Can randomness be random?

In mathematics, a true random number generator it's impossible, because any formula defines a process that, however complex, is not random.

A random event must be unrelated to any cause or condition, and therefore cannot be causal. It is a brute fact par excellence.

If I draw up a list of all the possible conditions, I can say that a random event is outside of it. But isn't this a rule that determines the conditions of a random event?

Edit: this question was wrong, I was confusing 'causal' with 'deterministic'.

• Random means unpredictable. There are uncountably many TRNGs. So, knowing that RNG is TRNG does not tell you much about given RNG. And, well, since there are uncountably many RNGs you hardly can draw a list of all possible events. And you can't distinguish between PRNG and TRNG in countable amount of time. – rus9384 Sep 25 '18 at 11:21
• According to current Computer Science theory it is unpredictable for any Turing Machine. Zeno Machine, for example, could predict something a Turing Machine can't. But, well, as there is no largest cardinal, there is no RNG that is not predictable for all machines. RNG that is not predictable for a machine A must be implemented on a machine B which is strictly more powerful than A. So, yes, from the position of modern CS it is relative. There might be people who disagree with this definition of RNG, though. – rus9384 Sep 25 '18 at 13:00
• @DavidThornley, you agree with me that TRNG produces unpredictable numbers. So, if we take a black box and see that any TM of the set of all TMs gives a prediction, it is not TRNG. Therefore, TRNG is not computable and there is no TM that can run an RNG procedure. And even more there is no such a procedure written on any TM language. – rus9384 Sep 25 '18 at 17:57
• You do have a hint of what will happen, it's either heads or tails. Quantum double slit experiment is identical in this regard, one slit or the other. And there are standard ways of converting random variables with one distribution into ones with another by algebraic transformations, even if the probabilities weren't equal. So your random/statistical distinction is pointless even if it could be made, which it can't. – Conifold Sep 25 '18 at 20:14
• Forget all that quantum qrap. The answer's that a truly random process wouldn't be describable by any computable function (hence, not "describable" at all). Your so-called "list of all possible conditions" is just begging a diagonal argument refutation, which I'll let someone else construct in detail. The upshot is that if you're given an unending list of numbers generated by a truly random process, then no program, given input i can output the i^th number on the list. There are only a countably infinite number of programs, but an uncountably infinite number of N-->N functions. – John Forkosh Sep 26 '18 at 7:19

A true random number is one that is unpredictable, even knowing the state of the Universe beforehand. In the special case of a random series of numbers, each number has to be generated with probability independent of all the previous numbers. It's not possible to do this with a mathematical formula or computer program, but it is possible to use principles of quantum mechanics (assuming they hold) to make a physical one.

• Quantum TMs with unlimited memory are not more powerful than deterministic ones. – rus9384 Sep 25 '18 at 17:57
• Why it is possible? – Francesco D'Isa Sep 25 '18 at 18:38
• "A true random number is one that is unpredictable, even knowing the state of the Universe beforehand." -- Do you believe in such things? Isn't the randomness of QM part of an interpretation and not necessarily part of the theory itself? I don't know the answer, I'm wondering about that. A lot of people claim QM is a source of true randomness. How do you know? – user4894 Sep 25 '18 at 23:14
• Assuming that the principles of quantum mechanics actually hold is just as justifiable as assuming that pseudorandom number generators produce numbers indistinguishable to the modern observer from random numbers. – Carl Masens Sep 26 '18 at 6:27
• See use of nsandi.com/ernie for UK premium bonds so as to create random number for what is essentially a lottery. – Keith Oct 17 '18 at 5:01

In mathematics, a true random number generator it's impossible, because any formula defines a process that, however complex, is not random.

A mathematician wouldn't use a formula to generate a random number. He or she would simply stipulate the properties that a random number might satisfy: for example, to model a dice, one would ask for a random integer drawn from the set {1,2,3,4,5,6} and uniformly distributed.

Nevertheless, what you say is accurate when it comes to actually implementing such a requirement on a computer. Then we have to be more precise and actually specify an algorithm.

A random event must be unrelated to any cause or condition, and therefore cannot be causal. It is a brute fact par excellence.

If I draw up a list of all the possible conditions, I can say that a random event is outside of it. But isn't this a rule that determines the conditions of a random event?

This is one extreme, the purely random; the other is purely deterministic. A proper typology of would explore the possibilities in between.

Randomness is inversely proportional to the information you have. If you have right amount of information you can predict almost any thing. So if something is random then it is because your brain does not have enough information to predict the outcome.

• I made some minor edits which you may roll back or continue editing. You can see the versions by clicking on the "edited" link. If you have references of others taking a similar view this would strengthen your answer. Welcome to this SE! – Frank Hubeny Sep 27 '18 at 16:01
• "Randomness is inversely proportional to the information you have." I think you mean percentage of information that I have compared to all information. Well, that's true. But there possibly are things which no brain can comprehend. Not even an infinite one. – rus9384 Sep 27 '18 at 16:07