My attitude in these blogs has often been flavored with a soupçon of flippancy, occasionally falling well short of conventional adult seriousness that let’s one know where one is, what to expect, and signifies there is a shared notion of what it is to be a grownup. Starting right now I shall attempt to mend my ways for an extended period of time during which I shall exude gravitas. … There. Done. I hope that suffices.
Conway in the 24-dimensional celestial sphere
John Horton Conway died recently, sadly succumbing to the fucking covid-19 virus that hovers like the cloud belched forth from Mount Doom at Sauron’s behest over Mordor and surrounding counties. I don’t do heroes, but Conway comes very close.
A couple decades or more ago, having already spent a couple of prior decades demonstrating that the very resonant (Hurwitz) division algebras (C, H, O, with resonant dimensions 2,4,8) provide a natural architecture for the Standard Model of elementary particles, I happened upon Conway and Sloane’s book, Sphere Packings, Lattices and Groups. In this I discovered that in lattice theory the resonant dimensions are 2, 8, and 24 (1×2, 2×4, 3×8), associated with the lattices A₂, E₈, and Λ₂₄. That last one is the Leech lattice. It is remarkable, and there are a few people in the world who understand why, to some extent. I am not one of them. I’m like a fan of a sports team, not willing (able?) to put in the years of work needed to become a proficient player, but still able to write about the sport, which I did most recently here.
I never met Conway, nor ever communicated with him, but my readings in Sphere Packings, and elsewhere, convinced me that the fellow was beyond brilliant. In an online obituary I encountered this:
“During what Dr. Conway called his ‘annus mirabilis,’ roughly 1969 to 1970, he discovered what’s known as the Conway group, an entity in the realm of mathematical symmetry that inhabits 24-dimensional space. He discovered a new type of number, ‘surreal numbers.’ And he invented the cellular automaton Game of Life, which is among the most beautiful mathematical models of computation. He described it as a ‘no-player never-ending’ game.
“’In mathematics and physics there are two kinds of geniuses,’ Dr. Kochen said by phone from his home in Princeton, echoing something once said about the physicist Richard Feynman. ‘There are the ordinary geniuses — they are just like you and me but they are better at it; if we’d worked hard enough, maybe we could get some of the same results.
“‘But then there are the magical geniuses,’ he added. ‘Richard Feynman was a magical genius. And the same always struck me about John — he was a magical mathematician. He was a magical genius rather than an ordinary genius.’”
By the way, Conway wrote another book (with Derek A. Smith) – On quaternions and octonions: Their geometry, arithmetic, and symmetry, and quite a few years ago, at the request of an editor of The Mathematical Intelligencer, I wrote a review of that book, which appeared in that journal. Something odd happened as a consequence – a glitch in The Matrix – and if you do a search for that title you will sometimes see my name listed with Conway and Smith’s in a way that gives the impression I was a coauthor. I was not. In life Conway inhabited a more rarefied plane of existence than do I, and now – sadly – it is even more rarefied.
And speaking of being humble: WPP
Stephen Wolfram, polymath, genius, and a model of self effacement we would all benefit from emulating, has initiated “a project to find the fundamental theory of physics”: The Wolfram Physics Project.
Let’s look into his bona fides, and things he has let go to his head:
> He published his first physics paper when he was 15. I misremembered that number; I thought he had said it was 2, but upon rereading his backstory I discovered that that was just a general impression he was generating. The real number is 15, which is still impressive.
> He got seriously involved with physics in the 1970s. He evidently frequently rubbed elbows with Feynman, an association that he definitely let go to his head, and one that I suspect prevented him from thinking more out of the box at that time. “Not that I was trying to find a fundamental theory of physics back then. Like essentially all physicists, I spent my time on the hard work of figuring out the consequences of the theories we already had.” So, that phrase – “essentially all” – is, methinks, intentionally dismissive of any theorists – like Gürsey and Günaydin at Yale – who were not “figuring out the consequences of the theories we already had”, but were involved in a search for the mathematical roots of “theories we already had”. (Their work inspired mine, and although they surrendered their efforts when faced with mainstream ridicule, I participated in keeping the flame alive in the wilderness, and although it took decades, that flame is now pretty much self sustaining.) Anyway, it’s just a guess, but the problem with being young, bright, AND a friend with the likes of Feynman, is that such friendships dampen the potential of youth and brilliance, leading one to avoid any thoughts or actions that could jeopardize the friendships and diminish the perceived respect those theory gods might have for your malleable young mind.
> The siren call of computers became un-ignorable at some point, and SW succumbed to the allure of its dulcet tones. He founded Wolfram Research, and used the Wolfram Language (See? Humble.) programming language to create the kernel of Mathematica, a now nearly ubiquitous STEM computing research tool. But he never forgot physics.
> And now we have the WPP. Ideas gleaned from decades of thinking digitally have led to this inherently digital approach to solving some – or all – of the big mysteries of physics. According to wikipedia it was Abraham Maslow, writing in the book The Psychology of Science, who used the phrase: “I suppose it is tempting, if the only tool you have is a hammer, to treat everything as if it were a nail.” I suspect that the WPP is a classic example of this. That doesn’t mean it’s wrong, or misconceived, but … well, you know, if it doesn’t involved the C, H, O trio at some point in its architecture, then even if the WPP succeeds in explaining everything, I won’t care, and I’ll lose all respect for Mother Nature. So, that’s my opinion; it’s not humble. I don’t do humble opinions. And just to be clear, the WPP has to do more than succeed; it has to explain why its success – and the particular path to that end – was inevitable.
> SW also says: “But what about other approaches to finding a fundamental theory of physics? Realistically I think the landscape has been quite barren of late. There’s a steady stream of people outside the physics community making proposals. But most of them are deeply handicapped by not connecting to quantum field theory and general relativity. Yes, these are mathematically sophisticated theories that pretty much take a physics PhD’s worth of study to understand. But they’re [QFT and GR] the best operational summaries we have right now of what’s known in physics, and if one doesn’t connect to them, one’s basically throwing away everything that was achieved in 20th-century physics.” It may be just my own ego speaking, but this blanket dismissal of things that do not include QFT and GR might include my own work. Keep in mind, I’d be happy to have another lifetime with which to shoehorn those notions into my work, and I’ve never dismissed their utility, but … I’m reminded of the excitement I felt years ago when I saw I was cited in a Roger Penrose tome, only to discover he felt my work, by involving division algebras beyond C, was thereby rendered unworthy. But maybe Wolfram is unaware of my dabblings. Is that better? Maybe I’m just being sensitive. Sniff.
Anyway, Stephen, good luck with that. It’d be nice were a ToE developed before I die, even if my efforts are beside the point. But please hurry.
More ABC conjecture humility
Although Peter Woit’s latest blog has taken a small step away from the ABC conjecture proof controversy, the debate that played out in his ABC blog comments – by some of the preeminent mathematicians in the field – was fascinating. I can’t pretend to understand much more than a small fraction …
“What my post above attempts to show, is: if passing to poly-isomorphism has the effect of doing no gluing/no identification of ring structures (arising from π₁, just the gluing from the actual log map), then the only gluing left is the actual log map, which gives one global chart, and no transition functions needed, essentially(?) since just one chart. I before never really seriously considered that full poly-isomorphism could have the effect of ‘no gluing arising from this part’ (instead of ‘choose your favourite gluing’), but a similar thing is (I think) asserted for the case of the theta-link in …”
… Holy crap. Anyway, Peter W is adamant that the purity of mathematics would be sullied by the publication of a “proof” of the conjecture that is viewed by many as flawed. (My comment that physicists publish flawed work all the time did not go over well, and it was dismissed.) I don’t disagree with PW, but I do wonder if this >500 page proof – the semantics of which is not understood by its critics, according to its author – has entered a zone of near unknowable-ness. I suggest we create an AI that can not only look into this vexing problem far more quickly than can a human, but also far more dispassionately.
You know, I’m only half kidding. Actually I’m not at all kidding. We then should turn the AI’s attention to physics, and turn the attention of mathematicians and physicists – who’ll no longer be needed doing what they’d been doing – to other areas, like farming, and web design.