[FOM] John Baez on David Corfield's book

[Stephen G Simpson]

John Baldwin Tue, 30 Sep 2003 18:21:03 -0500 (CDT) writes:

 > It seems to me that this illustrates Corfields point rather than
 > Simpson's. 

What do you think Corfield's point was, and why do you think general
topology illustrates it?

Corfield's actual point was a negative one:

  "f.o.m. research has no bearing on choice between rigorously defined
  concepts" 

(this is a quote from Corfield).

My point is that, historically, set-theoretical f.o.m. research
influenced mathematicians to study general topological spaces.  Don't
you agree with this point?  I presented this as a counterexample to
Corfield's assertion.

 > While the concepts of general topology remain of considerable interest to
 > logicians and especially set theorists, [...]

This comment is misleading.  The concepts of general topology also
remain of considerable interest to the majority of mathematicians.
Indeed, these concepts are almost universally required as part of the
standard curriculum for mathematics students at the advanced
undergraduate and beginning graduate level.

 > the general thrust of mathematics has been towards more `tame
 > spaces'.

There is some truth to this.  General topology, though still popular
as part of a standard framework for advanced mathematics, is now much
less in vogue as a research area than it used to be.  Topologists now
focus much more of their research effort on algebraic topology,
differential topology, etc.

 > Maclane told me that this distinction was already apparent in the
 > 20's 

Certainly the distinction between general topological spaces and
"tame" topological spaces (manifolds, simplicial complexes, etc) was
clear from the beginning.

My point is that, under the influence of set-theoretic f.o.m. dating
from the early 20th century, many mathematicians viewed the concept of
general topological space and associated concepts as elegant and
valuable.  Consequently, general topology was a lively research area
through most of the 20th century.  This refutes Corfield's assertion.

 > the analysis of why 3 dimensional manifolds are of intense interest
 > and the normal Moore space conjecture is not seems to be an
 > important methodological question not amenable to fom methods.

Why do you think this kind of question is not amenable to
f.o.m. methods?

On the contrary, f.o.m. research from the 1960's onward showed that
many questions of general topology (among them the normal Moore space
conjecture) have different answers in different models of ZFC.  This
contributed to a re-evaluation of general topology, by convincing
mathematicians that such questions cannot be answered using standard
mathematical methods.  Don't you think this is part of the reason why
topologists turned their attention away from general topological
spaces and toward "tame" spaces?

This appears to be another instance where f.o.m. research had a role
in influencing mathematicians to play down certain concepts and play
up others.

But this is something that Corfield apparently would never conceive
of, because of his anti-f.o.m. filter.

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