[FOM] foundational? more

Harvey Friedman friedman at math.ohio-state.edu
Wed Jul 9 12:15:05 EDT 2003

Continuation of my posting "foundational".

>A example of the kind of foundational issues which arise is, to recall an old
>Fom chestnut - Chow's Theorem.

I reproduce Marker's interesting posting here.

"There has been some discussion of  Chow's theorem.
While I would not call Chow's theorem a result in the foundations of
mathematics, I do believe it has some foundational significance.

First, a precise statement. Let M be a complex manifold. We say that a
subset X of M is "analytic" if for all m in M there is U an open
neighborhood of m and analytic functions f_1,...,f_n on M such that
X intersect U is the set of common zeros of f_1,...f_n. (Analytic sets
are locally the zero sets of systems of analytic equations.)

Let P^n be complex projective n-space. A subset X of P^n is "algebraic" if
there is a finite set of homogeneous polynomials p_1,...,p_m such that
X is the set of common zeros of p_1,...,p_m.

Chow's theorem asserts that every analytic subset of P^n is algebraic.
This implies the same holds for any projective algebraic variety.

Is this a result about the foundations of mathematics? I would say no.
But I think it does have some foundational significance.

Certainly one of the early problems in the foundations of mathematics
is to understand the relationship between algebraic and transcendental
methods. Chow's theorem says that in one important case anything that can
be done by transcendental methods can already be done by algebraic
I think this is a foundational point that Hilbert would have appreciated.

Dave Marker"

Notice the ambivalence of Marker. Do you have a vision of 
"foundations" for which this Theorem from core mathematics is 
"foundational"? And how does model theory enter the "foundational" 
picture here?

One of the great hallmarks of classical foundations of mathematics is 
its general intellectual interest that has captured the imagination 
of the wider intellectual community. Obviously, this is rather hard 
to claim for Chow's theorem as it stands. Do you have a way of 
looking at Chow's theorem and some related model theoretic insights 
that is of general intellectual interest? Or is that asking too much 
for model theory?

>>>One of the important discoveries of the middle 20th century is the
>>>futility of a trying to find a general foundations of mathematics.
>Shipman asks:
>>>In what sense is the usual foundations via ZFC "futile"?  Is there 
>>>a part of mathematics it cannot provide a foundation for?  Or does 
>>>it have some other irreparable inadequacy?
>Baldwin replies:
>ZFC is quite successful has providing a groundwork for mathematics. 
>But rather than providing a context for investigating `foundational 
>in mathematics as practiced it has become another branch of 
>mathematics (replace `model theory' by `set theory' in Friedman's 
>characterization of
>Baldwin's view of model theory.)

You say "it has become...", where "it" refers to "ZFC". However, this 
does not make any sense to me, since ZFC is not a subject, but rather 
a formal system.

ZFC has provided a context for investigating foundational questions 
in mathematics as practiced at least in the sense that it is used to 
make general statements about what can be achieved in practice. E.g.,

1) one is not going to settle the continuum hypothesis through the 
generally accepted practice of mathematics;

2) one never has to appeal to the continuum hypothesis or its 
negation, or even the axiom of choice or its negation, in a proof of 
a number theoretic statement or a variety of kinds of statements, 
through the generally accepted practice of mathematics.

>The general explanation by model theorists of the Borel Tits Theorem is a more
>direct contribution to the foundations of algebraic geometry and 
>group representations than results about measurable cardinals.
>(See Poizat: Stable groups  Chapter 4.)

To examine just what you mean by "foundations" here, could you 
explain further? Is there information of a "foundational" character 
about algebraic geometry and group representations that comes from 
this work in model theory that the algebraic geometers and group 
representation theorists understand and did not know?

I would agree that under our present knowledge, measurable cardinals 
have no more to say about the foundations of algebraic geometry and 
group representations than they have about mathematics generally.

>(Incidentially, some one more knowkedgeable could easily come up 
>with example from recent work in descriptive set theory which  is 
>just as good an example of a contribution to the foundations of 
>analysis or group representations.)

Again, I would like to see just what is meant by "foundational" here.

There IS one aspect of things at least with very strong connections 
with model theory that I instantly recognize as foundational - in a 
relatively obvious sense. And this is the matter of tameness versus 
wildness. The great original tameness theorems are the tameness of

the ordered group of integers, the ordered group of rationals, the 
field of real numbers, and the field of complex numbers.

This tameness is of obvious importance for the Grand Unification I 
spoke about in recent postings.

Harvey Friedman
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