Date: Apr 1, 2013 10:40 AM Author: fom Subject: Re: Mathematics and the Roots of Postmodern Thought On 4/1/2013 9:03 AM, Dan wrote:

> 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 .

One can find a critic for everything:

"A thought experiment is no substitute for a real experiment, he

claimed, and should be forbidden in science, including science

education." (A paraphrase of Duhem)

The next sentence, quite naturally being the statement:

"However, in view of the important role of actual thought experiments in

the history of physics ? from Galileo's falling bodies, to Newton's

bucket, to Einstein's elevator ? it is unlikely that anyone will feel or

should feel much sympathy for Duhem's strictures."

http://plato.stanford.edu/entries/thought-experiment/#DebOveThoExp

> I also recall

> someone mentioning "The Unreasonable Effectiveness of Mathematics in

> the Natural Sciences" .

>

Perhaps it is the unreasonable effectiveness of thought experiments.