If you don't believe me, read the chapter. Early probability theorists (e.g., von Mises, Kolmogorov) literally started thinking about randomness in order to define probability.
EDIT: And, I don't suppose it's worth pointing out that pseudorandomness is not at all the same thing as randomness. The fact that you seem to use them interchangeably is not a good sign IMHO.
EDIT 2: Why the unexplained downvote, HN? :(
Pseudorandomness is not the same thing as randomness but most algorithms today work on pseudorandom numbers so the concept is important. My impression was that that's what you were referring to.
PS :- FYI I didn't downvote your comment. Actually upvoted as your post made me discover some new math (various notions of randomness by kolmogorov, von mises, martin-lof) :)
re: Pseudorandomness, the point of pseudorandomness is the following.
1. A lot of algorithms use randomness to make pathologically bad cases extremely unlikely. For example, choosing a random pivot in quicksort makes the worst case very unlikely.
2. But in a lot of cases, this leads to huge amounts of space consumption. For example, most frequency moment estimations involve a matrix of random numbers. So if you're getting those numbers from a "truly random" source, then you have to store the entire matrix, which can be huge.
3. So, a better solution is to use a pseudorandom number generator! That way you can store a seed of s bits, and do something clever, like deterministically re-generate the matrix as you need it, rather than storing it outright.
Notice though, that this is not independent of the notion of randomness! In fact they are quite intimately tied together.