FOM: extent of agreement with Simpson

[Colin McLarty]

        In a post on

>My perspective on what has been going on the last few days:

Steve Simpson wrote about Mattes's list of 9 subjects which might or might
not be considered foundational:

>my answer is that I wouldn't *automatically*
>consider these subjects foundational.  In each case, someone would
>first need to explain clearly and honestly, starting with concepts
>that are of obvious general intellectual interest, exactly how subject
>X is related to the most basic mathematical concepts, and exactly what
>is the general intellectual interest of subject X.  This is probably
>what Harvey means by a foundational exposition of subject X.  Once
>this is done, we would have a basis for deciding whether subject X
>itself is foundational.

        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.

>McLarty wants to say that Chow's lemma is foundational.  At first
>McLarty tried to justify this by invoking a vague analogy between
>Chow's lemma and the Montgomery-Zippin solution of Hilbert's 5th
>problem.

        Actually, I referred to Hilbert's efforts on his 5th problem, and
Brouwer's solution for dimensions 1 and 2. These do not use bothersome
p-adic analysis, and were considered terrific work on foundations by
Hilbert. I further argued that Hilbert's view was justified according to
Steve's posted definition of "foundations". I don't know the
Montgomery-Zippin solution myself.
 
>Next, in response to pointed
>questions from Harvey, McLarty tried a little harder and came up with
>an undergraduate-level explanation of (a consequence of?) Chow's
>lemma: "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 line before that one gave a full statement of the theorem, which
I also explained. I then cut down to the version Steve quotes, which I will
try out on my barber this afternoon.

  On the basis of these remarks, McLarty then said: "I
>claim Hilbert and Brouwer would have called Chow's question
>foundational."  I don't think McLarty has justified his claim.  I
>would say that McLarty's second try was a substantial improvement, but
>it still wasn't good enough.  The standards for a foundational
>exposition are very high.  Hilbert's Appendix IV is an example of what
>a real foundational exposition looks like.

        I would like to persuade Steve (and others) of the point, of course.
And I have benefitted from the argument myself. But I don't plan a full
scale exposition of Chow's theorem in foundations of geometry. To me it is
one topic among many that meets Steve's posted definition of "foundational"
yet has drawn little attention from foundationalists. I am currently writing
at length about others--some of which even involve p-adic numbers, so I
won't try to bring them up on this list. 

        Shipman had claimed "Obviously Chow's theorem is not foundational in
anything like Steve's sense of the term (and he's the boss)" so I suppose he
is equipped to discuss the theorem specifically. I would be interested in
reading his objections to my arguments.