[FOM] When is it appropriate to treat isomorphism as identity?
Monroe Eskew
meskew at math.uci.edu
Tue May 19 02:33:44 EDT 2009
On Sat, May 16, 2009 at 12:29 PM, Andrej Bauer <andrej.bauer at andrej.com> wrote:
> Nevertheless, too much reliance onclassical set theory has its negative effects, such as:
> - it makes topologists blind to the topological notion of overtness,which is just as important as compactness (Paul Taylor has said >moreabout this in a parallel post)
Of course, a classical mathematician (set theorist or not) would find
anti-classical notions to be either meaningless or trivial when
interpreted from their viewpoint, and would thus not consider this
blindness.
> - it makes analysts blind to the possibility of axiomatizing theirsubject directly in terms of nilpotent infinitesimals. Luckily,physicsts >never much liked the epsilons and deltas and have kept thetradition alive (and built radio, TV, transistors, lasers and GPSusing the >infinitesimals that do not exist according to orthodoxmathematics)
Almost all engineers and scientists are trained in classical rather
than anti-classical mathematics, and all these examples can be
adequately understood with classical logic (and standard analysis, and
Robinson's nonstandard analysis). I would be interested to see an
example of an invention or scientific discovery that came about due to
the person's use of anti-classical mathematics.
> - it makes set theorists somewhat of a community of outcasts becausethey shake the foundation all the time by considering silly >thingssuch as large cardinals. Luckily Harvey Friedman is making excellentefforts in persuading the mathematicians that ZFC is not >the end ofthe story.
I don't know what you mean by shaking foundations. The theory of
large cardinals is done within ZFC.
> - it makes the lives of computer scientists difficult by shatteringthe connection between computaton and logic to pieces, so that >theyhave to bolt on computability as an afterthought (as a friend of mineonce said).
I don't know what you mean. Set theory does not contradict computer
science. As I said, set theory is designed to be inclusive of all
classical mathematics. People may study any aspect of logic they
wish, and that they do. There is a huge and active discipline called
"theoretical computer science."
Best,
Monroe
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