# Can true randomness come out of mathematical rules?

For example, prime numbers, they seem very random, and they are defined by a simple set of rules. I can't see how real randomness could exist in the real world, but what about mathematics?

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What is true randomness? What is a mathematical rule? What does it mean for the former to come out of the latter? – Qiaochu Yuan Sep 2 '12 at 19:52
Just as a further comment on "true randomness", there is a whole spectrum of random numbers distributed according to various distributions. Calling a particular form of randomness "true" should be followed by the name of the distribution you are talking about. Are you referring to the uniform distribution? – Kartik Audhkhasi Sep 2 '12 at 20:10
Do you have actually ever seen real randomness in real life? Are you sure you're not just too slow computing what's going to happen? ;-) – stefan Sep 2 '12 at 23:43
@Ajasja it remains conceivable that a hidden variable theory underlies quantum mechanics, in which case the "true randomness" of QM is no more random than anything else we are unable to predict with existing knowledge and technology. – James Sep 3 '12 at 10:02
@Ajasja: I hadn't heard of that work, it looks interesting. The important assumption they make is that QM is correct - it is surely possible that QM is just a very good approximation to a more accurate theory that is deterministic (or at least more deterministic) in the same way that Newtonian mechanics is an approximation to QM. Whether the universe is deterministic is really more of a philosophical question than a scientific or mathematical one, and either way, there will probably always be things we are unable to predict in practice, so that they are effectively random. – James Sep 3 '12 at 17:14

Mathematics doesn't purport to generate randomness from scratch.

Probability theory is a branch of mathematics dedicated to describing randomness, in the sense that if you ever find a source of real randomness somewhere, probability can help you analyze and predict its aggregate behavior.

Beware that "real randomness" in this context is essentially just code for "randomness that behaves in the way the theory assumes that real randomness will behave". But there seems to be plenty of real-world sources of randomness that agree with this assumption, at least well enough to allow casinos and insurance companies to make a profit.

You might argue that the apparent randomness of a coin flip is really just a result of the fine details of the starting conditions being unknown and uncontrollable, which is (or could be argued to be) different from randomness in principle. On the other hand, physicists claim to have solid experimental proof that the fundamental laws of the universe at a quantum level do produce mathematically "real" randomness.

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Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin. - John von Neumann

From: Various techniques used in connection with random digits", Applied Mathematics Series, no. 12, 36–38 (1951).

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One kind of randomness that cannot come from "rules" is Kolmogorov–Chaitin randomness. That follows immediately from its definition.

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I'm not sure what kind of answer you are expecting, but randomness is a thing in mathematics. For example http://en.wikipedia.org/wiki/Random_variable and http://en.wikipedia.org/wiki/Kolmogorov_complexity

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The binary digits of $\pi$ may look random (and it is conjectured that a property called being a "normal" number, which is related to this indeed holds). However, if you knew that the successive results of a coin flip were based on walking through the binary expansion of $\pi$, you could make a fortune by calculating ahead and betting accordingly. Determinately making a fortune is not possible with betting on a truly fair random coin.

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I claim that the function $f(n) =$ the digit in the hundredth decimal place of the decimal expansion of $\sqrt{n\pi}$ is a random function from $\mathbb{Z}$ to the set $\{0, 1, \dots, 9\}$, but I would be definitely impressed by anyone who can prove that this function is not random.

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You'll have to define "random" and what kind of random distribution you claim this function has. Uniform, I presume? Also, most definitions of randomness reject your claim effortlessly - information theory claims there is zero entropy in your function, therefore your "random" numbers are 100% predictable - which is true. You could argue that you can start from a random value for $n$ so that people can't evaluate the function at the same time as you, but where do you get that random "seed"? – Thomas Sep 3 '12 at 1:52

Once some ppl believed cellular automaton contained randomness. But in the meanwhile those opinions have been wheakened alot since they proved rule 30 is not such a good pseudorandom generator afterall.

It also depends on your definition of randomness.

Are you talking about Kolmogorov complexity ? The so called normal numbers ?

I believe there is some randomness in number theory in particular related to primes.

I personally see randomness as in 'unpredictable and balanced' although i realize thats not a solid definition.

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