[FOM] John Baez on David Corfield's book

[Stephen G Simpson]

Charles A Stewart writes:

 > empiricist or pragmatist about mathematical entities); rather while
 > I guessed that Stephen would dislike Corfield's work, I expected
 > him to dislike its "list-2" mindset.

I'm not sure that I would dislike Corfield's book.  I would probably
find something of value in it.  From Corfield's web site, it appears
that the book contains extended quotations from distinguished
contemporary mathematicians.  This material could be quite
interesting, e.g., for the history of mathematics, to document for
future generations how these mathematicians view their own work and
that of their colleagues.

Furthermore, I don't think Corfield represents what I have called the
"list 2 mindset".

Let me explain about "list 2".  

Back in the Golden Age, I defined f.o.m. as the study of the logical
structure and most basic concepts of mathematics, with an eye to the
unity of human knowledge.  I then made a tentative list of the most
basic mathematical concepts.  The list that I came up with was:
number, shape, set, function, algorithm, mathematical definition,
mathematical proof, mathematical axiom, mathematical theorem.  Call
this List 1.

Some distinguished applied model theorists replied that List 1 is too
narrow and ought to be supplemented by other mathematical concepts
which they claimed to regard as equally basic: cohomology, Riemannian
manifolds, etc etc.  This is List 2.

This idea seemed absurd to me.  The concepts on List 2, while
interesting and important, are of a very different character than
those on List 1.

Furthermore, it seemed to me that, by inflating and distorting the
notion of "basic mathematical concept" as used in f.o.m., the applied
model theorists were trying to subvert and strangle f.o.m. by allowing
it no room to exist as a subject distinct from pure mathematics.  I
referred to this as the "list 2 mindset".

It appears that Corfield's deprecation of f.o.m. is of a somewhat
different and less subtle nature.  His idea is to blatantly assert
that f.o.m. is irrelevant to "the pulse of contemporary mathematics".
As crude evidence for this, he points to the fact that nobody in
f.o.m. has won a Fields Medal subsequent to Cohen in 1964.  Of course
he is overlooking the fact that many top mathematicians are quite
interested in things like the G"odel incompleteness phenomenon.

I think I know why the applied model theorists are hostile to f.o.m.,
but I have no idea why Corfield is.

 > my interpretation is not that Corfield is trying to put across a
 > new set of fundamental mathematical concepts,

Yes, I agree.  In fact, Corfield doesn't appear to be trying to
introduce any new mathematical concepts or mathematics of any kind.

 > but rather is saying that philosophers of mathematics are ignoring
 > philosophically interesting work because it is not about
 > fundamental concepts.  

OK.  You are saying that Corfield is not attacking f.o.m. as such.
Rather, he is criticizing certain philosophers of mathematics, for
paying too much attention to f.o.m. and not enough to other
mathematical work which he claims is philosophically interesting.

Could you please provide an example of the kind of non-foundational
mathematical work that Corfield is talking about, and explain why it
is allegedly philosophically interesting?  In particular, what are the
philosophical questions that it addresses?

 > I think also the inflammatory nature of the book for some FOMers
 > would be reduced if we were to read "core mathematics" for "real
 > mathematics".

Not really.  It wouldn't make any difference.  We f.o.m. researchers
are by now quite used to hearing ignorant people deprecate our
mathematical work by saying that it is not "core mathematics" or "real
mathematics".  This kind of carping doesn't matter.

----

Stephen G. Simpson
Professor of Mathematics
Pennsylvania State University
http://www.math.psu.edu/simpson/