From: tycchow@riesz.mit.edu (Timothy Y. Chow)
Subject: Re: You know, the integers (was: Re: Stupid question about FLT)
Organization: None.  This saves me from writing a disclaimer.
Date: Tue, 21 Jul 92 18:33:05 GMT

In article <9601.Jul2112.44.3692@virtualnews.nyu.edu> brnstnd@nyu.edu (Dan Bernstein) writes:
>In article <1992Jul21.034140.10920@galois.mit.edu> tycchow@riesz.mit.edu (Timothy Y. Chow) writes:
>> But wait a second, someone will say.  Why can't we just take the syntactic
>> entities of ZFC to BE our sets?  We just DEFINE a set to be a syntactic
>> entity of ZFC.  Won't this solve our problems?  We don't have to worry about
>> MODELS of ZFC, which we don't even know exist.  We DO have the syntactic
>> entities, so why not just take those to be our sets.  Then math will be
>> reduced to syntax as per plan.
>
>Since that is exactly what mathematicians have always done and will
>always continue to do, what are you worried about?

Is it what mathematicians have done?  Texts on set theory will begin with a
bunch of axioms for sets.  In general, the term "set" is not defined, much
less defined to be a syntactic entity.  In fact, "syntactic entity" is
defined in terms of sets, while sets remain undefined.

Ontological questions like "What IS a set?" are generally restricted to
a preface and are not addressed in mathematics texts.  For the purposes
of DOING mathematics, it doesn't really matter what you think a set is,
as long as you can produce correct proofs about sets.  This I think you
will agree with.  But you go further, concluding from this fact that a
set IS a syntactic entity, i.e., taking a particular ontological stance
on the nature of sets.  I recall from an earlier discussion on this
forum that you said something like, "Goldbach's conjecture is the
statement that, in a certain set theory under a suitable logic, ..."
Well, most mathematicians would omit the phrase about set theory and
make the purely mathematical statement that Goldbach's conjecture is
the statement that every even number > 2 is the sum of two primes.  For
your approach presupposes a philosophical statement, that sets ARE
syntactic entities, whereas most mathematicians leave such ontological
proclamations alone and just do mathematics.

Nor is the statement that "sets are syntactic entities" trivial, because
in mathematics today it's the other way around.  Syntactic entities are
defined in terms of sets.  As I said before, take any logic textbook and
note that it begins by saying, "An alphabet is a set of symbols..."  In
an earlier article I complained that taking syntactic entities as our
starting point instead of sets seemed artificial.  Well, I take that back,
and say now that it is not really artificial; we seem to understand what
an "axiom" or a "string of symbols" is about as well as we understand what
a "set" is.  But I do maintain that I see no reason to give up the way we
do things now, i.e., basing everything, including syntactic entities, on
sets.

Earlier in this thread I asked, what does it take before we can say we
"know" what a mathematical entity is.  Experience with trying to
capture the properties of the integers (a perfectly respectable
mathematical object) with first-order logic suggests that we cannot
necessarily "define," or characterize uniquely, mathematical objects by
means of first-order axiomatization.  But you may say, we can
characterize all other objects in terms of sets, and then take some
first-order axiomatization of set theory.  To this I say, what
confidence do we have that first-order logic will succeed with set
theory, something much more complicated than the integers?  (In case
you're tempted to say, but sets just ARE syntactic objects of a
first-order logic, see above paragraphs.)  We don't have any assurance,
but I say we don't have to worry about this.  There is no need to
say that we don't know what a mathematical object "is" just because
its properties cannot be exhausted by a first-order axiomatization.

Note: although I have only talked about first-order axiomatizations,
the philosophical points made here apply for the most part for second-order
and other logics.
-- 
Tim Chow     tycchow@math.mit.edu
Where a calculator on the ENIAC is equipped with 18,000 vacuum tubes and weighs
30 tons, computers in the future may have only 1,000 vacuum tubes and weigh
only 1 1/2 tons.                               ---Popular Mechanics, March 1949