Date: Apr 1, 2013 10:03 AM
Subject: Re: Mathematics and the Roots of Postmodern Thought
On Apr 1, 3:40 pm, Frederick Williams <freddywilli...@btinternet.com>
> 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" .