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