[FOM] John Baez on David Corfield's book

Stephen G Simpson simpson at math.psu.edu
Fri Oct 3 13:06:49 EDT 2003

David Corfield, Thu, 25 Sep 2003 09:07:44 +0100 writes:

 > 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. 

I think you are misrepresenting what happened here.  Penrose tilings
did not "falsify" topology, neither heuristically nor otherwise.  The
concepts and theorems of topology remain as valid as ever.

Furthermore, you are again attacking a straw man, because nobody ever
claimed that "our topological resources" -- i.e., the concepts and
methods of topology -- suffice to "give us information about ANY
SPACE."  (My caps.)  The purpose of topology is to study topological
aspects of topological spaces, not non-topological aspects of
non-topological spaces.

Did you intend Penrose tilings as an example of a non-f.o.m. piece of
mathematics which is philosophically interesting?  If so, could you
please explain the alleged philosophical interest?

 > A big debate now, and surely one to interest philosophers, is
 > whether to follow Connes' or Grothendieck's conceptions about
 > mathematical space.

Your word "debate" is somewhat misleading.  There is no logical
contradiction between the allegedly "debated" concepts of Connes and
Grothendieck.  They refer to different classes of entities, both of
which can be studied and compared within the normal, standard,
rigorous, set-theoretic, foundational framework which is accepted and
used by the vast majority of mathematicians.

Therefore, when you say "debate", I assume you are referring to a
debate over how to set research priorities, allocate government
research funds for mathematics, or whatever.  Is that what you are
referring to?

I think such a debate might be of interest for mathematicians.  But
why do you think such a debate would be of interest for philosophers?
What philosophical questions do you think such a debate might address?
Are you claiming that such a debate might contribute to compelling
answers to core philosophical questions?

 > 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. 

I have already pointed out how this kind of shrill,
anti-f.o.m. statement is radically false in that it overlooks the
rigorous, logical, foundationalist, definition-theorem-proof framework
which most mathematicians employ in order to organize their
mathematical ideas.

 > Does anyone on this list believe that a topic such as the
 > mathematical conception of space should be ignored by philosophy?

It depends on what exactly you may mean by "the mathematical
conception of space"?  Mathematics embraces a large variety of
space-like concepts.  It may be that some of them are philosophically
interesting, and some not.  To determine their philosophical interest,
we would have to examine them on a specific, case-by-case basis.

If your "mathematical conception of space" holds out no prospect of
addressing significant philosophical questions, then surely philosophy
ought to ignore it.  Don't you agree?

 > Does anyone believe that proof theory can do the job?

What job?


Stephen G. Simpson
Professor of Mathematics
Pennsylvania State University

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