gjm

Mathematician. Reader. Fiftysomething. Cruciverbalist. Not very good at thinking of witty things to put in short bios.

I wasn't on Twitter before that nice Mr Musk suggested that everyone go to Mastodon, and I don't expect to be very active here, but who knows?

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Last Notes

2026-04-17 10:29:32 UTC

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@nprofile…tkx6 On the other hand, the image itself is also largely unnecessary given the text it's accompanying; all it does is to give a bit of evidence that the statement in the text is true and to show, for those able to see it, what it actually looks like.
Actually-useful alt-text for this image would pretty much just duplicate the information that's already there in the original text.
(I think there is such a thing as an image that can't usefully be alt-texted because its _only_ purpose is to provide a visual impression of something that's already been explained textually as well as the person writing the text knows how to do. Such images should in fact still have alt-text, to let visually impaired readers know what sort of image it is. "Image of a female character from The Elder Scrolls, from near ground level, showing the smiley face described in the text" or something.)
I suspect that the LLM-based auto-alt-text generators could in fact do this sort of thing pretty well if given the accompanying text as well as the image, and a suitable prompt.
[EDITED to add:] I had a look at altbot, which I think is the thing that does this. Its prompt specifically tells the LLM just to describe what it sees without interpreting -- which may well be the _correct_ thing to do when there is actual information in the image not already present in the text, even though in cases like this where there really isn't the result is something that wears its uselessness on its sleeve.
(Actually, _is_ it completely useless? If I had read the text and couldn't see the image, it feels like this auto-alt-text might tell me about as much about the image as could usefully be conveyed in text.)

2026-03-10 13:38:06 UTC

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@nprofile…tkx6 I wonder how much alcohol bought in shops is for _solo_ drinking as opposed to _drinking with one's family_ and how Anon Opin feels about the latter relative to "social drinking" and "private drinking", which it seems kinda intermediate between.
I also wonder whether social drinking is a social good, or whether _socializing_ is a social good and drinking is just a thing that tends to accompany it. If alcohol in restaurants were cheaper, people would probably eat out about the same amount, although they might go to marginally fancier restaurants or something. Cheaper alcohol in pubs probably _would_ mean more people going for a drink in a pub, but would it mean more socializing or just make some people switch from other social activities?
(It could well be that social drinking as such _is_ beneficial -- being a bit more relaxed and disinhibited probably helps with making and building friendships, and maybe it helps more on balance than having slightly more drunken arguments and fights hurts.)

2026-01-19 16:08:11 UTC

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@nprofile…g0zc You can use +n or -n and have it still work, no? Just offset the value of n by 1.

2026-01-19 16:03:49 UTC

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@nprofile…g0zc That's very clever!
(It's Euler's 𝑛²+𝑛+41 thing in disguise. And of course it just happens to break down immediately after that 01601 block.)

2026-01-14 12:34:10 UTC

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@nprofile…g0zc Huh, how come the car's red lights are in the shape of the word OBEY?

2025-12-18 18:41:11 UTC

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@nprofile…l4xy Schramm was a damn good mathematician. It's very sad that he died so young. (I guess it would be very sad that he died so young even if he'd been a rotten mathematician, but you know what I mean.)

2025-10-09 14:47:54 UTC

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@nprofile…yu7x By "fairly small" I meant on the order of 10^-10, which is the sort of thing I would expect to be fairly easily achievable in multiple ways by things looking like linear combinations of spherical shells.

2025-10-09 12:49:30 UTC

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@nprofile…yu7x Note that the gravitational field of _any_ spherically-symmetrical thing is the same (up to a scaling of the total mass) _everywhere_ outside the sphere containing it. So it doesn't particularly matter whether the sample points are all on a single sphere.
Of course, that's what's true for an _actually literally spherically symmetrical_ mass density, and what you've got here is just an approximation to that. I don't think I'd have predicted ahead of time that the determinant would be so very small ... though maybe what's going on isn't that there are _incredibly small eigenvectors_ but that there are _quite a lot of fairly small eigenvectors_.
Further conjecture: if you move your sample points further out, the determinant will get smaller; if you move them further in, it will get larger. (On the grounds that I expect the spherically-symmetrical approximation to get better at larger distances.)

2025-10-09 12:22:50 UTC

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@nprofile…yu7x Have you looked to see what the almost-nullspace looks like? That is, what not-very-short vectors are mapped almost-to-zero? Or, to put it differently: your empirical finding shows that some distributions of mass have almost the same effect on your 83 red dots as others; what are they?
My best guess, which I don't trust much: your 83 red dots are all at a single distance, so any distribution of mass that looks "enough" like a spherical shell of any size smaller than that will look (up to scaling the masses) exactly like any other, and perhaps your nearly-nullspace vectors look like linear combinations of spherical shells.

2025-08-22 12:48:25 UTC

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@nprofile…yu7x Some years ago, in the UK there was an election which the Liberal Democrat candidate was expected to win, with the Conservative candidate a close second. So the Conservatives got someone to run as a "Literal Democrat", he picked up a bunch of votes from people who didn't read carefully, and the Conservatives won. (The election regulations got changed after that so we didn't get the otherwise-inevitable deluge of "Conversatives" and "Literal Democrats" fighting to see who could confuse more voters.)
Are these people hoping to attract parents who think they're getting a Montessori school? It seems kinda unlikely, so I guess they either don't know or don't care about the similarity in name. Neither is very encouraging.

2025-08-06 14:15:16 UTC

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@nprofile…yu7x By a curious coincidence, RFK Jr. _is_ an evil troll whose actions are likely to make some children sneeze non-stop until they drop dead, or at least something in the same conceptual vicinity.

2025-08-02 18:38:45 UTC

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@nprofile…fa4k @nprofile…95xs I hope they made it not too difficult for people to leave early. I'm going to guess you weren't the only ones.

2025-08-02 12:26:50 UTC

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@nprofile…fa4k Good luck! (I like Tavener but I don't think I would cope well with 8 hours of one Tavener piece at a stretch.)

2025-08-02 10:30:57 UTC

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@nprofile…fa4k Pedantic note: I think you mean not "The Protecting Veil", a famous Tavener piece but only ~45 minutes long, but "The Veil of the Temple", a newer piece by the same composer that is, indeed, 8 hours long.

2025-07-22 00:04:07 UTC

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@nprofile…fa4k @nprofile…yu7x This is a slightly atypical case because usually when something's named after X it's because someone else discovered the thing first and then X wrote about it later[1], whereas in this case IIUC Moessner really did get there first but didn't _prove_ it.
[1] Though my actual favourite example is Pell's equation, which so far as I can tell Pell never worked on _at all_.

2025-07-06 16:30:17 UTC

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@nprofile…cas5 I certainly hope it would believe it was created by an intelligent being different from itself.

2025-06-16 01:04:39 UTC

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@nprofile…saev I have seen "directly congruent" (and, likewise, "directly similar"). It's fairly easy to guess what those mean if you don't happen to know.
I really don't like "conguent", which will leave readers who know the distinction you're making unsure whether you're making it or have just made a typo, and readers who _don't_ just thinking you made a typo.
(I'm not a big fan of "rngs" and "rigs" and the like, either.)

2025-06-15 13:21:05 UTC

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@nprofile…fa4k @nprofile…mqrg @nprofile…6cfc @nprofile…hszg @nprofile…mlph That paper also contains the line "the d-ism of Leibniz overtook the dotage of Newton" which is rather magnificent. Knuth puts it in quotes but doesn't say who it's from; it seems to be Charles Babbage, though his wording was a little different.

2025-06-12 20:57:06 UTC

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@nprofile…saev @nprofile…e8p2 @nprofile…6al5 Every day is @pigworker appreciation day.

2025-06-11 20:02:13 UTC

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@nprofile…hr6z Aperiodic _tilings_ but not aperiodic _tiles_. That is, their tiles can tile the plane periodically as well as aperiodically.
My understanding is that what's claimed to be new is something about the particular aperiodic patterns these tiles can tile the plane in.

2025-05-29 13:37:21 UTC

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@nprofile…hr6z Looks like you're writing it in COBOL.
TOKEN SECTION.
CALL 'INITIALIZE-PARSER'.
ADD ONE TO CHARACTER-COUNT.
...
END OF TOKEN SECTION.

2025-05-14 21:07:22 UTC

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@nprofile…saev Further, a comment on the MSE answer (whose correctness I have made no attempt to check) says that the 48-scalar-multiplication 4x4 matmul described there _assumes the matrix entries are in a commutative ring_ which means you can't use it recursively to get a better exponent than Strassen, which (assuming you can for the Deep Mind one, which I also haven't checked) seems like a very important distinction.

2025-04-30 20:36:08 UTC

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@nprofile…59k6 Well, that's why I gave groups as an example -- the notion doesn't really need any prior knowledge, and you ought to be able to get sufficient motivation from the idea by thinking about geometry, which the ancient Greeks were very keen on.
(Though historically group theory came out of thinking about roots of polynomials, not so much an ancient Greek thing although there's a certain amount in common between "constructible with straightedge and compass" and "solvable by radicals".)

2025-04-30 15:16:48 UTC

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@nprofile…59k6 There are similar things in pure mathematics, which by definition isn't technology-limited (although some things in pure mathematics are _motivated_ by empirically-influenced scientific advances, and some are _impractical_ without high-speed automated calculating devices). For instance, there's no particular reason why the ancient Greeks shouldn't have discovered/invented group theory, but they didn't.
Obviously "group theory" and "role-playing games" are quite different kinds of thing! But they _are_ both examples of good ideas that could have arisen much earlier than they did.

2025-02-27 21:08:58 UTC

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@nprofile…59k6 How does that work out, geometrically?
The usual presentation of Borromean rings has them all roughly coplanar, but they need to wiggle a bit to avoid intersecting one another. That doesn't seem like it would work, especially as they'd all be way off centre from the central star.
There are nice symmetrical ways to put them well apart from one another in 3D (variations on the theme of "three mutually perpendicular rectangles whose corners are the vertices of a regular icosahedron") but then the rings have to be some way from circular to avoid intersecting one another. That also doesn't seem like it would work.
The above doesn't mean I think it wouldn't work. I don't understand the idea well enough to have any idea whether it would work. I bet you do. What's the plan? (And, when can we start building?)

2025-02-22 17:55:23 UTC

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@nprofile…cas5 @nprofile…fqqu @nprofile…q8c7 Correct! Ordinary primes of the form 4n-1 are Gaussian primes; ordinary primes of the form 4n+1 split into two different Gaussian primes (e.g., 5=(2+i)(2-i)); and 2 is (a unit times) the square of a prime: (1+i)^2 = 2i.

2025-02-22 17:20:45 UTC

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@nprofile…cas5 @nprofile…fqqu @nprofile…q8c7 Not quite! Gaussian integers _do_ have unique factorization. But other similar "rings of algebraic integers" often don't. Kummer thought he had a proof of FLT but was wrong because he implicitly assumed unique factorization.
(But a lot of correct and important mathematics came out of Kummer's work on FLT, even though he didn't have a correct proof.)
If you look at "rings of integers in imaginary quadratic number fields", which almost but not quite means looking at things of the form a + b sqrt(-d) with a,b varying integers and d a fixed integer, it turns out that there are only a few that have unique factorization. The "last" of them has d=163, and this is intimately connected (in ways I don't understand well, but plenty of more expert people do) to the fact that exp(pi sqrt(163)) is startlingly close to an integer.

2025-02-22 15:22:55 UTC

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@nprofile…cas5 @nprofile…fqqu @nprofile…q8c7 Yeah, but e.g. n^2 is always at least as good as n, so you can still get unboundedly many representations of a square as a sum of two squares.

2025-02-22 13:50:16 UTC

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@nprofile…cas5 @nprofile…fqqu @nprofile…q8c7 This is because 5 factors (as an ordinary integer) as 5 . 13 . 17, and because all those prime factors are 1 mod 4 each of those factors further as a Gaussian integer: (2+i)(2-i).(2+3i)(2-3i).(4+i)(4-i), and now you can take any of the 8 ways of picking one factor from each of those pairs -- e.g., (2+i)(2-3i)(4+i) -- and multiply that out (in this case getting 32-9i, so 32^2+9^2=1105). That gives you 4 rather than 8 genuinely different cases because if you take all the conjugates, (2-i)(2+3i)(4-i), you just get the conjugate of the original answer, 32+9i.

2025-02-22 00:24:48 UTC

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@nprofile…cas5 @nprofile…fqqu @nprofile…q8c7 There are plenty of solutions to just 𝑎²+𝑏²=…=𝑀² -- the tricky thing is getting those extra two conditions to hold as well.
(Solving the equations without the extra two conditions basically comes down to finding numbers with "enough" factors in the so-called Gaussian integers, things of the form a + ib where i^2 = -1. Any time you have (𝑎+𝑖𝑏)(𝑎−𝑖𝑏)=𝑀² you also have 𝑎²+𝑏²=𝑀² . It's possible to get many many such factors, i.e., for that circle to have lots of rational points on it. But it's more difficult when we have the extra constraints relating those different (a,b) pairs.)

2025-02-21 17:50:52 UTC

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@nprofile…hujc "Enough incorrect guesses along the way" appears to mean "literally any suboptimal keystrokes"; e.g., if you mistype one character then you get a not-perfect score and therefore not the top possible rank.
It's a fun game.

2025-02-08 13:50:57 UTC

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@nprofile…59k6 @nprofile…yu7x I think the characteristic of Gödel that's relevant to the story is less "genius mathematical logician" and more "person inclined to treat things as puzzles in mathematical logic".

2024-11-23 14:58:16 UTC

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@nprofile…hujc I dunno; looks like a mixed bag. Because the _other_ outcome is that some writers will decide to write a novel that features indigenous people, will remember this case, and will do more research than Griffiths did, and will write novels that represent those indigenous people less inaccurately.
So we'll probably get fewer books with inaccurately represented indigenous people, and (by a smaller amount) more books with more-accurately represented indigenous people.
It's not at all obvious to me exactly how that tradeoff will go, or whether it will actually be better on balance for indigenous people or indigenous peoples. I think it _could_ be better.
Analogy: Imagine a world where Christianity never became the big deal it did in our world and Christians are a small minority (but their opinions, religious practices, etc., are much as they are in our world). Someone publishes a book in which fair treatment of Christians is a major theme and an important character is Christian ... which shows up in the fact that he worships a prophet called Muhammad, meets with his fellow-believers for a ritual where one of them spills their own blood and the others drink it, and pledges never ever ever to have sex with another man. Do you think many Christians would be glad of such "representation"?