[FOM] John Baez on David Corfield's book
jbaldwin@math.uic.edu
jbaldwin at math.uic.edu
Tue Sep 30 19:21:03 EDT 2003
Simpson writes:
> An example is the pervasive concept of topological space, i.e., an
> ordered pair (X,t) where X is a nonempty set and t is a collection of
> subsets of X containing X and the empty set and closed under unions
> and finite intersections. This concept occurs very frequently in the
> 20th and 21st century mathematical literature, especially in textbooks
> for advanced mathematics students. Now, my point is that 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.
>
It seems to me that this illustrates Corfields point rather than Simpson's.
While the concepts of general topology remain of considerable interest to
logicians and especially set theorists, the general thrust of mathematics has
been towards more `tame spaces'. Maclane told me that this distinction was
already apparent in the 20's but I took this bit of hindsight with a grain of a
salt.
The point is that 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. Indeed they might suggest
the importance is in the other direction.
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