I see dead physics

Pop physics stories on Flipboard have recently been dominated by conjectural fluff about black holes, wormholes, time travel, and finding planets in other galaxies. Consider that last one. Why is that a thing? Why wouldn’t there be planets in other galaxies? The probability that some nearby galaxy should lack planets is 0. The probability that there are scads of them is 1. The probability that one of those planets should be habitable is closer to 1 than 0. Should we discover that one is more than habitable, maybe even paradisiacal, the probability that we should ever be able to travel there and make it non paradisiacal, maybe even uninhabitable, is 0. Not that we couldn’t easily destroy its ecosystem; we just will never have the technology to make the trip. We will never have the technology to send humans to other stars in our own galaxy – or to wormholes, should these highly conjectural figments even exist, and even if in entering one you wouldn’t be crushed … So, no, we don’t live in a universe in which any sci fi plot device is possible, requiring only a little funding and effort. Looking for planets in other galaxies is like looking for sand on a beach. Take a plastic shovel and a bucket, et voilà.

Anyway, one thing I almost never see in pop physics stories anymore is any mention of high energy physics (HEP), and elementary particles. The vast majority of exceptions to this dearth are paeans of praise for the plethora of physics advances to which the 20th century gave birth. These stories extoll the virtues of the likes of Feynman, Dirac, and Heisenberg, virtually patting humanity on the back for the intellectual achievements of this ilk, while ignoring the what-have-you-done-lately meme. The science to which I devoted most of my research life is not dead, but certainly comatose. Well, it might be dead.

And speaking of the 20th century, 87 year old S Weinberg recently won a 3 million dollar Breakthrough Prize. Not that he doesn’t deserve prizes, but his breakthroughs occurred half a century ago. Although I may not live long enough to see it, when all the Standard Model Nobelists are gone, to whom will such prizes be given?

Arguably the last glory decade for HEP was the 1970s, and those involved in making it glorious are now more likely to appear in obituaries than in arxiv. Four ensuing decades of work on creating a viable theory of quantum gravity produced lots of interesting mathematics, but by and large fizzled as regards physics and reality. Wide-eyed youths who surrendered to the clarion call of this effort became lost in a swamp, and its viscous bogginess only relatively recently became apparent. They still dot the arxiv with matters supersymmetric, or stringy, especially as regards how these faded glories relate to black holes – but now far less often than earlier. Of course, the arxiv gatekeepers allow this dross into hep-th, only relegating to gen-phys ideas that do not relate to beloved failed dogmas. Sniff.

Still, no one cares anymore. Forbes main physics guy, Ethan Siegel, recently published an article entitled: Why Are Scientists So Cruel To New Ideas? This is filled with carefully crafted hurdles over which novel ideas must leap if they are to be taken seriously. But …

You know how in the olympics pole vaulters have to vault over ever higher bars, and with each raise of the bar vaulters get eliminated, and how sometimes the bar is set so high that no one can achieve victory? Well, the failure of the LHC vaulter to provide unequivocal evidence of new physics beyond the Standard Model has doomed the mainstream’s coterie of young theoretical vaulters to face a bar so inconceivably high it is invisible. And the slew of erstwhile enthusiastic pop sci onlookers have become disenchanted with this particular enterprise and mostly drifted away.

In Ethan’s article there is a picture of Bohr and Einstein lounging contentedly while they discuss and debate ideas that will be testable in their lifetimes. This picture exemplifies the notion that dead physicists are the best, as is dead physics. But it has utterly nothing to do with the milieu in which young physicists find themselves, and especially the mavericks amongst that group.

What to do? How do you avoid the slow decline of rigor into an almost spiritual acceptance of whatever notions fit your fancy, and evidence be damned? Because – let’s face it – evidence in any historically conventional sense is hard to find anymore. Is there a viable alternative to experimentally based progress?

Well, in my oh so humble and self-effacing opinion (IMosHaSEO) there is a way of assigning potential value to new ideas. It’s not the first time I’ve suggested this, but here is my notion of rigor based on mathematics:
1. Start with mathematics, and at least convince yourself that your chosen mathematics is unavoidable, resonant, and special;
2. Make your work on this mathematics unassailable;
3. Make the interpretation of this work as a contextual foundation of some part of theoretical physics unavoidable, or as much so as possible;
4. Ignore your own biases and let the mathematics speak – follow it – do not lead it or push it.

You know, as an example of applied mathematics that irks me, at the core of much of quantum mechanics is the mathematical notion of Hilbert space. There are infinitely many Hilbert spaces, but surely infinitely many of those are physically irrelevant. The problem is, Hilbert space is not really a space, nor even a mathematical object; it is an abstract collection of properties from which spaces and mathematical objects can be constructed. It is an entirely too flexible tool that suited the needs of 20th century theorists (now mostly dead) trying to make sense of things quantum. Again, IMosHaSEO Nature is not that nonspecific nor flexible. Nature is in fact not at all nonspecific. If you require yourself to use a Hilbert space, pick one, damn it. And make your mathematical application of it unassailable; and blah blah blah. And you might want to base it on the parallelizable spheres … if you want it to have anything to do with the universe we live in.

And by the way, Hilbert is dead too – beatified, sure, but … Even in science we are prone to thinking religiously.

Number theory for complete beach puppies

Not only optimal, but intended.

Caveat: I’m lazy, so grains of salt should be on hand while reading this blog. Due diligence and researching precedents are not my fortes. I depend on google for that, but I do not always avail myself of its services. If you’ve read my previous blogs, then you’re likely already prepared.

Let’s recap. These are a few of my favorite mathematical things:

Ⓐ Primes numbers;

➁︎ Parallelizable spheres in dimensions 1,2,4,8;

Ⅲ︎ Pretty pretty laminated lattices in dimensions 1,2,8,24.

In this episode I want to discuss prime numbers, and how I perceive my thinking about primes differs from – well, this is unclear. Let’s just say I’ve not encountered views similar to those I’m about to extoll, but then, this is not my field. I am merely an enthusiastic dilettante, like a puppy at the beach.

So, I recently happened upon an article suggesting that mathematicians will never stop finding novel ways to prove the prime number theorem. Which is what? Well, if you plot π(n) (the number of primes less than or equal to n) vs n, the plot rises in a manner that looks like it may relate to ln(n). This gave rise to the following approximation:

π(n) ~ n / ln(n).

This is not a very good approximation. I don’t care if the step function π(n) and the smooth function [n / ln(n)] cross each other an infinite number of times (I don’t know that they do), if you look at the table here comparing these two functions, it is clear that Nature is largely uninterested in the correspondence. Still, the ratio of these two functions has been proven to converge to 1, and that is the prime number theorem, which, according to that aforementioned article, is a thing that mathematicians make a hobby of repeatedly proving. Never mind that they diverge in an arithmetic sense; but how about that ratio!

Still, better approximations abound. Maybe [1 / ln(x)] is the density of primes at x, in which case the integral of that from 2 to n, denoted Li(n), should be a good approximation to π(n). In fact, this is a better approximation, but while it is true that π(n) / Li(n) converges to 1, Li(n) does not at all behave like it is Nature’s intended smooth approximation to π(n).

Does Nature have intended smooth approximations to number theory step functions? Is this a thing? Yes, it is. Never mind how I know; just take my word for it.

Evidently Riemann found an exact form for π(n), which is discussed here. Riemann was there first for quite a bit of modern mathematics, and the zeros of his famous zeta function play a part in his exact form. As this involves the as yet unproven Riemann Hypothesis, I’m going to pretend Riemann never existed and carry on discussing π(n) from my own peculiar perspective.

Peculiar perspective

To begin with, there is a problem with the notion of just counting primes as they pop up. What you really want is a measure of prime occurrences. I mean, consider the integers from 1 to 10. There are 4 primes in that collection: 2, 3, 5, 7. But there are more occurrences of primes than just those primes. What I mean by that is this: the prime 2 occurs 3 times, because 2³ = 8 is less than 10. And the prime 3 occurs twice, because 3² = 9 is less than 10. But how do we measure that? In my puppy on a beach investigations I decided upon lcm(n) (least common multiple of the integers from 1 to n). So, for example, lcm(10) = 2³ x 3² x 5 x 7 = 2520.

If you plot lcm(n) it looks a lot like a step function version of en. Did you see? Therefore without a shadow of a doubt Nature’s intended smooth approximation to ln[lcm(n)] is a simple linear function:

ln[lcm(n)] ~ n-1.

Why subtract 1? Well, I did that to enforce equality at n = 1. Nature concurred.

Want another test? Well, if

lcm(n) ~ en-1,


[lcm(n)]1/(n-1) ~ e.

Plot that. Take your time.

Anyway, I will go to my grave (in the not too distant future, unfortunately) fully convinced that these smooth approximations of these number theory step functions are Nature’s intended approximations. And I just don’t understand why in all my beach puppy readings in number theory – back in my beach puppy days – I never encountered this sort of thing. I’m not the only person to have noticed this. If I ever write a book entitled Number Theory for Complete Beach Puppies, this lcm(n) stuff will form its core.

Well, so I used this Nature’s intended approximation to derive a really really good (Nature’s intended) approximation to π(n). And that’s my story. You’ll find that I will be sticking to it like a limpet on a rock.

Pollyanna at the Church of Jordan and Penrose

Just Ignore Me

Pascual Jordan developed Jordan algebras. He’s famous. So are von Neumann and Wigner. Together with Jordan, they were able to classify all finite dimensional Jordan algebras. They published their results in 1934.

This troika, in the years after Solvay, achieved a renown that elevated them to such a high celestial sphere that it is tantamount to heresy to question the worth of anything they produced. This includes Jordan algebras.

But what motivated Jordan? We get a good idea in this John Baez blog post. I quote:

“… in traditional quantum mechanics, self-adjoint n×n complex matrices count as observables. These aren’t closed under matrix multiplication. Instead, they’re closed under linear combinations and the commutative operation a∘︎b = ½(ab + ba).”

This Jordan product of observable operators forces the result of the product to in turn be an observable. So, yippee. We want that don’t we? Don’t we?

Well, I’ve never been anything approaching 100% sure, but I’m a maverick – not a pariah, maybe, but … Was I alone in thinking Jordan algebras were little more than mathematically pretty human constructs that Nature herself can do without? I did some googling.

I googled Jordan algebra critiques, and – quelle surpris – topping the list was a Luboš Motl blog post with which I found myself largely in agreement. Luboš – by way of providing context – is an aging (but aren’t we all) pit bull blogger on all things HEP and a great deal else. He falls short of being politically correct by a margin that dwarfs the Grand Canyon, but my reaction to some of his more outré pronouncements – and I haven’t read many – has generally been of the eye rolling sort. Sometimes he lacks tact. For example, he claimed most vehemently, and numerous times in the same pair of blog posts, that I (and many others) am a crackpot, and, more pointedly – because this same accusation was leveled at no one else – that I am not Lance Dixon. Well, he’s right about at least half of that. But the point – from which I have strayed by a goodly bit – is that he is not a fan of Jordan algebras. So I am not alone. Yay.

Decades of Hope

In Lie group theory there are infinite sequences of classical Lie groups associated with the division algebras R, C, and H. There is also a finite sequence of exceptional Lie algebras associated with the last division algebra, the octonions, O.

The same sort of thing occurs with finite dimensional real Jordan algebras: infinite sequences associated with R, C, and H; and a finite sequence associated with O. The last of these, usually called the exceptional Jordan algebra, is 𝔥₃(O), the set of 3×3 self-adjoint octonion matrices. This set is 27-dimensional, but if you take away multiples of the identity, which doesn’t really participate in this Jordan algebra, you’re left with 26 traceless elements of 𝔥₃(O). And in fact, the automorphism group of this Jordan algebra is the 52-dimensional Lie group, F₄, the 2nd of the 5 exceptional Lie groups, whose fundamental representation is 26-dimensional.

John Baez has a theorem: The subgroup of F₄ that preserves the splitting of each off-diagonal octonion into complex scalar and complex vector parts and preserves a copy of 𝔥₂(O) in 𝔥₃(O) is isomorphic to (SU(3)×SU(2)×U(1))/ℤ₆, which he calls “the true gauge group of the Standard Model”. (The ℤ₆ is a sort of anal flourish which makes rigorous mathematicians happy.)

This is the kind of result that has kept many people plugging away at this algebra for decades. It’s cool, but is it that cool? First, in a previous blog I mentioned the difference between what I called Ubjects and Mools. Ubjects (short for mathematical ur-objects) are things that in a sense exist without the need of any sentience to invent them. My partial list of Ubjects includes my favorite things: prime numbers; the parallelizable unit spheres in dimensions 1, 2, 4 and 8; and the exceptional laminated lattices in dimensions 1, 2, 8 and 24.

Mools are mathematical tools, like quantum field theory. QFT, while incredibly useful within our potentially limited lights, needed to be invented, as opposed to discovered. The same can be said of Jordan algebras, but in this case matters are worse, because they are not incredibly useful. Few people use them at all, although many people study them.

But what about the standard symmetry inside F₄? Surely that’s something. Well, I have a few things to say about this that explain why this leaves me less than excited. The restrictions on 𝔥₃(O) that lead to this subgroup are unmotivated and rather artificial. And F₄ mixes the 26 traceless elements of 𝔥₃(O), which are essentially 1-dimensional. Fermions cannot be represented in this way. You’ve got to attach Dirac spinors somehow, thereby increasing the artificiality. (Contrast this to my work on the tensored division algebras: the standard symmetry, when derived, automatically acts on Dirac spinors, and that’s because the theory begins with a spinor space, not some algebra or Lie group). And what about the whole algebra of observable thing that got the mathematical physics gods so excited back in the 1930s? Well, that’s been largely – or entirely – discarded, at which point 𝔥₃(O) is just some algebra someone happened upon that has properties that appear relevant to physics, and, hey, there’s that number 3, so maybe it could explain the 3 family generation problem. Who knows?


Peter Woit continues to blog, and his assessment of the future of HEP funding is very 2020. Basically, the science to which I have devoted most of my life (and it will always be most, as I have metastatic prostate cancer and will be dead within 5 years), but anyway, that science also appears to have an incurable disease, and researchers who desire funding are streaming to quantum computing and AI.

Peter, taking advantage of this chaotic decline, has decided to get daring, while few people are looking anymore, and develop ideas related to quantum gravity using Penrose’s twistors. Penrose, of course, is another mathematical physics god, whose thoughts and pronouncements, however outré, are generally considered beyond criticism.

Twistors live in a universe based upon the complex algebra, C, and if his one reference to my work is any indication, he was not at all interested in incorporating H and O. They are merely irrelevant oddities.

Peter is not so narrow minded, and is willing to consider some role for those two higher dimensional division algebras. Carlos Perelman (or Castro, depending on his mood) has already developed quantum gravity ideas based on T = CHO. I don’t believe these involve twistors (I confess myself unable to judge the ideas of either Peter or Carlos), but they do involve all the relevant division algebras, so my seal of approval is upon them. (I never carried my work to quantum gravity, but following the mathematics – and listening solely to the mathematics – did lead to a solution to the missing antimatter problem, which I consider excellent, like a multifaceted diamond.)

Twistors, by the way, are Mools, but I suspect that if you are a devotee they seem like Ubjects. I’m not withholding judgment on the matter, but am disinclined to debate it. My judgement is not firm enough.


So, there you are. Everything is changing: for me personally, having learned I have an expiration date; for the human population, beset by disease, political unrest, and economic distress, painfully giving birth, inevitably, to a new world order that is being resisted across the political spectrum; and let’s not forget the environment, transcending all of that, and looming ever larger as a threat to old ways of thinking, being, and doing.

Everything is changing, and all in this year. What a year.

I thought my last blog was to be my last. I was wrong. I hope to keep being wrong for as long as possible.

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.