Consider the mathematical number Pi. It is a number that extends numerically into infinity, it has no end and has no repeating pattern to its digits. Currently we have computers that can calculate Pi out to many thousands of digits but at a certain point we reach a limit. Beyond that limit those numbers are unknown and essentially do not exist until they are observed. With that in mind, my question is this, if we could create a more powerful computer that could continue to calculate Pi beyond the current limit, and we started at exactly the same time to compute Pi out beyond the current limit on two identical computers, would we observe the computers generating the same numbers in sequence. If this is the case would that not infer that reality is deterministic in that unobserved and unknown numbers only become “real” upon being observed and that if identical numbers are generated those numbers have been, somehow, predetermined. Alternatively, if our reality was non-deterministic would that not mean that the two computers would generate potentially different numbers at each iteration as it moved forward into unobserved infinity inferring that unobserved reality is not set and therefore we live in a reality defined by free will?

You're no doubt right that any computers we happen to have available will only compute π to a finite number of digits, though as far as I know, there's nothing to stop a properly-designed computer from keeping up the calculation indefinitely (or until it wears out.) But you add this:

"Beyond that limit those numbers are unknown and essentially do not exist until they are observed"

Why is that? Let's suppose, for argument's sake, that we'll never build a computer that gets past the quadrillionth entry in the list of digits in π. Why would than mean that there's no fact of the matter about what the quadrillion-and-first digit is? What does a computer's having calculated it or (at least as puzzling) somebody having actually seen the answer have anything to do with whether there's a fact of the matter?

To be a bit more concrete: the quadrillion-and-first digit in the decimal expansion of π is either 7 or it isn't. If it's 7, it's 7 whether anyone ever verifies that or not. If it's not 7, then it's something else whether or not anyone every figures out what. It would take a lot of arguing to give a reason why we should think otherwise.

This is related to another of your questions. Yes: if two properly-programmed, properly-functioning computers kept spitting out the digits in π, they would print out the same digit when they got to the quadrillion-and-first entry. π is the ratio of the radius of a circle to its circumference. This number is the same for all circles not as a matter of incidental empirical fact but because of what it is to be a circle. That ratio is a specific number, and if two computers disagree about one of the digits, at least one computer is wrong.

But this doesn't help us with the question of determinism. Let's suppose that the world is indeterministic. (For all any of us knows, it is.) This doesn't mean that every physical process is also indeterministic. It just means (roughly) that from the laws and the total state of the world at one time, the total state of the world at other times doesn't follow. But the word "total" matters here. Even if quantum processes are indeterministic at some level, the output of a computer running certain sorts of programs isn't. If a computer fits the requirements for being a Turing machine, its output is deterministic. Of course, any physical machine can break down, and it could be that what makes some particular machine crap out has a chance element. But that doesn't mean that all particular physical processes are indeterministic and in any case, doesn't make mathematics mushy.

Finally, on free will. The connection between free will and physical determinism is actually not as simple as it seems. If you're interested in thinking about that, there is a recent book by Jenann Ismael called How Physics Makes Us Free. It's accessible, engaging and rigorously argued. I recommend it highly.

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