Thanks for replying, I think I now understand your position a bit better. Okay, so if your concern is around measurements only being finitely precise, then my exactly-zero example is not a great one, because I agree that it’s impossible to measure the universe as being exactly flat.
Maybe a better example: if the universe’s large-scale curvature is either zero or negative, then it necessarily follows that it’s infinite.
—(I didn’t give this example originally because of the somewhat annoying caveats one needs to add. Firstly, in the flat case, that the universe has to be simply connected. And then in the negatively curved case, that our universe isn’t one of the unusual finite types of hyperbolic 3-manifold given by Mostow’s rigidity theorem in pure math. (As far as I’m aware, all cosmologists believe that if the universe is negatively curved, then it’s infinite.))—
I think this new example might address your concern? Because even though measurements are only finitely precise, and contain uncertainty, you can still be ~100% confident that the universe is negatively curved based on measurement. (To be clear, the actual measurements we have at present don’t point to this conclusion. But in theory one could obtain measurements to justify this kind of confidence.)
(For what it’s worth, I personally have high credence in eternal inflation, which posits that there are infinitely many bubble/pocket universes, and that each pocket universe is negatively curved—very slightly—and infinitely large. (The latter on account of details in the equations.))
I agree it makes sense to have some probability mass on the universe having null or negative local curvature. However, I think there is no empirical evidence supporting a null or negative global curvature:
One can gather empirical evidence that the observable universe has a topology which, if applicable to the entire universe, would imply an infinite universe.
Yet, by definition, it is impossible to get empirical evidence that the topology of the entire universe matches that of the observable universe.
Inferring an infinite universe based on properties of the observable universe seems somewhat analogous to deducing an infite flat Earth based on the obervable ocean around someone in the sea being pretty flat.
Wikipedia’s page seems to be in agreement with my 2nd point (emphasis mine):
If the observable universe encompasses the entire universe, we might determine its structure through observation. However, if the observable universe is smaller [as it would have to be for the entire universe to be infinite], we can only grasp a portion of it, making it impossible to deduce the global geometry through observation.
I guess cosmologists may want to assume the properties of the observable universe match those of the entire universe in agreement with the cosmological principle. However, this has only proved to be useful to make predictions in the observable universe, so extending it to the entire universe would not be empirically justifiable. As a result, I get the impression the hypothesis of an infinite universe is not falsifiable, such that it cannot meaningly be true or false.
However, this has only proved to be useful to make predictions in the observable universe, so extending it to the entire universe would not be empirically justifiable.
Useful so far! The problem of induction applies to all of our predictions based on past observations. Everything could be totally different in the future. Why think the laws of physics or observations will be similar tomorrow, but very different outside our observable universe? It seems like essentially the same problem to me.
As a result, I get the impression the hypothesis of an infinite universe is not falsibiable, such that it cannot meaningly be true or false.
Why then assume it’s finite rather than infinite or possibly either?
What if you’re in a short-lived simulation that started 1 second ago and will end in 1 second, and all of your memories are constructed? It’s also unfalsifiable that you aren’t. So, the common sense view is not meaningfully true or false, either.
Why think the laws of physics or observations will be similar tomorrow, but very different outside our observable universe? It seems like essentially the same problem to me.
I agree it is essentially the same problem. I would think about it as follows:
If I observed the ground around me is pretty flat and apparently unbounded (e.g. if I were in the middle of a large desert), it would make sense to assume the Earth is larger than a flat circle with a few kilometers centred in me. Ignoring other sources of evidence, I would have as much evidence for the Earth extending for only tens of kilometers as for it being infinite. Yet, I should not claim in this case that there is empirical evidence for the Earth being infinite.
Similarly, based on the Laws of Physics having worked a certain way for a long time (or large space), it makes sense to assume they will work roughly the same way closeby in time (or space). However, I should not claim there is empirical evidence they will hold infinitely further away in time (or space).
Why then assume it’s finite rather than infinite or possibly either?
Sorry for the lack of claririty. I did not mean to argue for a finite universe. I like to assume it is finite for simplicity, in the same way that it is practical to have physical laws with zeros even though all measurements have finite precision. However, I do not think there will ever be evidence for/against the entire universe being finite/infinite.
Hmm, okay, so it sounds like you’re arguing that even if we measure the curvature of our observable universe to be negative, it could still be the case that the overall universe is positively curved and therefore finite? But surely your argument should be symmetric, such that you should also believe that if we measure the curvature of our observable universe to be positive, it could still be the case that the overall universe is negatively curved and thus infinite?
My answer to both questions would be yes. In other words, whether the entire universe is finite or infinite is not a meaningful question to ask because we will never be able to gather empirical evidence to study it.
Thanks for replying, I think I now understand your position a bit better. Okay, so if your concern is around measurements only being finitely precise, then my exactly-zero example is not a great one, because I agree that it’s impossible to measure the universe as being exactly flat.
Maybe a better example: if the universe’s large-scale curvature is either zero or negative, then it necessarily follows that it’s infinite.
—(I didn’t give this example originally because of the somewhat annoying caveats one needs to add. Firstly, in the flat case, that the universe has to be simply connected. And then in the negatively curved case, that our universe isn’t one of the unusual finite types of hyperbolic 3-manifold given by Mostow’s rigidity theorem in pure math. (As far as I’m aware, all cosmologists believe that if the universe is negatively curved, then it’s infinite.))—
I think this new example might address your concern? Because even though measurements are only finitely precise, and contain uncertainty, you can still be ~100% confident that the universe is negatively curved based on measurement. (To be clear, the actual measurements we have at present don’t point to this conclusion. But in theory one could obtain measurements to justify this kind of confidence.)
(For what it’s worth, I personally have high credence in eternal inflation, which posits that there are infinitely many bubble/pocket universes, and that each pocket universe is negatively curved—very slightly—and infinitely large. (The latter on account of details in the equations.))
Thanks for elaborating!
I agree it makes sense to have some probability mass on the universe having null or negative local curvature. However, I think there is no empirical evidence supporting a null or negative global curvature:
One can gather empirical evidence that the observable universe has a topology which, if applicable to the entire universe, would imply an infinite universe.
Yet, by definition, it is impossible to get empirical evidence that the topology of the entire universe matches that of the observable universe.
Inferring an infinite universe based on properties of the observable universe seems somewhat analogous to deducing an infite flat Earth based on the obervable ocean around someone in the sea being pretty flat.
Wikipedia’s page seems to be in agreement with my 2nd point (emphasis mine):
I guess cosmologists may want to assume the properties of the observable universe match those of the entire universe in agreement with the cosmological principle. However, this has only proved to be useful to make predictions in the observable universe, so extending it to the entire universe would not be empirically justifiable. As a result, I get the impression the hypothesis of an infinite universe is not falsifiable, such that it cannot meaningly be true or false.
Useful so far! The problem of induction applies to all of our predictions based on past observations. Everything could be totally different in the future. Why think the laws of physics or observations will be similar tomorrow, but very different outside our observable universe? It seems like essentially the same problem to me.
Why then assume it’s finite rather than infinite or possibly either?
What if you’re in a short-lived simulation that started 1 second ago and will end in 1 second, and all of your memories are constructed? It’s also unfalsifiable that you aren’t. So, the common sense view is not meaningfully true or false, either.
Thanks for jumping in, Michael!
I agree it is essentially the same problem. I would think about it as follows:
If I observed the ground around me is pretty flat and apparently unbounded (e.g. if I were in the middle of a large desert), it would make sense to assume the Earth is larger than a flat circle with a few kilometers centred in me. Ignoring other sources of evidence, I would have as much evidence for the Earth extending for only tens of kilometers as for it being infinite. Yet, I should not claim in this case that there is empirical evidence for the Earth being infinite.
Similarly, based on the Laws of Physics having worked a certain way for a long time (or large space), it makes sense to assume they will work roughly the same way closeby in time (or space). However, I should not claim there is empirical evidence they will hold infinitely further away in time (or space).
Sorry for the lack of claririty. I did not mean to argue for a finite universe. I like to assume it is finite for simplicity, in the same way that it is practical to have physical laws with zeros even though all measurements have finite precision. However, I do not think there will ever be evidence for/against the entire universe being finite/infinite.
Hmm, okay, so it sounds like you’re arguing that even if we measure the curvature of our observable universe to be negative, it could still be the case that the overall universe is positively curved and therefore finite? But surely your argument should be symmetric, such that you should also believe that if we measure the curvature of our observable universe to be positive, it could still be the case that the overall universe is negatively curved and thus infinite?
My answer to both questions would be yes. In other words, whether the entire universe is finite or infinite is not a meaningful question to ask because we will never be able to gather empirical evidence to study it.