Real as can be, after all?
It is possible to consider that the value of π is not infinite because it is recursively enumerable. You can make as many correct digits as you like using finite means. I think some scientists are thinking this means that π doesn’t “actually” have an infinite number of digits, because there are simple rules for getting as many as you need, without limit.
However, there is another arena well known to physicists where an “actual infinity” is much harder to exorcise, and that is the randomness that seems to be built in at a basic level in quantum mechanics. It is now known, thanks to recent results, that it is possible to certify that a sequence of bits produced by quantum processes are truly, irreducibly random. It is also known that it is not possible to compute an irreducibly random sequence of indefinite length using a program of finite size. So as far as I can see, this means that the universe is not any sort of computer, because if it were it would not be possible to physically certify that a random sequence is random.
If my layman’s understanding is correct, the result is essentially that the Kolmogorov complexity of the universe was shown to be at least “equivalent” to the “size” of the universe, which rules out the possibility of it being the result of computation. (Contrasting with the fact that π can be computed because, even though it is a non-repeating infinite series in the decimal system, its Kolmogorov complexity is yet very modest. (That is not hard to conceive. A trivial way to create a non-repeating infinite series is to emit a series of 1s separated by an always-increasing number of 0s. Cycle length: ∞; Kolmogorov complexity: ~0.))
If this is correct, and true, I find it very exciting. I always found the concept of the universe as a kind of simulation profoundly unsatisfying, due to the infinite regress that is immediately invoked: is that computer itself simulated, in turn? If so, does the chain end? If it does, why/how there? It all seems like multiplication of entities that achieves nothing in exchange for opening up unlimited arbitrariness potential.