[FOM] John Baez on David Corfield's book
Alexander M Lemberg
sandylemberg at juno.com
Wed Oct 8 17:20:44 EDT 2003
On Mon, 29 Sep 2003 17:09:15 -0400 Stephen G Simpson
<simpson at math.psu.edu> 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."
Ironically, this remark of yours seems to be in the spirit of Lakatos et
al to which you so vehemently object. I do not agree with it.
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.
Questions of limits and continuity were addressed by Cauchy and Bolzano
in the early 19th century. Bolzano's "Rein analytischer Beweis" was
about the topological property of connectedness. The "Heine-Borel
Theorem" was about compactness, another topological property. These ideas
were floating around without an adequate framework decades before the
development of set theory . 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.
It could be argued that the investigation of the continuum had more to do
with ordered sets than with topology, but I think that would be msguided.
Not only is the flavor of the foundational questions of 19th century real
analysis quite topological, but multidimensional geometric considerations
are also topological in nature. For example, the geometric and complex
analytic thinking of Gauss and Riemann clearly anticipated the
topological category, and could not be accommodated in the framework of
ordered sets.
Sandy
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