The page http://cstheory.stackexchange.com/questions/32202/the-complexity-of-finding-... is relevant, but does not address my concern, which might be phrased as, Is there some sense in which the n=2 case is computationally harder than the n=1 case (aka the Intermediate Value Theorem)? Perhaps I am looking for something more in the spirit of descriptive set theory (or effective descriptive set theory). If f is a "nice" map from S^2 to R^2, can a "nice" curve in R^2 have a preimage in S^2 that is "nasty" in a way that impedes the intuitive proof? Or maybe the way to morally prove that Borsuk-Ulam is deeper than Vsauce's "proof" suggests would be to show that if you replace two-dimensional continuity of f by the weaker condition that f is one-dimensionally continuous on all great circles, the conclusion fails. You can see that I'm flailing about a bit, which is why I'm posting here and not on MathOverflow. The MathOverSeers discourage this kind of vagueness. Jim Propp On Tuesday, October 4, 2016, James Propp <jamespropp@gmail.com> wrote:
On a more mathematically substantive topic, can the proof of Borsuk-Ulam on S^2 that Vsauce presents be made rigorous without a lot of extra machinery?
Neither of the proofs at https://en.m.wikipedia.org/ wiki/Borsuk–Ulam_theorem is as simple as one might hope.
Jim Propp
On Tuesday, October 4, 2016, James Propp <jamespropp@gmail.com <javascript:_e(%7B%7D,'cvml','jamespropp@gmail.com');>> wrote:
Michael at Vsauce, in his superb video on fixed points ( https://m.youtube.com/watch?v=csInNn6pfT4), asserts that the reason you eventually end up with 4 if you iterate the operation "spell the word and count the letters" is that 4 is the only fixed point of this map. This neglects the possibility of cycles. Can anyone find a language in which the spell-and-count map contains one or more cycles of length greater than 1?
There is some ill-definedness of the spell-and-count map in English, and probably other languages too; e.g., 101 is both "one hundred one" and "one hundred and one". There's also the matter of the British billion vs. the American billion. All such variants are legitimate for purposes of my question.
Jim Propp