If we could take every "non-deterministic" event that ever occurred, and represent them neatly "in order" (whatever that means; let's assume also that there's some kind of natural way of ordering such events), it'd be very "obvious" that we were dealing with something fully deterministic if the numbers happened to form the binary representation of pi. But from the standpoint of what we can observe, we can only actually read one out of every [very large number] bits, and we've already missed [very larger number] of the first bits and will likely never be able to figure them out.

And so I claim that even if the "non-deterministic" bits in the universe are actually just bits from some well-known number that has certain nice properties, we would not practically be able to distinguish this from true "non-determinism." Whether this claim is true or not, it certainly avoids Bell's theorem: pi would be a hidden variable, but it could not have locality as the impact of pi is not limited by the speed of light.

I am also not aware of any sort of tests, statistical or otherwise, that would allow bits of pi selected in the manner suggested in the previous paragraph (that is, the limited selection that we can observe, assuming that pi "governs" the entire universe) to be distinguished from "true" non-determinism. In any case, I think you'd have a very difficult time indeed arguing that the universe doesn't answer to

*any*of infinitely many forms of (still fully deterministic) pseudo-randomness. Like I said, invisible-god's book doesn't have to be "random" in any meaningful sense, it merely has to be good enough to fool us into thinking it is such, and I claim this is actually a fairly easy feat given that we know very little about the macroscopic impact of quantum "non-determinism" and have measured a really very limited amount of it.

Now, I admit that I don't have a deep knowledge of information theory, but as far as I know there's simply no absolute test of "randomness." That is, there's no (finite length, finite running-time) program that can distinguish "random" bits from the bits of an arbitrary (but well-defined) irrational constant. Sure, simple tests that compare frequency of digits will be able to distinguish

*some*constants from "randomness" (for instance, the string of all zeroes will not be mistaken for random), but again, I don't think this can be done in general. The test you're proposing would work on periodic numbers and numbers with non-uniform frequencies but, for instance, I think it would be fooled by pi or sqrt(2). You could of course check for such things explicitly in your program, but there are (countably) infinitely many such constants, and your program needs to be finite.

Even if I grant you that random numbers exist, as you mentioned, proving that the universe is driven by them is still impossible as long as we only have a finite set of data.

Obviously in this post I made a lot of claims that I did not prove and I'd love to see counterexamples to some of them, but I think my overall point still stands: we can't say for sure that the universe is governed by "non-determinism" as distinct from a sufficiently clever but still well-defined pseudo-random algorithm.