My primary response to this comment will take the form of a post, but I should add that I wrote: “I will provide informal hints on how surreal numbers could help us solve some of these paradoxes, although the focus on this post is primarily categorisation, so please don’t mistake these for formal proofs”.
Your comment seems to completely ignore this stipulation. Take for example this:
“Of course, your solution seems to involve implicitly changing the setting to have surreal-valued time and space… You might want to make more of an explicit note of it, though”
Yes, there’s a lot of philosophical groundwork that would need to be done to justify the surreal approach. That’s why I said that it was only an informal hint.
I’m going to assume, since you’re talking about surreals and didn’t specify otherwise, that you mean exp(s log 2), using the usual surreal exponential
Yes, I actually did look up that there was a way of defining 2^s where s is a surreal number.
Let’s accept the premise that you’re using a surreal-valued probability measure instead of a real one
I wrote a summary of a paper by Chen and Rubio that provides the start of a surreal decision theory. This isn’t a complete probability theory as it only supports finite additivity instead of countable additivity, but it suggests that this approach might be viable.
I could keep going, but I think I’ve made my point that you’re evaluating these informal comments as though I’d claimed they were a formal proof. This post was already long enough and took enough time to write as is.
I will admit that I could have been clearer that many of these remarks were speculative, in the sense of being arguments that I believed were worth working towards formalising, even if all of the mathematical machinery doesn’t necessarily exist at this time. My point is that justifying the use of surreals numbers doesn’t necessarily involve solving every paradox; it should also be persuasive to solve a good number of them and then to demonstrate that there is good reason to believe that we may be able to solve the rest in the future. In this sense, informal arguments aren’t valueless.
My primary response to this comment will take the form of a post, but I should add that I wrote: “I will provide informal hints on how surreal numbers could help us solve some of these paradoxes, although the focus on this post is primarily categorisation, so please don’t mistake these for formal proofs”.
You’re right; I did miss that, thanks. It was perhaps unfair of me then to pick on such gaps in formalism. Unfortunately, this is only enough to rescue a small portion in the post. Ignoring the ones I skipped—maybe it would be worth my time to get back to those after all—I think the only one potentially rescued that way is the envelope problem. (I’m still skeptical that it is—I haven’t looked at it in enough detail to say—but I’ll grant you that it could be.)
(Edit: After rechecking, I guess I’d count Grandi’s series and Thomson’s lamp here too, but only barely, in the sense that—after giving you quite a bit of benefit of the doubt—yeah I guess you could define things that way but I see absolutely no reason why one would want to and I seriously doubt you gain anything from doing so. (I was about to include god picking a random integer here, too, but on rechecking again, no, that one still has serious other problems even if I give you more leeway than I initially did. Like, if you try to identify ∞ with a specific surreal, say ω, there’s no surreal you can identify it with that will make your conclusion correct.))
The rest of the ones I pointed out as wrong (involving surreals, anyway) all contain more substantial errors. In some cases this becomes evident after doing the work and attempting to formalize your hints; in other cases they’re evident immediately, and clearly do not work even informally.
The magic dartboard is a good example of the latter—you’ve simply given an incorrect proof of why the magic dartboard construction works. In it you talk about ω_1 having a first half and a second half. You don’t need to do any deep thinking about surreals to see the problem here—that’s just not what ω_1 looks like, at all. If you do follow the hint, and compare the elements of ω_1 to (ω_1)/2 in the surreals, then, as already noted, you find everything falls in the first half, which is not very helpful. (Again: This is the sort of thing that causes me to say, I suspect you need to relearn ordinals and probably other things, not just surreals. If you actually understand ordinals, you should not have any trouble proving that the magic dartboard acts as claimed, without any need to go into the surreals and perform division.)
Meanwhile the paradox of the gods is, as I’ve already laid out in detail, an example of the former. It sounds like a nice informal answer that could possibly be formalized, sure; but if you try to actually follow the hint and do that—switching to surreal time and space as needed, of course—it still makes no sense for the reasons I’ve described above. Because, e.g., ω is a limit ordinal and not a successor ordinal (this is a repeated mistake throughout the post, ignoring the existence of limit ordinals), because in the surreals there are no infima of sets (that aren’t minima), because the fact that a surreal exponential exists doesn’t mean that it acts like you want it to (algebraically it does everything you might want, but this problem isn’t about algebraic properties) or that there’s anything special about the points it picks out.
In addition, some of the things one is expected to just go with would require not just more explanation to formalize (like surreal integration) but to even make even informal sense of (like what structure you are putting on a set, or what you are embedding it in, that would make a surreal an appropriate measure of its size).
In short, your hints are not hints towards an already-existing solution (or at least, not one that anyone other than you would accept); they’re analogy-driven speculation as to what a solution could look like. Obviously there’s nothing wrong with analogy-driven speculation! I could definitely go on about some analogy-driven speculation of mine involving surreals! But, firstly, that’s not what you presented it as; secondly, in most of your cases it’s actually fairly easy (with a bit of relevant knowledge) to follow the breadcrumb trail and see that in fact it goes nowhere, as I did in my reply; and, thirdly, you’re purporting to “solve” things that aren’t actually problems in the first place. The second being the most important here, to be clear.
(And I think the ones I skipped demonstrate even more mathematical problems that I didn’t get to, but, well, I haven’t gotten to those.)
FWIW, I’d say surreal decision theory is a bad idea, because, well, Savage’s theorem—that’s a lot of my philosophical objections right there. But I should get to the actual mathematical problems sometime; the philosophical objections, while important, are, I expect, not as interesting to you.
Basically, the post treats the surreals as a sort of device for automatically making the infinite behave like the finite. They’re not. Yes, their structure as an ordered field (ordered exponential field, even) means that their algebraic behavior resembles such familiar finite settings as the real numbers, in contrast to the quite different arithmetic of (say) the ordinal or cardinal numbers (one might even include here the extended real line, with its mostly-all-absorbing ∞). But the things you’re trying to do here often involve more than arithmetic or algebra, and then the analogies quickly fall apart. (Again, I’d see our previous exchange here for examples.)
My primary response to this comment will take the form of a post, but I should add that I wrote: “I will provide informal hints on how surreal numbers could help us solve some of these paradoxes, although the focus on this post is primarily categorisation, so please don’t mistake these for formal proofs”.
Your comment seems to completely ignore this stipulation. Take for example this:
Yes, there’s a lot of philosophical groundwork that would need to be done to justify the surreal approach. That’s why I said that it was only an informal hint.
Yes, I actually did look up that there was a way of defining 2^s where s is a surreal number.
I wrote a summary of a paper by Chen and Rubio that provides the start of a surreal decision theory. This isn’t a complete probability theory as it only supports finite additivity instead of countable additivity, but it suggests that this approach might be viable.
I could keep going, but I think I’ve made my point that you’re evaluating these informal comments as though I’d claimed they were a formal proof. This post was already long enough and took enough time to write as is.
I will admit that I could have been clearer that many of these remarks were speculative, in the sense of being arguments that I believed were worth working towards formalising, even if all of the mathematical machinery doesn’t necessarily exist at this time. My point is that justifying the use of surreals numbers doesn’t necessarily involve solving every paradox; it should also be persuasive to solve a good number of them and then to demonstrate that there is good reason to believe that we may be able to solve the rest in the future. In this sense, informal arguments aren’t valueless.
You’re right; I did miss that, thanks. It was perhaps unfair of me then to pick on such gaps in formalism. Unfortunately, this is only enough to rescue a small portion in the post. Ignoring the ones I skipped—maybe it would be worth my time to get back to those after all—I think the only one potentially rescued that way is the envelope problem. (I’m still skeptical that it is—I haven’t looked at it in enough detail to say—but I’ll grant you that it could be.)
(Edit: After rechecking, I guess I’d count Grandi’s series and Thomson’s lamp here too, but only barely, in the sense that—after giving you quite a bit of benefit of the doubt—yeah I guess you could define things that way but I see absolutely no reason why one would want to and I seriously doubt you gain anything from doing so. (I was about to include god picking a random integer here, too, but on rechecking again, no, that one still has serious other problems even if I give you more leeway than I initially did. Like, if you try to identify ∞ with a specific surreal, say ω, there’s no surreal you can identify it with that will make your conclusion correct.))
The rest of the ones I pointed out as wrong (involving surreals, anyway) all contain more substantial errors. In some cases this becomes evident after doing the work and attempting to formalize your hints; in other cases they’re evident immediately, and clearly do not work even informally.
The magic dartboard is a good example of the latter—you’ve simply given an incorrect proof of why the magic dartboard construction works. In it you talk about ω_1 having a first half and a second half. You don’t need to do any deep thinking about surreals to see the problem here—that’s just not what ω_1 looks like, at all. If you do follow the hint, and compare the elements of ω_1 to (ω_1)/2 in the surreals, then, as already noted, you find everything falls in the first half, which is not very helpful. (Again: This is the sort of thing that causes me to say, I suspect you need to relearn ordinals and probably other things, not just surreals. If you actually understand ordinals, you should not have any trouble proving that the magic dartboard acts as claimed, without any need to go into the surreals and perform division.)
Meanwhile the paradox of the gods is, as I’ve already laid out in detail, an example of the former. It sounds like a nice informal answer that could possibly be formalized, sure; but if you try to actually follow the hint and do that—switching to surreal time and space as needed, of course—it still makes no sense for the reasons I’ve described above. Because, e.g., ω is a limit ordinal and not a successor ordinal (this is a repeated mistake throughout the post, ignoring the existence of limit ordinals), because in the surreals there are no infima of sets (that aren’t minima), because the fact that a surreal exponential exists doesn’t mean that it acts like you want it to (algebraically it does everything you might want, but this problem isn’t about algebraic properties) or that there’s anything special about the points it picks out.
In addition, some of the things one is expected to just go with would require not just more explanation to formalize (like surreal integration) but to even make even informal sense of (like what structure you are putting on a set, or what you are embedding it in, that would make a surreal an appropriate measure of its size).
In short, your hints are not hints towards an already-existing solution (or at least, not one that anyone other than you would accept); they’re analogy-driven speculation as to what a solution could look like. Obviously there’s nothing wrong with analogy-driven speculation! I could definitely go on about some analogy-driven speculation of mine involving surreals! But, firstly, that’s not what you presented it as; secondly, in most of your cases it’s actually fairly easy (with a bit of relevant knowledge) to follow the breadcrumb trail and see that in fact it goes nowhere, as I did in my reply; and, thirdly, you’re purporting to “solve” things that aren’t actually problems in the first place. The second being the most important here, to be clear.
(And I think the ones I skipped demonstrate even more mathematical problems that I didn’t get to, but, well, I haven’t gotten to those.)
FWIW, I’d say surreal decision theory is a bad idea, because, well, Savage’s theorem—that’s a lot of my philosophical objections right there. But I should get to the actual mathematical problems sometime; the philosophical objections, while important, are, I expect, not as interesting to you.
Basically, the post treats the surreals as a sort of device for automatically making the infinite behave like the finite. They’re not. Yes, their structure as an ordered field (ordered exponential field, even) means that their algebraic behavior resembles such familiar finite settings as the real numbers, in contrast to the quite different arithmetic of (say) the ordinal or cardinal numbers (one might even include here the extended real line, with its mostly-all-absorbing ∞). But the things you’re trying to do here often involve more than arithmetic or algebra, and then the analogies quickly fall apart. (Again, I’d see our previous exchange here for examples.)