I think there is something fundamentally interesting about the challenge of creating a small system with a finite internal state that is capable of producing an output stream that
- Appears to be independent, identically distributed
- Is not periodic or repeating and can not be compressed by normal algorithms
- Has the highest possible entropy (when the state is unknown), and is unpredictable
- Can not be predicted from any length of its output
It's not exactly trivial to make such a system, but it is not all that difficult to do rather well either.
The things that is most fascinating to me is that such apparent complexity is possible from a few lines of computer code.
In a way, it is similar to requiring atom smashers that operate at higher and higher energies, or CMB detectors that are ever more sensitive : if you make finer and finer uses of the random numbers as part of complicated experiments, you want to remove as much as possible the chance that the small effects you are looking to detect are due to an artefact of your system, and in the case of using random numbers to aid simulations you want them to behave so randomly that it is as if nature were creating the simulation for you. Then test the parts of the experiment you are interested in.
Something awesomely cool is that, if it is possible to create a large subset of all possible messages using these random number generators that operate from a small finite state, would not it someday be possible to find the state that generates any possible sequence, say the sequence of bits in a video?
Compression down to the tuple of a small state, and a particular algorithm chosen from a family of algorithms could exceed the savings indicated by entropy calculations based on the symbol distribution, since it is possible to create sequences of many megabytes in length that, based on their symbol distribution have maximum entropy, yet were created with 64 bits of state and a few lines of computer code.
That's some of the reasons I think random numbers are cool and why searching for better ways of creating them, or better ways to understand the ways we create them is awesome.
Perhaps the best analogy for why this is a very interesting pursuit is that the output of these generators is really a metaphor for the universe as a whole ( or at least, what we hope the universe is like ) : a system of immense apparent complexity, but with an internal state that we can try ever harder to measure and rules we can hope to deduce.