infinity, which interprets Z but does not satisfy it.

  The
>various foundations would seem to start to diverge at ZF, Z plus
>Fraenkel's axiom of replacement (don't press me on this, but I think
>topos-theoretic foundations while equivalent to Z are not known to be
>equivalent to ZF).

They are not nearly strong enough.  Usually, they are not even as
strong as Z.  They are as strong as type theory, which is strictly
weaker than Z (a result of Kemeny which has never been published).
Type theory is equivalent to Z (or ZF) with all quantifiers restricted
to sets in comprehension (and replacement).

  By the time you get to ZFC, ZF plus the Axiom of
>Choice, you don't have to look to category theory to see that things
>have come apart at the seams.  You have the choice of having Choice or
>doing without and adopting some other axiom contradicting Choice such
>as the Axiom of Determinacy (which assumes properties of certain
>games).

You are _known_ to be free to choose Choice or not; this is analogous
to Euclidean vs. non-Euclidean geometry before it was known what the
geometry of space was.  (personally, I find Choice reasonable in
platonic terms, but systems without choice interesting to study).
Choosing AD is much more dangerous, since it requires the existence of
many measurable cardinals.

>
>So Platonism is on very shaky ground.  It only seems firm because we
>are trained to believe it is firm, and because we walk on it all the
>time without noticing any tremors, reinforcing our lifelong belief in
>the absoluteness of mathematics.  But in reality the reality of
>mathematics is an illusion.

There are none so blind as those that will not see.

>
>You say you are a serious Platonist.  What is the official Platonic
>position today on which parts of mathematics exist?  Do the
>consequences of Z exist?  If so, what about ZF?  (Although Fraenkel's
>Replacement Axiom looks innocent enough, ZF is pretty wild when you
>think about it.)  And if you think all of ZF exists, then what about
>ZFC?  Do you believe in Stone spaces, and are they dual to Boolean
>algebras?  This depends on ZFC.  Well, maybe that's too close to logic
>to be considered concrete mathematics, it might be one of those
>self-referential sentences in disguise.  So let's go for something a
>Platonist would have to agree was absolute reality.  Do you believe
>that an Abelian group G has to be free whenever it turns out that every
>epimorphism h into G with kernel the group of integers splits?  (g
>splits when there exists h with gh = 1.)  As a Platonist you would say
>it is either true or false, right?  Well, here's a shocker.  It is
>independent of ZFC.  You can take either it or its negation as an
>axiom, either one when added to the axioms of ZFC yields a consistent
>foundations for mathematics.

I would say that it is or is not the case, but I don't know; I can
model either situation, but my intuition isn't adequate to decide the
issue.  Experience will teach us (maybe) -- but perhaps we will
_never_ know.  It is not necessary for me to know something or to know
even how to find out the answer to believe that it is either true or
false.  Epistemology is subordinate to metaphysics.


>
>Mathematics is not just hot and chaotic, it's ephemeral to boot.  The
>work of the devil indeed, St. Augustine was quite right.

See the _City of God_, Book XII, chapter 19, where St. Augustine
demonstrates the existence of the actual infinity of natural numbers
and strongly suggests that "omega" exists.  (I am NOT kidding).

>
>-- 
>======================================================| God found the positive
>Vaughan Pratt   pratt@cs.Stanford.EDU   415-494-2545  | integers, zero was
>======================================================| there when He arrived.

Cantor has entered his paradise; Kronecker is still counting in his
grave.
-- 
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.physics