We were discussing on the radio today how random numbers are generated, and how could it be proved  to the satisfaction of a mathematician  that the number really is random? Evan Stanbury explains... Listen Now Download as mp3
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CommentsTruly random numbers can only be generated by radioactive decay or some such mechanism. The computer algorithms only produce pseudorandom numbers. These are tested for randomness  an equal probability for each number, within a statistically determined variance, and no theoretical reason why any particular sequence should be absent. Some computer algorithms even allow you to set the first seed, so that you can reproduce the same pseudorandom sequence again, but it is more usual to use a clock reading as the first seed. John von Neumann, computer pioneer said: A reply sent to me from the programme in question by Reenen Laurie: A very deep question, lot of great answers here. I've done a few simulations using computer generated “pseudorandom” numbers and the Monte Carlo method at Los Alamos many years ago. We needed a random number with a Gaussian distribution and we had a uniform random number generator over the region 0 to 1. So we added 6 of these together to get an approximation to a Gaussian distribution by way of the central limit theorem. The ends of the distribution are cut off at 0 and 6 but for many purposes it worked just fine. It was easy enough to test the numbers for mean, standard deviation, correlation or whatever parameters might be important in the simulation and insure they were “random enough” for the problem at hand. Of course there are built in functions in many math packages that do this now, yes I am that old, started out using punch cards. Thought this might be of interest here, my rather old 2 cents worth! distimpson, Fri, 3rd May 2013 There's an app for that ... Thanks for the great answers so far. It's very difficult. The toughest application of pseudorandom number generators is probably in cryptography. Whether randomness exists is a metaphysical question that I've yet to see addressed, convincingly. The best we can do is unpredictable sequences of maximal entropy (no identifiable pattern). Even perfectly well defined sequences can be effectively random. E.g. if you started listing the digits of pi, beginning with the 100 trillionth digit... the sequence would appear random to anyone who didn't realize the source of the info. AndroidNeox, Sun, 12th May 2013
It might take a fair amount of computing power to calculate something like e \pi \sqrt{2} to the some very long digit sequence. But there would be no shortage of irrational numbers that one could calculate to a long digit sequence. Perhaps a bit more complex if one wanted multidigit numbers, although I suppose with zeros in the sequence, one could just say take 10 digit blocks from your decimal expansion of your favorite irrational number.
You cannot prove that a number is really random, if someone picks out a numbered card from a pack, that is not random, because whoever shuffled the pack, left them in a certain order, however chaotic, and the person picking out a card does this in some kind of order, wether it be the top, the bottom, the middle or somewhere inbetween. A computer cannot randomly select a number, because it has to start it's next random shuffle from a set point, which is the point it stopped at on the previous random selection. I suppose the more random tries before a random selection might allow you to gaining a more random number, but in reality it is still not random. confusious says, Tue, 14th May 2013
Interesting, I looked at the link you gave. 8D confusious says, Thu, 16th May 2013 There's quite a few articles like that out there. One shows two squares, one scattered with dots at random, and one with correlations between the dots that make them appear relatively smooth and homogeneous. Most people, when asked to select the random square, see the clumps and squiggles of dots in the random one and so pick the square with the smoother distribution insead (see Points & Poisson). The author says "The point about random processes, even simpler ones like repeated tossing of a coin, is that coincidences happen much more frequently than one might suppose". This goes some way to explaining why so many people believe the coincidences they encounter in daily life have paranormal significance  they seriously underestimate the expected frequency of those coincidences. Thanks to Evan, who submitted a lovely piece of audio to make his case.
The Copenhagen Interpretation is dependent upon concepts, like simultaneity, that have no physical reality (action at a distance). Also, all science is dependent upon the assumption of causality... when physics discarded causality we should have realized we'd gotten off track. We should have realized that observation doesn't alter the observed. Observation alters the observer. All physics is local. All observable reality is purely causal. 
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