Date: Apr 1, 2013 10:03 AM
Subject: Re: Mathematics and the Roots of Postmodern Thought

On Apr 1, 3:40 pm, Frederick Williams <>
> 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[1] who invented Hilbert space, then it seems it
> was invented in order to give quantum mechanics a rigorous underpinning.
> [1] 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 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 , 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" .