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Sewing-Machine comments on Open Thread: July 2010, Part 2 - Less Wrong
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By your definition, is the empirical fact that one tenth of the digits of pi are 1s emergent behavior of pi?
I may not understand the work that "low-level" and "high-level" are doing in this discussion.
On the length of derivations, here are some relevant Godel cliches: System X (for instance, arithmetic) often obeys laws that are underivable. And it often obeys derivable laws of length n whose shortest derivation has length busy-beaver-of-n.
(Uber die lange von Beiwessen is the title of a famous short Godel paper. He revisits the topic in a famous letter to von Neumann, available here: http://rjlipton.wordpress.com/the-gdel-letter/)
Just a pedantic note: pi has not been proven normal. Maybe one fifth of the digits are 1s.
I'll stick to it. It's easier to perform experiments than it is to give mathematical proofs. If experiments can give strong evidence for anything (I hope they can!), then this data can give strong evidence that pi is normal: http://www.piworld.de/pi-statistics/pist_sdico.htm
Maybe past ten-to-the-one-trillion digits, the statistics of pi are radically different. Maybe past ten-to-the-one-trillion meters, the laws of physics are radically different.
The later case seems more likely to me.
I was just thinking about the latter case, actually. If g equalled G * (m1 ^ (1 + (10 ^ -30)) * (m2 ^ (1 + (10 ^ -30))) / (r ^ 2), would we know about it?
Well, the force of gravity isn't exactly what you get from Newton's laws anyways (although most of the easily detectable differences like that in the orbit of Mercury are better thought of as due to relativity's effect on time than a change in g). I'm not actually sure how gravitational force could be non-additive with respect to mass. One would have the problem of then deciding what constitutes a single object. A macroscopic object isn't a single object in any sense useful to physics. Would for example this calculate the gravity of Earth as a large collection of particles or as all of them together?
But the basic point, that there could be weird small errors in our understanding of the laws of physics is always an issue. To use a slightly more plausible example, if say the force of gravity on baryons is slightly stronger than that on leptons (slightly different values of G) we'd be unlikely to notice. I don't think we'd notice even if it were in the 2nd or 3rd decimal of G (partially because G is such a very hard constant to measure.)
I can't disagree about what you want but I myself don't really see the point in using the word emergent for a straightforward property of irrational numbers. I wouldn't go so far as to say the term is useless but whatever use it could have would need to describe something more complex properties that are caused by simpler rules.
This isn't a general property of irrational numbers, although with probability 1 any irrational number will have this property. In fact, any random real number will have this property with probability 1 (rational numbers have measure 0 since they form a countable set). This is pretty easy to prove if one is familiar with Lebesque measure.
There are irrational numbers which do not share this property. For example, .101001000100001000001... is irrational and does not share this property.
True enough. it would seem that irrational number is not the correct term for the set I refer to.
The property you are looking for is normalness to base 10. See normal number.
ETA: Actually, you want simple normalness to base 10 which is slighly weaker.
Any irrational number drawn from what distribution? There are plenty of distributions that you could draw irrational numbers from which do not have this property, and which contain the same number of numbers in them. For example, the set of all irrational numbers in which every other digit is zero has the same cardinality as the set of all irrational numbers.
Yes, although generally when asking these sorts of questions one looks at the standard Lebesque measure on [0,1] or [0,1) since that's easier to normalize. I've been told that this result also holds for any bell-curve distribution centered at 0 but I haven't seen a proof of that and it isn't at all obvious to me how to construct one.
Well, the quick way is to note that the bell-curve measure is absolutely continuous with respect to Lebesgue measure, as is any other measure given by an integrable distribution function on the real line. (If you want, you can do this by hand as well, comparing the probability of a small bounded open set in the bell curve distribution with its Lebesgue measure, taking limits, and then removing the condition of boundedness.)
Excellent, yes that does work. Thanks very much!
IMO, that would be emergent behaviour of mathematics, rather than of pi.
Pi isn't a system in itself as far as I can see.
I have in mind a system, for instance a computer program, that computes pi digit-by-digit. There are features of such a computer program that you can notice from its output, but not (so far as anyone knows) from its code, like the frequency of 1s.