FOM: ReplyToHarvey Friedman <friedman@math.ohio-state.edu>

[Colin McLarty]

>Put bluntly, if you think that some material is foundational; i.e.,
>properly in foundations of mathematics, then the appropriate action for you
>to take for this group would be to give a novel, interesting, and creative
>foundational exposition of that material. This would be very welcome. If
>you want examples of such, look at various articles of Godel in his
>Collected Works. Also look in Putnam, Benacceraf, Philosophy of
>Mathematics, and also von Heijenoort, From Frege to Godel. Also various
>collections of works by Quine. There is also collected works of von
>Neumann. Also Hilbert's Foundations of Geometry, Open Court Classics, is a
>good classic. There are many many more places to look for foundational
>expositions - or partially foundational expositions.

        Good. We agree that Hilbert's Foundations of Geometry is a
foundational work. Taking nothing for granted, I specify that I include
Appendix IV of the Open Court edition as foundational (that is, the essay
"Foundations of Geometry" taken from Math. Annalen 1902).

        And will you agree that the many published papers on reverse
mathematics are foundational? I take them too as paradigms of foundational
thought. But notice they are extensive, and sometimes difficult reading even
for logicians who don't happen to specialize in proof theory. So I suppose
you agree that a foundational exposition can be demanding, and can take some
time?

>McLarty writes:
>
>>Hilbert and Brouwer both thought issues like Chow's lemma were
>>paradigms of foundational problems.
>
>Can you give us some real documentation? Can you give a foundational
>exposition of Chow's lemma?
>
>>I assume the
>>comparison [OF HILBERT'S AND BROUWER'S WORK ON LIE GROUPS] to Chow's lemma
is obvious.
>
>Why don't you make it obvious for us? Start at the beginning, since many
>people on this group are not primarily mathematicians.

        Sophus Lie in the 1880s could axiomatize Euclidean and other spaces
by their groups of symmetries--reflections, translations et c. But he had to
assume the symmetry maps involved were differentiable. Neither he nor anyone
else interested in foundations of geometry liked this assumption, but the
theory had such practical uses that he went ahead with the version he could
manage. Hilbert's Fifth Problem in his talk on "Mathematical Problems" in
Paris was this: Can purely algebraic and topological assumptions on
symmetries replace the assumption of differentiability, and thus serve as a
foundation for geometry?

        Hilbert is explicit, his first six Problems are "only questions
concerning the foundations of the mathematical sciences". See p. 15 of
reprinted translation of "Mathematische Probleme" in

CONFERENCE   Symposium in Pure Mathematics (1974 : Northern Illinois
                University)
 TITLE        Mathematical developments arising from Hilbert problems :
                [proceedings] / [edited by Felix E. Browder]
 IMPRINT      Providence : American Mathematical Society, 1976.  

In the article "Foundations of Geometry" cited above, Hilbert attempts a
solution to the Fifth Problem at least for the Euclidean plane. The first
accepted solution came from Brouwer a few years later (see Yang's article in
the Conference volume just cited).

        Brouwer solved the problem in dimensions 1 and 2, giving axioms for
a variety of spaces. For example, he gives group-theoretic axioms for the
Euclidean, hyberpolic, and elliptic planes on pp.34-36 of his dissertation
"On the Foundations of Mathematics" reprinted in 

AUTHOR       Brouwer, L. E. J. (Luitzen Egbertus Jan), 1881-1966.
 TITLE        Collected works / L. E. J. Brouwer.
 PUBLISH INFO Amsterdam : North-Holland Pub. Co. ; New York : American Elsevier
                Pub. Co., 1975-1976.
 DESCRIPT'N   2 v. : port. ; 26 cm.
  CONTENTS     v. 1. Philosophy and foundations of mathematics, edited by A.
                Heyting.

He discusses Hilbert's work on this problem pp.39ff, but most of the first
55 pages are variations on this theme.

        Steve's definition of foundations is that (among other requirements)
they deal with "basic" concepts needed for any "more-or-less systematic
analysis of the most basic or fundamental" facts about a subject. Lie,
Klein, Hilbert, Brouwer, and others were led to this problem as soon as they
tried to relate the traditional style of point/line axioms for geometry to
the function theoretic methods (such as differential equations) actually
used for any serious geometric problem. I claim that this problem not only
seemed "foundational" to Hilbert and Brouwer, it is "foundational for
geometry" in Steve's sense.        

>Are you prepared to show us any "more-or-less systematic analysis of the
>most basic or fundamental facts of geometry?" If you can do this to a
>suitably high standard, that would be valuable for this group.

        It already exists, see Hilbert. 

>>In the case of Chow's lemma: as soon as you think
>>seriously about curves defined by polynomials. Indeed Descartes DID run up
>>against it in his GEOMETRY--though he was no closer to a clear statement of
>>Chow's lemma than Fermat was to a clear statement of the Matjesevich result.
>
>Please document this.


        On to Chow's lemma, or more commonly "Chow's theorem".

        I assume people know what is meant by n-dimensional complex space
C^n and n-dimensional complex projective space P^n and how these naturally
arise from considering polynomials, even if you assume real integer
coefficients. I can expand on this if anyone needs more information. 

        Now when you study subspaces of P^n defined by polynomials you are
quickly led to use more complicated functions, specifically analytic
functions such as log and exponential. Clearly you will want to use square
roots and other roots, and you will want to use inverses 1/x and you will
want to use differentiation and integration--and integrating 1/x will give
you a logarithm. So the question arises right at the start of the subject:
What relation can I assume between my spaces defined by polynomials and all
the spaces I could define using these further functions?

        Descartes hit this question already in his "Optics", even before
coming to his "Geometry" in the order of publication--though surely the
problems all arose at once for him in fact. What means of mathematical
description do I need for the curves and surfaces I can define by mechanical
means, including optical definitions such as "the lens surface that truly
focusses light" (which is certainly not a spherical surface)? How far will
polynomials suffice? And within the "Geometry" there is Descartes's own
uncertainty about just what is involved in his method of 'moving two lines'
to trace out the curve of their intersections--which he explicitly relates
to the problem of "geometry" versus "mechanics". (See the openning passage
of the "second book" of the "Geometry". I'm using an old out of print Bobbs
Merrill edition so i won't bother with publication data.)  

        Of course the question remains inchoate in Descartes, but it is
there, and the answer is in Chow's theorem. 

CHOW'S THEOREM: Any subspace of P^n definable by analytic functions, is
definable by polynomials. 

        This means any subspace of real Euclidean space R^n which can be
defined by analytic functions--in such a way that the functions do not act
weird even for imaginary values or at infinity--can be defined by polynomials.

        The proof is hard, for me anyway. I am not primarily a
mathematician. But foundational results can be hard. The question inevitably
arises in any "more-or-less systematic analysis of the most basic or
fundamental" facts even about integer polynomials in two variables--i.e.
facts about compact Riemann surfaces.

        Explicitly, the analogy is: Hilbert's Fifth Problem asked "Among the
spaces Lie has axiomatized using differentiable groups, which can be defined
assuming only continuity of the groups?" And the answer is "all of them".
Chow answered the question "Among the spaces we can define using analytic
functions, which can be defined by purely algebraic means" and the answer is
again "all".

        I claim Hilbert and Brouwer would have called Chow's question
foundational, just as they did Hilbert's Fifth problem. Further, I claim
they would be justified by Steve's posted definition of foundations.