From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt)
Subject: Re: Report on Philosophies of Physicists
Organization: Computer Science Department,  Stanford University.
Date: Sat, 12 Sep 1992 02:21:32 GMT


In article <1992Sep11.152100.438@guinness.idbsu.edu> holmes@opal.idbsu.edu (Randall Holmes) writes:
>>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.

This is contradicted by the existence of multiple mathematical
realities in which one postulates either sign for each of a set of
independent sentences like the group theory example.  One can do
perfectly good and even attractive mathematics in any of those
realities without addressing the question of which reality is the one
the Platonists believe in.  (This begs the question of whether all
Platonists believe in the same reality, which seems to me like asking
whether all monotheists believe in the same deity.)  Deciding which
reality is the Platonic reality is a different activity from simply
working in one or more arbitrarily chosen realities.

>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.

Yes, and moreover it looks like you're more on top of these
alternatives than I am---I get lost easily in the maze of alternative
foundations.  Since Platonism can be defended against without venturing
outside the set theory fortress, those of us foolhardy enough to
venture outside during an attack are easily driven back inside in this
way.  Inside, one finds alternative realities by choosing different
axioms all within the one language of set theory.  Alternative
foundations aren't needed to construct alternative realities.

But while strict Platonism is easily defended against (well, I feel
like I'm still alive :-) there may be less strict forms of Platonism
that are harder to defend against.  Different realities may differ by
certain criteria, e.g. their mathematical fertility.  A consensus could
emerge as to the best reality, which would seem to be a de facto win
for Platonism provided all Platonists agreed that the reality all saw
as flowing with milk and honey was the true one.  In this situation an
alternative foundation becomes relevant if it induces another ranking
whose top-ranked reality makes the milk in set theory's winner seem
sour.  I'd be interested in hearing from any topos theorists out there
who believe they are remotely near this happening and can make their
case clearly, succinctly, and non-topos-theoretically.

>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).

Right, Cantor himself mentions this chapter in a long and rambling
letter to Hermite in 1895 about Bacon, Shakespeare, St. Augustine, and
the nature of numbers.  He would have loved Usenet.

In article <TORKEL.92Sep11100055@isis.sics.se> torkel@sics.se (Torkel Franzen) writes:
>In article <1992Sep10.205022.15408@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.
>Stanford.EDU (Vaughan R. Pratt) writes:
>
>   >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.
>
>  This is not historically accurate. Set theory caught on chiefly
>because of its role in the creation and development of such subjects
>as point set topology (which Poincare considered a "disease") and
>measure theory. And, of course, the simple but striking applications
>of the basic ideas such as the uncountability of the set of
>transcendental numbers.  The idea of set theory as a foundation for
>mathematics came much later.

I don't think there was any time in its first hundred years when it
could be said that set theory was found attractive for other than
mainly foundational reasons.  (More recently the new math has been
peddling it on pedagogical grounds, but that's another debate.)  You
may be interpreting my "workable foundation for mathematics" too
narrowly.  Granted the modern concept of ZF-based set theory as the FOE
(Foundation Of Everything) evolved later, but at no time were either
Cantor's supporters or his opponents in any doubt as to the
foundational role being proposed by Cantor for his set theory,
regardless of how strongly they felt for or against it.  Cantor
intended his creation as a foundation for general mathematics from the
beginning.  It was accepted most quickly by those most urgently in need
of secure foundations, such as the people you mention, but they saw
their work as being in the mainstream if not the very center of
mathematics.  The "role of set theory in the creation and development
of point set topology" was unquestionably foundational, with topology
in turn constituting a major strengthening of the foundations of
analysis.

>
>   >You can take either it or its negation as an
>   >axiom, either one when added to the axioms of ZFC yields a consistent
>   >foundation for mathematics.
>
>  What do you mean, "yields a consistent foundation for mathematics"?
>What you say applies equally to the statement "ZFC is consistent".

Had this come from a non-Platonist I would just have amended the above
to read "as consistent a foundation".

As it was I had to think for a while about whether Platonists can raise
this picky point without denying their faith by admitting the
possibility that ZFC (hence ZF) might be inconsistent, hence false (of
the Platonic reality).  I concluded they could provided ZFC remained
for them only a working hypothesis about reality, one that the evidence
may one day contradict.  In that respect Platonists are no different
from anyone else: we are all at risk of someone proving not-F (the
denial of Replacement) in Z (the same thing as ZF turning out to be
inconsistent), and having to scrap a whole ton of math (the trees, my
god).

But then I wonder, what is the point of Platonism?  Is it a sort of
religion from which one derives comfort in the thought that out there
somewhere is a reality, regardless of whether one's adopted axioms
happen to be false of that reality?  I guess I can see that, but I sure
don't need it for myself.  (Now why do I have this feeling that
lightning could strike me for saying that? :-)

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