One topic in mathematics and computer science that always confused me were random numbers. I tried searching for the exact meaning but it feels kind of abstract and incomplete.
Random numbers are numbers that occur in a sequence such that two conditions are met: (1) the values are uniformly distributed over a defined interval or set, and (2) it is impossible to predict future values based on past or present ones.
I have no problem with 2nd point but 1st feels kind of abstract because we can't specify the defined interval or set. Not just that but if all the values are to be equally distributed it kind of isn't truly random any more.
Supposing true random number generator(generating numbers between 1-9) generates 100 numbers with no 9s. In this case we can say that the probability of 9 appearing is high in the next 10 numbers. But if the probability is high how are the numbers truly random ?
Also supposing a true random number generator(generating numbers between 1-9) generates 1million consecutive 3s. Is this random ? Most people will say probably not, but what if that same generator generated over 100 million numbers and found uniform distribution among numbers between 1 and 9. This comic illustrates this point.
For me the first point is pointless as we can't define an interval or set. This basically means that for truly random numbers the probability of 1 million consecutive 3 is as equally likely as an equal distribution of numbers.
I guess in the real world you would define numbers as random only as long you can't find a sequence to tell what the next number will be. Many articles say the digits inside irrational(√2, pi) numbers can be treated as random. In my opinion they are only treated as random as long as we don't know the right sequence. That essentially means no numbers are truly random, we just don't know their sequence.
TLDR How does the mathematical world define random numbers ?