# 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. Sep 25, 2018 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. Sep 25, 2018 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. Sep 25, 2018 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. Sep 25, 2018 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.
– user19423
Sep 26, 2018 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. Sep 25, 2018 at 17:57
• Why it is possible? Sep 25, 2018 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? Sep 25, 2018 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. Sep 26, 2018 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. Oct 17, 2018 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! Sep 27, 2018 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. Sep 27, 2018 at 16:07

Statistical randomness is a mathematical property of a series. It means both the lack of any pattern and the unpredictability of the next item in the series.

Statistical randomness does not make any distinction between true and pseudo-randomness.

Philosophical (true) randomness is a property of a single item. It means that the item is not deliberately selected by anyone or a product of an algorithm (which has to be selected along with the seed values).

Truly random = Unintentional

Pseudo-random = Intentional

It seems to me the problem here, might be a conflicting ontology. True randomness can't be modelled, things that can't be modelled are considered not fully understood/understandable by physics.

The Many Worlds interpretation of quantum mechanics provides an interesting counterpoint. We do not find ourselves in the universe with a random outcome, but in every universe with every outcome. Determinism, and yet every version looking at every other, trying to work out why they are on this branch in particular.

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

I'm going to point out that this question and its answer here suffer from a lack of any clear description of what constitutes a "random number generator" in mathematics, since "random number generator is not a technical term. But also, it has no obvious technical meaning.

The answer presented in this quote suggests that the random number generator is supposed to be defined by a formula. But what kinds of formulas count as formulas? This is left unstated, even though it would be possible to try to answer the question only if this is specified.

That addresses how to define what kinds of generators might or might not ever be random.

But also: What does random mean here? What is the criterion that a generator must satisfy in order to be classified as random? This, too, is not so easy to define. Suppose in the simplest case our random number generator started outputting a string of 0's and 1's, and the ideal case is the mythical fair coin flipped repeatedly in an unending sequence of (1/2,1/2) Bernoulli trials. Roughly speaking, the best definitions of a "random" sequence seem to require that for every natural number K, every string of 0's and 1's of length K occurs with the expected frequency 1/2^K.

The trouble is that to know this frequency, a) it is necessary to know an entire countable sequence of outputs, and b) for any finite initial string of outputs, that string can be ignored and the countable sequence will be just as random (or not) as it was with the finite initial string.

So for instance if for n = 1, 2, 3, ... we let the nth output of the generator be

F(n) = 0 if p(n+1) = 1 (modulo 4),

F(n) = 1 if p(n+1) = -1 (modulo 4)

(where p(n) denotes the nth prime number)

then this would probably satisfy my definition above.

But that "generator" is entirely deterministic.