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?

This is discussed very nicely in Volume 2 of Knuth's The Art Of Computer Programming. The executive summary is that randomness is a mathematical concept that can be defined in mathematics but not easily. 
