[FOM] Neil Tennant's Questions

[F.A. Muller]

> 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