Date: Apr 1, 2013 10:03 AM Author: dan.ms.chaos@gmail.com Subject: Re: Mathematics and the Roots of Postmodern Thought 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[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

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" .