[FOM] John Baez on David Corfield's book

[Jon Williamson]

David Corfield's book raises an important question for philosophers of
maths: why focus so strongly on so few questions about maths (typically
concerning mathematical truth, metaphysics and foundations), when there are
so many other interesting questions to be addressed? E.g. what counts as a
good concept in maths? what role does analogy play in mathematical
reasoning? what role does uncertainty play?

Stephen G Simpson:
> Could you please [1] provide an example of the kind of non-foundational
> mathematical work that Corfield is talking about, and [2] explain why it
> is allegedly philosophically interesting?  In particular, [3] what are the
> philosophical questions that it addresses?

Taking the three questions in reverse order:
3.  E.g. what counts as a good concept in maths? what role does analogy play
in mathematical reasoning? what role does uncertainty play?
2. One would hope that the study of mathematical conceptualisation and
mathematical reasoning would be of interest to philosophers of maths.
1. I recommend that you take a look at the book.

Philosophers of maths have devoted much of their attention to foundational
issues. A hundred years ago there was very good reason for this - it looked
like intuitively compelling foundations of maths might be feasible (simple
logical axioms, simple set-theoretic axioms or a formalist consistency
proof). For a while now it has become clear that any philosophical
foundations that arise out of FOM are not going to be very intuitive and are
likely to satisfy few people. FOM has moved on - the subject is now largely
studied for its own intrinsic interest and to elucidate more subtle logical
relationships between mathematical propositions than those required for
intuitively compelling philosophical foundations of maths. Why haven't
philosophers of maths moved on too? Why don't they study interesting new
questions about maths? Corfield appears to be one of the very few who are
trying to open up new areas.

Surely no FOMer would claim that their subject should be the sole focus of
philosophers of maths?

Jon
-------------------
Jon Williamson
Department of Philosophy, King's College, Strand, London, WC2R 2LS, UK
http://www.kcl.ac.uk/jw


----- Original Message ----- 
From: "Stephen G Simpson" <simpson at math.psu.edu>
To: <fom at cs.nyu.edu>
Sent: Friday, September 26, 2003 2:03 PM
Subject: [FOM] John Baez on David Corfield's book


> 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/
>
>