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 <>
> 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

> 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 .

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."

> I also recall
> someone mentioning "The Unreasonable Effectiveness of Mathematics in
> the Natural Sciences" .

Perhaps it is the unreasonable effectiveness of thought experiments.