FOM: Mattes, McLarty, Hilbert, Yang, Chow, Pratt, Chu, Sazonov

[Stephen G Simpson]

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

Josef Mattes gave a list of 9 diverse mathematical subjects and asked
me whether I would consider them foundational, i.e. within the scope
of f.o.m.  I'm not sure what the point of Mattes' question is, since I
don't know whether Mattes is interested in f.o.m. at all.  If he
isn't, then I guess the only purpose of the question was to disrupt
the discussion of f.o.m.  But let's give Mattes the benefit of the
doubt.  In that case, 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.  If subject X is only remotely connected to
the most basic mathematical concepts, then it's not foundational.  Of
course everything in mathematics is related to the most basic
mathematical concepts in some way or another; that's part of why
they're called basic.  But I wouldn't consider subject X foundational
unless someone shows how subject X arises inevitably in close relation
to or tells us something significant about the most basic mathematical
concepts qua basic mathematical concepts.  If subject X is nothing
more than a relationship between two non-basic concepts, then it's
hard to see why subject X should be viewed as foundational.

Colin McLarty mentioned Hilbert's 5th problem and Appendix IV of
Grudlagen der Geometrie (reprinted from Mathematische Annalen 1902).
Hilbert's discussion is indeed foundational; he gives natural axioms
for planar motions and proves, in a rather satisfying way, that any
system obeying his axioms is either Euclidean or Bolyai-Lovachevskian
plane geometry.  Contrast this with, for example, Yang's article on
Hilbert's 5th problem (in the Hilbert problem symposium volume, 1976),
which is a boring discussion of technical results and conjectures in
transformation groups, of no foundational interest whatsoever that I
can see.  A typical sentence from Yang's article: "In order to ask
definite questions suggested by Hilbert's fifth problem, we take for
granted some basic definitions which are often used in study of
transformation groups, as seen in [15] and [3]."  One of the questions
is some obscure thing about p-adic transformation groups.  If this has
any interest whatsoever to more than about 2 or 3 specialists, Yang
doesn't explain it.  Obviously a huge intellectual chasm separates
Yang from Hilbert.

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.  Obviously that wasn't good enough to qualify as a
foundational exposition of Chow's lemma.  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."  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.

If McLarty or somebody else comes up with an honest foundational
exposition of Chow's lemma, then at that point in time it will be
appropriate to consider whether or not Chow's lemma is part of genuine
f.o.m.  This may be a useful exercise.  Even if it turns out that
Chow's lemma isn't f.o.m. as I suspect, then this may still turn out
to be an example of how expositions of non-f.o.m. mathematics can
benefit from the foundational perspective.

Vaughan Pratt has provided voluminous vague claims and information
about his current research interest, Chu spaces and (a fragment of?)
linear logic.  The postings present a lot of musings and half-formed
thoughts.  In all of these lengthy postings, I can't find anything
that has anything to do with foundations of mathematics.  Here of
course I'm referring to the mathematics of human beings on Earth, not
fictional critters on Quasar 9.  Vaughan draws an analogy with
intuitionistic logic.  My response would be that intuitionistic logic
has received a clear foundational exposition in terms of Brouwer's
critique of non-constructive mathematical existence proofs and
Heyting's attempt to make Brouwer's ideas rigorous.  Vaughan has not
provided any remotely similarly clear f.o.m. motivation for linear
logic, or for Chu spaces.

Vaughan has said:
 > Mathematics is like art and fashion, it is defined by its producers
 > and consumers and it has its trends and staples.  Everything
 > changes in mathematics, some faster than others.

Elsewhere, Vaughan allowed as how some standard mathematical concepts
such as e and pi may have a longer half-life.  I wonder if Vaughan
would care to estimate the half-life of Chu spaces.  Is there any
reason to think that it will exceed 5 years?  I'm reminded of a fad
from the 80's, called dilators.  Perhaps not coincidentally, the
inventor of dilators was none other than the inventor of linear logic.
Does anybody remember what a dilator is?  I'm not trying to denigrate
linear logic; rather, I'm trying to goad Vaughan into explaining the
general intellectual interest of Chu spaces and linear logic, if they
have any.

To end on a positive note, I'll remark that Sazonov's posting on
feasible numbers represents genuine foundational thought, with a clear
foundational motivation.

-- Steve