Surreal numbers are useless for all of these paradoxes.
Infinitarian paralysis: Using surreal-valued utilities creates more infinitarian paralysis than it solves, I think. You’ll never take an opportunity to increase utility by x because it will always have higher expected utility to focus all of your attention on trying to find ways to increase utility by >ωx, since there’s some (however small) probability p>0 that such efforts would succeed, so the expected utility of focusing your efforts on looking for ways to increase utility by >ωx will have expected utility >pωx, which is higher than x. I think a better solution would be to note that for any person, a nonzero fraction of people are close enough to identical to that person that they will make the same decisions, so any decision that anyone makes affects a nonzero fraction of people. Measure theory is probably a better framework than surreal numbers for formalizing what is meant by “fraction” here.
Paradox of the gods: The introduction of surreal numbers solves nothing. Why wouldn’t he be able to advance more than 2−ω miles if no gods erect any barriers until he advances 2−n miles for some finite n?
Two-envelopes paradox: it doesn’t make sense to model your uncertainty over how much money is in the first envelope with a uniform surreal-valued probability distribution on [1n,n] for an infinite surreal n, because then the probability that there is a finite amount of money in the envelope is infinitesimal, but we’re trying to model the situation in which we know there’s a finite amount of money in the envelope and just have no idea which finite amount.
Sphere of suffering: Surreal numbers are not the right tool for measuring the volume of Euclidean space or the duration of forever.
Hilbert hotel: As you mentioned, using surreals in the way you propose changes the problem.
Trumped, Trouble in St. Petersburg, Soccer teams, Can God choose an integer at random?, The Headache: Using surreals in the way you propose in each of these changes the problems in exactly the same way it does for the Hilbert hotel.
St. Petersburg paradox: If you pay infinity dollars to play the game, then you lose infinity dollars with probability 1. Doesn’t sound like a great deal.
Banach-Tarski Paradox: The free group only consists of sequences of finite length.
The Magic Dartboard: First, a nitpick: that proof relies on the continuum hypothesis, which is independent of ZFC. Aside from that, the proof is correct, which means any resolution along the lines you’re imagining that imply that no magic dartboards exist is going to imply that the continuum hypothesis is false. Worse, the fact that for any countable ordinal, there are countably many smaller countable ordinals and uncountably many larger countable ordinals follows from very minimal mathematical assumptions, and is often used in descriptive set theory without bringing in the continuum hypothesis at all, so if you start trying to change math to make sense of “the second half of the countable ordinals”, you’re going to have a bad time.
Parity paradoxes: The lengths of the sequences involved here are the ordinal ω, not a surreal number. You might object that there is also a surreal number called ω, but this is different from the ordinal ω. Arithmetic operations act differently on ordinals than they do on the copies of those ordinals in the surreal numbers, so there’s no reasonable sense in which the surreals contain the ordinals. Example: if you add another element to the beginning of either sequence (i.e. flip the switch at t=−2, or add a −1 at the beginning of the sum, respectively), then you’ve added one thing, so the surreal number should increase by 1, but the order-type is unchanged, so the ordinal remains the same.
Thanks for your feedback. I’ll note that these are only informal hints/thoughts on how surreal numbers could help us here and that I’ll be providing a more developed version of some of these thoughts in a future post.
Infinitarian paralysis: I consider Pascal’s Mugging to be its own seperate problem. Indeed Pascal’s Mugging type issues are already present with the more standard infinities. In any case, the measure theory solution is dependent on an individual being a finite fraction of the agents in the universe. While this is an extremely plausible assumption, there doesn’t seem to be any principled reason why our solution to infinite paralysis should depend on this assumption.
Paradox of the Gods, Banach-Tarski: Your complaint is that I’m disallowing the a sequence consisting of all finite inverses/all finite integers. I actually believe that actual and potential infinity models of these problems need to be treated separately, though I’ve only outlined how I plan to handle actual infinities. Hopefully, you find my next post on this topic more persuasive.
Two-envelopes paradox: “The probability that there is a finite amount of money in the envelope is infinitesimal”—Hmm, you’re right. That is a rather significant issue.
Sphere of suffering: “Surreal numbers are not the right tool for measuring the volume of Euclidean space or the duration of forever”—why?
St Petersberg Paradox: Ah, but you have an infinitesimal chance of winning a higher infinity. So it becomes an even more extreme version of Pascal’s Mugging, but again that’s its own discussion.
Magic Dartboard: Yes, I’m aware that rejecting the existence of magic dartboards could have far-reaching consequences. It’s something I hope to look into more.
Parity: See my response to gjm. Ordinal numbers lack resolution and so can’t properly describe the length of sequence.
Indeed Pascal’s Mugging type issues are already present with the more standard infinities.
Right, infinity of any kind (surreal or otherwise) doesn’t belong in decision theory.
“Surreal numbers are not the right tool for measuring the volume of Euclidean space or the duration of forever”—why?
How would you? If you do something like taking an increasing sequence of bounded subsets that fill up the space you’re trying to measure, find a formula f(n) for the volume of the nth subset, and plug in f(ω), the result will be highly dependent on which increasing sequence of bounded subsets you use. Did you have a different proposal? It’s sort of hard to explain why no method for measuring volumes using surreal numbers can possibly work well, though I am confident it is true. At the very least, volume-preserving transformations like shifting everything 1 meter to the left or rotating everything around some axis cease to be volume-preserving, though I don’t know if you’d find this convincing.
“Indeed Pascal’s Mugging type issues are already present with the more standard infinities.”
Right, infinity of any kind (surreal or otherwise) doesn’t belong in decision theory.
But Pascal’s Mugging type issues are present with large finite numbers, as well. Do you bite the bullet in the finite case, or do you think that unbounded utility functions don’t belong in decision theory, either?
The latter. It doesn’t even make sense to speak of maximizing the expectation of an unbounded utility function, because unbounded functions don’t even have expectations with respect to all probability distributions.
There is a way out of this that you could take, which is to only insist that the utility function has to have an expectation with respect to probability distributions in some restricted class, if you know your options are all going to be from that restricted class. I don’t find this very satisfying, but it works. And it offers its own solution to Pascal’s mugging, by insisting that any outcome whose utility is on the scale of 3^^^3 has prior probability on the scale of 1/(3^^^3) or lower.
There’s definitely a part of me wondering if infinities exist, but before I even consider tackling that question, I need to figure out the most consistent interpretation of infinities assuming they exist.
“At the very least, volume-preserving transformations like shifting everything 1 meter to the left or rotating everything around some axis cease to be volume-preserving, though I don’t know if you’d find this convincing”—Well there are non-measurable sets that do this without surreals, but do surreals add more examples?
I’ll have to read more about how surreals apply to volumes. It may be hard getting convergence to the exact infinitesimal, but I don’t know if the problems will extent beyond that.
(Also, the ability to integrate is mostly besides the point. Instead of the sphere of suffering, we could have defined the expanding cube of suffering. This will then let us solve some special cases of the sphere of suffering)
There are measurable sets whose volumes will not be preserved if you try to measure them with surreal numbers. For example, consider [0,∞)⊆R. Say its measure is some infinite surreal number n. The volume-preserving left-shift operation x↦x−1 sends [0,∞) to [−1,∞), which has measure 1+n, since [−1,0) has measure 1. You can do essentially the same thing in higher dimensions, and the shift operation in two dimensions ((x,y)↦(x−1,y)) can be expressed as the composition of two rotations, so rotations can’t be volume-preserving either. And since different rotations will have to fail to preserve volumes in different ways, this will break symmetries of the plane.
I wouldn’t say that volume-preserving transformations fail to preserve volume on non-measurable sets, just that non-measurable sets don’t even have measures that could be preserved or not preserved. Failing to preserve measures of sets that you have assigned measures to is entirely different. Non-measurable sets also don’t arise in mathematical practice; half-spaces do. I’m also skeptical of the existence of non-measurable sets, but the non-existence of non-measurable sets is a far bolder claim than anything else I’ve said here.
I’m happy to bite that bullet and destroy the symmetry. If we pick a random point and line in the universe, are there more unit points to the left or right? Well, that depends on where the point is.
It’s a bad bullet to bite. Its symmetries are essential to what makes Euclidean space interesting.
And here’s another one: are you not bothered by the lack of countable additivity? Suppose you say that the volume of Euclidean space is some surreal number n. Euclidean space is the union of an increasing sequence of balls. The volumes of these balls are all finite, in particular, less than n2, so how can you justify saying that their union has volume greater than n2?
“Its symmetries are essential to what makes Euclidean space interesting”—Isn’t the interesting aspect of Euclidean space its ability to model our world excluding relativity?
Well, I just don’t think it’s that unusual for functions to have properties that break at their limits. Is this any different from 1/x being definable everywhere except 0? Is there anything that makes the change at the limit particularly concerning.
I don’t follow the analogy to 1/x being a partial function that you’re getting at.
Maybe a better way to explain what I’m getting at is that it’s really the same issue that I pointed out for the two-envelopes problem, where you know the amount of money in each envelope is finite, but the uniform distribution up to an infinite surreal would suggest that the probability that the amount of money is finite is infinitesimal. Suppose you say that the size of the ray [0,∞) is an infinite surreal number n. The size of the portion of this ray that is distance at least r from 0 is n−r when r is a positive real, so presumably you would also want this to be so for surreal r. But using, say, r:=√n, every point in [0,∞) is within distance √n of 0, but this rule would say that the measure of the portion of the ray that is farther than √n from 0 is n−√n; that is, almost all of the measure of [0,∞) is concentrated on the empty set.
As I understand it, there is not yet a good theory of integration on the surreals. Partial progress has been made, but there are also some negative results establishing limitations on the possibilities. Here is a recent paper.
Surreal numbers are useless for all of these paradoxes.
Infinitarian paralysis: Using surreal-valued utilities creates more infinitarian paralysis than it solves, I think. You’ll never take an opportunity to increase utility by x because it will always have higher expected utility to focus all of your attention on trying to find ways to increase utility by >ωx, since there’s some (however small) probability p>0 that such efforts would succeed, so the expected utility of focusing your efforts on looking for ways to increase utility by >ωx will have expected utility >pωx, which is higher than x. I think a better solution would be to note that for any person, a nonzero fraction of people are close enough to identical to that person that they will make the same decisions, so any decision that anyone makes affects a nonzero fraction of people. Measure theory is probably a better framework than surreal numbers for formalizing what is meant by “fraction” here.
Paradox of the gods: The introduction of surreal numbers solves nothing. Why wouldn’t he be able to advance more than 2−ω miles if no gods erect any barriers until he advances 2−n miles for some finite n?
Two-envelopes paradox: it doesn’t make sense to model your uncertainty over how much money is in the first envelope with a uniform surreal-valued probability distribution on [1n,n] for an infinite surreal n, because then the probability that there is a finite amount of money in the envelope is infinitesimal, but we’re trying to model the situation in which we know there’s a finite amount of money in the envelope and just have no idea which finite amount.
Sphere of suffering: Surreal numbers are not the right tool for measuring the volume of Euclidean space or the duration of forever.
Hilbert hotel: As you mentioned, using surreals in the way you propose changes the problem.
Trumped, Trouble in St. Petersburg, Soccer teams, Can God choose an integer at random?, The Headache: Using surreals in the way you propose in each of these changes the problems in exactly the same way it does for the Hilbert hotel.
St. Petersburg paradox: If you pay infinity dollars to play the game, then you lose infinity dollars with probability 1. Doesn’t sound like a great deal.
Banach-Tarski Paradox: The free group only consists of sequences of finite length.
The Magic Dartboard: First, a nitpick: that proof relies on the continuum hypothesis, which is independent of ZFC. Aside from that, the proof is correct, which means any resolution along the lines you’re imagining that imply that no magic dartboards exist is going to imply that the continuum hypothesis is false. Worse, the fact that for any countable ordinal, there are countably many smaller countable ordinals and uncountably many larger countable ordinals follows from very minimal mathematical assumptions, and is often used in descriptive set theory without bringing in the continuum hypothesis at all, so if you start trying to change math to make sense of “the second half of the countable ordinals”, you’re going to have a bad time.
Parity paradoxes: The lengths of the sequences involved here are the ordinal ω, not a surreal number. You might object that there is also a surreal number called ω, but this is different from the ordinal ω. Arithmetic operations act differently on ordinals than they do on the copies of those ordinals in the surreal numbers, so there’s no reasonable sense in which the surreals contain the ordinals. Example: if you add another element to the beginning of either sequence (i.e. flip the switch at t=−2, or add a −1 at the beginning of the sum, respectively), then you’ve added one thing, so the surreal number should increase by 1, but the order-type is unchanged, so the ordinal remains the same.
Thanks for your feedback. I’ll note that these are only informal hints/thoughts on how surreal numbers could help us here and that I’ll be providing a more developed version of some of these thoughts in a future post.
Infinitarian paralysis: I consider Pascal’s Mugging to be its own seperate problem. Indeed Pascal’s Mugging type issues are already present with the more standard infinities. In any case, the measure theory solution is dependent on an individual being a finite fraction of the agents in the universe. While this is an extremely plausible assumption, there doesn’t seem to be any principled reason why our solution to infinite paralysis should depend on this assumption.
Paradox of the Gods, Banach-Tarski: Your complaint is that I’m disallowing the a sequence consisting of all finite inverses/all finite integers. I actually believe that actual and potential infinity models of these problems need to be treated separately, though I’ve only outlined how I plan to handle actual infinities. Hopefully, you find my next post on this topic more persuasive.
Two-envelopes paradox: “The probability that there is a finite amount of money in the envelope is infinitesimal”—Hmm, you’re right. That is a rather significant issue.
Sphere of suffering: “Surreal numbers are not the right tool for measuring the volume of Euclidean space or the duration of forever”—why?
St Petersberg Paradox: Ah, but you have an infinitesimal chance of winning a higher infinity. So it becomes an even more extreme version of Pascal’s Mugging, but again that’s its own discussion.
Magic Dartboard: Yes, I’m aware that rejecting the existence of magic dartboards could have far-reaching consequences. It’s something I hope to look into more.
Parity: See my response to gjm. Ordinal numbers lack resolution and so can’t properly describe the length of sequence.
Right, infinity of any kind (surreal or otherwise) doesn’t belong in decision theory.
How would you? If you do something like taking an increasing sequence of bounded subsets that fill up the space you’re trying to measure, find a formula f(n) for the volume of the nth subset, and plug in f(ω), the result will be highly dependent on which increasing sequence of bounded subsets you use. Did you have a different proposal? It’s sort of hard to explain why no method for measuring volumes using surreal numbers can possibly work well, though I am confident it is true. At the very least, volume-preserving transformations like shifting everything 1 meter to the left or rotating everything around some axis cease to be volume-preserving, though I don’t know if you’d find this convincing.
But Pascal’s Mugging type issues are present with large finite numbers, as well. Do you bite the bullet in the finite case, or do you think that unbounded utility functions don’t belong in decision theory, either?
The latter. It doesn’t even make sense to speak of maximizing the expectation of an unbounded utility function, because unbounded functions don’t even have expectations with respect to all probability distributions.
There is a way out of this that you could take, which is to only insist that the utility function has to have an expectation with respect to probability distributions in some restricted class, if you know your options are all going to be from that restricted class. I don’t find this very satisfying, but it works. And it offers its own solution to Pascal’s mugging, by insisting that any outcome whose utility is on the scale of 3^^^3 has prior probability on the scale of 1/(3^^^3) or lower.
There’s definitely a part of me wondering if infinities exist, but before I even consider tackling that question, I need to figure out the most consistent interpretation of infinities assuming they exist.
“At the very least, volume-preserving transformations like shifting everything 1 meter to the left or rotating everything around some axis cease to be volume-preserving, though I don’t know if you’d find this convincing”—Well there are non-measurable sets that do this without surreals, but do surreals add more examples?
I’ll have to read more about how surreals apply to volumes. It may be hard getting convergence to the exact infinitesimal, but I don’t know if the problems will extent beyond that.
(Also, the ability to integrate is mostly besides the point. Instead of the sphere of suffering, we could have defined the expanding cube of suffering. This will then let us solve some special cases of the sphere of suffering)
There are measurable sets whose volumes will not be preserved if you try to measure them with surreal numbers. For example, consider [0,∞)⊆R. Say its measure is some infinite surreal number n. The volume-preserving left-shift operation x↦x−1 sends [0,∞) to [−1,∞), which has measure 1+n, since [−1,0) has measure 1. You can do essentially the same thing in higher dimensions, and the shift operation in two dimensions ((x,y)↦(x−1,y)) can be expressed as the composition of two rotations, so rotations can’t be volume-preserving either. And since different rotations will have to fail to preserve volumes in different ways, this will break symmetries of the plane.
I wouldn’t say that volume-preserving transformations fail to preserve volume on non-measurable sets, just that non-measurable sets don’t even have measures that could be preserved or not preserved. Failing to preserve measures of sets that you have assigned measures to is entirely different. Non-measurable sets also don’t arise in mathematical practice; half-spaces do. I’m also skeptical of the existence of non-measurable sets, but the non-existence of non-measurable sets is a far bolder claim than anything else I’ve said here.
Well shifting left produces a superset of the original, so of course we shouldn’t expect that to preserve measure.
What about rotations, and the fact that we’re talking about destroying a bunch of symmetry of the plane?
I’m happy to bite that bullet and destroy the symmetry. If we pick a random point and line in the universe, are there more unit points to the left or right? Well, that depends on where the point is.
It’s a bad bullet to bite. Its symmetries are essential to what makes Euclidean space interesting.
And here’s another one: are you not bothered by the lack of countable additivity? Suppose you say that the volume of Euclidean space is some surreal number n. Euclidean space is the union of an increasing sequence of balls. The volumes of these balls are all finite, in particular, less than n2, so how can you justify saying that their union has volume greater than n2?
“Its symmetries are essential to what makes Euclidean space interesting”—Isn’t the interesting aspect of Euclidean space its ability to model our world excluding relativity?
Well, I just don’t think it’s that unusual for functions to have properties that break at their limits. Is this any different from 1/x being definable everywhere except 0? Is there anything that makes the change at the limit particularly concerning.
I don’t follow the analogy to 1/x being a partial function that you’re getting at.
Maybe a better way to explain what I’m getting at is that it’s really the same issue that I pointed out for the two-envelopes problem, where you know the amount of money in each envelope is finite, but the uniform distribution up to an infinite surreal would suggest that the probability that the amount of money is finite is infinitesimal. Suppose you say that the size of the ray [0,∞) is an infinite surreal number n. The size of the portion of this ray that is distance at least r from 0 is n−r when r is a positive real, so presumably you would also want this to be so for surreal r. But using, say, r:=√n, every point in [0,∞) is within distance √n of 0, but this rule would say that the measure of the portion of the ray that is farther than √n from 0 is n−√n; that is, almost all of the measure of [0,∞) is concentrated on the empty set.
As I understand it, there is not yet a good theory of integration on the surreals. Partial progress has been made, but there are also some negative results establishing limitations on the possibilities. Here is a recent paper.