I think all the evidence for infinity is coming from having some weight on infinity in our prior. Empirical evidence can take us from a very large universe to an arbitrarily large universe (for an arbitrarily large amount of evidence), but never to an infinite universe? An arbitrarily large universe would still be infinitely smaller than an infinite universe, so I would say the former would provide no empirical evidence for the latter. If this is so, I am confused about why discussions about infinite ethics often mention there is empirical evidence pointing to the existence of infinity. From a footnote of your post (emphasis mine):
Bostrom (2011): āRecent cosmological evidence suggests that the world is probably infinite. [continued in footnote] In the standard Big Bang model, assuming the simplest topology (i.e., that space is singly connected), there are three basic possibilities: the universe can be open, flat, or closed. Current data suggests a flat or open universe, although the final verdict is pending. If the universe is either open or flat, then it is spatially infinite at every point in time and the model entails that it contains an infinite number of galaxies, stars, and planets. There exists a common misconception which confuses the universe with the (finite) āobservable universeā. But the observable partāthe part that could causally affect usā would be just an infinitesimal fraction of the whole. Statements about the āmass of the universeā or the ānumber of protons in the universeā generally refer to the content of this observable part; see e.g. [1]. Many cosmologists believe that our universe is just one in an infinite ensemble of universes (a multiverse), and this adds to the probability that the world is canonically infinite; for a popular review, see [2].ā
In contrast, in maths, there is the axiom of infinity, which I assume points to the fact infinity as to be assumed from the onset rather than deduced.
[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 itās 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: āthe falloff of light intensity /ā gravity etc is very close to being proportional to 1d2, but presumably the exponent isnāt exactly 2 since our distribution over k for dk is continuousā.
all the evidence for infinity is coming from having some weight on infinity in our prior.
āAllā 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 thatās 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 canāt 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 donāt know where I come down on this but this is a case where Iām unusually sympathetic to such critiques (which I associate with Popper/āDeutsch[1]).
[Continuing email] I do agree that āthe universe is infinite in extentā (made precise) is different from āfor any size, we canāt rule out the universe being at least that bigā, 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 donāt 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) ZFCā¹axiom of infinity youād be forced to believe in an actual physical infinity. But that has the same suspect āsynthetic a prioriā character as ontological arguments for Godās existence, and is moot in any case because (ii) is false!
For what itās worth, as a complete outsider I feel a surprised by how little serious discussion there is in e.g. astrophysics /ā 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 donāt think these views would have much constructive to say about how much credence to put on the universe being infinite, since theyād 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 Iām misrepresenting the view.
Hi Joe,
I think all the evidence for infinity is coming from having some weight on infinity in our prior. Empirical evidence can take us from a very large universe to an arbitrarily large universe (for an arbitrarily large amount of evidence), but never to an infinite universe? An arbitrarily large universe would still be infinitely smaller than an infinite universe, so I would say the former would provide no empirical evidence for the latter. If this is so, I am confused about why discussions about infinite ethics often mention there is empirical evidence pointing to the existence of infinity. From a footnote of your post (emphasis mine):
In contrast, in maths, there is the axiom of infinity, which I assume points to the fact infinity as to be assumed from the onset rather than deduced.
[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 itās 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: āthe falloff of light intensity /ā gravity etc is very close to being proportional to 1d2, but presumably the exponent isnāt exactly 2 since our distribution over k for dk is continuousā.
āAllā 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 thatās 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 canāt 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 donāt know where I come down on this but this is a case where Iām unusually sympathetic to such critiques (which I associate with Popper/āDeutsch[1]).
[Continuing email] I do agree that āthe universe is infinite in extentā (made precise) is different from āfor any size, we canāt rule out the universe being at least that bigā, 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 donāt 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) ZFCā¹axiom of infinity youād be forced to believe in an actual physical infinity. But that has the same suspect āsynthetic a prioriā character as ontological arguments for Godās existence, and is moot in any case because (ii) is false!
For what itās worth, as a complete outsider I feel a surprised by how little serious discussion there is in e.g. astrophysics /ā 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 donāt think these views would have much constructive to say about how much credence to put on the universe being infinite, since theyād 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 Iām misrepresenting the view.