[FOM] John Baez on David Corfield's book

David Corfield david.corfield at philosophy.oxford.ac.uk
Sat Sep 27 11:44:22 EDT 2003

Stephen G Simpson wrote:

David Corfield writes:

 >> By the pulse of contemporary mathematics I mean the development of the
 >> kind of mathematics relating to Fields Medals: [...]

 >> Presumably, we agree that your work on foundations has no bearing
 >> on the way mathematicians view the proper organisation of
 >> mathematical ideas.

>I don't agree.

>The vast majority of recent and contemporary mathematicians, including
>Fields Medalists, choose to organize mathematical ideas in terms of a
>logical, hierarchical structure of axioms, basic concepts,
>definitions, lemmas, theorems, proofs.  Contemporary f.o.m. research,
>including my own f.o.m. research, is a study of this logical,
>hierarchical structure.  The purpose of such research is to develop
>precise answers to fundamental questions such as: What are the basic
>concepts of mathematics?  What are the appropriate axioms for
>mathematics?  What is the role of definitions in mathematics?  Etc
>etc.  So, yes, I would say there is some bearing.

I meant f.o.m. research has no bearing on choice between rigorously
defined concepts. You don't get to pass your PhD exam
just for producing logically consistent work. I don't need to invoke
non-rigorous work which does get rewarded (e.g., Ed Witten's) as
a contrast. Surely, we can agree that there are theories expressible
in systems of precisely the same logical strength, some of which
play a vital role in the life of mathematics, and some of which never

>We could turn the question around.  With your talk of "the pulse of
>contemporary mathematics," are you claiming that your own work has
>some bearing on how Fields Medalists organize their mathematical
>ideas?  If so, what bearing?

Quite straightforward really. Mathematicians are frequently called upon
to make decisions as to whether a piece of work is 'important'.
Presumably, you'd agree that these decisions are not totally irrational
affairs and would hope that they are based on good reasons. I believe
there's a job of work for philosophers to tease out the kinds of
consideration at stake. For example, the term 'natural' appears
extraordinarily frequently when mathematicians write about developments
dear to them. I hope my analysis of its use is of interest to mathematicians
(it was to Baez anyway) and also to philosophers. The situation is very
similar to one studied by philosophers of science. Does a term refer to
a natural kind (like copper), a useful kind (like bush), or a gerrymandered
kind (like that old philosophical chestnut the 'emerose' - a curious blend
of emerald and rose). The groupoid concept is treated in precisely these
ways (see chap. 9).

>But this is beside the point.  My real question for you is, why are
>you apparently so hostile to "foundationalism"?  And, what exactly do
>you mean by "foundationalism"?

I don't recall using the word 'foundationalism'. The 'foundationalist
I object to is 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,
theory. Notice that this does NOT imply that these theories are of no
philosophical interest.

>Of course, I haven't read your book.

Feel free to try it.

David Corfield

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