The first thing I noticed as I tried to do my research on randomness on the Internet is how difficult it is to find trustworthy resources and definitions.
I spent some time searching the web and trying to understand what is the general consensus on randomness. I have come to the conclusion that, generally, it is perceived as patternless.
No matter what definition of “pattern” we choose, it seems impossible to exclude the concept of repetition from it. So if something doesn’t get repeated, does it mean there is no pattern?
Usually, we think of repetition as a relationship between at least two entities. It is usually either in space or in time that two entities are different, yet they share some other form of similarity. However, it seems difficult to imagine something that won’t get repeated, since if something doesn’t get repeated in time, we would never become aware of it, because even to remember is to repeat something in time. If it doesn’t get repeated in space, it also wouldn’t make much sense, since our pattern matching abilities wouldn’t pick it up, since if it were to match some shape or color, it would be a form of repetition.
However, it is clearly not patternless because we can recognize it, even if as just something nonsensical to us.
It seems that the consensus has it that randomness is a state of highest informational entropy, which essentially means that every outcome in the set of all outcomes is equally likely. (I’ve made some remarks on entropy in this article.)
However, if we take a set of natural numbers, then if we consider each number as an outcome, then this set will have the highest entropy as the numbers don’t repeat themselves, however, it is clearly not random.
Or is it? If there is no requirement for it to be ordered, and if it was just a magic bag from which we are to pick one, then it would be quite random.
But how to ensure that the outcomes are equally likely? Is it possible to create an algorithm generating random numbers? It seems that the determination of whether something is random or not can only be done in retrospect, where we can keep track of the likelihood of the outcomes. But if we start generating a sequence from seed and keeping track of the likelihoods, we will realize that our generation algorithm gives numbers preferences and there is a relationship between previous and consecutive numbers.
It seems impossible to generate something truly random, but as with all things calculus, we can probably infinitely approach it by making the relationships in the sequence more and more subtle ad infinitum.
So, in conclusion, I think randomness is not patternless, but the pattern behind it is very subtle, and I believe that even though true randomness is only an ideal concept, some things are more “random” than others.