“Mathematics is the language of God.”--Galileo
Regarding the Infinite, Pt. 1
For most people on the planet, some Supreme Being is the given arbiter of all subjects with no beginning or ending, and most frequently referred to in those terms, or term limits. (Which is the case for Supreme Court Justices as well, and who really should be under the latter aegis.) However, for mathematicians, physicists and others for whom infinity is a factor (meaning: some value to add into a subject in order to come to a conclusion—more or less, when approached via higher disciplines), while the subject is often discussed in terms which may approximate religious zeal, agape wonderment or prayer (if reworking equations over and over might be said to be comparable to davening, dervish dancing or holy rolling), the brutal fact of the matter is: numbers count, and nothing else does.
As much as we may want to get all mystical about it, or even metaphysical (like calling something “eternal”, which implies more than it means), the idea of a “transcendental figure” is best served by the Greek letter of pi, rather than anything conjured up by the Maharishi at Rishikesh. But that doesn’t mean there isn’t a lot to conjure with there; some people go all nuts over rolling it out for a couple million digits, looking for some pattern. It is only because of our particular schooling—that we have an innate teleological frame of mind (that being the belief that everything has a reason) from poor education into the nature of systems—that the vast majority would rather think about the Old Guy on the Clouds instead of the awe and majesty of 1) a discovery of a physical, universal truth of the universe, and 2) that the human mind could actually do this without divine intervention.
So, whist in the midst of this sun-dazed contemplation, I get back to the Kindle and turn the page on another story from the Philip K. Dick collection and what do I find? “The Infinites.”
For real. This was NOT arranged by me, OR the powers-that-be, to fit into the “flow” as much as you teleologists might think. The reason I cited Dick in part one was to explain how amazing his thought processes were well before the late 1960’s body of work which made his rep, and which has become something of a staple for Hollywood in recent years. More, after watching “The Twilight Zone” marathon over the last Independence Day celebrations, one cannot read these tales and help but think how well Philip would’ve fit in with Rod. Morality plays, O. Henry-like endings, scientific or psychological twists on space operas, smattering of sardonic humor pieces—even today, these are still reminders of how a sense of wonder could be encapsulated in stories under 5,000 words. And in Serling’s anthology format they couldn’t’ve failed, up there next to Robert Bloch, Earl Hamner, Jr. and the rest of his stable of pens.
But that this title would surface so neatly…begins to, in essence, creep one out. Teleologically speaking. I am equally suspicious of auspicious tarot cards, fortune cookies and I-Ching coin tosses. However, by this point in the day, the sun has begun to turn everything to burnt copper in long shadows; the brainbake from noon now producing a mind rind with the consistency of canola-crisped calamari, thoughts drift back to the topless towers of Illium and another beach, the one at Troy where Paris said to Pelias, “Hey, dad, why argue? We’ll burn it as an offering to the gods. Tomorrow.”
(Tempus Fugit, son.)
. "We've evolved," Eller said. "The radiation from the asteroid speeded up cell growth, like cancer. But not without design. There's purpose and direction to these changes, Blake. We're changing rapidly, moving through centuries in a few seconds."
. Blake stared at him.
. "It's true," Eller said. "I'm sure of it. The enlarged brain, diminished powers of sight, loss of hair, teeth. Increased dexterity and tactile sense. Our bodies have lost, for the most part. But our minds have benefited. We're developing greater cognitive powers, greater conceptual capacity. Our minds are moving ahead into the future. Our minds are evolving."
. "Evolving!" Blake sat down slowly. "Can this be true?"
. "I'm certain of it. We'll take more X-rays, of course. I'm anxious to see changes in the internal organs, kidneys, stomach. I imagine we've lost portions of our --"
. "Evolved! But that means that evolution is not the result of accidental external stresses. Competition and struggle. Natural selection, aimless, without direction. It implies that every organism carries the thread of its evolution within it. Then evolution is ideological, with a goal, not determined by chance."
. Eller nodded. "Our evolution seems to be more of an internal growth and change along distinct lines. Certainly not at random. It would be interesting to know what the directing force is."
. "This throws a new light on things," Blake murmured. "Then we're not monsters, after all. We're not monsters. We're -- we're men of the future."
. Eller glanced at him. There was a strange quality in Blake's voice. "I suppose you might say that," he admitted. "Of course, we'll still be considered freaks on Terra."
. "But they'll be wrong," Blake said. "Yes, they'll look at us and say we're freaks. But we're not freaks. In another few million years the rest of mankind will catch up to us. We're moving ahead of our own time, Eller."
Ok. This is late 50’s/early ‘60s science-fiction, even if it is PKD. Admittedly, as compelling as the ideas are, the story ain’t that far from pulp fiction. After the lab rats—literally—reverse the process, Eller gets the gal lab technician, so not that far off from the girl/girl stuff anyways.
But the point here is the main thesis. The view of Blake, that evolution—which, according to Darwin was a combination of environmental factors and benefits to the organism—shows itself to have a purpose, is teleological. Can’t remember how often that plot device was used on TV: in the “Zone”, “Outer Limits”, “Star Trek” and countless other futuristic utopian fictions.
And yet, it was only today, getting zoned by the sun's spectrum of visible frequencies (plus alpha-, beta-, gamma-, delta-, UV- and X-rays) and Carl Wilson, spaced by the horizon line where the water meets the sky, that this one singular revelation finally broke through. Evolution doesn’t have a purpose, and endpoint or such…but, perhaps, genes may have a potential for such, under such extreme mutations.
This could, then, be seen as an argument for Richard Dawkins’ “selfish gene”, but it also applies to extreme math—a catchall term I use for anything involving numbers…and letters that stand-in for numbers. Because I have an aversion to formulae, I freely admit that what follows is simply a “naturalist’s” point of view, which is to say, ultimately a heuristic journey. If you want to find out about something, Wikipedia is a great place to start. Then just follow the other links and you can deduce a lot about your subject. (I am well aware of the ongoing mis-, dis- and mal-information which propogates all the jokes about what you get on search engines. Which is probably true—if you are looking for stories on Lee Harvey Oswald or Area 51. However, higher math? I cannot conceive of a prankster who would go to the trouble of creating articles on geometry in order to hoax the scientific world. First, such fallacies would be exposed by the first person of knowledge to click the button. Second, elaborate plots of this sort are reserved for things people might be interested in; urban legends, conspiracy theories, Hawaiian birth certificates. And, as you are about to observe, what follows here is going to put anyone but me—and the odd cruiser from CalTech if one should appear—dead to sleep in moments.)
The first thing you notice is the reduced formulae—good old semi-simple English. What this makes you realize is that these things need more explanation than even Field Medal specialists can immediately comprehend, via the traditional routes. And this get even better when you consider the Infinite. Perhaps the most telling quote comes from Georg Cantor. In 1907, he wrote the “Theory of Transfinite Numbers”, the well-nigh primer on the infinite series, and how to calculate something by adding in nothing (as far as I can figure, yes: that is an accurate statement). When he came to his first conclusions on the subject, he wrote: “To see is not to believe.” That should give you a good idea of what even the experts think about this. And one good reason is that: You can't get there from here.
(Remember, as Saint McLuhan said: The Medium is the Message.)
Which is how one might develop a keen sense of vertigo contemplating the macro and the micro in the universe. If I know one thing, it is that if Mandelbrot was right, and everything points to that, then the closer you look, the further you go into minute examination of any material substance, the more likely you are to find your own eye staring right back at you.
Exactly what is it that makes this sound absurd? Doubtless it is the idea that "you" are at the other end of, say, the Hubble telescope raised to the sky, to whatever the exponential of "nth" is. Sure, that's what sounds dumb: like a universe so massive might be something else...on another scale. (Why does that word keep coming up?) Let's try a metaphor from one of the world's leading physicists, Lee Smolin. A favorite quote about him is from the Nobel-prize winning discoverer of the quark itself, Murray Gell-Mann: "Is he the guy with all the crazy ideas? He might not be wrong." And of these, the craziest and most fascinating--also called (by Richard Dawkins) "gloriously Darwinian"--is his solution to The Goldilocks Problem. (You can figure out that one, right? No? Check out Martin Rees book "Just Six Little Numbers" and you'll see more that you can imagine.)
It goes like this: Universes give birth to baby universes in black holes. (So? Why not? All that matter and energy that goes in and nothing, not even information, comes out. It has to go somewhere, right?) And the "daughter" universes inherit the fundamental laws and constants of the parental physics. (Makes sense.) However, in the birth process, mutations do occur, giving rise to heterogenous populations of universes. Lineages of varying universes are subject to the kind of natural selection in favor of whatever traits assist survival. (Some last longer than others, giving them time to reproduce. And some are more likely to generate black holes than others.) And the process goes on.
So, this is crazy? Stop and add in what we have learned from Chaos Theory: that in all regimes of turbulence, the point at which surface tension breaks down into swirls of random energy, patterns begin to form within them and that demonstrates the principle of self-organizing units. There is no reason to think that this principle is any different on the marcocosmic scale (again) than any other.
Then, not so absurd, speculatively, eh?
Ok, what about absolutely scientifically, yeah? Something with numbers or measurements.
Perhaps, then, is it the idea that there is something which could be at two places at the same time? The dilemma here is Classical Physics vs. Quantum Physics. We know that both obey their laws in their realms enough so that we can predict, with some certainty, how matter and energy will behave, and what is possible and what isn't. In classical physics--no, you can't be in two places at the same time (and you can say that because you can use a word like "time" in Classical). In Quantum physics, however, if you can't tell where you are twice at the same elapse-of-photon-based-measurement-message-return (because you can't use a word like "time" in Quantum), whatever constitutes "you" (at the level where a Planck length would be the size of a tree...if an atom were the size of the known universe) just could (meaning: under the laws of probability) be.
Ok. Stupid pet tricks abound when one tries to combine these two. But, given the parameters of the question, you have to envision a tipping point between these realms and, though it is so far into the minimum existence you'd need a souped-up cyclotron to get any data, accept the fact that all things being equal--
And there's the real crux of the problem: all things ARE equal. Even if you DO use a Large Hadron Collider, there's no reason to think that either end of a scale differs substantively, especially when they are numbers. But it appears to be the same when speaking of the languages of geometry as well. One of the major points of “The Elegant Universe” is how string theory seems to solve a lot of the contradictions between the language of the Standard Model (Quantum) and General Relativity (everything else), and that, when using the term “language” is freely accepted as methods of defining both, the major “misunderstanding” between them is pretty much called “syntax”—that which we in the grammatical world say is "the study of the principles and processes by which sentences are constructed in particular languages". (Definition admittedly ripped off from Wiki.) The reason for the previous plural is that one of the things that has come to light in my investigation is that there really are different systems of syntax; whole entities unto themselves used to describe such discrete elements as nested planes, curved space, rotations and such things as "rolling" one system along another via special linkages. If you are like me, you probably stopped with Euclid and squares, triangles and circles...because when you got to the idea of number pi (aforementioned) you went wugga-wugga-woo.
Which is pretty much how Wilson's central solo sounds. It is said that Carl eschewed the central tenet of rock and didn't plug into an amp in later recordings; he just jacked into the mix and wailed into headphones. And here on the beach, that's perfect: a private experience of turbulence erupting into chaos the way it replicates the grind from the pluck as it soars up the shore in the tidal bore, extending the light in silver slivers of white, ending in the eager eagre that resolves into furze frizzle foam which crashes into sand and disappears forever. Small wonder people pay so much to retire to oceanfront properties; you'd have to be pretty dim to ever get tired of this.
And that leads me to someone who, probably, shares the same vision as I, and maybe Carl, but on a much more significant level. There's this physicist who surfs named Garrett Lisi. Or—a surfer who does physics named Garrett Lisi. It’s a tossup which he prefers, having spent as much time on the board as studying for Boards, it would appear (from his press at least). As much as I can figure, he's fluent in a number of geometric languages and, like some utility translator at the United Nations, he has heard the babble and seen, amid the mess of the signal-to-noise ratio, a message. It went something like this. Ten years or so ago he was working on spinor fields (mathematical representations of particles like electrons) which were traditionally expressed algebraically. Only he wondered how they would look in geometric terms.
Hold that thought.
In some distant pre-X system in the past, Apple included an app called a graphic calculator. Sure, it could do sums at tax time just fine, but it also had the additional feature that if you put in a formula with x, y, and z as coordinates, it would show you not only how the graph looked, but how it looked in 3-D! You could see saddles and puckers and ellipses and hyperbolas and, oh, a lot of neato shapes. Hours of fun for the whole family. It has been said often, but most pointedly by the string theory adherent and science-TV popularizer Michio Kaku, that one of the barriers to us understanding higher dimensions is that we lack the physical equipment—i.e., five senses and ONLY binocular vision—to see it. Even to process the visualization. Until computer graphics came along. This meant that what we could think of—more or less—we could make pictures of.
So, Lisi’s spinor fields actually fit in very nicely with a mathematical language for Clifford Bundles, usually used to describe rotations. Their most common use? Computer graphics.
And now you may see.
The rest of the terms are just as complex but they do indicate something very key to the whole so hold on. Curiosity getting the better of him, Lisi wonders what Clifford Bundles would “look like” (more as a speculation than an actual image, though) used to describe strong/weak/electromagnetic forces. To do this, he uses a 30-year-old description of gravity called the MacDowell-Mansouri Approach. (Too detailed to go into here. The MacDowell–Mansouri action is a mathematical object (an action) that is used to derive Einstein's field equations of general relativity. [Wiki theft] The upshot is that it incorporates into the flow from Euclidean Geometry to Riemannian geometry and Klein Geometry by adding into the chain Cartan Geometry—the last being, again, a way of taking maths and putting them into versatile shapes, but adding in point-to-point connections not previously accounted for.) Where this led was to the properties of exceptional Lie groups, and, in particular, one called E8, famous for usage in descriptions of higher-level symmetries. Lisi then zeroed in on the E8 group—described as a spiderweb cloud with thousands of strands exploding from hubs of concentric circles. Yes. Takes a minute.
It is easy to bandy about a lot of obscure physics-meets-math doubletalk and expect people to accept what you say on trust. I would greatly enjoy going into the reams of readings to back up my proofs, but, as this isn’t for MENSA, or a grad degree, I will state only that this world is not for the faint of heart or the easily dissuaded. However, the yield from such exploration is rather astonishing.
Consider JUST the case of Lisi, who—right or wrong—has this strangely elegant formulation which feels like sense. He had to go through four, or maybe five, languages, each with their own syntax, before getting at something that even looked like a resolve. But, even for the individual following the progression of the various geometries aforesaid, to arrive at the end and realize just HOW you got there? That’s intoxicating. And why, you may say, does this Lie Group E8 look more like “sense” than anything else? Go to the Wiki page for it and look at it. Even as it comes in (if you’re lucky, it loads slowly) you can see the lines developing much as they probably did in Photoshop or Illustrator layers. And when it finally assembles into the finished picture, if it doesn’t ring that bell of “supersymmetry”…well, I can’t help you further.
Of course, that doesn’t mean anything without some further proofs does it? It could all be a pure fiction of imagination, I’ll grant.
Did Newton get hit on the head by an apple? No. Did he see one fall and make the connection? Both are apocryphal but the latter is reputed as having some validity.
Does Lisi catching glassies mean anything? The more you watch waves, the more they say to you. I can only wonder that riding them might give even greater insight. He has come to the place where gravity is seen as a manifestation of force fields, and he did it before we got the announcement from CERN that--
--takes us back to the Higgs Boson.
(to be continued)