
Re: Mathematics and the Roots of Postmodern Thought
Posted:
Apr 1, 2013 11:09 AM


On Apr 1, 5:40 pm, fom <fomJ...@nyms.net> wrote: > 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#Historythe 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 ï¿½ from Galileo's falling bodies, to Newton's > bucket, to Einstein's elevator ï¿½ 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.
While we can use thought experiments to reason about the physical world, my point is that mathematics is 'the domain of thought' , and as such , the only kind of experiment you can have in mathematics is a 'thought experiment' . Falsifiability in mathematics is a category error .

