How to fit an elephant (2011)(johndcook.com) |
How to fit an elephant (2011)(johndcook.com) |
Otherwise, it's just too easy to hide a lot of complex machinery inside a "single parameter". In fact, from this perspective it's arguable that an arbitrary real number is actually a (countably) infinite set of parameters, since it takes that many bits to uniquely specify any real number.
But this (set) isomorphism between R and C is not continuous. Indeed one can show that there exists no continuous epimorphism f: R^n -> R^m, where m > n, since for every such continuous map f the image f(R^n) has a measure of 0 with respect to the Borel measure in R^m.
"When posed with a variant of this question involving a fly and two bicycles, John von Neumann is reputed to have immediately answered with the correct result. When subsequently asked if he had heard the short-cut solution, he answered no, that his immediate answer had been a result of explicitly summing the series (MacRae 1992, p. 10; Borwein and Bailey 2003, p. 42)."
One parameter at a time.
https://www.google.at/url?sa=t&source=web&rct=j&url=https://...
"the trains take one hour to collide (their relative speed is 100 km/h and they are 100 km apart initially). Since the fly is traveling at 75 km/h and flies continuously until it is squashed (which it is to be supposed occurs a split second before the two oncoming trains squash one another), it must therefore travel 75 km in the hour's time."
So if von Neumann was solving it by explicitly summing the series, as the anecdote claims, then he was doing it wrong :)
The fly travels 3/2 times the speed of a train, so every bounce the fly travels 3/5 of the remaining track and leaves 1/2 * 2/5 = 1/5 track to travel, so we just compute the geometric sum
3/5 * \sum 1/5^r = 3/5 * 1/(1 - 1/5)
= 3/5 * 1/(4/5)
= 3/5 * 5/4
= 3/4.
Really? Then what is a https://en.wikipedia.org/wiki/Space-filling_curve?
Moreover OP's argument specifically proves too much, because space-filling curves (as described in the article) have a range with positive Borel measure.
This is indeed possible - but this is clearly not the topology that "ordinary people" and physicists mean when talking about continuity of functions from R^n to R^m.