How to fit an elephant (2011)
johndcook.comI can't help but feel like a complex number is two parameters (real&imag / mod&arg) - so really this is 8 parameters.
This is a good point. If we're posing Occam's Razor-like arguments against models for having "too many free parameters", then we should probably really be comparing something more precise, like the models' Kolmogorov complexities.
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.
Cantor showed that these sets have the same cardinality. You can represent a complex (in the form of two reals) by interleaving digits or using a space-filling curve for example.
> Cantor showed that these sets have the same cardinality. You can represent a complex (in the form of two reals) by interleaving digits or using a space-filling curve for example.
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.
> there exists no continuous epimorphism f: R^n -> R^m, where m > n
Really? Then what is a https://en.wikipedia.org/wiki/Space-filling_curve?
Right from that article you linked: "A non-self-intersecting continuous curve cannot fill the unit square because that will make the curve a homeomorphism from the unit interval onto the unit square (any continuous bijection from a compact space onto a Hausdorff space is a homeomorphism). But a unit square has no cut-point, and so cannot be homeomorphic to the unit interval, in which all points except the endpoints are cut-points."
Yeah, a non-self-intersecting map cannot, but OP didn't specify that, only "epimorphic" which certainly can.
Moreover OP's argument specifically proves too much, because space-filling curves (as described in the article) have a range with positive Borel measure.
Sure, but did we need continuity? Also, if you want to be awkward, you can get around this by using the discrete topology, I don't think we needed the metric structure of R^n.
> Sure, but did we need continuity? Also, if you want to be awkward, you can get around this by using the discrete topology
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.
So, it's all actually just one parameter?
Pairing functions only work on countable sets. This is funny because Cantor is the same person who proved real numbers are uncountable, and that there is no pairing function between 1 real number and naturals, let alone 2.
I'm claiming a "pairing function" between single reals and pairs of reals. They have respective cardinalities 2^N0 and 2*2^N0=2^N0 where N0<2^N0 is the cardinality of the naturals.
You're right. I was wrong. I found an explanation of how to make interleaving method work. https://math.stackexchange.com/a/183383
He could have used an octonion, then it was 1 parameter.
The only thing missing here is a little anecdote about how Von Neumann, when challenged on this, did it in his head and started rattling off the parameters. For arbitrary animals.
Source please. Must learn more.
Von Neumann was renowned for his great prowess at mental maths. A famous (if also not entirely serious) story:
"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)."
The article you linked to provides a trivial solution:
"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 :)
Well, if you're familiar with sums, then the "shortcut" isn't all that much faster; it's mostly a function of how quickly you can grok the salient points:
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
One nice thing about this sum is that it encodes a bit more insight about the fly's flight path than the shortcut method.3/5 * \sum 1/5^r = 3/5 * 1/(1 - 1/5) = 3/5 * 1/(4/5) = 3/5 * 5/4 = 3/4.I assume you're trying to beat the original responder in brazen literalism - now we have a citation request for a throwaway joke, an explanation (with citation) and a literal interpretation of the joke in the citation.
How do you fit an elephant?
One parameter at a time.
Now can you add one more and make him wiggle his trunk?
In the actual paper they do exactly that
https://www.google.at/url?sa=t&source=web&rct=j&url=https://...