[FOM] John Baez on David Corfield's book

David Corfield david.corfield at philosophy.oxford.ac.uk
Thu Sep 25 04:07:44 EDT 2003

Charles A Stewart wrote:

> Despite his attachment to Lakatos, I didn't get the impression that
Corfield was
> anti-foundationalist in the sense of Tymoczko and Hersh (ie. was some some
> empiricist or pragmatist about mathematical entities); rather while I
> that Stephen would dislike Corfield's work, I expected him to dislike its
> "list-2" mindset.

The way to get Lakatos's quasi-empiricism is via the notion of
developing an idea properly. Evidence that you have gone astray
comes from the bottom-up - heuristic falsifiers. People get confused
by what Lakatos is saying because he can't see how this process
can occur once you have arrived in a rigorous setting, so makes
disparaging comments about the advent of rigour. Of course, it
is perfectly possible to have heuristic falsifiers in an axiomatic setting:
we thought our topological resources were sufficient to give us information
about any space. However, as far as classical topology goes the only open
subsets of the space of Penrose tilings are the whole space and the empty
space. Either you pass the space off as pathological or, like Connes, you
develop the concepts of topology. A big debate now, and surely one to
interest philosophers, is whether to follow Connes' or Grothendieck's
conceptions about mathematical space. You can even try for both -
see Pierre Cartier's 'Mad Day's Work' article for an attempted

As for foundations, it may just be the case that they have naturally
evolved to become completely divorced from the question of the proper
organisation of mathematical concepts. From recent revisions of Frege's
views, it is clear that he would have been disappointed about this state
of affairs. I can see why people promote category theory as a way of
bringing them back together. But I'm not going down that well-beaten
track here.

> 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".

The are several allusions in place here, one of them being G.H.Hardy's
famous comment. 'Inflammatory' is an inflammatory word here. I just
want to draw attention to what I see is an important oversight on the part
of philosophers. Does anyone on this list believe that a topic such as the
mathematical conception of space should be ignored by philosophy?
Does anyone believe that proof theory can do the job?

David Corfield

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