fom
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Re: Mathematics and the Roots of Postmodern Thought
Posted:
Apr 1, 2013 10:40 AM


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/thoughtexperiment/#DebOveThoExp
> I also recall > someone mentioning "The Unreasonable Effectiveness of Mathematics in > the Natural Sciences" . >
Perhaps it is the unreasonable effectiveness of thought experiments.

