On Apr 1, 3:40 pm, Frederick Williams <freddywilli...@btinternet.com> wrote: > Dan wrote: > > Real mathematicians do their own thing ... no physicist thought > > Hilbert spaces or Riemannian geometry would have any real application > > when they first appeared . > > That's a big claim to make. It seems likely that when they (Hilbert > spaces and Riemannian geometry) first appeared, not every physicist > voiced an opinion that has come down to us. > > If it was von Neumann who invented Hilbert space, then it seems it > was invented in order to give quantum mechanics a rigorous underpinning. > >  von Neumman, _Mathematical foundations of quantum mechanics_, > Princeton UP. > > -- > When a true genius appears in the world, you may know him by > this sign, that the dunces are all in confederacy against him. > Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
https://en.wikipedia.org/wiki/Hilbert_space#History the concept of Hilbert Space was developed prior to the realization of its utility within quantum mechanics, although Von Neumann was the first to give a completed axiomatic formulation, specifically for this purpose . My point was that ,while I do believe set theory to be excessive, this is not so for anything up to second order arithmetic . Furthermore , as mathematicians , we should not let ourselves be constraint by the narrow vision of what empiricists believe as legitimate. Leibniz , Euler , and Russell used infinitesimals in developing their results . The same empiricist stigma was once manifest against the 'fictions quantities' we now refer to as imaginary numbers . Imagine doing modern physics without imaginary numbers. While 'empirical exploration of numbers' may sometimes give us hints (and sometimes false ones http://en.wikipedia.org/wiki/Graham%27s_number , as the counterexamples are too far of to be determined empirically ) , mathematics isn't about empiricism, it's about rational proof . If we proved Fermat's theorem true , we need not check every number for counterexamples . Furthermore , doing so would be a futile endeavor . No one has ever "seen the numbers" , or "performed an experiment on the numbers" , unless it was fundamentally a 'thought experiment' .The essential difference between 'thought experiment' and 'empirical experiment' should be the theme of this discussion . I also recall someone mentioning "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" .