Randomness as a function of the observer's knowledge

[N123]

I found a couple of closed threads related to the definition of randomness, but my question is slightly different.

Is there a mathematical way to express the fact that randomness is the eye of the beholder?
For example, if I give a sequence of 1000 numbers such that you cannot predict the 1001th number using your existing knowledge. So it is a random sequence as far as you are concerned.
Now I tell you that in fact those were the digits 3000...3999 in the decimal expansion of pi. Now suddenly, the sequence is not random.
(I understand that we may use an absolute definition saying the sequence was never random, but the fact is that for you it was, for a while.)
Since mankind's knowledge is finite and expanding, collectively we will come across sequences which used to be random and are not random anymore. How do you say "Sequence X is random if your amount of knowledge is Y"?

 

[andrewkirk]

A sequence is never random, because there is no mathematical definition of a random sequence, or of randomness. There is a definition of a random process, every realization of which is a sequence. But that definition is of no use in most real-life cases, because we usually only can see one realization - eg the sequence of prices of Cisco stock on the NYSE at close of trading, ever since it was first listed. The other realisations live in alternative universes, if there are such things, and can never be known by us.
The closest random-ish concept that applies to a sequence is that of a normal sequence, aka normal number. The wiki article on normal numbers is quite good. The sequence of digits of pi is conjectured to be normal, but there is no proof that they are.
Regarding the broader question of whether all randomness is simply a function of the observer's knowledge (ie epistemological), I think a good case can be made that that is true. You may be interested in this note about that issue.

 

[jbriggs444]

Not entirely helpful, but there is a notion of randomness which is based on ideas of computability or Kolmogorov complexity. Roughly speaking, a sequence is random in this sense if it is incompressible. The decompressor of choice is a universal Turing machine.

This allows for the notion of randomness to be applied not just to distributions of sequences, but to sequences themselves.