I’ve realised that I’m slightly more confused on this topic than I thought.
As non-logically omniscient beings, we need to keep track of hypothetical universes which are not just physically different from our own, but which don’t make sense—i.e. they contain logical contradictions that we haven’t noticed yet.
For example, let T be a Turing machine where we haven’t yet established whether or not T halts. Then one of the following is true but we don’t know which one:
(a) The universe is infinite and T halts
(b) The universe is infinite and T does not halt
(c) The universe is finite and T halts
(d) The universe is finite and T does not halt
If we then discover that T halts, we not only assign zero probability to (b) and (d), we strike them off the list entirely. (At least that’s how I imagine it, I haven’t yet heard anyone describe approaches to logical uncertainty).
But it feels like there should also be (e) - “the universe is finite and the question of whether or not T halts is meaningless”. If we were to discover that we lived in (e) then all infinite universes would have to be struck off our list of meaningful hypothetical universes, since we are viewing hypothetical universes as mathematical objects.
But it’s hard to imagine what would constitute evidence for (or against) (e). So after 5 minutes of pondering, that more or less maps out my current state of confusion.
I think you’re confused if you think the finitude of the universe matters in answering the mathematical question of whether T halts. Answering that question may be of interest for then figuring out whether certain things in our universe that behave like Turning machines behave in certain ways, but the mathematical question is independent.
Your confusion is that you think there need to be objects of some kind that correspond to mathematical structures that we talk about. Then you’ve got to figure out what they are, and that seems to be tricky however you cut it.
I agree that the finitude of the universe doesn’t matter in answering the mathematical question of whether T halts. I was pondering whether the finitude of the universe had some bearing on whether the question of T halting is necessarily meaningful (in an infinite universe it surely is meaningful, in a finite universe it very likely is but not so obviously so).
Surely if the infinitude of the universe doesn’t affect that statement’s truth, it can’t affect that statement’s meaningfulness? Seems pretty obvious to me that the meaning is the same in a finite and an infinite universe: you’re talking about the mathematical concept of a Turing machine in both cases.
Conditional on the statement being meaningful, infinitude of the universe doesn’t affect the statement’s truth. If the meaningfulness is in question then I’m confused so wouldn’t assign very high or low probabilities to anything.
Essentially:
I have a very strong intuition that there is a unique (up to isomorphism) mathematical structure called the “non-negative integers”
I have a weaker intuition that statements in second-order logic have a unique meaningful interpretation
I have a strong intuition that model semantics of first-order logic is meaningful
I have a very strong intuition that the universe is real in some sense
It’s possible that my intuition might be wrong though. I can picture the integers in my mind but my picture isn’t completely accurate—they basically come out as a line of dots with a “going on forever” concept at the end. I can carry on pulling dots out of the “going on forever”, but I can’t ever pull all of them out because there isn’t room in my mind.
Any attempt to capture the integers in first-order logic will permit nonstandard models. From the vantage point of ZF set theory there is a single “standard” model, but I’m not sure this helps—there are just nonstandard models of set theory instead. Similarly I’m not sure second-order logic helps as you pretty much need set theory to define its semantics.
So if I’m questioning everything it seems I should at least be open to the idea of there being no single model of the integers which can be said to be “right” in a non-arbitrary way. I’d want to question first order logic too, but it’s hard to come up with a weaker (or different) system that’s both rigorous and actually useful for anything.
I’ve realized one thing though (based on this conversation) - if the universe is infinite, defining the integers in terms of the real world isn’t obviously the right thing to do, as the real world may be following one of the nonstandard models of the integers. Updating in favor of meaningfulness not being dependent on infinitude of universe.
I’ve realised that I’m slightly more confused on this topic than I thought.
As non-logically omniscient beings, we need to keep track of hypothetical universes which are not just physically different from our own, but which don’t make sense—i.e. they contain logical contradictions that we haven’t noticed yet.
For example, let T be a Turing machine where we haven’t yet established whether or not T halts. Then one of the following is true but we don’t know which one:
(a) The universe is infinite and T halts
(b) The universe is infinite and T does not halt
(c) The universe is finite and T halts
(d) The universe is finite and T does not halt
If we then discover that T halts, we not only assign zero probability to (b) and (d), we strike them off the list entirely. (At least that’s how I imagine it, I haven’t yet heard anyone describe approaches to logical uncertainty).
But it feels like there should also be (e) - “the universe is finite and the question of whether or not T halts is meaningless”. If we were to discover that we lived in (e) then all infinite universes would have to be struck off our list of meaningful hypothetical universes, since we are viewing hypothetical universes as mathematical objects.
But it’s hard to imagine what would constitute evidence for (or against) (e). So after 5 minutes of pondering, that more or less maps out my current state of confusion.
I think you’re confused if you think the finitude of the universe matters in answering the mathematical question of whether T halts. Answering that question may be of interest for then figuring out whether certain things in our universe that behave like Turning machines behave in certain ways, but the mathematical question is independent.
Your confusion is that you think there need to be objects of some kind that correspond to mathematical structures that we talk about. Then you’ve got to figure out what they are, and that seems to be tricky however you cut it.
I agree that the finitude of the universe doesn’t matter in answering the mathematical question of whether T halts. I was pondering whether the finitude of the universe had some bearing on whether the question of T halting is necessarily meaningful (in an infinite universe it surely is meaningful, in a finite universe it very likely is but not so obviously so).
Surely if the infinitude of the universe doesn’t affect that statement’s truth, it can’t affect that statement’s meaningfulness? Seems pretty obvious to me that the meaning is the same in a finite and an infinite universe: you’re talking about the mathematical concept of a Turing machine in both cases.
Conditional on the statement being meaningful, infinitude of the universe doesn’t affect the statement’s truth. If the meaningfulness is in question then I’m confused so wouldn’t assign very high or low probabilities to anything.
Essentially:
I have a very strong intuition that there is a unique (up to isomorphism) mathematical structure called the “non-negative integers”
I have a weaker intuition that statements in second-order logic have a unique meaningful interpretation
I have a strong intuition that model semantics of first-order logic is meaningful
I have a very strong intuition that the universe is real in some sense
It’s possible that my intuition might be wrong though. I can picture the integers in my mind but my picture isn’t completely accurate—they basically come out as a line of dots with a “going on forever” concept at the end. I can carry on pulling dots out of the “going on forever”, but I can’t ever pull all of them out because there isn’t room in my mind.
Any attempt to capture the integers in first-order logic will permit nonstandard models. From the vantage point of ZF set theory there is a single “standard” model, but I’m not sure this helps—there are just nonstandard models of set theory instead. Similarly I’m not sure second-order logic helps as you pretty much need set theory to define its semantics.
So if I’m questioning everything it seems I should at least be open to the idea of there being no single model of the integers which can be said to be “right” in a non-arbitrary way. I’d want to question first order logic too, but it’s hard to come up with a weaker (or different) system that’s both rigorous and actually useful for anything.
I’ve realized one thing though (based on this conversation) - if the universe is infinite, defining the integers in terms of the real world isn’t obviously the right thing to do, as the real world may be following one of the nonstandard models of the integers. Updating in favor of meaningfulness not being dependent on infinitude of universe.