[FOM] John Baez on David Corfield's book

Stephen G Simpson simpson at math.psu.edu
Wed Oct 1 20:05:44 EDT 2003

David Corfield, Wed, 1 Oct 2003 15:51:08 +0100 writes:

 > I'm not objecting here to foundationalism, but the view that
 > philosophers should treat mathematics solely via the foundational
 > disciplines.

Nobody has that view.  You are objecting to a straw man.  Can you name
even one person who has said that?

 > [...] I still maintain that there's more to mathematics of
 > philosophical interest than that f.o.m picks up.

OK.  It doesn't seem out of the question for you to say that there is
some non-f.o.m. mathematics that is of philosophical interest.  But,
why aren't you giving any examples?

I am asking you to please give just one good example of a piece of
non-foundational mathematics that you think is of philosophical
interest, and explain its philosophical interest.

You spoke of

 > >  > [...] the conception common to the majority of philosophers of
 > >  > mathematics that the ONLY aspects of mathematics of
 > >  > philosophical interest can be detected by proof theory, model
 > >  > theory, set theory, recursion theory.

I replied:

 > > [...] Have any of them actually said this, or is this merely an
 > > inference on your part?  Aren't you putting words into their
 > > mouths?

You replied:

 > The problem is an institutional one. [...]

OK, so you admit that you can't name even one philosopher of
mathematics who explicitly holds that "the ONLY aspects of mathematics
of philosophical interest can be detected by proof theory, model
theory, set theory, recursion theory."  Evidently your earlier
statement was merely an inference on your part.  You were attacking a
straw man.  I'm glad we have finally got that straw man out of the

 > Graduate training requires huge investment in learning logic, set
 > theory, etc.

I think it is reasonable to require graduate students in philosophy of
mathematics to know a reasonable amount of mathematical logic and set
theory, since these f.o.m. subjects are of evident philosophical

 > with no encouragement to find out what a vector bundle is.

Apparently you think vector bundles are of philosophical interest.
What do you think is the philosophical interest of vector bundles?

 > As for funding, I don't know how the money for philosophy of math
 > is distributed in your country, but over here, a huge chunk goes to
 > the Neo-Fregean programme.

OK, so I guess your real complaint is that, according to you,
Lakatosian philsophy in the UK doesn't get enough government funding,
and f.o.m. gets too much.

I am not familiar with the government funding situation for
philosophers of mathematics in the US or in the UK.  Does anybody here
on the FOM list have any insight into this?

One thing I can say is, Lakatosian philosophers in both the US and the
UK would almost surely get more government funding than they do now,
if they were to address core philosophical questions in a more
original and compelling way.  Is there any prospect of this happening?

 > The two small glimmers of hope that it's possible to do something
 > different that gets noticed are those doing history of philosophy
 > of mathematics, who tell us we've got a lop-sided view of Frege,
 > Hilbert, etc., [...]

Obviously anybody who bad-mouths Frege and Hilbert will get some
attention.  But the real question is, what do you think is lop-sided
about the conventional view of Frege and Hilbert as towering
f.o.m. heroes?

 > I have written in another post that foundational and methodological
 > issues were interwoven in the 1880-1930 period and that they have
 > since come apart, [...]

When did you say this on the FOM list?  I can't find it by searching
for 1880 in the FOM archives.

 > My claim was obviously time restricted then.

OK, so you were not claiming that f.o.m. has *always* been irrelevant
to mathematics.  You are "only" claiming that f.o.m. since 1930 has
been irrelevant to mathematics.

But, to bring up my earlier example, general topology (what you are
calling point set topology) has been an active area of research for
most of the 20th century, not only prior to 1930.  If you grant my
claim that general topology was influenced by set-theoretic f.o.m.,
then it seems to me you have to back down again.

By the way, what did you think of my other point, in reply to Baldwin,
that f.o.m. research from the 1960s onward showed that various
questions of general topology are independent of ZFC and so forced a
re-evaluation of that subject?  This seems to be a counterexample to
an even more restricted version of your claim.

 > What I doubt is the profound impact of the f.o.m activity of the
 > past twenty or so years on the conceptual decision making of:
 > Connes, Deligne, Drinfeld, Kontsevich, Manin, Arnold, Yau,
 > Donaldson, Givental, Gromov, Gowers, Borcherds, Thurston, Chern,...

OK, so now you are restricting it to the last 20 years.  The scope of
your anti-f.o.m. claims seems to be shrinking rapidly.

When Connes visited here 5 or 6 years ago to give a series of four
lectures, he spent his entire first lecture discussing the G"odel
incompleteness phenomenon, including some modern references.

Recently another prominent mathematician, Shmuel Weinberger, visited
here, and he spent his first lecture on the impact of Turing's
f.o.m. research (Turing computability, etc) in his own field,
Riemannian geometry.

By the way, is there any evidence that Lakatosian philosophy has had
any impact on mathematics?

 > So, concerning physics, we have discussions of time, cause,
 > symmetry, etc.  The kinds of things we might look in math are space
 > [...] symmetry, and duality (from projective geometry and Fourier
 > analysis right up to mirror symmetry). The only vaguely plausible
 > way at present of treating these kind of concepts in a formal,
 > linguistic framework is category theory. [...]

I have no idea what you are talking about.  Could you please explain
more clearly?

Stephen G. Simpson
Professor of Mathematics
Penn State University

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