How do we check Randomness? Let's imagine a guy who claims to possess a machine that can each time produce a completely random series of 0/1 digits (e.g. $1,0,0,1,1,0,1,1,1,...$). And each time after he generates one, you can keep asking him for the $n$-th digit and he will tell you accordingly.
Then how do you check if his series is really completely random?
If we only check whether the $n$-th digit is evenly distributed, then he can cheat using:

$0,0,0,0,...$
  $1,1,1,1,...$
  $0,0,0,0,...$
  $1,1,1,1,...$
  $...$

If we check whether any given sequence is distributed evenly, then he can cheat using:

$(0,)(1,)(0,0,)(0,1,)(1,0,)(1,1,)(0,0,0,)(0,0,1,)...$
  $(1,)(0,)(1,1,)(1,0,)(0,1,)(0,0,)(1,1,1,)(1,1,0,)...$
  $...$

I may give other possible checking processes but as far as I can list, each of them has flaws that can be cheated with a prepared regular series.
How do we check if a series is really random? Or is randomness a philosophical concept that can not be easily defined in Mathematics?

Leaving aside the theoretical aspect of your question, there are also pragmatic answers to it because there are real world uses for high-quality random generators (whether hardware or algorithmic). For statistical uses, "statistical randomness" is a used. For example you can use these "diehard tests" or TestU01.
