[FOM] Neil Tennant's Questions
F.A. Muller
F.A.Muller at phys.uu.nl
Thu Feb 19 06:04:21 EST 2004
> Put in a nutshell (or two):
> (a) Is our existing mathematics complete enough for all the possible
> demands of natural science?
> (b) Is a surprisingly weak fragment of existing mathematics complete
> enough for all the possible demands of natural science?
I would be extremely surprised if the sciences needed more
mathematics than can be proved on the basis of ZC. You
then have a tower of sets of ordinal level up to but not
including omega + omega. Replacement (F) is needed to get
the entire hierarchy and some theorems of descriptive
set-theory no one will ever apply (about Borel-games),
or so I conjecture.
What about Choice? For a general presentation of say quantum
mechanics (QM), it is indispensible. Often one finds expressions
in QM-textbooks like `Choose a basis of some Hilbert-space.'
To prove that every vector-space has a basis you need Choice.
Whether this is really needed for applications I am unable to
tell. When calculating probabilities in QM, one often has a
fixed Hilbert-space, usually L^{2}(R^{3}), and an explicit
basis (`special functions': Legendre polynomials, Bessel-functions,
etc with some exponential factor added), of which you can prove
they form a basis of L^{2}(R^{3}). Choice is not needed for that.
Whether Choice can be left alone for all actual applications,
I don't know. Someone must be able to say which often
applied theorems in Analysis rely on Choice.
I know of talks at LMPS-meeting in Florence (1994?) where
mathematicians investigated how far you can get in
science without Choice. I don't remember who they are.
-> F.A. Muller
Utrecht University
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