[Copied from an email exchange with Vasco, slightly embellished]
I think the probability of a flat universe is ~0 because the distribution describing our knowledge about the curvature of the universe is continuous, whereas a flat universe corresponds to a discrete curvature of 0.
Sure, if you put infinitesimal weight on a flat universe in your prior (true if your distribution is continuous over a measure of spatial curvature and you think it’s infinite only if spatial curvature = 0), then no observation of (local) curvature is going to be enough. On your framing, I think the question is just why the distribution needs to be continuous? Consider: “the falloff of light intensity / gravity etc is very close to being proportional to 1d2, but presumably the exponent isn’t exactly 2 since our distribution over k for dk is continuous”.
all the evidence for infinity is coming from having some weight on infinity in our prior.
‘All’ in the sense that you need nonzero non-infinitesimal weight on infinity in your prior, but not in the sense that your prior is the only thing influencing your credence in infinity. Presumably observations of local flatness do actually upweight hypotheses about the universe being infinite, or at least keep them open if you are open to the possibility in the first place. And I could imagine other things counting as more indirect evidence, such as how well or poorly our best physical theories fit with infinity.
[Added] I think this speaks to something interesting about a picture of theoretical science suggested by a subjective Bayesian attitude to belief-forming in general, on which we start with some prior distribution(s) over some big (continuous?) hypothesis space(s), and observations tell us how to update our priors. But you might think that’s a weird way to figure out which theories to believe, because e.g. (i) the hypothesis space is indefinitely large such that you should have infinitesimal or very small credence in any given theory; (ii) the hypothesis space is unknown in some important way, in which case you can’t assign credences at all, or (iii) theorists value various kinds of simplicity or elegance which are hard to cash out in Bayesian terms in a non-arbitrary way. I don’t know where I come down on this but this is a case where I’m unusually sympathetic to such critiques (which I associate with Popper/Deutsch[1]).
[Continuing email] I do agree that “the universe is infinite in extent” (made precise) is different from “for any size, we can’t rule out the universe being at least that big”, and that the first claim is of a different kind. For instance, your distribution over the size of the universe could have an infinite mean while implying certainty that the universe has some finite size (e.g. if that distribution over the size of the universe is 1sizek where k>1).
That does put us in a weird spot though, where all the action seems to be in your choice of prior.
I don’t know how relevant it is that the axiom of infinity is independent of ZFC, unless you think that all true mathematical claims are made true by actual physical things in the world (JS Mill believed something like this I think). Then you might have thought you have independent reason to believe (i) the ZFC axioms, and if so believing that (ii) ZFC⟹axiom of infinity you’d be forced to believe in an actual physical infinity. But that has the same suspect “synthetic a priori” character as ontological arguments for God’s existence, and is moot in any case because (ii) is false!
For what it’s worth, as a complete outsider I feel a surprised by how little serious discussion there is in e.g. astrophysics / philosophy of physics etc around whether the universe is infinite in some way. It seems like such a big deal; indeed an infinitely big deal!
Though I don’t think these views would have much constructive to say about how much credence to put on the universe being infinite, since they’d probably reject the suggestion that you can or should be trying to figure out what credence to put on it. Paging @ben_chugg since I think he could say if I’m misrepresenting the view.
[Copied from an email exchange with Vasco, slightly embellished]
Sure, if you put infinitesimal weight on a flat universe in your prior (true if your distribution is continuous over a measure of spatial curvature and you think it’s infinite only if spatial curvature = 0), then no observation of (local) curvature is going to be enough. On your framing, I think the question is just why the distribution needs to be continuous? Consider: “the falloff of light intensity / gravity etc is very close to being proportional to 1d2, but presumably the exponent isn’t exactly 2 since our distribution over k for dk is continuous”.
‘All’ in the sense that you need nonzero non-infinitesimal weight on infinity in your prior, but not in the sense that your prior is the only thing influencing your credence in infinity. Presumably observations of local flatness do actually upweight hypotheses about the universe being infinite, or at least keep them open if you are open to the possibility in the first place. And I could imagine other things counting as more indirect evidence, such as how well or poorly our best physical theories fit with infinity.
[Added] I think this speaks to something interesting about a picture of theoretical science suggested by a subjective Bayesian attitude to belief-forming in general, on which we start with some prior distribution(s) over some big (continuous?) hypothesis space(s), and observations tell us how to update our priors. But you might think that’s a weird way to figure out which theories to believe, because e.g. (i) the hypothesis space is indefinitely large such that you should have infinitesimal or very small credence in any given theory; (ii) the hypothesis space is unknown in some important way, in which case you can’t assign credences at all, or (iii) theorists value various kinds of simplicity or elegance which are hard to cash out in Bayesian terms in a non-arbitrary way. I don’t know where I come down on this but this is a case where I’m unusually sympathetic to such critiques (which I associate with Popper/Deutsch[1]).
[Continuing email] I do agree that “the universe is infinite in extent” (made precise) is different from “for any size, we can’t rule out the universe being at least that big”, and that the first claim is of a different kind. For instance, your distribution over the size of the universe could have an infinite mean while implying certainty that the universe has some finite size (e.g. if that distribution over the size of the universe is 1sizek where k>1).
That does put us in a weird spot though, where all the action seems to be in your choice of prior.
I don’t know how relevant it is that the axiom of infinity is independent of ZFC, unless you think that all true mathematical claims are made true by actual physical things in the world (JS Mill believed something like this I think). Then you might have thought you have independent reason to believe (i) the ZFC axioms, and if so believing that (ii) ZFC⟹axiom of infinity you’d be forced to believe in an actual physical infinity. But that has the same suspect “synthetic a priori” character as ontological arguments for God’s existence, and is moot in any case because (ii) is false!
For what it’s worth, as a complete outsider I feel a surprised by how little serious discussion there is in e.g. astrophysics / philosophy of physics etc around whether the universe is infinite in some way. It seems like such a big deal; indeed an infinitely big deal!
Though I don’t think these views would have much constructive to say about how much credence to put on the universe being infinite, since they’d probably reject the suggestion that you can or should be trying to figure out what credence to put on it. Paging @ben_chugg since I think he could say if I’m misrepresenting the view.