# FOM: foundations of geometry; set-theoretic foundations; Chow]

Colin Mclarty cxm7 at po.cwru.edu
Thu Nov 6 21:20:44 EST 1997

```Reply to message from simpson at math.psu.edu of Thu, 06 Nov

>Colin McLarty writes:
> > Here I agree with Steve in good part--except that you can't ask
> > every poster to start from 0 and build every subject they refer
> > to. You have to let the list find its level through some give and
> > take.
>
>Yes, of course, there is no need to spell out a full-scale
>foundational exposition, provided you can sketch one in a convincing
>way.
>
>Let's take it for granted that you did that for Chow's lemma.  We then
>have to note that complex projective varieties are a high-level,
>technical mathematical concept, the connection to "shape" or other
>basic mathematical concepts being remote and requiring many steps
>(complex numbers, points at infinity, polynomials, etc.)

Surely complex numbers are no high level technical concept, nor
are roots of polynomials. And you could easily explain Complex projective
space to any high school honors class in mathematics. In the full statement
of Chow's theorem, the only notion that even uses calculus is "analytic
function". Stating the theorem is not hard--proving it is another matter
and I'll come back to that.

Also, this
>technical notion is only one of many competing technical notions:
>others are topological space, Riemannian manifold, simplicial complex,
>etc, and Chow's lemma is just one of many technical results relating
>these technical notions to each other.  So I don't regard Chow's lemma
>as foundational, neither f.o.g. nor f.o.m.  A touchstone for this is
>that Chow's lemma doesn't have any general intellectual interest.
>The contrast with G"odel's incompleteness theorem is quite stark.
>
I wouldn't say Chow's theorem is as important to f.o.m. as
Goedel's incompleteness. But if we only set the standard that high I
don't know what we will talk about besides Goedel.

I do think math is a vast rich field and there really are a lot
of foundational issues in it. But neither would I lose Chow's theorem in
the sea of all possible approaches to geometry. Chow's theorem compares
integer polynomials to log and trig functions. Pretty basic stuff.

>About the analogy with Hilbert's 5th problem:
>
>Hilbert viewed Hilbert's 5th problem as foundational because of its
>f.o.g. aspect, as in Appendix IV where he considers axioms for planar
>motions.  This f.o.g. aspect has been completely lost in, for
>instance, Yang's paper on modern mathematical developments arising
>from Hilbert's 5th problem.  I can see that Chow's lemma is loosely
>analogous to this Yang-like, non-f.o.g. aspect of Hilbert's 5th
>problem, but I don't see that Chow's lemma has any f.o.g. aspect of
>its own.

I come back to the question of proof: Specifically, suppose
a question deals with simple concepts, prominent in the basics of
some part of math with wide intellectual interest. Is that enough to
make it foundational? Or must there also be a simple proof of the
answer? Do we want to say that difficult, sophisticated proofs are