Thread by @getjonwithit

The Rabin-Scott theorem is one of the (philosophically) deepest mathematical results I know. When properly understood, I claim that it can't help but alter your view of reality in a fairly foundational way. Yet its typical textbook presentation obscures much of this depth. (1/8)

Suppose you have a non-deterministic computer (formally, a non-deterministic finite automaton) which can perform a whole tree of possible actions from a given input state, and I have a deterministic one, which can only follow a single path. Clearly yours is better, right? (2/8)

Well, no. The Rabin-Scott theorem says that I can still simulate yours by defining each state of my computation to be a *collection* of possible states of yours. So the state space of my computation is simply the power set (i.e. the set of possible subsets) of yours. (3/8)

My state space is thus exponentially larger (of size 2^n) than yours, but still, the point stands: everything you can compute, I can compute too. Okay, that's cool, but I think there's a much deeper lesson about the nature of (non-)determinism to be extracted here. (4/8)

Most people are familiar with the idea that, if you have a deterministic model of something, you can always make it non-deterministic by deleting some information or coarse-graining over certain degrees of freedom. But far fewer people are familiar with the converse. (5/8)

Namely, if you have a non-deterministic model of something, you can always make an equivalent deterministic model (yielding identical predictions) whose space of states is just the power set of the original. This fact has been (largely) ignored in the philosophy of science. (6/8)

The conclusion? Determinism and non-determinism are not properties of *systems* but properties of *models*. So it simply doesn't make sense to ask "Is the universe deterministic?" or "Is quantum mechanics non-deterministic?" or even "Do humans have free will?" (7/8)

Sometimes it's useful to model these things deterministically, sometimes it's not. But never forget that these notions of determinism and non-determinism are human concepts that *we* have introduced through our choice of models. The universe, fundamentally, doesn't care. (8/8)