On Jun 25, 10:38 am, Paul Rubin <http://phr...@nospam.invalid> wrote: > Robin Becker <ro...@reportlab.com> writes: > > someone once explained to me that the set of systems that are > > continuous in the calculus sense was of measure zero in the set of all > > systems I think it was a fairly formal discussion, but my > > understanding was of the hand waving sort. > > That is very straightforward if you don't mind a handwave. Let S be > some arbitrary subset of the reals, and let f(x)=0 if x is in S, and 1 > otherwise (this is a discontinuous function if S is nonempty). How > many different such f's can there be? Obviously one for every > possible subset of the reals. The cardinality of such f's is the > power set of the reals, i.e. much larger than the set of reals. > > On the other hand, let g be some arbitrary continuous function on the > reals. Let H be the image of Q (the set of rationals) under g. That > is, H = {g(x) such that x is rational}. Since g is continuous, it is > completely determined by H, which is a countable set. So the > cardinality is RxN which is the same as the cardinality of R. > > > If true that makes calculus (and hence all of our understanding of > > such "natural" concepts) pretty small and perhaps non-applicable. > > No really, it is just set theory, which is a pretty bogus subject in > some sense. There aren't many discontinuous functions in nature. > There is a philosophy of mathematics (intuitionism) that says > classical set theory is wrong and in fact there are NO discontinuous > functions. They have their own mathematical axioms which allow > developing calculus in a way that all functions are continuous. > > > On the other hand R Kalman (of Bucy and Kalman filter fame) likened > > study of continuous linear dynamical systems to "a man searching for > > a lost ring under the only light in a dark street" ie we search > > where we can see. Because such systems are tractable doesn't make > > them natural or essential or applicable in a generic sense. > > Really, I think the alternative he was thinking of may have been > something like nonlinear PDE's, a horribly messy subject from a > practical point of view, but still basically free of set-theoretic > monstrosities. The Banach-Tarski paradox has nothing to do with nature.
I'll take the Banach-Tarski construct (it's not a paradox, damn it!) over non-linear PDEs any day of the week, thankyouverymuch. :) -- http://mail.python.org/mailman/listinfo/python-list
