Implications of finding there is order in total randomness?

With "total randomness" I mean that every number generated has exactly the same probability of appearing and there is no way to predict which number will come next in a set. With order I mean that a pattern appears in a sequence of totally random numbers, and that pattern is consistent with every set of totally random numbers. What would be the implications of finding that? Is that a paradox?

EDIT 1

There isn't a final test of total randomness, some tests have been developed and possibly more will come. So it might be possible using a given process to be developed, that we find there is a pattern in total randomness. This is just a mental experiment but still possible. If we find such pattern we could create an algorithm that cancels the pattern, then we would have a natural predictable randomness and an artificial unpredictable randomness.

EDIT 2

Example of possible pattern: http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.html (thanks to @Chris Degnen who answered below)

• If you could genuinely know that the numbers were being generated randomly, then the "pattern" is just a coincidence. A more likely explanation is to doubt that your "random" is truly random. But no paradox arises. Mar 29 '17 at 2:22
• Can you give an example of such a pattern?
– user18800
Mar 29 '17 at 2:54
• You might be interested in computable numbers, en.wikipedia.org/wiki/Computable_number. A real number (between 0 and 1 for simplicity) can be thought of as an infinite sequence of fair coin flips, interpreted as a binary decimal. There are uncountably many real numbers but only countably many Turing machines over a finite or countable alphabet. So most bit patterns are not the output of any algorithm. They're noncomputable. You might enjoy this article that outlines the mathematical theory of complexity and randomness. en.wikipedia.org/wiki/Algorithmic_information_theory Mar 29 '17 at 3:21
• It may be a meaningless question, because outside of quantum mechanics, there has been no source of actual random data in the universe... just data of what we are to some degree ignorant of the generating algorithm. If you do want to include quantum mechanics, well, just don't put too much confidence in anybody claiming that we know exactly what's happening. Randomness is really just a statistical measure of lack of knowledge. Mar 29 '17 at 16:49
• Actually, sequences of random numbers exhibit many patterns, one of them is called the law of large numbers. The problem with your "definition" of "total randomness" is that it is incoherent in the same way that "everything that can be conceived" or "the set of all sets" are incoherent. For "there is no way to predict" to be meaningful you have to specify precisely what counts as a "way". Without that your mental experiment can not even begin. Mar 29 '17 at 19:38

What you would have found is a logical inconsistency. If a series of numbers is random, it is random. If there is a pattern beyond the distribution of numbers that series is drawn from, then it is no longer random, by definition.

Of course, if you consider infinite strings of random numbers, it is highly likely that every finite series will show up somewhere in every infinite string.

The more interesting question would be what it means if a particular string doesn't appear. It's not forbidden for a string of numbers to never appear in a sequence of random numbers, but it is peculiar, and frustrating in that you cannot prove it is meaningful!

• But there isn't a final test of total randomness, some tests have been developed and possibly more will come. So it might be possible using a given process to be developed, that we find there is a pattern in total randomness. This is just a mental experiment but still possible. If we find such pattern we could create an algorithm that cancels the pattern, then we would have a natural predictable randomness and an artificial unpredictable randomness. Mar 29 '17 at 15:32
• I think you're running into an interesting problem of trying to create a "X which does not have the properties of X," which I would not consider to be a paradox. The situation you define is an Independent Identically Distributed series of draws which is not independent, which is merely an inconsistent set of assumptions. I suppose technically that's a paradox, but no more paradoxical than a system with 0=1. There might be a more interesting problem to be had if we add a layer, saying "What if all processes which were believed to have the IID property actually did not have that property." Mar 29 '17 at 16:06
• As for canceling out "predictable" randomness, we already do that with our hardware random number generators. For example, one common HRNG is to listen to a noise source and take the low bit of the signal, but that can be subject to a bias term. von Neuman figured out a way to remove that simple bias term, substantially increasing the apparent randomness of the signal by taking two draws instead of one, and only accepting the first draw if the two were different, discarding both if they were the same. Mar 29 '17 at 16:14
• @Cort Ammon: really? so a sequence of size n where all digits are 1 could not be randomly generated? don't think so. "if a series of numbers is random, it is random" is just silly.
– user20153
Mar 29 '17 at 21:41
• @mobileink I explicitly state "it is highly likely that every finite series will show up somewhere in every infinite [random] string," which should satisfy the case you mention. The ability to generate a series where all digits are one is built into the distribution of the random variable. Knowing that distribution, we can even calculate the probability that a particular subsequence appears out of a sequence of length x Mar 29 '17 at 22:00

Then it won't be a totally random string, under the definition of randomness in computational complexity theory. Let's do a reductio ad absurdum argument:

Here's a nice computational definition of a random string: (Axiom R) a string S is totally random iff there does not exist a program string P such that S =C(P) and |P| < |S|. In other words, a for a string to be random, it is necessary and sufficient that there is no computable function that takes in a string P, which is shorter in length than S, and outputs the string S. In computational complexity theory, we call this concept of program-size length the Kolmogorov Complexity of S. Let's assume two things:

1. S is totally random
2. There is a pattern in S

Now, we should realize that there will always exist a P that is equal in length to S. That is exactly the case P = S, where C is the identity function over strings. Intuitively, when we say that there is a "pattern" in a string, we mean that we can simplify/encode the contents of the string into a form that is less complex, so that when we apply our decoding mechanism, we obtain our original string. Now just take simplification to mean reducing string length, the simplified string to be P, and the decoding mechanism to be C. Now that we've found a pattern in the string, we have found a new totally random string P such that C(P) = S and |P| < |S|. But this is contradictory to axiom R, therefore either S is not totally random or there is not a pattern in S. Since we had stipulated that S must have a pattern, S must not be totally random.

• "Let's assume three things". I see two listed. What did I miss? Mar 29 '17 at 23:18
• Oops, I meant to write two. Nice catch. Will fix. Mar 31 '17 at 1:03