So far as anyone knows, no actual processes in the actual world are accurately described by surreal numbers. If not, then I suggest the same goes for the “nearest possible worlds” in which, say, it is possible for Mr Trump to be faced with the sort of situation described under the heading “Trumped”. But you can have, in a universe very much like ours, an endless succession of events of order-type ω. If the surreal numbers are not well suited to describing such situations, so much the worse for the surreal numbers.
And when you say “I don’t think the ordinary notion of sequence makes sense”, what it looks like to me is that you have looked at the ordinary notion of sequence, made the entirely arbitrary choice that you are only prepared to understand it in terms of surreal numbers, and indeed not only that but made the further arbitrary choice that you are only prepared to understand it if there turns out to be a uniquely defined surreal number that is the length of such a sequence, observed that there is not such a surreal number, and then said not “Oh, whoops, looks like I took a wrong turn in trying to model this situation” but “Bah, the thing I’m trying to model doesn’t fit my preconceptions of what the model should look like, therefore the thing is wrong”. You can’t do that! Models exist to serve the things they model, not the other way around.
It’s as if I’d just learned about the ordinals, decided that all infinite things needed to be described in terms of the ordinals, was asked something about a countably infinite set, observed that such a set is the same size as ω but also the same size as 1+ω and ω2, and said “I don’t think the notion of countably infinite set makes sense”. It makes perfectly good sense, I just (hypothetically) picked a bad way to think about it: ordinals are not the right tool for measuring the size of a (not-necessarily-well-ordered) set. And likewise, surreal numbers are not the right tool for measuring the length of a sequence.
Don’t get me wrong; I love the surreal numbers, as an object of mathematical study. The theory is gorgeous. But you can’t claim that the surreal numbers let you resolve all these paradoxes, when what they actually allow you to do is to replace the paradoxical situations with other entirely different situations and then deal with those, while rejecting the original situations merely because your way of trying to model them doesn’t work out neatly.
Maybe I should re-emphasise the caveat at the top of the post: “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. I’m also aware that simply noting that a formalisation provides a satisfactory solution doesn’t philosophically justify its use, but this is also not the focus of this post.”
You wrote that I “made the entirely arbitrary choice that you are only prepared to understand it in terms of surreal numbers”. This choice isn’t arbitrary. I’ve given some hints as to why I am taking this approach, but a full justification won’t occur until future posts.
So far as anyone knows, no actual processes in the actual world are accurately described by surreal numbers. If not, then I suggest the same goes for the “nearest possible worlds” in which, say, it is possible for Mr Trump to be faced with the sort of situation described under the heading “Trumped”. But you can have, in a universe very much like ours, an endless succession of events of order-type ω. If the surreal numbers are not well suited to describing such situations, so much the worse for the surreal numbers.
And when you say “I don’t think the ordinary notion of sequence makes sense”, what it looks like to me is that you have looked at the ordinary notion of sequence, made the entirely arbitrary choice that you are only prepared to understand it in terms of surreal numbers, and indeed not only that but made the further arbitrary choice that you are only prepared to understand it if there turns out to be a uniquely defined surreal number that is the length of such a sequence, observed that there is not such a surreal number, and then said not “Oh, whoops, looks like I took a wrong turn in trying to model this situation” but “Bah, the thing I’m trying to model doesn’t fit my preconceptions of what the model should look like, therefore the thing is wrong”. You can’t do that! Models exist to serve the things they model, not the other way around.
It’s as if I’d just learned about the ordinals, decided that all infinite things needed to be described in terms of the ordinals, was asked something about a countably infinite set, observed that such a set is the same size as ω but also the same size as 1+ω and ω2, and said “I don’t think the notion of countably infinite set makes sense”. It makes perfectly good sense, I just (hypothetically) picked a bad way to think about it: ordinals are not the right tool for measuring the size of a (not-necessarily-well-ordered) set. And likewise, surreal numbers are not the right tool for measuring the length of a sequence.
Don’t get me wrong; I love the surreal numbers, as an object of mathematical study. The theory is gorgeous. But you can’t claim that the surreal numbers let you resolve all these paradoxes, when what they actually allow you to do is to replace the paradoxical situations with other entirely different situations and then deal with those, while rejecting the original situations merely because your way of trying to model them doesn’t work out neatly.
Maybe I should re-emphasise the caveat at the top of the post: “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. I’m also aware that simply noting that a formalisation provides a satisfactory solution doesn’t philosophically justify its use, but this is also not the focus of this post.”
You wrote that I “made the entirely arbitrary choice that you are only prepared to understand it in terms of surreal numbers”. This choice isn’t arbitrary. I’ve given some hints as to why I am taking this approach, but a full justification won’t occur until future posts.
OK! I’ll look forward to those future posts.
(I’m a big surreal number fan, despite the skeptical tone of my comments here, and I will be extremely interested to see what you’re proposing.)