The Wikipedia ordinal arithmetic story
Oct. 6th, 2024 02:21 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
sniffnoySo here's something unusual that happened recently. I got an email from some guy who wanted to ask me about something that had been added to the ordinal arithmetic Wikipedia page. He says, hey, someone added to the article this alternate recursion for natural multiplication, but didn't include a citation for it... you seem to be a person who knows about ordinal arithmetic; is this alternate definition correct? If so, do you know a citation for it?
I took a look, and, huh, this alternate definition was one I'd never seen before! So, if it was correct, I certainly didn't know a cite for it!
But, was it correct? The claim was that α⊗β was the supremum of all (α'⊗β)⊕β for α'<α together with all (α⊗β')⊕α for β'<β.
This recursion actually works for (α,β)<(ω2,ω2), but it isn't correct; at (ω2,ω2), it fails. The natural product of ω2 and ω2 is of course ω²4, but applying this recursion instead yields ω²3. So I wrote back and said all this and the guy removed it from the article (I didn't feel like getting involved myself).
But, huh! What an odd thing to happen. I wonder who added it -- OK, I guess I can go check that myself if I really care. But I wonder where it came from? Was it just a bit of incorrect original research? Like this person came up with what they thought was an alternate definition, just stuck it on Wikipedia like you're not supposed to do, and then it turned out not to be correct? Probably! But it would have been quite interesting had it actually turned out to be copied from somewhere...
There's a few other notable things here. First off, I'm flattered that someone thought I was the person to ask about ordinal arithmetic! I mean, I can't disagree that I'm a good person to ask, I've spent quite a bit of time examining ordinal arithmetic, not just the ordinary operations and the natural operations and some intermediate operations (that's the paper that caught his attention), but also weirder ones... have I never discussed [the operation that, having invented, I have named] Nim-Jacobsthal multiplication here?? Huh, I'll have to write about sometime. (No idea when I'd ever actually write that up for publication... it's somewhere near the bottom of the pile...) Still, I'm not who I would expect someone to go to first for such a thing as the semi-outsider that I am, even if I am a good choice!
The other thing is that I didn't notice this change myself. This page is actually on my watchlist, you see; the thing is that I haven't actually regularly checked my Wikipedia watchlist in years. And, wow, it looks like I haven't written about this here either.
Basically, some years ago, I got into a stupid edit war over Allan Lichtman's "Keys to the White House". I say it was a "stupid" edit war because it wasn't over the validity of the system or anything (which is obviously terrible!) -- it was purely about phrasing. At some point I just got too worked up about it and couldn't continue further, and, well, it led to me just not really checking my watchlist anymore.
Recently, though, I went and finally removed that page from my watchlist (after reading an article about how the "The Keys to the White House" isn't just bad, it's actually dishonest -- Lichtman goes back and changes things after the fact so that he appears to be always right) so I can start checking it again without encountering that. So far, well, I haven't actually done that. But maybe I will!
(And maybe I'll write sometime here about Nim-Jacobsthal multiplication, but if I do, you'll see why it's near the bottom of the pile publication-wise. :P But I guess that means a blog post is more suited to it...)
I took a look, and, huh, this alternate definition was one I'd never seen before! So, if it was correct, I certainly didn't know a cite for it!
But, was it correct? The claim was that α⊗β was the supremum of all (α'⊗β)⊕β for α'<α together with all (α⊗β')⊕α for β'<β.
This recursion actually works for (α,β)<(ω2,ω2), but it isn't correct; at (ω2,ω2), it fails. The natural product of ω2 and ω2 is of course ω²4, but applying this recursion instead yields ω²3. So I wrote back and said all this and the guy removed it from the article (I didn't feel like getting involved myself).
But, huh! What an odd thing to happen. I wonder who added it -- OK, I guess I can go check that myself if I really care. But I wonder where it came from? Was it just a bit of incorrect original research? Like this person came up with what they thought was an alternate definition, just stuck it on Wikipedia like you're not supposed to do, and then it turned out not to be correct? Probably! But it would have been quite interesting had it actually turned out to be copied from somewhere...
There's a few other notable things here. First off, I'm flattered that someone thought I was the person to ask about ordinal arithmetic! I mean, I can't disagree that I'm a good person to ask, I've spent quite a bit of time examining ordinal arithmetic, not just the ordinary operations and the natural operations and some intermediate operations (that's the paper that caught his attention), but also weirder ones... have I never discussed [the operation that, having invented, I have named] Nim-Jacobsthal multiplication here?? Huh, I'll have to write about sometime. (No idea when I'd ever actually write that up for publication... it's somewhere near the bottom of the pile...) Still, I'm not who I would expect someone to go to first for such a thing as the semi-outsider that I am, even if I am a good choice!
The other thing is that I didn't notice this change myself. This page is actually on my watchlist, you see; the thing is that I haven't actually regularly checked my Wikipedia watchlist in years. And, wow, it looks like I haven't written about this here either.
Basically, some years ago, I got into a stupid edit war over Allan Lichtman's "Keys to the White House". I say it was a "stupid" edit war because it wasn't over the validity of the system or anything (which is obviously terrible!) -- it was purely about phrasing. At some point I just got too worked up about it and couldn't continue further, and, well, it led to me just not really checking my watchlist anymore.
Recently, though, I went and finally removed that page from my watchlist (after reading an article about how the "The Keys to the White House" isn't just bad, it's actually dishonest -- Lichtman goes back and changes things after the fact so that he appears to be always right) so I can start checking it again without encountering that. So far, well, I haven't actually done that. But maybe I will!
(And maybe I'll write sometime here about Nim-Jacobsthal multiplication, but if I do, you'll see why it's near the bottom of the pile publication-wise. :P But I guess that means a blog post is more suited to it...)
