From: holmes@opal.idbsu.edu (Randall Holmes)
Newsgroups: sci.physics,sci.math
Subject: Re: Report on Philosophies of Physicists
Date: 14 Sep 92 18:59:19 GMT
Organization: Boise State University Math Dept.

In article <1992Sep12.022132.4581@CSD-NewsHost.Stanford.EDU> pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>
>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.

No, it is not.  Either "the jury is still out" on the question of
which models are closest to reality, or it has been decided that both
models are intrinsically interesting as _parts_ of reality (or views
of the whole).  This situation fits just fine with what I was saying.

  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 is true, but does not mean that there is no such question to be
addressed.


  (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.)

Hmmm...

  Deciding which
>reality is the Platonic reality is a different activity from simply
>working in one or more arbitrarily chosen realities.

And both are legitimate activities from my standpoint; either can be
used to assist the other.

>
>>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 the "alternative realities" are simply different ways of looking
at the one reality, if each can encode the others.

>
>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.

I'm not sure I understand what you mean here.  Quine has proposed an
ontology (not entirely seriously!) which would put you yourself (and
the whole physical universe) inside the universe of ZFC.  My position
is that the physical universe is a subset of the mathematical one.

  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.

This is not correct; the conflicts between people using different
theories can be compared to battles between people convinced of the
merits of different coordinate systems.  Even intuitionism turns out
to be intertranslatable with classical mathematics!  Very weak systems
of foundations sometimes translate to subsets of the usual systems and
question the legitimacy of the rest, of course, but this is a special
case of the same phenomenon; we are really all talking about the same
thing!


  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.

Don't hold your breath.  Topos theory turns out to be roughly
equivalent to intuitionistic type theory, which is neither a new
entrant nor attractive for general mathematical use.  It seems to be
good for computers, though.

>>[...]
>>   >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).

Eh?  We can admit that it is _consistent_ with ZFC that the
arithmetical sentence expressing "ZFC is not consistent" is true (this
is the case, after all), without admitting that it is possible that it
is true "in 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).

Godel's second theorem shows that we must always take this "empirical"
attitude.  But I don't doubt that ZFC is consistent.  But Godel's work
also shows that we understand mathematical reality (of numbers, at
least) _before_ we axiomatize, or no one would understand what he
meant when he said that his sentence was true in arithmetic, although
not provable.  No one does arithmetic in PA + ~Godel sentence.



>
>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? :-)

The point is metaphysical, not mathematical.

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


-- 
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