From: Helger Lipmaa <helger@saturn.tcs.hut.fi> Subject: [math-fun] Parallel Universes
http://www.sciam.com/article.cfm?articleID=000F1EDD-B48A-1E90-8EA5809EC58800...
"In infinite space, even the most unlikely events must take place somewhere."
A nice article but that "infinite" word jars a bit. I wish the Cantor diagonalization process had been discussed. I suppose it is considered in with-math version of the article, but that article is so long that it could not be fit within the 2^N size of an issue of SciAm or the 2^M size of a reader's mind. Given countable number of physical constants in binary choose new constant differing from each one previous, etc. Or any other measurement of context difference. Voila - a universe not matching any of the existing. Actually we have to imagine "voila" but the "whole" idea is unobservable imaginings so that's consistent. Some of the article's phraseologies are stimulating... "Imagine moving planets to random new locations, imagine having married someone else, and so on. At the quantum level, there are 10 to the 10118 universes with temperatures below 108 kelvins. That is a vast number, but a finite one." If the imaginations must be embedded in finite imaginers, then there are only a finite number of imaginings too. "All of physics is ultimately a mathematics problem: a mathematician with unlimited intelligence and resources could in principle compute..." Same difficulty. There exists "in the world" no such mathematician. My favourite version is a restating of Hamlet: "There are more philosophies dreamt in heaven than exist in our earth". "Mathematical structures have an eerily real feel to them. They satisfy a central criterion of objective existence: they are the same no matter who studies them." That I like. There might be an interesting analogue to the recent discussion of deriving a "fair coin" probability distribution by tossing a weighted coin. At least an opportunity to do some more math-stuff! Apropos of "the" set of natural numbers, author says... "Which is simpler, the whole set or just one number? Naively, you might think that a single number is simpler, but the entire set can be generated by quite a trivial computer program, whereas a single number can be hugely long. Therefore, the whole set is actually simpler." The whole set is simpler only if we believe that the algorithm is sufficient to create a representative of such a set in the physical universe. Otherwise the single number is simpler. Even that may be trouble, since "twoness" requires a recognizer capable of all the complexity of mathematics. (Though I do know a very young man who presently counts only "2, 2, 2..." in a somewhat monotonous fashion, he is ultimately capable of more nuanced mathematical complexity.) The best mathematical construct I have encountered to adapt to the operational role of "universal" thinking, is that of dual space of functions. If the physical universe exists, the mappings of physical onto some flatness (eg paper, via scribbling or photographs) is a universe of imaginings, with enhanced structure. It is also an optimistic outlook, since an I-universe need not be subject to 2nd law of thermodynamics in the same way as a P-universe. Only its implementation and its output representations in P-universe have that particular constraint. It is conceivable, if time is directional, that information increases in I-universe though P-universe becomes less thermodynamically rich. That counting algorithm describing the generation of the set of all natural numbers can exist in various P-universe representations (Peano, computer pgm etc), but can also be implemented in I-universe and has its fullest output there, which can then be selectively projected onto P-universe. The problem is not with the math but with ourselves. A pleasant entertainment for the start of this weekend which celebrates the spring season of rebirth. Best to all, Ken Roberts