FOM: definition of f.o.m.: reply to Davis
Stephen G Simpson
simpson at math.psu.edu
Tue Jan 13 23:04:13 EST 1998
Martin Davis writes:
> What I claimed to be crucial for fom is THE PROBLEMATIC CHARACTER
> OF MATHEMATICAL TRUTH AT A (SHIFTING) BOUNDARY ...
>
> In ordinary mathematical research, one is working at the boundary
> of what is known, but in a context where the legal methods and
> acceptable techniques are understood and agreed upon. Foundational
> problems arise when this understanding breaks down because of new
> developments.
Martin, thanks for that clarification. To a certain extent, I agree
with both your original statement and your clarified statement. But I
have some extremely serious reservations. We need to be very cautious
here.
For instance, in the light of your clarified statement, what do you
make of Lou's example, Riemann surfaces? The history is that there
was something of a crisis in function theory (= the theory of analytic
functions of one complex variable) because people were using
questionable techniques involving multi-valued functions, etc. Then
Weyl came along and cleaned this up by giving a rigorous,
set-theoretic definition of the concept of a Riemann surface. Would
you call this a contribution to f.o.m.? Lou would. I wouldn't. I
have no problem calling it a contribution to f.o.f.t., foundations of
function theory. But I definitely wouldn't call it a contribution to
f.o.m.
> From my point of view, this historical understanding is more to the
> point than attempting an ad hoc delimitation of fom.
Maybe you and Sol don't care whether Weyl's clarification of Riemann
surfaces is called f.o.m. or f.o.f.t. But I care passionately. The
reason I care passionately is because I want f.o.m. to keep its own
identity as a subject. I don't want f.o.m. to be obliterated by being
viewed inappropriately as indistinguishable from high-level pure math.
It's absolutely crucial to insist on a sharp distinction between, on
the one hand, foundations of math, and on the other hand, foundations
of particular branches of math, for instance f.o.f.t.
The way to state the distinction sharply and clearly is by noting that
the focus of f.o.m. is the most basic mathematical concepts. Riemann
surfaces etc etc etc are NOT among the most basic mathematical
concepts. That's the point. I feel that your statement about
f.o.m. (are you proposing it as a *definition* of f.o.m.?) is woefully
inadequate, because it doesn't make this essential distinction between
basic and non-basic mathematical concepts.
In other words, I find your statement about f.o.m. to be much too
friendly to those of the "list 2 mind-set" persuasion, who want to
obliterate f.o.m. as a subject.
Best regards,
-- Steve
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