May I have this Resonance

Ad Nauseam

Subjective resonance is that fuzzy feeling you get when an idea clicks. That is, prior to ever encountering said idea, you are psychologically inclined to react positively to it, maybe because you are on the verge of developing its concepts yourself, or possibly because it meshes well with ideas of which you’ve already formed very positive opinions. It resonates.

Subjective resonance requires a mind capable of making judgements.

Objective resonance is why the Tacoma Narrows Bridge collapsed. Yes, it was designed (poorly) and built by humans, but once built, even if all the humans magically disappeared shortly thereafter, it was doomed eventually to experience a catastrophic resonant response to wind energy, and collapse.

Again, many experiments designed to detect some phenomenon do so by measuring quantity Y as some other quantity X is varied, the hope being that around some value of X the value of Y will spike above the background, indicating that Y resonates there – the meaning of this depending upon the context. This resonant reaction of Y at said value of X exists even in the absence of whatever theory is being tested, and even in the absence of the experimental apparatuses designed and built by the helpful experimentalists.

In mathematics there are also resonances, and these too exist with or without sentient beings to discover them. The Euclidean dimensions 1, 2, 4 and 8 are resonant (this is our X). In support of this contention I would yet again tediously point out the following gases that bubble up (the Y) from these dimensions:
• The parallelizable spheres …
• The Hurwitz division algebras, R, C, H and O
• All the classical Lie groups …
• Jordan algebras …
• Clifford algebras …
• Oh, hell, the list goes on and on …
Clearly I find subjectively resonant the notion that these dimensions are objectively resonant. Let’s move on.

These dimensions are associated with different resonances in dimensions 1, 2, 8 and 24, pertaining to lattice theory. Given its conceptual simplicity, lattice theory is extraordinarily complicated and difficult, resisting almost every effort to apply induction to make predictions about lattices in dimension n+1 from those in dimensions 1 through n.

This devilish complexity, however, markedly diminishes in dimensions 1, 2, 8 and 24, and specifically regarding the laminated lattices in those dimensions (which can be nicely represented by points in R¹, C¹, H², and O³). These lattices are, Λ₁ (equally spaced dots), Λ₂ (also denoted A₂), Λ₈ (also denoted E₈), and Λ₂₄ (the Leech lattice).

Nice lattices are more easily thought of as sphere packings, and even visualizable as such in dimensions 1, 2 and 3. A major lattice theory problem is finding the densest packing of spheres in any dimension. As it turns out, in dimensions 1, 2, 3, 8 and 24, we know that the densest sphere packings are associated with the laminated lattices in those dimensions (very little is known about all the other dimensions; for example, 49, and 343 … I could go on, but we don’t have time … ok, just one more: 117649). The dimension 3 is the outlier in the group, and the only reason anyone bothered proving this result (extremely difficult) is that we live in that dimension, and we have a vested interest in knowing the tightest way of packing oranges in a crate.

Of those other four dimensions, 8 and 24 are the most interesting, for they are relatively high dimensions, and knowledge of what goes on in neighboring dimensions is extremely limited, and likely to remain so. Proofs that the laminated lattices in dimensions 8 and 24 yield the tightest possible sphere packings can be found in The sphere packing problem in dimension 8, by Maryna S. Viazovska, and in The sphere packing problem in dimension 24, by Henry Cohn, et al.

Disclaimer 1: My own work in this area is much lower lying fruit than those two references, and my interest stems from my total conviction that our physical reality arises from resonant mathematics. Having demonstrated this in regard to dimensions 1, 2, 4 and 8, I spent some years dabbling with lattices in dimensions 1, 2, 8 and 24, finding a collection of relatively simple structures and results that had been missed in more highfalutin mathematics. I made a brave effort to connect this work to physics, but I’ll never know if I was on the right track.

Still, these resonant lattice dimensions crop up in physics from time to time (frequently as regards dimension 8), even in the mainstream. One example, and the reason I decided to write this blog, can be found in a recent article entitled, Sphere packing and quantum gravity.

Disclaimer 2: While I likely could, given the inclination to partake in months of study, reach a point where I came close to fully understanding Maryna S. Viazovska’s original 8-dimensional lattice proof, I have always found it difficult to go from interest, to inclination, to action. The initial hurdle is when I encounter modular forms, which are defined on the upper half of the complex plane. If any reason for this restriction has ever been proffered in any of the places I’ve encountered modular forms, I was too lazy to see it. If I dug deeper I’d figure it out. But I never have. Still, I gave a semi-educated skim of her entire paper, and was left with an overwhelming impression of mondo coolness.

However, I did not look over the paper extending Maryna’s proof to the Leech lattice in 24-dimensions (What would be the point? It’s clearly also mondo cool.), but I did look over the Sphere packing and quantum gravity paper, which looks at both the 8-d and 24-d lattice cases, and establishes “a precise relation between the modular bootstrap, used to constrain the spectrum of 2D CFTs, and the sphere packing problem in Euclidean geometry.” Keywords employed in that paper: “AdS-CFT Correspondence, Conformal Field Theory, Black Holes, Conformal and W Symmetry”.

Disclaimer 3: From early on in my so-called career as a theoretical physicist I avoided immersion in quantum field theory. I am aware that lots of attention has been paid to anti de Sitter (AdS) space and conformal field theory (CFT), not because it’s necessarily physically relevant (our universe is not anti de Sitter), but because it is more tractable than the de Sitter alternative, allowing the use of theoretical tools that people who study this kind of thing really like to use. In one of Peter W’s blogs the comments included much discussion of AdS/CFT, and I had to ask: “Am I correct in thinking that this surfeit of AdS/CFT talk is tantamount to looking for a nail in the dark under a lamppost because the light is better there, and anyway all you’ve got is a hammer, so the solution to your problem had better require a nail? Maybe two.” Peter replied: “Yes.”

So, anyway, the relevance of the Sphere packing and quantum gravity paper to reality is unimportant. What’s really cool is the remarkable correspondence between the modular bootstrap (don’t ask – it’s a thing used by a certain kind of theorist) and the sphere packing problem in 8 and 24 dimensions. It highlights something I said in Division Algebras, Lattices, Physics, Windmill Tilting: “… any theory, however bizarre, will find that things work best when the resonant dimensions, 1,2,4,8 and/or 24, are involved.” (Speaking of bizarre theories, the number of transverse dimensions in super string theory and bosonic string theory are 8 and 24, respectively. This is not unconnected to the quantum gravity paper cited above.)

So, that’s my take. You’ll find a nifty connection of the dimensions 8 and 24 to the division algebras and Clifford algebras in my last paper: Division Algebras, Clifford Algebras, Periodicity.


I think I may end my blogging here, as it can serve no further purpose (“further” … ha!). I’ve been spewing forth blather about mathematical resonance for decades now, but the notion has gained no traction. It’s stuck in the mud, a viscous, slippery stuff arising from a scientific culture that far more highly values the toolmaker Feynman than the architect Dirac.

Anyway, here’s a link to my presently favorite music video, the lyrics of which summarize my place in the world of theoretical physics. I want to be able to do that dance, but at my age I’d likely hurt myself.



I want to make a prediction, albeit one about which I harbor profound doubts that it will ever be verified by experiment or experience (it may simply never be recognized). Ok, focus (talking to myself here – I have a propensity to ramble … and there I go again).

So, right, here’s the deal. Let’s suppose some deity, or maybe the purported conscious universe communicating via a rock or burning bush or someat – or maybe just a super advanced bunch of aliens hovering in a spaceship above the earth, their scientific acumen and trustworthiness attested to by the fact that they’ve navigated to the earth from somewhere far far away … so, anyway, one of these things – or something else the opinions of which we should find it difficult to contest – it hands us on a silver platter a correct and complete TOE, blithely assuming we have the background to understand it, and are willing to give it a go.

So, right, here’s the deal – the real deal. I predict that by the vast majority of theoretical physicists this TOE would not be understood. Furthermore, while some few might have a notion of what it means, there is a 100% chance the TOE will contain concepts at odds with their cherished prejudices about how the universe should be described. The end result of all this generalized befuddlement and disaffection will be a tacit agreement to ignore this new TOE, to carry on tapping away at things that by mutual consent we label legitimate veins of truth.

Some might grab bits of the TOE from the silver platter on occasion and incorporate them into their own work, but without attribution, for the deity, or consciousness, or alien race, will have made the huge mistake of not publishing in a respected journal, and even worse, will not have the TOE listed in the arXiv, either because they didn’t bother trying, or because the arXiv gatekeepers saw their unusual and far too mathematical notions as unworthy and disruptive. (After all, the purported TOE doesn’t have a single Feynman diagram (all hail QFT!), so, crikey … there you are.)

Thanos was an idiot

So, let’s assume the TOE provider is an alien with the temperament of Thanos, and as punishment for our oblivious adhesion to orthodoxy decides to turn half of humanity to dust. The actual fictional Thanos did this to all life throughout the universe, and 5 years after having done so we saw an aerial view of a sports stadium – I think it was a baseball stadium in New York, so who cares … but anyway. The stadium looked unused, overgrown and rundown, for having lost half of humanity has caused … what did it cause?

1972. In the year 1972 the population of the earth was half what it is today. Was there no baseball in 1972? I seem to recall that there was baseball. I recall it. It was only 48 years ago. I was alive then. So, had the Avengers not mucked up his plans, Thanos’s big finger snap biocide would on the earth, within his lifetime, have been completely undone and the population back to where it is now. What an idiot. And anyway, he went off afterward to a seemingly uninhabited planet and became a contented farmer at peace with the universe. He could have done that without the whole finger snap incident. (And those 5 years later humanity was exceptionally mopey and depressed. Really? Think back to World War I and the 1918/19 flu epidemic that followed. Tens of millions died. Was all that horror followed by the Mopey 20s? Nay nay, it was followed by the Roaring 20s, a time of exuberance and invention. Anyway, just saying – human nature, right?)

In any case, if you’re handed a TOE on a silver platter by an alien – or even a deity – with the temperament of Thanos, act obsequious and grateful, and don’t go ignoring it until later, when the entity is well out of earshot.

All your hearts now seem so far from me
It hardly seems to matter now
And the nurse will tell you lies
Of a kingdom beyond the skies
But I am lost within this half-world
It hardly seems to matter now
Play me my song
Here it comes again
Play me my song
Here it comes again
Just a little bit
Just a little bit more time
Time left to live out my life
Play me my song
Here it comes again
Play me my song
Here it comes again

Discontented blather

Ethan Siegel, science guy for Forbes magazine, wrote an article entitled: No, The Universe Is Not Purely Mathematical In Nature. I couldn’t let this pass, so supplied the following comment everywhere I encountered the article, and comments were allowed:

“This by and large relates to the interface between reality and sentience – in this case, human. Delete all sentience from the universe and very little changes; the universe goes on as it had, despite there being nothing in the universe trying to understand. There are rules, and they are not based on magic. There are mathematical concepts that transcend any requirement of sentience to think them up. Prime numbers is a simple example. The universe isn’t going to quit working just because no sentient being is testing its validity, and mathematics underlies how it works, and even why it exists.”

I generally like Ethan’s posts, but he is mainstream, accustomed to thinking … well, here’s a quote from Sydney Coleman I encountered on LinkedIn:

“The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction.”

The poster agreed with this, as did many of the commenters. The point is, Ethan’s view of what constitutes physics, and what constitutes mathematics, … ah, I see I’m about to go down a very well trodden path. So, uh, yeah … Thanos. What an idiot.

Here’s a quote from a recent Scientific American blog about scientific advances – especially the big ones:

“In fact, it may be necessary for pioneers to face the headwind of rejection for a while, or their idea might eventually be credited to the mainstream. An innovator has to persevere through an initial denial phase, as Weiss did, during which the mainstream rejects the idea so publicly that the proposer can later wear the rejection as a badge of honor. Under more typical circumstances, when a new idea is simply ignored, there is a real danger that mainstream proponents will claim it for themselves after introducing some cosmetic variations to its presentation.”

Over the last 20 years I’ve seen ominous signs that history is trending in this direction regarding my own work in theoretical physics. Is this hubris? Pshaw! My ideas are fundamentally correct, so no. (My PhD advisor used to say I had a hotline to god, so there you are. I have the silver platter to prove it.) People aware of my work are occasionally inspired, seemingly, not to build on its intractable ideas, but inspired by its Gestalt – by the idea that there could be an algebraic foundation to a TOE hitherto undiscovered and unincorporated – so they construct their own. It becomes a hobby among many fringe theorists – like being a ham radio enthusiast, or a computer hacker. Whatever. Cosmetic variations indeed.

How does one prepare oneself psychologically for this looming chunk of ineluctability? What are we even talking about? Who says things like “looming chunk of ineluctability”, and what does this refer to? I’m leaving. My Victory Garden isn’t going to grow itself.

Louche Universe

Shared delusion

Ok, bear with me. Make what judgements you will – it won’t affect what I’m about to say – but I get quite a lot of my news from the Flipboard app on my iPad. Under the heading of “Physics” I encountered an article with the following title: WHY IS SCIENCE GROWING COMFORTABLE WITH PANPSYCHISM (“EVERYTHING IS CONSCIOUS”)? So, I have some thoughts and opinions on this idea. Assume it’s true, and that the universe is awash with consciousness. And I’m willing to take the notion a large step further. Let’s further assume that this ubiquitous consciousness can interact with matter; that it can be a causative agent in a desultory and not necessarily benevolent manner. This would go a long way to explaining a number of bizarre occurrences that have featured in my life, and the life of SWMBO, and our lives as a couple.

For example, we have observed over the years that if we really like something – often a foodstuff – our display of enthusiasm would sometimes be followed by that something disappearing from the shop shelves forever. This happened with those excellent arancini; and that super superior chocolate ice cream; and enough other things to make it statistically noticeable.

And then – although this happens less often nowadays – the statistically noticeable frequency with which streetlights would blink off when one, the other, or both of us would pass by.

And how about all those near death occurrences that have so marked my travels, as recorded here? What could all that mean, or portend? Or could it be the universe is simply louche.

Louche: “disreputable or sordid in a rakish or appealing way”. That sums up the conscious, causative universe that I have experienced. But there’s more. (Oh, goodie. I was just getting interested.)

More outré-ness

So, there I was, a youngish physics graduate student (how that came about … well, that story is in the cloud if you’re interested), and I encountered, and became enamored with, the quaternion algebra, H. It helped me get through my advanced exam when I was asked an EM question and told the assembled professors that in order to answer their question I’d have to write Maxwell’s equations in a way they’d never seen before. Eyebrows were raised, and I was told to pray proceed. This I did, and true to my word, they’d never seen anything like my way, because they didn’t look upon H as useful.

But not too long after that a collaborator of Feza Gürsey (Yale) came up and gave a talk at Harvard. I trekked into Cambridge and learned from this talk of yet another division algebra: the octonions, O. The louche universe had thrown these things in my way, like some sort of intellectual opium, and kept me addicted from that time to this – and, I predict, all future times in which I participate.

This led to the my constructing the spinor kernel, T = CHO, which I used to explain where the standard model comes from, and why the universe is made of matter, and where is all the concomitant antimatter. So, yay, cool, and oh, the universe failed to mention that none of that mattered, because it wasn’t relevant to the meaty work of “real” physicists: using QFT to get numbers that could be compared to experiment, leading to clumsy attempts at high-fives at numerous conferences, on the assumption that the QFT numbers matched the experiment numbers.

Soporific QFT

QFT, as commonly used in physics, is basically mathematical alchemy. As exploited in QED and QCD it has proven to be a stellar addition to any ostensibly legitimate theorist’s toolkit. But these successes warped physics for decades, as new hypothetical fields were thrown willy nilly into the mix and new predictions made (sometimes). This could happen because the architecture underlying theoretical physics was insufficiently rigid. After all, you might distill urine and end up discovering phosphorus. It could happen.

The last serious bit of real theoretical architecture was Dirac’s spinors and algebras. All that can be built from P = CH, and from this starting point you also get U(1)xSU(2) as an internal symmetry. Cool. Then – being inspired by Gürsey’s work – you wonder what happens if you expand the architectural underpinnings to T = CHO. Et voila, U(1)xSU(2)xSU(3), and all that other stuff.

But where’s the QFT? Well, it’s not there – yet. Oh? Well, come back when it is.

More Louche-ness

I understand that attitude. That’s not to say I approve. There is a hell of a lot of human behavior I understand but disapprove of, occasionally quite strenuously. But the universe had a new twist up its sleeve: it allowed the LHC to dash the hopes of the majority of alchemists, and many of them are drifting away from the merry TOE playground. Witten, for example (or so it is rumored – I don’t pay strict attention to these goings on, so, you know, grain of salt and all that), has moseyed over to black holes and qubits.

One commenter on a recent Peter W blog post wrote: “For many of us, it was pretty clear a long time ago that pursuing the search of a final TOE was a complete waste of time, for the simple reason that… it does not exist.”

So, the universe, in a perversely indolent manner, having led me down this … well (and yes, it’s all about me):

I shall be telling this with a sigh
Somewhere in ages and ages hence:
Two roads diverged in a wood, and I –
I took the one less traveled by,
and that has made all the difference.

You see, the mainstream is not just taking their ball and going home; they are taking the very concept of the game that required the ball. You can keep the ball, but there is no game that anymore requires it. Make a new game, but good luck finding players willing to learn a whole new set of rules. The old players are being shifted to more amenable fields … like croquet and crochet. And where does that leave me? Sitting on an ice rink with a tennis racket and a rugby ball. Alone. Sniff.

But the universe wasn’t done toying with me. To ensure the world would turn its attention elsewhere, it cancelled all conferences and made everyone stay at home and … well, I have little idea what others are doing, but it’s no longer partaking in self congratulatory confabs in places like Corsica or the Canadian Rockies.

On the other hand, it may be that the universe is not conscious, and all this is just stuff that happened. And that it is not aimed explicitly at me … as difficult as that is to believe.

Have horn; Will toot

Rambling context

The fear of losing it someday, since retiring I have poured forth into a kind of cloud – in the form of several books, and this blog – the contents of my mind. It’s why I’m so comfortable with the idea that none of my humorous memoirs will be read by others, nor my discovery of a significant piece of the mathematical architecture underlying the makeup of the universe will ever receive attention from the diverse community of theorists. It’ll all be out there, stored, enabling me to free up portions of mind for other purposes. No more need to struggle to keep all that detritus sensible and neat.

In my youth – as is the wont of youthful idealism – I saw an imperfect world and yearned to change it. I still yearn for it to change, as I yearn to be bitten by a radioactive spider and instead of dying a horrible death, have it endow me with hidden superpowers, a vastly extended lifespan, and somehow lead to a Bruce Wayne level of monetary comfort.

A sand castle held together by the moisture of the sea will eventually dry out, and succumb to wind and gravity. So it is with youthful dreams. Human herds will always drift into corruption. Our reason for existing will always be to produce another generation that is likely to produce another generation that is likely to … Well, thankfully there exists as well a yearning to transcend the muck – although admittedly its manifestation is often merely a means to improve our chances to breed – but not always. I hope not always.

Yeah, well, this is probably just the covid-19 cloud talking. A smattering of Weltschmerz. Wash your hands and mind after reading.

The world catching up, then claiming credit

So, let’s get just a wee bit technical. Penguins have been spreading rumors for several months now that ANITA has encountered events that could lead to “going beyond the Standard Model” (SM), a theoretical mantra stemming from the desire of brainy people to stay relevant and potentially cutting edge. So, you folks have anything more than that to corroborate your dreams?

Well, yes they do. At the Perimeter Institute (PI) some folks have decided to look into the consequences of assuming CPT symmetry applies to the whole universe – to all space and all times. Interesting stuff.

A possibly erroneous impression to give some context. The PI, in my limited experience, if allowed to, would ignore all good ideas from other people at other places, refurbish them, and claim them as PI originals, and attribution be damned. A blatant example if this occurred several years ago relating to my own work, and I did not take it well. Now, age having given me a slap in the face, reminding me that death is coming, and that eftsoons, I give fewer shits than previously. It’s just inevitable herd corruption. Still, this ANITA/CPT thing has appeared in the pop sci press several times in recent days. For a good example of this, I direct your attention here.

Here are some quotes, which I’ll discuss at bottom.

1. Our universe could be the mirror image of an antimatter universe extending backwards in time before the Big Bang. So claim physicists in Canada, who have devises a new cosmological model positing the existence of an “antiuniverse” which, paired to our own, preserves a fundamental rule of physics called CPT symmetry.

2. “There is this frame of mind that you explain a new phenomenon by inventing a new particle or field,” he [Neil Turok of PI] says. “I think that may turn out to be misguided.”

3. They asked themselves whether there is a natural way to extend the universe beyond the Big Bang – a singularity where general relativity breaks down – and then out the other side. “We found that there was,” he says. … The answer was to assume that the universe as a whole obeys CPT symmetry.

4. Instead, says Turok, the entity that respects the symmetry is a universe–antiuniverse pair. The antiuniverse would stretch back in time from the Big Bang, getting bigger as it does so, and would be dominated by antimatter as well as having its spatial properties inverted compared to those in our universe – a situation analogous to the creation of electron–positron pairs in a vacuum, says Turok.

5. Turok adds that quantum uncertainty means that universe and antiuniverse are not exact mirror images of one another – which sidesteps thorny problems such as free will.

6. Turok says that the new model provides a natural candidate for dark matter. This candidate is an ultra-elusive, very massive particle called a “sterile” neutrino hypothesized to account for the finite (very small) mass of more common left-handed neutrinos. According to Turok, CPT symmetry can be used to work out the abundance of right-handed neutrinos in our universe from first principles.

7. Turok describes that mass as “tantalizingly” similar to the one derived from a couple of anomalous radio signals spotted by the Antarctic Impulsive Transient Antenna (ANITA).

8. Turok, however, points out a fly in the ointment – which is that the CPT symmetric model requires these neutrinos to be completely stable. But he remains cautiously optimistic.

The promised horn tooting

Ok, let’s get real. The title of this blog is not “Every idea is beautiful in its own way”; it’s “3 spheres to rule them all”, a kind of LotR reference to the three parallelizable spheres, leading to the division algebras, C, H and O, and thence to the spinor kernel, T = CHO – a bit of mathematical architecture that gives a natural framework on which to hang the SM.

There are a couple of T-maths theoretical consequences that are relevant to the physicsworld quotes above. Let’s discuss.

1. 8 years ago I realized that my T-maths architecture had a very important consequence. There had to exist a mirror antiuniverse, and that the verse and the antiverse are linked by an extra 6 dimensions that carry SU(3) charges. So the PI people, starting from a very physicsy/analytical/computational place, have predicted a big part of my prediction … so, wait, is their idea then a prediction, or a corroboration? Well, it’s a prediction as long as we “ignore all good ideas from other people at other places”, so I guess it’ll remain a prediction then. (BTW, 7 years ago I presented my idea at a conference, and a year after that it was published, forcing the arxiv gatekeepers to list it, although with the dreaded gen-phys brand, a mark of shame and revilement. Goodness. So, anyway, you can find it here, and a longer version here.)

2. I like their thinking here. Most of the papers in the arxivs that get the much vaunted hep-th brand are of this sort. These papers remind me of a well-known quote (attributed to Einstein, but I suspect it’s older than that): “The definition of insanity is doing the same thing over and over again and expecting a different result.” That rather nicely summarizes the majority of the last 40 years of mainstream theoretical physics.

3. As a physicsy/analytical/computational idea this is rather nice. And it hadn’t been considered before, why?

[Note added, 2020.04.28] Well, in fact, a version of this had been considered before, John Gonsowski recently reminded me. I do not know how widely known Sakharov’s work is, but as it appeared over 50 years ago, deep in the Cold War, originating from the USSR, it was likely widely ignored in the west. Alas, I have but the dimmest memory of having encountered it.

4. And here they’ve caught up to me. Welcome aboard … wait, what? Why am I being forced to walk that plank. HEY! There are sharks down there! Sigh.

5. Well, that’s a relief. I’d hate to think this blog was simultaneously being written in the antiverse by the anti-me.

6. So, yeah, right-handed neutrinos are an essential part of my T-maths architecture, as are neutrino masses. Yeah yeah, walk the plank. I’m going. Just be patient. (I mean, what choice do I have? Physicsworld called their work a “new model”.)

7. Well, that’s cool.

8. No comment that wouldn’t require a new technical paper, and I’m done with that crap.

So, hey, bartender, I’d like a Weltschmerz martini, shaken, not stirred, with the merest touch of Lebensangst. 3 olives … to rule them all.

The art of being a humble adult

Brevis Gravitas

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.

Be careful, Alice

Follow the White Rabbit

Let’s assume you know as little of the intricacies of the abc conjecture proof controversy as I do. Actually, let’s not assume that; forget I said that. However, before I carry on, a caveat: I very likely have no idea what I’m going to be talking about, and likely would be well advised to refrain from offering an opinion. However, having no one but myself to provide said advice, and being personally disinclined to offer it …

So, I decided to go as far as I could and was able, or as far as my motivation to do so would take me, to understanding what the hubbub was all about. And to begin, here is one wiki-way of expressing the conjecture:

If a, b, and c are coprime positive integers such that a + b = c, it turns out that “usually” c < rad(abc). The abc conjecture deals with the exceptions. Specifically, it states that:

ABC conjecture. For every positive real number ε, there exist only finitely many triples (a, b, c) of coprime positive integers, with a + b = c, such that
c > rad(abc) (1 + ε)

Let’s parse this. So, coprime means a, b and c have no prime factors in common. For example,

a = 125 = 5³;
b = 91 = 7×13;
c = 216 = 2³×3³.

So, what is rad(abc), the “radical” of this integer? Well,

abc = 5³×7×13× 2³×3³.

To get rad(abc) we take that product of primes with exponents and replace all the exponents by 1. So,

rad(abc) = 5×7×13×2×3 = 2730.

Evidently, we are assured, “usually” c < rad(abc), and in this case this is true.

216 < 2730.

Keep in mind, a week ago I knew nothing about this conjecture, but I became a number theorist dilettante in my early teens, so you can trust me.

It should be fairly obvious that if c is prime, then certainly

c < rad(abc) = (something > 1)×c.

This suggests that if we want to find a counterexample to the “usually”, then maybe c should be very un-prime. For example,

a = 27;
b = 5;
c = 32.

In this case,

c = 32 > rad(abc) = 3×5×2 = 30.

[Note added 2020.04.12]
Regarding that positive real number ε:
What if ε = 0? Well, for all positive integers k, if
log2 3k is very close to an integer m,
then set a equal to the lesser of 3k and 2m, and c the greater. b is the difference.
In this case, if a and c are sufficiently close, then
c > rad(abc) = 6×rad(b).
In this way an infinite number of positive integer triples (I believe) can be obtained for which
c > rad(abc).
This means for all positive ε, if the conjecture is correct, there are infinitely many triples satisfying
rad(abc) < c < rad(abc) (1 + ε),
and finitely many outside that range. That is, satisfying
rad(abc) < rad(abc) (1 + ε) < c.
And that is indeed interesting.
[Note added 2020.04.16]
Of course, there is a simple example of that kind of thing. For every positive real ε, there are infinitely many fractions on the form 1/k between 0 and ε, but finitely many greater than or equal to ε, where k is a positive integer. Anyway, …

Drink the magic potion

If you are like me – or if you are me (at least one of us is) – then you are wondering, WTF? In particular, these are the thoughts bubbling in my witches cauldron of a brain:

A. Yes, like Fermat’s last theorem, this conjecture is fairly easy to write down and comprehend;

B. Personally, were I asked to prove this conjecture I wouldn’t know where to start, except finding a book in my library with the widest possible margins in which to do my work;

C. And, again, being asked to prove this conjecture my initial response would likely be something of the form, “I’m pretty busy at the moment, what with counting bits of bellybutton lint and staring vacantly at the ceiling and all, so no, find someone else.”

Not being a highly trained PhD number theorist, I’m probably missing something important, but at my dilettantish level I can’t conceive of any reason one would need to nail this conjecture down with a proof. As mentioned, I spent years playing with prime numbers, which resulted in my conviction that the gods and Mother Nature have set

ln(lcm(n)) =~ n-1,

(natural log of the least common multiple of the integers from 1 to some positive integer n is best fitted (approximated) by n-1). This leads to a really great continuous approximation to π(n), again receiving Mother Nature’s seal of approval. And how did I reach this stunning conclusion? By fussing about with primes and graphing things.

Consequently, I can’t convince myself that the abc conjecture originated with anything more profound than a similar kind of fussing with numbers. My thinking is, someone famous and highly respected suggested this, and it became a longstanding mathematics meme. Proving it became something mathematicians could busy themselves with, and agreement on that became justification enough. However, I wouldn’t take my word for any of this.

Turmoil in the rabbit hole

So why do I care at all? Because the popular STEM media have recently been unignorably full of the following controversy: A Japanese mathematician (Shinichi Mochizuki) claims to have a proof (more than 500 pages long); Other mathematicians, including one Fields Medal winner (Peter Scholze) have questioned the validity of the proof; Ordinarily resolving their doubts would be required before the proof would be eligible for publication in a respected journal; It was published anyway in Publications of the Research Institute for Mathematical Sciences (RIMS); Mochizuki is a lead – if not the head – editor of RIMS.



I mean, really, what are the odds someone is going to happen upon this 500 page publication, have the time to look it over, have the chops to understand a significant portion of it, and not be familiar with the controversy. Has this person been living in a cave? (I think I know this person. I must have a word with them.)

Rabbit hole asphyxiation

Unable to restrain myself, I decided to look into this matter and see how far I could get before … Well, it turns out, not very far. Maybe it’s just the kinds of things that I find interesting, but notions kept popping up in my reading that I’ve encountered before. Like the Langlands program, which wiki says “is a web of far-reaching and influential conjectures about connections between number theory and geometry”. My gut feeling is that the Langlands program is very cool stuff, but the communication between my gut and my higher brain functions is a little sparse, and likely to remain so.

Another concept that pops up in much of my casual STEM reading is elliptic curves, which also figured in Andrew Wiles’ proof of Fermat’s Last Theorem. I gather that elliptic curves are an integral part of the Langlands program. And what are they?

“In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form

y² = x³ + ax + b

which is non-singular; that is, the curve has no cusps or self-intersections.” And evidently this may employ a mathematical field different from the real numbers. (In particular, there is a finite field of order pk for every prime p, and every positive integer k (numbers of this form play an important part in my excellent conjecture:
ln(lcm(n)) =~ n-1).)

And this is where my eyes start to glass over, and I start thinking I should maybe go outside and play. I mean, that equation is just so damned specific. And although there are evidently some nifty things that come out of studying elliptic curves, …

Anyway, reading more deeply into the controversy – and evidently it remains a controversy because the proof is so dense, and involves many concepts invented by its author, so few experts in this field understand it – so, yeah, anyway, I eventually encounter references to Category Theory and Functors, at which point I fall into a deep coma from which I am only able to awaken upon hearing some soothing Ibiza music. As a graduate student of mathematics, decades ago, my reaction was similar.

Bad rabbit

Although I despair of ever being able to offer a cogent opinion on the proof – and indeed, were that even remotely possible my mathematical interests and proclivities would prohibit the attempt – I do have an opinion about the scandalous publication of the insufficiently refereed thing: I don’t care. All STEM communities take themselves way too seriously, and this small community is no exception (which I precluded by the use of the word “all”). I do not foresee an imminent collapse of the community’s social order arising out of this event. This isn’t covid-19, after all.

Belaboring the obvious

Annus Mirabilis

Is there a silver lining to pandemics? No, not really. Knowing my skinny old ass could be dead of this #%&@4$ disease in the next 6 months tends to color my thinking on the matter. Still, pop sci authors, working from home, have been quick to point out a potential bright side.

In 1665 Cambridge University sent its students home to continue their studies in an effort to protect them from the plague (according to Wikipedia the bubonic plague stemmed from China – quelle surprise – in 1331). Among the students fleeing Cambridge in 1665 was one Isaac Newton, then in his early 20s. Over the next year+, referred to as his annus mirabilis, he revolutionized our understanding of the universe. The key ingredients leading to this revolution were:

> Genius (well, duh);

> Isolation (no cheery faces poking in the door wondering if you want to join the crew for a trip to their favorite pub);

> Focus (same as above – I mean, what else are you going to do out in the boonies? Molest sheep?);

> Ripe times (an international atmosphere brimming with ideas, all waiting for the right brain with enough time to make sense of them);

> A la Feynman, a willingness to disregard the opinions of others (much easier to disregard the tyranny of other voices when they are stilled by distance; but you also need to carry an independent streak with you, for without it, intellectual pollutants clinging to your mind from your non-quarantined life will mar the brew);

> A population of thinkers and researchers in your field not overwhelmingly large, powerful and persuasive (directed cacophony is difficult to ignore, even if one is in seclusion).

Absent any of these ingredients and you run the risk of succumbing to the herd, and producing nothing remotely original. Newton’s annus mirabilis is the archetypal example of genius in isolation producing a paradigm shift, but there are many others. Hell, just an obstinate proclivity to yield to one’s own maverick instincts can lead to a variety of isolation from the herd. The herd has a kind of inbuilt antibody response to nonconformist thinking. This manifests as a cloud of nudge-nudge-wink-wink derogatory remarks aimed at the offending individual, rapidly followed by a circling of the wagons, with all eyes steadfastly focused only on ideas and people within the circle. Newton’s great advantage was being both isolated for a year, and having the kind of bona fides others in the field could not ignore; and the time was right. Just ask Herr Leibniz. (Keep in mind, pointing out self evident human foibles will not bring about change, for they are self evident in being part of human nature.)

Anus Mirabilis

Speaking of humanity and its imperfections, let’s pause for a second and discuss homicidal tyrants, with which our history is replete. (This is always fun.) Accounts of their atrocities will generally lead off with something like this: “Stalin killed millions”. (I just did a google search and found an article with that in its title.) Ok, pause for another second. Look at a fluffy cloud, if one is available, and cooly reflect. Did Stalin kill millions? No, of course not. No one kills millions. There isn’t time. They need to eat, poop, sleep, and in general brush up on their copies of “12 Easy Rules to be an Effective Tyrant”. No, the writer of the article put the blame on Stalin because the truth would come dangerously close to blaming the writer him or her self. See, the actual deaths were caused by minions, many of whom were just average Joes, malleable and compliant when confronted by Stalin, and the times that produced him. It was minions (people) who imprisoned Galileo for heresy, tortured thousands during the Inquisition (again with the heresy; a pattern is forming), and sent millions to be retrained in the name of Mao. Anyway, as Pogo summed up more than half century ago: “Yep, son. We have met the enemy, and he is us.” Tyrants are shepherds; people – the herd – they’re the actual freedom deniers and death dealers.

Newton’s FOMO

Ok, gosh. Humanity is imperfect. Scarcely an indigestible notion, or new, so let’s carry on.

Among the things listed above as – well, not really required for, but certainly aiding and promoting the possibility of some genius having an annus mirabilis while isolated and otherwise socially distanced during the present pandemic – focus on the last one. There are just too many people doing physics nowadays, and the clamorous tyranny of their bleating is enough to dull the maverick tendencies of all but very very few. And even when not, their collective voices – appearing as journal articles and pop sci pieces – will easily drown out any voice calling plaintively from the wilderness.

But now, late in March, 2020, that is the least of the things missing from the list. You can go live in the country, separating yourself physically from the rest of humanity for however long you want, but if you don’t shut off the internet, you will never be truly isolated, free from the pollution of the herd. Yes, academics the world over are presently physically more isolated than they’ve ever been, but intellectually they are no more isolated than the Borg, and like the Borg they each are subservient to the whole – to groupthink. The internet makes originality extremely unlikely.

Had Newton had the internet, a blessing and a curse – well, the mind boggles. Failing to develop calculus and a theory of why things fall down, we’d likely have fallen back on the notion that all things that don’t fall down long ago drifted into space, ergo … Personally I’d be happy in a world that believed that. But it’s wrong. Probably.

Anyhow, Newton didn’t have the internet. He had his brain, and time. However, had he come back from isolation to the presently huge population of physicists, all of whom were content with the status quo, his ideas would likely not have failed, but gaining traction may have taken much longer.

Ok, so, he didn’t have those problems, and he became a titan. History records it thus. The end. Another pointless screed in which I say in slightly different terms things I’ve been saying for years. Perhaps I should get help. One or two sessions per week should suffice.

Deus ex dystopia

The genre

It can’t have escaped anyone’s notice that cultures the world over have become increasingly intrigued by the notion of dystopia. The horrors of the first half of the 20th century – let’s recap some of it: World War I (the war to end all wars); 1918 flu; 10 years of giddy creative euphoria the excesses of which led inevitably to; the Great Depression; and in our desperation to be free of that we embraced Hitler, Stalin, Mussolini, Franco, and watched in horror (but not disbelief) as they murdered 10s of millions of civilians and started World War II; and let’s not forget The Bomb – so, yeah, small wonder then that writers like George Orwell were inspired to write the archetypal dystopian novels that formed the rich soil from which more modern dystopian fiction has sprung.

Personally I’m a fan of the genre – parts of it, anyway. Big fan of both Blade Runners; and The Matrix (number 1) blew me away. Avengers Endgame began dystopic, and it was great. I’ve mentioned Girls’ Last Tour already, which starts out almost as dystopic as you can get, but then in a very weird way descends even further – in fact, all the way. (So, it’s odd, but really, think about it: dystopia requires elements of humanity struggling against heavy odds to be worthy of the name; once humanity is gone, it’s not dystopia anymore. Put a colony of humans on the surface of Mercury, and you’ve got instant dystopia. But until then, it’s just a rock in space.)

And then there are the immersive computer games. Fallout 4 could hardly be more dystopic, but in my opinion it is surpassed by The Last of Us, which even when finished – when won, as it were – leaves one with a feeling of hopelessness. There is no real victory in that game. (Not sure I’ll play The Last of Us 2 when it comes out; the creeping dread of number 1 lingers still. As an antidote to dystopia I’m now playing No Man’s Sky, and while one can “die” in the game, that requires some ineptitude; once one learns how to avoid that, the game is all about space exploration, building bases, and … sure, it gets repetitive, and there are hints that you may only be a computer simulation, but I’ve taken the Blue Pill, and for now I am content; I’ve just entered galaxy number 4 (the game is to all extents and purposes infinite).)

All bets are losers

Many people harbor dark thoughts about how the world – or even just civilization – may end. What passes for science on television these days spends more time discussing how your daily routine might be disrupted should the earth fall into a black hole … pfft. You know, in the early days of the public tv science series, NOVA, as a poor graduate student I sometimes showed my support by sending them a check. However, although better able to afford such philanthropy now, I no longer do so. Maybe it’s me – which I doubt – but the density of real science information on the show has diminished over the years. It has become lurid, more likely to discuss death by tsunami, earthquake, asteroid, venom, disease – whatever – and all backed up by musical scoring more appropriate to the MCU than … Anyway, I’m more crotchety than I was as a graduate student, so maybe it is just me. Pfft.

So, the point is there’s a steady stream of apocalypse in our lives now, and my significant other (and, whoa, she’s very significant indeed), myself, and a couple of our friends, have a friendly wager going on as to how the apocalypse will come about. We’ve labeled the subsequent dystopic ages of humanity according to cause. For example, the Dougocene is caused by anthropogenic environmental decay, assumed to lead to the loss of much of what makes life worth living now, like Paris cafes (which covers about 90% of what I’d miss most). Wars, and other human on human acts of communal violence, might lead to the Denisocene (she’s not really playing, but we’ve given her this honorary -ocene just because she’s a friend). The scariest -ocene is the Suzocene, arising from fucking microbes.

The last -ocene is the Geocene, which is mine. You know how in Harry Potter they have this game, Quidditch, played on flying brooms? And the majority of players are flying around making points in what is essentially aerial basketball? And how none of their points ultimately matter because there are special players, one on each team, who are trying to locate the elusive Golden Snitch, and whichever of these players manages to catch the Snitch – well, their team instantly wins, so what was the point of all that flying basketball? You know what I’m talking about? Well, the Geocene is the Golden Snitch of apocalyptic events. These include all nonbiological catastrophes, like a comet impact, or the eruption of the Yellowstone super volcano. Of course, the Geocene needs to be careful; in particular, an asteroid impact runs the risk surpassing dystopia and turning the earth into just a rock in space – or several rocks. In that case I’d have to surrender my -ocene trophy and declare the game winner-less; then head to a Paris cafe to mope on the unfairness of it all.

No one expects the Spanish Inquisition!

It’s all good fun until someone gets hurt. I mean, ideally one wants to be on one’s deathbed, and minutes before expiring learn that shortly after demise the apocalypse will commence. That’s the ideal apocalypse.

In a previous blog post I mentioned the curse: “May you live in interesting times.” I also mentioned that upon first encountering the phrase I did not understand it to be a curse; I did not understand that “interesting” was a euphemism for disruptive upheaval, like either of the world wars, or the Spanish flu of 1918, or the plague of earlier ages … or the Fall of Rome, or … well, that word, “interesting”, I was all like, yeah, man, lay it on me; I like being interested. Alas …

Asinine animal husbandry practices, in a country I shall not name, have given rise to yet another global viral infestation, and this one is offering an unpleasant foretaste of dystopia – of Suzocene. SWMBO and I had planned a springtime trip to Paris and Milan. Now we’re leaning towards checking out the cafe scene in uninhabited portions of Canada.

I was born after WWII, and I experienced the Cuban missile crisis, which left my grammar school aged brain unfazed. My life was not disrupted. The neighborhood kids still played (literally) sandlot baseball. Or hockey on Johny’s Pond. In short, my entire life has left me comfortably inured to the notion that the heavily disruptive, “interesting”, times through which my ancestors may have lived – well, that was all in the past, relegated now to history books, and movies. Like most people, I did not expect the Spanish Inquisition.

The universe uses irony

So, one of the -ocene friends has a Star Trek calendar. Each month has a picture drawn from some episode of original Trek. It is now March, 2020, and this month’s picture is from the episode in which disease has ravaged a planet, leaving only children to carry on. This is disturbingly ironic, given that this present viral pandemic is over 10 times as likely to kill Boomers as Gen Alphas (if that’s what those younger than Gen Zs are to be called). As to me, although in many ways I have the intellectual and emotional depth of a ten year old, I doubt that the virus will take that into account. Damn it.

Nerds on alert

In an effort to slow down the progression of the infestation, events worldwide that were expected to attract large gatherings of people have been canceled. MIT – as tech savvy an institution of higher education as imaginable – decided to shift to online learning. Many other universities and colleges have followed suit, including where I got my PhD (I get emails), and where SWMBO (oh, come on: She Who Must Be Obeyed) is a professor.

Closer to home, while I am retired and unlikely to attend physics conferences anymore, should I even want to now, it is presently even more unlikely, because they’re being canceled in droves. Still, my blogs on physics and physicists, you may have noticed, have not been commendatory, and this enforced physics community “time out” cannot help but benefit in the long run. They should all sit in the corner and think about what they’ve done – and failed to do. And why.

Just seconds ago this notification from Bloomberg flashed atop my iPad: “Coronavirus Will Change How We Shop, Travel and Work for Years”. This, ironically, may lead to the threat of Dougocene (anthropogenic environmental disruption) being diminished. Satellites have detected reductions in pollutants in many places in the world. So, Suzocene and Dougocene, while both can ultimately be laid at the feet of overpopulation, the former is likely to diminish the latter, whereas the latter could enhance the former. Anyway, just saying.

On the bright side (ok, but less dim, anyway), the reputations of science and scientists (non-physicists), which have been in decline over the last couple of decades, should improve as the many-headed come to realize that their safety may be in the hands of nerds. Prayer might work, but my money is on nerds.

Fait accompli

Winding down

Forbes, a business magazine, publishes a surprising number of articles on science topics that are not half bad. These are generally (always?) attributed to Ethan Siegel, and his recent contribution in the perennial struggle to counter human doofiness was devoted to string theory. He had words of praise for the theory, but there was a soupçon of eulogizing to the whole thing.

Naturally Peter latched onto the article. More versed on the subject than Ethan, Peter was able to diminish the praise, and point out that no further nails were needed in that coffin.

I’ve been following Peter’s blog (Not Even Wrong) for years. During all that time he cogently made the case that the emperor had no clothes. It was great fun. But then, as we all know, the LHC finished doing its smashing, and found nothing new. Well, in particular, it found nothing that anyone other than the most deluded optimist could interpret as supporting string theory. This gave the emperor a bad flu, and the poor fellow died. The naked corpse still sits on the throne, but it no longer pontificates and is largely ignored.

What to do? Should Peter change his blog title from “Not Even Wrong” to “Not Even There”? I mean, really … Forbes? It provided but the most meager excuse to post a blog in response. Peter, it would seem, once a Samurai, is now a Ronin – at least as it relates to string theory. Still, string theory is not the only idea that is/was not even wrong. There are many other pretenders to the throne in similar states of advanced dishabille.

Voice in the wilderness

Well, speaking of titles, I chose to call this blog “Three Spheres to Rule Them All” for a reason. That it exists at all has a lot to do with “Not Even Wrong”. Peter occasionally struggles to keep comments added to his entries on topic. He has rules, and they are enforced. They are reasonable. Violations simply do not appear; they are expunged. Many of my comments have met this fate, although half of those were submitted knowing full well they would be rejected; sometimes I just feel naughty, and poke the bear.

Still, starting my own blog was in some measure a response to rules I found it difficult to abide. In my own blog I could say whatever I wanted to, even to the point of extolling the virtues of a particular set of theoretical ideas, viz., my own. And I could do it without the onerous burden of accepting or rejecting comments. All comments are by default rejected. I don’t handle criticism well, and I don’t want to see it. I handle praise only marginally better. And, truth be told, I don’t really care if anyone reads this blog. As I get older, I write more, but mostly just to and for myself, a most appreciative audience; kind of a Gandalf thing going on there.

Humanity is capable of all sorts of heinous behavior, from the hanging of the decapitated corpses of one’s enemies on the underside of bridges, to the less egregious, and somewhat less annoying, Tammany Hall style corruption into which unfettered elites invariably slide if there is no equally powerful countervailing … well, Animal Farm anyone?

TP elites have been Tammany Hall for the last several decades, and many of my blog entries have excoriated their narrow focus and wrong thinking. But, so, ok, we’ve covered that territory. It’s time to …

Close the circle

Unlike Peter, whose leery attitude to the mainstream arises from a highly educated understanding of their foibles, I base my negative assessment of the last 40 or so years of TP theorizing on one simple idea: my ideas are fundamentally correct – in their essentials – and ideas at odds with mine are therefore wrong. As I mentioned somewhere back in the depths of this blog site, the fact that those 40 years of theorizing came up empty was to all extents and purposes a prediction of my own work. Let’s talk about this.

I’m going to coin a new word. I can’t help myself; it’s really neat. So, in mathematics there are Ur-objects, which I’ll call, Ubjects (See? Neat, huh?); and there are tools (let’s get crazy and call them Mools (mathematical tools)). Ubjects are things that in some sense exist without the intervention of sentient creatures – should the universe ever produce any. Prime numbers and parallelizable spheres are examples of Ubjects. Mools are things invented to help the inventors better understand Ubjects. Ubjects are objective – invariable from species to species; Mools are subjective – and should we ever encounter an alien source of Mools, differences in approach will likely give rise to schisms and intergalactic wars. (The search for ET, as a consequence, should be squashed, and that posthaste.)

As regards my own work, the underlying Ubjects are the (connected) parallelizable spheres, which exist only in dimensions 2, 4 and 8 (so the spheres have dimensions 1, 3 and 7). I am thoroughly convinced that there exists some (spinor) field theoretic Ur-reason that these spheres are an essential part of the design of our universe – any universe. I have only the vaguest notion how one would go about proving this, but, hey, give me a break. Hundreds of some of the brightest people on the planet spent decades mucking about with string theory, so … you know, give me a break.

Associated with these Ubjects – the parallelizable spheres – is an equally finite collection of Mools – the Hurwitz division algebras: C (complex numbers), H (quaternions), and O (octonions). The quantity and importance of the mathematical concepts to which these algebras give rise and/or link can not be exaggerated. They are extremely resonant.

Ok, so, the Pauli algebra, from which is built the Dirac algebra, is isomorphic to P=CH. You can get an SU(2) doublet of Dirac spinors by stacking a pair of elements of P, and voilà, all the algebraic elements with which to build a U(1)xSU(2) Yang-Mills theory are there and ready to be exploited.

So what happens if you start from T=CHO instead? In this case the expanded collection of elements is ready to be turned into:

> A U(1)xSU(2)xSU(3) gauge theory (this occurs naturally);
> There are distinctions in the ways SU(2) and SU(3) manifest, and these explain why …
> SU(2) is broken and chiral;
> SU(3) is exact and non-chiral;
> The elements of the SU(2)xSU(3) multiplets are ordinary Dirac spinors;
> Neutrinos are Dirac, and massive;

In my first book I built the various Lego bits into a semi-coherent model, essentially the Standard Model. In my second book, I threw a bunch of the Legos on the floor for people to either play with, or step on barefoot in the dark on the way to the kitchen to have a midnight snack. Subsequently I wrote a paper that I consider my pièce de résistance (there is also a longer, fuller version). From all this one gets:

> The hyperspinor built from T contains a full family and antifamily of leptons, quarks, antileptons, and antiquarks;
> The spacetime associated with a model built from P is 1,3-dimensional; the expanded model built from T has an associated 1,9-dimensional spacetime;
> Projectable from this 1,9-dimensional spacetime is a 6-dimensional subspace (carrying color charges);
> And a 1,3-dimensional sub-spacetime that manifests as matter, or antimatter;
> We live in the matter version; we simply don’t see the antimatter version (but this mirror universe also exists).

That last bit, if it has escaped your attention, provides an explanation of one of the biggest mysteries in TP. Pop sci articles, like this one, will not mention my work on this mystery. But then, if a pop sci journalist asks mainstream theorists their thoughts on a maverick idea X, the mainstreamer will provide a dismissive response, with an underlying threat to withhold their wisdom in future should the journalist even mention idea X. Sadly, this is understandable behavior, and it is difficult to fault.

The list of niceness arising from T is longer than this, but I’ve begun to forget a lot of it, so screw it. Anyway, it is inconceivable that all of this is mere coincidence. I mean, really, don’t even try to conceive that; you’ll hurt yourself. Some of the most important and resonant Ubjects in all of mathematics give rise to Mools from which the Standard Symmetry, multiplets of Dirac spinors, and all the rest … I mean, sure, it could be coincidence. Nature might be that stupid and perverse. Sure.

Not included in the model building, as I envisioned it, are:

> Axions;
> Wimps and any other of the panoply of particles and fields people have predicted, but for which there is no evidence.

And that’s why the fact that 40 years of efforts to find the Next Big Thing, and win the Next Nobel Prize … the fact that all that came up empty is not at all surprising to me. And it’s why I am now perfectly content to sit back and watch the future of TP unfold as a nonparticipant. I do not expect to be surprised. Well, there’s always quantum gravity, which may yet surprise (but not without T); and astrophysics has still more secrets, I’d wager, if it can avoid talk of wormholes (which is just goofy in the absence of a coherent theory of quantum gravity).

Wrapping up

So, I’m not sure I’m going to contribute many – if any – more blog posts devoted to TP. I don’t have the depth of knowledge of Peter or Sabine. Really, I’m only an expert in ideas to which I gave rise, and as they are not integral to any of the directions in which mainstream TP is moving itself, I’d only embarrass myself if I tried.

But no matter …

All the generally unflattering attributes that define us as humans, however violent and/or underhanded, can not alter the reality of Truth, only its perception. Physical Truth is immune to prejudice and self interest. It is what it is, and even if it is in the interest of a coterie – or even cabal – to create a fiction at variance with the Truth, as a means of maintaining influence and power, in Science, far more than in politics, … well, in politics, if an idea lives long enough, it becomes perceived Truth. In science, an idea that varies from Truth will always be fiction. Although this may never be understood. Unlike in politics, you can not change the context and make it true.

In my last blog I discussed the anime, “Girls’ Last Tour”. Recently I shared a further thought on it in an email (a fitting way of ending this blog entry):

“The anime is a work of art on several levels.  Visually, certainly, but the story of two doomed tweens who use each other’s company to stave off gloom is enchanting.  And for me the most magical moment comes when they encounter a now defunct art museum.  They aren’t 100% sure what it’s all about, but there is a replica of a cave drawing, essentially humanity’s first work of art, on a wall.  One of the girls draws, and she pins her last drawing – humanity’s last work of art – next to humanity’s first.  Then they wander off, and neither work will ever be seen again – at least not by a human.”

TP’s Last Tour

Sometimes play is less fun, more traumatic

Avid readers of this blog – which, admittedly, is probably just me (and, yes, I do avidly read my own writing) – will recall that I more than once suggested to Tony Smith that choosing a more playful activity than obsessing about the cool to frigid reception (when received at all) of his theoretical physicists (TP) ideas by the Establishment might improve his state of mind. Since his passing I came to the conclusion I should have just kept my advice to myself. It could be likened to trying to convince a tiger to go vegetarian. It’s simply not in the tiger’s nature to thrive on beans and kale, and their health would suffer in making the effort.

In my Tony Smith eulogy blog I quoted from several of his emails, one of which ended with this:

“PS – As to Girls Last Tour – if it or its author is suicidal then I do not need to get into it. From what the web says of the Manga ending it seems as though they die without either clear victory or defeat with respect to their life goals.”

Let’s talk about this.

I have been a fan of animation from early childhood through to right now, where that “right now” will be every “right now” from this “right now”, at this very moment, until I breath my last and end experiencing any more “right nows” (although I am not averse to having more after breathing my last, but my hopes are not high on that score). I mentioned to Tony at one point that my latest animation fixation was an anime series call Girls’ Last Tour, and I was finding it a quietly enthralling work of art – the anime; I have since gone through the entire manga series. The former is video; the latter a series of graphic novels.

For many of you, I’m guessing, anime viewing is not a proper adult pastime, and it conjures images of large-eyed teens emoting dramatically, and perhaps carrying big-ass swords that may be used on monsters. This is a signal to noise problem. In this case the noise is the surfeit of animes devoted to big eyed teens with magical powers, or just moping about in school experiencing angst. The signal, if you will, are those far less frequently encountered gems, like almost anything from Studio Ghibli. (Oh, and if it isn’t clear by now, let it be known that I am not a proper adult, and proper adult pastimes I often find soporific.)

Ghibli animations involve very inventive stories that evolve in ways that I find thematically unWestern. One frequently encounters characters who early on behave in ways that my rearing in the USA has inured me to think of as villainous. One disapproves, and one expects these characters to get a heaping helping of comeuppance in the final act. This very rarely if ever happens in Ghibli stories. Not only is there a dearth of comeuppance, but the purported villain may actually in the end move in with the central characters as part of a congenial family unit. It’s all oddly refreshing.

Anyway, Girls’ Last Tour is none of the above. Yes, its central characters are school age, and yes, their eyes are on the large size. But they aren’t in school, because there are no more schools. There are hardly any other people at all, and you only ever encounter two, and neither of these are central to the story. As to that, there is very little story in any conventional sense. It’s just these two young girls wandering through a post apocalyptic cityscape in a mechanical conveyance called a Kettenkrad. At one point, we are informed in flashbacks, there was a world of adults at war. A parental figure, wanting to save the girls, sent them away from the war zone on the Kettenkrad. They were evidently told to get to the highest part of the city.

In episode 1 of the animation the warring adults are gone. Exterminated. We don’t know this yet. We just see the Kettenkrad and its occupants tootling along in a dark interior space. This space is an old industrial space full of pipes and fans and dripping and loose screws. And dark. So dark.

Initially you have no idea who the kids are, whither they are traveling, and why. As to the where – the milieu – this future city is ultimately revealed to be very bizarre, with whole conventional cityscapes built in levels stacked very very high above each other, held up and connected by gigantic columns. It’s hyper-industrial, and much of it is in decay. This is due to the war – now some years in the past – and the subsequent lack of maintenance.

At the end of the first episode I felt confused and bemused. What was this? Did anything happen? The kids traveled, ate (food is a running theme), and argued. Is there a story arc? I wasn’t sure, and for some time I didn’t watch any more. But two things brought me back finally: the artwork was grand in a darkly mysterious way; and I needed to know what I had watched. Oh, and I lack the maturity to forego.

The anime does not take the girls as far as the manga, although it is rumored there will be a season 2 following the final journey of the potatoes (which is the endearing term applied to the girls by fans of this work of art). Season 1 of the anime ends on a note that is potentially upbeat. Things have happened. They’ve had encounters with two other people, and some things that weren’t people, but all these people and things are out of the picture at the end, and the potatoes continue their travels, heading where we believe they were told to go by the parental figure long ago. Well, I was now hooked, so …

Then I bought the manga books and finished the story. Dark and mysterious. Metaphorically dark. Existentially dark. In fact, they do get to the highest level, and it is a flat snow covered plain with a single feature: a (possibly) concrete cube thing with markings under the outermost layer (that we do not see at first). It’s not big – maybe six feet high or so. To get to this point the girls have lost or left behind almost everything that was of value to them. But the view is spectacular. They have a snowball fight, look at the sunset, then sit next to the cube and eat the last of their food. Then they decide they should get some sleep.

The End.


As Tony wrote after looking online for more information, “… it seems as though they die without either clear victory or defeat with respect to their life goals.” Well, we never see them die, and in the last frame of the manga the girls are gone, and a portion of the cube has broken off revealing glyphs on an inner layer. But many have argued that this open end is devoid of hope. They had just enough food to get to the top, and now to escape that place they’d need to walk, for the Kettenkrad is irreparably broken. Moreover, the city itself – oh yes, the city – the world – well … but I’ve said too much.

Tony (as well as the creator of Girls’ Last Tour, online gossip would lead us to believe) fought depression for many years. And Tony died like the potatoes, without either clear victory or defeat with respect to his life goals. I myself expect to die like that, but in the meantime I’m going to emulate the potatoes, watch the sunset, have a snowball fight when the opportunity affords itself, and in general bathe in the warm waters of my immaturity.

Yet another metaphor

So, yes, the TP mainstream (to which I’ll assign an endearing moniker: the tomatoes), spent over 40 years tootling in their Kettenkrad through a huge dark and mysterious cityscape. Mixing metaphors, the tomatoes had created a theoretical speculative bubble, buying on margin, and hoping JP Morgan (the LHC) would bail them out and prevent the collapse of their businesses. But JP Morgan didn’t bail them out, and those bright young souls drifted disconsolately away to other things – intellectual hobos, as it were. Once cutting edge, they are now the old guard, loitering outside the local candy store hoping for a handout.

String theory, in particular, very rarely gets discussed in the pop sci press anymore, nor are there very many colloquia on the topic – at least in the Boston area. Its two most prominent spokespeople – Michio Kaku and Brian Greene (both of whom were in the past involved with some quite saccharine video content extolling the wonders of string theory), have drifted into podcasts and books with a more generic sort of content. Still, they’re bright and have creative energies that need an outlet or two, so I’ve no problem with any of that.

Ishii’s story

Ok, that’s enough for the nonce of TP’s Last Tour. I’m not done discussing Girls’ Last Tour. You can build your own metaphors.

One of the two adults the potatoes encounter in their tour is a woman named Ishii. She has been living by herself and is trying to make an airplane – sort of a Wright brothers like machine. She wants to use it to escape the city. She needs help, and promises the potatoes that she will fix their wonky Kettenkrad in return for help getting her plane together. All this done, she hops on board, flies off, and the plane seems to be working fine. But then in the distance the potatoes see a wing break, and Ishii is forced to bail out and parachute down to a much lower part of the city. We focus on her face, see her wan smile as she drifts down. And she says (or maybe just thinks): “But, well, once you fail, you feel so carefree.” She drifts out of sight, and we never see her again. One review online had this to say:

“Ishii’s story was one of my favorites in the comic and in the anime, there’s something really beautiful about a little story centered around the highly optimistic concept of ‘Even if you fail, it’s still okay’ framed against the actual apocalypse. The last Pilot on earth crashed, and it’s okay.”

I hope string theorists, and supporters of other TOE ideas – ideas that will likely fail to escape the city in their lifetimes – can achieve that carefree feeling. I wish that Tony had. I hope that I do. Sometimes I think I’m there.

So … snowball fight, anyone?