Cards on the table, I affirm the A-theory and I'm against against against the infinite (read: I don't affirm the existence of actual infinites).
Reason 4: Can you elaborate on why denying an actual infinite would require a smallest unit of space?
Reason 6: Theists who hold to the A-theory would just say that the future is potentially infinite and not actually infinite because the A-theory is pretty plausible and accepted by many smart people.
Reason 7: This statement is just false and potentially a category mistake. I don't even know what it means for God's knowledge or power to be infinite in a mathematical sense. Infinity, when applied to God, shouldn't be viewed as a mathematical or quantitative concept, but rather qualitative, i.e. God has all the great making properties and has them to the maximum degree.
Maybe you could define infinity at the beginning of the article to help clarify your position? Are you always using it as a mathematical concept and as an actual infinite substantiated in reality?
Thanks for the post! This is one of my favorite topics.
4: Suppose space is infinitely divisible. There's an actual infinite--number of ways you can slice up space. Additionally, suppose I go two feet. Seems I made an actually infinite number of movements--going half way, a quarter of the way, and eighth of the way, and so on. Swinburne explains it well https://www.youtube.com/watch?v=4a4MHbXVIfs
6: Agreed! But if finitists must be A theorists while infinitists don't, that's a cost of finitism (it will strictly lower the probability).
7: God's knowledge is infinite in the sense that he knows an infinite number of things (he knows 1 is a number, 2 is a number, etc).
4: I would argue an actual infinite and an infinitely divisible line, distance, etc. are different. I can traverse two feet of infinitely divisible space and I can successively add up numbers to get to two feet; however, I cannot traverse an actual infinite many number of days, years, etc. or add up days to arrive at an actual infinite.
6: Makes sense.
7: I still don't think that's right though. I can conceive of the infinite in my mind, but I don't believe that's evidence of an actual infinite being substantiated in reality. Moreover, wouldn't God's knowledge of propositions just be potentially infinite and not actually infinite?
4. If there were an actual infinity of increasingly small segments, it would take an infinite amount of time to traverse them, by definition! A finite distance that is infinitely divisible is something wholly different from an actual infinite amount of space, time, etc.
7: The knowledge isn't potential vs actual; the knowledge is actual, but the amount of things known is only potentially infinite, i.e. inexhaustible. There aren't an actual infinite amount of things to know! Adding up a bunch of finite things never gets you to an infinite amount.
I wonder if you would say that we know an infinite number of things as well. After all, you also know that 1 is a number, 2 is a number, etc. Would you say that the difference between our knowledge and God's knowledge is that we can't actually hold an infinite amount of numbers in our head all at the same time and God can?
Something similar might apply to power. It seems like we as humans can do an infinite number of things. After all, I can move 1 inch, 2 inches, 1/2 inches, etc. Of course, it seems like God can do a lot more than we can, but I'm not exactly sure how to go about comparing infinities in this situation.
I agree the distinction between qualitative infinity and quantitative infinity is big when talking about God. God's knowledge is not (just) infinite in the sense that God is aware of infinitely many truths. God's knowledge is infinite in the sense that it is absolutely complete and not mediated by conceptual frameworks as ours is.
We might say something like: "God doesn't just know that 2 is prime, and 3 is prime, and 5 is prime, etc. Rather, for God, the idea of Quantity is apprehended fully in a way that precedes it's division into finite mathematical propositions."
Similarly for the Good, or whatever else God is meant to be in an unconditioned or infinite way.
Another interesting issue is that in Mathematics we have many measures for the same infinity. Take an square of side length 1 and other with side length 2. Their cardinals are equal to each other, (both have the cardinal or real numbers) but the lebesgue measure of the second is 4 times bigger than the first. Which measure shall you use for anthropics? Well, in probability we use lebesgue measures, so all this discussion on cardinals….
>Third, we have some evidence from physics that space is infinitely big.
You've been claiming this for some time, but the linked article effectively provides no evidence of this, and I would say it's effectively a trojan source. Let's critically evaluate some of the opinions of the interviewees.
Anna Moore says:
>Current measurements aren’t accurate enough for us to know whether the universe’s flat geometry is represented by a piece of paper, a cylinder, torus, or any other shape that permits the parallel passage of two beams of light... But for now we still don't know the shape of the universe, and therefore nor can we know its size.
She is essentially saying that current measurement methods in physics do not decide between the infinitude or finitude of space either way. This is true, if understated - I don't think there have been any proposed physical methods in principle that would allow us to determine whether space is finite or infinite.
Here is what Sara Webb says:
>I lean towards another possibility, which considers the rapid inflation that followed the Big Bang. There's a theory this inflation is actually eternal inflation, meaning it’s always occurring at one point or another in the universe — rendering the universe infinite.
Note she doesn't offer any empirical justification for eternal inflation, she just says, "being the sci-fi lover I am, how could I not want this to be true?" There is no currently accepted physical mechanism by which we could decide between eternal inflation and standard Big Bang cosmology that doesn't embed our observable universe in a larger ensemble of pocket universes.
Onto Tanya Hill:
>It’s not possible to see if the universe is finite, because we can’t see it all. Instead, we can tackle this question by exploring the universe's shape. While we don’t know the shape of all space, we do know our part of space is flat... A flat universe could be infinite... But it could also be finite... it’s possible inflation didn’t just seed our universe. Perhaps it also occurred elsewhere and is happening still. How big might that make the entire universe, or multiverse? It opens up such possibilities that, to my thinking, an infinite universe becomes easier to imagine than a finite one.
This again is not any sort of physics based evidence that the universe is infinite. It's the same point about our local inflationary epoch possibly being due to us being one pocket universe in a larger ensemble of pocket universes where inflation targets each pocket universe at different rates but occurs forever (hence, eternal inflation). Our observable universe doesn't even indicate an edge where matter just drops off - which is what eternal inflation postulates since there is effectively empty space between different pocket universes, so again, there is not any good evidence from existing physics that external inflation is true, and discovering any boundary lies beyond our observable universe, so it seems to be impossible in principle to accumulate physics-based evidence for the theory.
Here's what Sam Baron says:
>But there's another way for space to be finite. It could be a torus, which is spatially finite but edge-free, like a cosmic donut. If the universe is donut-shaped, then there’s a very natural scientific test that would reveal whether it is finite... While nothing conclusive has been found yet, who knows what we might uncover if we keep looking!
He describes a possible closed topology of the universe that would render it finite, and mentions that nothing conclusive has been found yet that would determine the question one way or the other. I will be the one to point out that the observable universe is estimated flat up to 0.4% margin of error, meaning that if it is a closed loop we would need the light rays we fire to travel ~250x the length of the observable universe to perform this experiment, meaning it's not physically possible since the observable universe is by definition the matter we can observe.
Here's Kevin Orrman-Rossiter:
>Let's propose, for a moment, that space is infinite. In a simple sense if this were the case and I set out in a spaceship in any direction, I would never reach a boundary. But there's a problem with this experiment: I would need to travel for an infinitely long period of time to ensure there isn’t a boundary “just a little further out”. It doesn’t matter what speed I travel. My voyage of proof would need to be infinite in order to prove my hypothesis that space is infinite. ...the current cosmological thinking is there will be an ending and the universe will not persist forever. It has a finite existence in time and, to return to the start of my argument, therefore I would propose that at some stage my spaceship voyage will reach an end.
The final interviewee proposes a conceptual challenge to determining whether space is infinite - namely that we would need to conduct a test that takes an infinite amount of time to complete to measure whether the universe is infinite. Current cosmology predicts that in a finite amount of time, space will expand faster than forces can keep matter together, so this test isn't possible in principle form our current understanding.
Basically, every single interviewer answered some variation of, "Not possible," "Not knowable," "Current physical theories predict against it," "Some untestable hypotheses assert an infinite inflationary period," or "I don't have evidence but I want it to be true." This is not good evidence for the universe being infinite, and is in fact evidence against the question being answerable. Definitely one of the more memeworthy articles you've linked.
Okay, explain how you do that in regards to the universe. Suppose each galaxy has 5 civilizations with property R and 1 with property B. What's the probability that our civilization has property R. Now, the natural answer is 5/6, but of course, this universe could eb reshuffled to make each galaxy have 5 Bs and only 1 R, because two infinites of the same cardinality are able to be put into one to 1 or 2 or 3 to 1 correspondence.
> Fourth, the view that the infinite is impossible requires space to have a smallest unit. But we have no evidence that space has a smallest unit and it seems impossible in principle for there to be a smallest unit of space. Can’t you just consider the right half of the smallest unit of space?
Let's say you live in a computer program. Your location in the 2D plane of this program is represented as two integers, e.g. (123,-456). If you move 1 up, then you go to 124. If you move 1.5 up, however, you can't go to 124.5 - the location is represented as two integers, so the system will round it off to either 124 or 125.
(This applies to floating points too - that's why some games like No Man's Sky or Outer Wilds exhibit strange behavior when you fly far enough away from the play zone.)
Okay, so suppose we have an infinitely good god. That means that there are infinitely many good things/utils in the universe, right? But there can also be, as should be pretty clear from living in the world, some number of evils. If the number of evils were infinite, then changing a single evil to be a good would have no impact on either the total number of evils or the total number of goods, because infinity - 1 or + 1 = infinity. So does this mean that there has to be a finite number of evils in the world and a commitment to some form of evil-reduction over good maximization for God to be able to work towards the good, since otherwise from God's perspective the amount of what matters in the world would be the same? Or is there some way in which an infinite God can also care about whether individual things that could be possibly good or evil are actually good or evil?
Ya finitism has issues. But that doesn’t mean you shouldn’t be weary of the infinite. Whenever possible it should be avoided in arguments.
Like with SIA, I take it the other way and see its implication that there must be infinite (of whatever flavor you prefer) other people as a knock on the theory.
You start with trying to get information from the fact of your own existence and you end up arguing that god must exist and he must have created a non-cardinally infinite number of people… your “this is getting really silly” warning light should be blinking red. Such a warning light is something many philosophers lack tbh.
“Hey do you think this god guy from this book the church keeps talking about exists?”, ends up with people talking about perfect simple beings, and whether traits can be infinitely instantiated. Like, we’re talking about the pillar of salt god, not whatever conceptual eunuch the philosophers seem to care about.
“Can we say true things about counterfactuals?”, turns into, “literally there is a world out that actually exists and consists entirely of 10 billions gallons of Asian elephant sperm suspended in a toroidal shape above a dark mountain”… learn to recognize silliness! Stop being silly!
I agree the infinite is weird. I think we have independent reason to think a correct theory of anthropics will tell us there are infinite people https://benthams.substack.com/p/the-self-indication-assumption-works, so this isn't a mark against SIA, but I agree infinite is weird.
Great article. There is only one part I disagree with, which is the part where you suggest that finite space may be inconceivable. I agree that an edge of space is extremely unlikely, but no one (or at least no physicist) who believes in a finite universe thinks there's an edge. Instead it's possible that the universe is a 3-sphere, a 3-torus, or some other finite manifold with no edge. I also think the idea that discrete space is impossible because you could just consider the right half of one of the smallest units is circular - there is no right half of a voxel if space is discrete. The only reason we think we can imagine the right half of one is because we inaccurately imagine them as spatially extended things.
So I do think that it's metaphysically possible for space to be finite in extent or discrete. Of course, the finitist has to claim that it's metaphysically necessary, which seems wildly implausible to me, and the physical evidence doesn't point to either of these things actually being true.
Also, you should go harder against finitists for rejecting B theory. I think B theory is logically necessary (more specifically, it's logically necessary that if time exists, B theory is true), and the case for eternalism (a slightly weaker claim than B theory which finitists also have to reject) is extremely overdetermined.
Existence in mind is different from existence in external reality. Infinite can be thought, and in that sense (the mathematical sense!) it exists and it is well defined. But what happens in your mind does not impose any obligation on external reality.
Cards on the table, I affirm the A-theory and I'm against against against the infinite (read: I don't affirm the existence of actual infinites).
Reason 4: Can you elaborate on why denying an actual infinite would require a smallest unit of space?
Reason 6: Theists who hold to the A-theory would just say that the future is potentially infinite and not actually infinite because the A-theory is pretty plausible and accepted by many smart people.
Reason 7: This statement is just false and potentially a category mistake. I don't even know what it means for God's knowledge or power to be infinite in a mathematical sense. Infinity, when applied to God, shouldn't be viewed as a mathematical or quantitative concept, but rather qualitative, i.e. God has all the great making properties and has them to the maximum degree.
Maybe you could define infinity at the beginning of the article to help clarify your position? Are you always using it as a mathematical concept and as an actual infinite substantiated in reality?
Thanks for the post! This is one of my favorite topics.
Thanks for the kind words.
4: Suppose space is infinitely divisible. There's an actual infinite--number of ways you can slice up space. Additionally, suppose I go two feet. Seems I made an actually infinite number of movements--going half way, a quarter of the way, and eighth of the way, and so on. Swinburne explains it well https://www.youtube.com/watch?v=4a4MHbXVIfs
6: Agreed! But if finitists must be A theorists while infinitists don't, that's a cost of finitism (it will strictly lower the probability).
7: God's knowledge is infinite in the sense that he knows an infinite number of things (he knows 1 is a number, 2 is a number, etc).
4: I would argue an actual infinite and an infinitely divisible line, distance, etc. are different. I can traverse two feet of infinitely divisible space and I can successively add up numbers to get to two feet; however, I cannot traverse an actual infinite many number of days, years, etc. or add up days to arrive at an actual infinite.
6: Makes sense.
7: I still don't think that's right though. I can conceive of the infinite in my mind, but I don't believe that's evidence of an actual infinite being substantiated in reality. Moreover, wouldn't God's knowledge of propositions just be potentially infinite and not actually infinite?
4: why are they different. How many increasingly small segments did you traverse? Answer: infinity.
7: Why is knowledge potential and not actual? Seems like God actually knows infinite things.
Again, I don't think either of those are right.
4. If there were an actual infinity of increasingly small segments, it would take an infinite amount of time to traverse them, by definition! A finite distance that is infinitely divisible is something wholly different from an actual infinite amount of space, time, etc.
7: The knowledge isn't potential vs actual; the knowledge is actual, but the amount of things known is only potentially infinite, i.e. inexhaustible. There aren't an actual infinite amount of things to know! Adding up a bunch of finite things never gets you to an infinite amount.
I wonder if you would say that we know an infinite number of things as well. After all, you also know that 1 is a number, 2 is a number, etc. Would you say that the difference between our knowledge and God's knowledge is that we can't actually hold an infinite amount of numbers in our head all at the same time and God can?
Something similar might apply to power. It seems like we as humans can do an infinite number of things. After all, I can move 1 inch, 2 inches, 1/2 inches, etc. Of course, it seems like God can do a lot more than we can, but I'm not exactly sure how to go about comparing infinities in this situation.
Yeah agreed!
I agree the distinction between qualitative infinity and quantitative infinity is big when talking about God. God's knowledge is not (just) infinite in the sense that God is aware of infinitely many truths. God's knowledge is infinite in the sense that it is absolutely complete and not mediated by conceptual frameworks as ours is.
We might say something like: "God doesn't just know that 2 is prime, and 3 is prime, and 5 is prime, etc. Rather, for God, the idea of Quantity is apprehended fully in a way that precedes it's division into finite mathematical propositions."
Similarly for the Good, or whatever else God is meant to be in an unconditioned or infinite way.
Another interesting issue is that in Mathematics we have many measures for the same infinity. Take an square of side length 1 and other with side length 2. Their cardinals are equal to each other, (both have the cardinal or real numbers) but the lebesgue measure of the second is 4 times bigger than the first. Which measure shall you use for anthropics? Well, in probability we use lebesgue measures, so all this discussion on cardinals….
Only for platonic metaphysics. If you understand that mathematics is a set of mental tools, this creates no problem!
>Third, we have some evidence from physics that space is infinitely big.
You've been claiming this for some time, but the linked article effectively provides no evidence of this, and I would say it's effectively a trojan source. Let's critically evaluate some of the opinions of the interviewees.
Anna Moore says:
>Current measurements aren’t accurate enough for us to know whether the universe’s flat geometry is represented by a piece of paper, a cylinder, torus, or any other shape that permits the parallel passage of two beams of light... But for now we still don't know the shape of the universe, and therefore nor can we know its size.
She is essentially saying that current measurement methods in physics do not decide between the infinitude or finitude of space either way. This is true, if understated - I don't think there have been any proposed physical methods in principle that would allow us to determine whether space is finite or infinite.
Here is what Sara Webb says:
>I lean towards another possibility, which considers the rapid inflation that followed the Big Bang. There's a theory this inflation is actually eternal inflation, meaning it’s always occurring at one point or another in the universe — rendering the universe infinite.
Note she doesn't offer any empirical justification for eternal inflation, she just says, "being the sci-fi lover I am, how could I not want this to be true?" There is no currently accepted physical mechanism by which we could decide between eternal inflation and standard Big Bang cosmology that doesn't embed our observable universe in a larger ensemble of pocket universes.
Onto Tanya Hill:
>It’s not possible to see if the universe is finite, because we can’t see it all. Instead, we can tackle this question by exploring the universe's shape. While we don’t know the shape of all space, we do know our part of space is flat... A flat universe could be infinite... But it could also be finite... it’s possible inflation didn’t just seed our universe. Perhaps it also occurred elsewhere and is happening still. How big might that make the entire universe, or multiverse? It opens up such possibilities that, to my thinking, an infinite universe becomes easier to imagine than a finite one.
This again is not any sort of physics based evidence that the universe is infinite. It's the same point about our local inflationary epoch possibly being due to us being one pocket universe in a larger ensemble of pocket universes where inflation targets each pocket universe at different rates but occurs forever (hence, eternal inflation). Our observable universe doesn't even indicate an edge where matter just drops off - which is what eternal inflation postulates since there is effectively empty space between different pocket universes, so again, there is not any good evidence from existing physics that external inflation is true, and discovering any boundary lies beyond our observable universe, so it seems to be impossible in principle to accumulate physics-based evidence for the theory.
Here's what Sam Baron says:
>But there's another way for space to be finite. It could be a torus, which is spatially finite but edge-free, like a cosmic donut. If the universe is donut-shaped, then there’s a very natural scientific test that would reveal whether it is finite... While nothing conclusive has been found yet, who knows what we might uncover if we keep looking!
He describes a possible closed topology of the universe that would render it finite, and mentions that nothing conclusive has been found yet that would determine the question one way or the other. I will be the one to point out that the observable universe is estimated flat up to 0.4% margin of error, meaning that if it is a closed loop we would need the light rays we fire to travel ~250x the length of the observable universe to perform this experiment, meaning it's not physically possible since the observable universe is by definition the matter we can observe.
Here's Kevin Orrman-Rossiter:
>Let's propose, for a moment, that space is infinite. In a simple sense if this were the case and I set out in a spaceship in any direction, I would never reach a boundary. But there's a problem with this experiment: I would need to travel for an infinitely long period of time to ensure there isn’t a boundary “just a little further out”. It doesn’t matter what speed I travel. My voyage of proof would need to be infinite in order to prove my hypothesis that space is infinite. ...the current cosmological thinking is there will be an ending and the universe will not persist forever. It has a finite existence in time and, to return to the start of my argument, therefore I would propose that at some stage my spaceship voyage will reach an end.
The final interviewee proposes a conceptual challenge to determining whether space is infinite - namely that we would need to conduct a test that takes an infinite amount of time to complete to measure whether the universe is infinite. Current cosmology predicts that in a finite amount of time, space will expand faster than forces can keep matter together, so this test isn't possible in principle form our current understanding.
Basically, every single interviewer answered some variation of, "Not possible," "Not knowable," "Current physical theories predict against it," "Some untestable hypotheses assert an infinite inflationary period," or "I don't have evidence but I want it to be true." This is not good evidence for the universe being infinite, and is in fact evidence against the question being answerable. Definitely one of the more memeworthy articles you've linked.
> If there are ℵ0 people with every property in the universe, how do I probabilistically reason about my odds of having different properties?
The same way mathematicians do when reasoning about the natural numbers. Half of them are even.
But yeah, you'll see arguments like yours crop up all the time https://www.reddit.com/r/badmathematics/comments/18tz1fb/according_to_this_groundbreaking_proof_there_are/ because outside of a math class people are imprecise.
Okay, explain how you do that in regards to the universe. Suppose each galaxy has 5 civilizations with property R and 1 with property B. What's the probability that our civilization has property R. Now, the natural answer is 5/6, but of course, this universe could eb reshuffled to make each galaxy have 5 Bs and only 1 R, because two infinites of the same cardinality are able to be put into one to 1 or 2 or 3 to 1 correspondence.
> Fourth, the view that the infinite is impossible requires space to have a smallest unit. But we have no evidence that space has a smallest unit and it seems impossible in principle for there to be a smallest unit of space. Can’t you just consider the right half of the smallest unit of space?
Let's say you live in a computer program. Your location in the 2D plane of this program is represented as two integers, e.g. (123,-456). If you move 1 up, then you go to 124. If you move 1.5 up, however, you can't go to 124.5 - the location is represented as two integers, so the system will round it off to either 124 or 125.
(This applies to floating points too - that's why some games like No Man's Sky or Outer Wilds exhibit strange behavior when you fly far enough away from the play zone.)
Okay, so suppose we have an infinitely good god. That means that there are infinitely many good things/utils in the universe, right? But there can also be, as should be pretty clear from living in the world, some number of evils. If the number of evils were infinite, then changing a single evil to be a good would have no impact on either the total number of evils or the total number of goods, because infinity - 1 or + 1 = infinity. So does this mean that there has to be a finite number of evils in the world and a commitment to some form of evil-reduction over good maximization for God to be able to work towards the good, since otherwise from God's perspective the amount of what matters in the world would be the same? Or is there some way in which an infinite God can also care about whether individual things that could be possibly good or evil are actually good or evil?
Ya finitism has issues. But that doesn’t mean you shouldn’t be weary of the infinite. Whenever possible it should be avoided in arguments.
Like with SIA, I take it the other way and see its implication that there must be infinite (of whatever flavor you prefer) other people as a knock on the theory.
You start with trying to get information from the fact of your own existence and you end up arguing that god must exist and he must have created a non-cardinally infinite number of people… your “this is getting really silly” warning light should be blinking red. Such a warning light is something many philosophers lack tbh.
“Hey do you think this god guy from this book the church keeps talking about exists?”, ends up with people talking about perfect simple beings, and whether traits can be infinitely instantiated. Like, we’re talking about the pillar of salt god, not whatever conceptual eunuch the philosophers seem to care about.
“Can we say true things about counterfactuals?”, turns into, “literally there is a world out that actually exists and consists entirely of 10 billions gallons of Asian elephant sperm suspended in a toroidal shape above a dark mountain”… learn to recognize silliness! Stop being silly!
I agree the infinite is weird. I think we have independent reason to think a correct theory of anthropics will tell us there are infinite people https://benthams.substack.com/p/the-self-indication-assumption-works, so this isn't a mark against SIA, but I agree infinite is weird.
Great article. There is only one part I disagree with, which is the part where you suggest that finite space may be inconceivable. I agree that an edge of space is extremely unlikely, but no one (or at least no physicist) who believes in a finite universe thinks there's an edge. Instead it's possible that the universe is a 3-sphere, a 3-torus, or some other finite manifold with no edge. I also think the idea that discrete space is impossible because you could just consider the right half of one of the smallest units is circular - there is no right half of a voxel if space is discrete. The only reason we think we can imagine the right half of one is because we inaccurately imagine them as spatially extended things.
So I do think that it's metaphysically possible for space to be finite in extent or discrete. Of course, the finitist has to claim that it's metaphysically necessary, which seems wildly implausible to me, and the physical evidence doesn't point to either of these things actually being true.
Also, you should go harder against finitists for rejecting B theory. I think B theory is logically necessary (more specifically, it's logically necessary that if time exists, B theory is true), and the case for eternalism (a slightly weaker claim than B theory which finitists also have to reject) is extremely overdetermined.
Existence in mind is different from existence in external reality. Infinite can be thought, and in that sense (the mathematical sense!) it exists and it is well defined. But what happens in your mind does not impose any obligation on external reality.