[FOM] John Baez on David Corfield's book

[Stephen G Simpson]

Below I reply to various messages from various people concerning
various topics: (1) alleged philosophical interest of non-foundational
mathematics, (2) alleged historical inevitability of general topology,
(3) mathematics as the grand laboratory of foundationalism.

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W.Taylor at math.canterbury.ac.nz 05 Oct 2003 writes:

 > It is: complex variables, why is it so incredibly, *unexpectedly* tidy?
 > 
 > [...]
 > Is this not a fit question for math philosophy?

Why do you think the tidyness of complex variables may be of interest
for philosophy?  What philosophical issues is it likely to address?

Or, are you perhaps suggesting that philosophers are likely to have
some special insight into the tidyness of complex variables?  If so,
what kind of insight?

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Alexander M Lemberg 5 Oct 2003, speaking of the alleged philosophical
interest of exotic R^4, writes:

 > I will admit that the present stage of philosophical progress on
 > what I have in mind is more or less at the level of numerology. But
 > that does not take away from the clear philosophical interest of
 > this discovery.

I find this statement baffling.  What philosophical progress are you
referring to?  I see no philosophical progress whatsoever.  If it is
only numerology, how can it be of "clear philosophical interest"?  I
am not a professional philosopher, but I think I have a good idea what
the core philosophical issues are, and numerology isn't one of them.
You can't infer anything from numerical coincidences.

 > In many cases, the philosophical import is in the body of
 > mathematics itself.

What does this mean?  Are you saying that philosophers ought to be
interested in (or perhaps dumbstruck by) the fact that there is a
large body of mathematics, without going into details about the
content?

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Referring to the general, abstract concept of topological space, which
is extremely widespread in the 20th and 21st century mathematical
literature, expecially in textbooks.  I said (29 Sep 2003):

 > the history of this concept of topological space seems to show that
 > it developed hand-in-hand with set-theoretical f.o.m. as pioneered
 > by f.o.m. researchers such as Zermelo.  If set-theoretical
 > f.o.m. were not in vogue, then mathematicians would surely have
 > chosen some other concept."

Lemberg replied (8 Oct 2003)

 > What other concept and for what purpose? I know of no other concept
 > that would have been developed as a framework for limits and
 > continuity.

Mathematics embraces plenty of other frameworks for limits and
continuity.  Some of them are: differentiable manifolds, Riemann
surfaces, simplicial complexes, etc etc.  All of these frameworks are
well known to mathematicians and play important roles in mathematics.
General topological spaces are an abstraction of certain common
features of all of these.  This story is well known to mathematicians.

 > It is clear to me that the invention of the topological category
 > was historically inevitable before set theory was available and
 > that set theory would be necessary for this invention to occur.

Why are you so sure that general topology and set theory were
"historically inevitable"?  This seems highly dubious.

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Timothy Y. Chow 6 Oct 2003

 > [...] may provide a model for finding a successful synthesis of
 > foundationalism and coherentism (using these terms the way
 > philosophers typically use them) and give insight into the
 > structure of human knowledge in general.  Mathematical knowledge
 > has the advantage of being highly precise and is thus suitable for
 > an initial case study.

I think you are more or less agreeing with my view of mathematics
(FOM, 3 Oct 2003) as "the grand laboratory of foundationalism".
Mathematics is indeed highly precise, and the precision is closely
tied up with the fact that mathematics is the one science where the
foundationalist program has been so highly successful.

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