You do raise a very good point, though surely a random number generator is inherently non-deterministic and so this would effectively be a non-replacement. Were you then referring to superdeterminism?

I think the problem is that we define randomness in such a way that it's not clear that the concept is even coherent. We define randomness in terms of what it isn't -- predictable -- but that doesn't actually mean there's anything left over. In any case, though, I think that by assuming that invisible-god already knows the outcome, we avoid "non-determinism" at least in the way most people would like to believe it. Note also that the numbers in the book don't actually need to be random; they just need to be "good enough" that anyone other than invisible-god can't figure out what the pattern is.

Since we're bringing up non-determinism anyway I might as well register again how unimpressed I am with the "dichotomy" of determinism and "free will." I've said this here before and been met with fairly universal skepticism but I may as well try again:

It's intuitively "obvious" that if it's possible to predict exactly which actions I will take in which situations, then I don't really have "free will." That's fine, I guess, if we define "free" in such a way that doesn't allow for this kind of predictability, but what about "will"? Let's say we introduce non-determinism (if such a thing even exists) to give me back my "freedom." That means that, at least some subset of my actions are completely unpredictable.

That is, nothing about the observable world has any bearing whatsoever on what I end up doing in such cases. This includes both the makeup of my brain itself as well as everything my brain knows. Is this "free"? Sure. But how can it be "will"? I claim that, at very least, my will has to have intentionality; that is, it has to be "about" something. It has to be about my environment, or it has to be about me. My choices have to have at least some basis in reality for them to be "choices" at all. And non-determinism completely strips me of this.

After all, if my "choices" are non-deterministic, how are they different from your "choices"? My "will" is just as much your "will" or the "will" of the God of Randomness or whatever else you want to call it, because, by virtue of being devoid of any meaningful relationship to the visible world, my "choices" no longer have anything to do with me.

To bring this back on-topic, at least sort of, this is roughly why I'm so confused by the notion of "qualia zombies" or the possibly-even-less-coherent "consciousness zombies." We seemingly must reject qualia as purely epiphenomenal simply by virtue of the fact that we can refer to it. If my use of the word "qualia" is not completely incidental to, well, qualia itself, (and this is sort of an assumption, but I'm not sure it's worth considering the possibility that I experience qualia and then, by complete coincidence, I talk about "qualia" in a way that has nothing to do with the real phenomenon) then qualia must have an impact on my behavior. (this, by the way is roughly the argument that recompile made earlier)

Now, from this, there are only two possible conclusions: the first possibility is that a sufficiently well-created functional model of me will have qualia in the same way that I do, as it will use the word "qualia" in the same way that I do and be describing the same thing. Note that this case doesn't assume that "qualia" is a real thing for me or the model; it need only be that the model of me have the same experience that I do, whatever that may be.

The other possibility is that the model will never experience qualia in the same way that I do, and thus, it won't be able to talk about qualia. But then where does qualia manifest for the "real" me? How does it impact my behavior? It cannot be modeled algorithmically (or else we'd be in the former case) but then can it be predictable at all? If not, how is it anything other than random? And if the only difference between me and the functional model of me is that I talk about "qualia" due to some process that impacts my behavior randomly, then it seems like "qualia" isn't really what we want it to be after all.

The short version of this is the following: it's very common to hear people try to justify some "unique" aspect of the human mind, be it free will, consciousness, soul, or whatever, through the proposal of some non-deterministic/quantum/random process that manifests itself in human thought, and, ultimately, human behavior. To me this is even less interesting then the purely functional model, though, because randomness is just that: random. It's not for, because of, or about anything. At least in the fully deterministic model of the human mind I can say that I make my choices based on the information that's available to me and the functional structure of my mind.

End rant.

I think you may be conflating determinism with local hidden variables: how about deterministic hidden variables that are not local? Let's assume the universe is discrete somehow, so that the output of our "non-deterministic" source can always be expressed as a stream of bits. Now let's imagine that, instead of random bits, we get the bits of, say, pi (obviously if the universe can't be expressed discretely we'd have to come up with a similar method of generating our continuous "non-deterministic" output but let's ignore that for simplicity).

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.

I've always assumed that we'd need a fully functioning quantum computers to get a computer to be near the level of an actual human brain.

I didn't even know that this was still going ;D