FOM: Faltings' theorem isn't basic
Stephen G Simpson
simpson at math.psu.edu
Wed Oct 22 16:55:51 EDT 1997
Lou van den Dries writes:
> Let me try to asnwer Simpson's question "What does Faltings'
> theorem say?"
...
> I agree that an honest asnwer does require notions that
> may be somewhat arcane from the FOM viewpoint.
Lou, thank you for acknowledging that. Yes, the notions in question
(genus, etc.) are indeed very arcane from the perspective of
foundations of mathematics. Would you also acknowledge the following:
They are even more arcane from a general intellectual viewpoint,
e.g. the perspective of a chemist or a novelist. This arcane quality
is a symptom of the fact that the statement of Faltings' theorem is
not foundational.
Dave Marker says he thinks that, from a general intellectual
perspective, the statement of Faltings' theorem is easier to
understand than that of Matijasevic's theorem. Lou, do you agree? I
certainly don't.
> In fact, a weakness of that viewpoint is that it admits certain
> questions as basic, but looses interest if the answer emerges in
> terms of notions not admitted as basic, even if that is the only
> reasonable way an answer can be given at all.
This isn't a weakness of f.o.m. It's a limitation, to be sure, but
then again all subjects, no matter how generally interesting or
generally useful, are limited in some sense.
Far from being a weakness, the ruthless insistence on basic concepts
is one of the great strengths of f.o.m. That's why many f.o.m. issues
are of general intellectual interest. That's why many people consider
G"odel's work in foundations to be deeper and more significant than
anything in 20th century pure mathematics.
To answer Lou's specific point, I would note that even the very
question addressed by Faltings' theorem (how many rational points on a
complex curve) is very far from basic. As I said before, rational
numbers may be basic, curves may be basic, but rational points on
curves are definitely not basic. (Here of course I mean "basic" in
the sense of my little essay at
www.math.psu.edu/simpson/Hierarchy.html, which provides the necessary
context for this discussion.)
To concretize this, try explaining rational points on curves to your
barber or your accountant.
> So, the basic question under discussion is how many rational solutions
> a given equation p(x,y) = 0 can have, where p is a polynomial with
> rational coefficients, p not a constant.
As I said above, this question isn't basic. It may be basic to number
theory, but it's not basic to mathematics or to general intellectual
life.
> "Basic" questions often have answers in terms of "arcane"
> notions. I believe the general scientific public accept s that.
Sure, the general scientific public accepts that principle. But
here's the key point: the general scientific public doesn't regard
questions about rational points on curves as "basic". It rightly
regards such questions as "arcane" (to use Lou's terminology).
Sincerely,
-- Steve
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