“L’imagination se lassera plutot de concevoir que la nature de fournir.” (“Imagination tires [in conception] before nature does.”)—Blaise Pascal (French mathematician who wrote on projective geometry and later corresponded with Pierre de Fermat on probability theory)
“Nature’s fundamental laws do not govern the world as it appears in our mental picture…but control a substratum where we cannot form pictures without irrelevancies.”—P.A.M. Dirac (English theoretical physicist and Nobel co-winner with Erwin Schroedinger)
“They go, stop, start again, mount, descend, mount again, without the least tendency toward immobility.”—Jean Perrin, 1909 (Nobel Prize-winner for founding work on Brownian Motion)
“This endless embed of this shape into itself gives us an idea of what Tennyson describes somewhere as the Inner Infinity… Such similarity between the whole and its parts leads us to consider the Triadic Koch Island as marvelous. Had it been given life, it would not be possible to do away with it without destroying it altogether for it would rise again and again.”—Ernesto Cesaro, 1905 (Italian mathematician and pioneer in differential geometry)
“Two things are infinite: the universe and human stupidity; and I'm not sure about the universe.”
"A human being is a part of a whole, called by us universe, a part limited in time and space. He experiences himself, his thoughts and feelings as something separated from the rest...a kind of optical delusion of his consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us. Our task must be to free ourselves from this prison by widening our circle of compassion to embrace all living creatures and the whole of nature in its beauty. —Albert Einstein
“Infinity behaves differently from anything else, and is best avoided.”—Galileo
Regarding the Infinite, Pt. 2
The top-heavy quote-load for this missive is just to remind one and all just what is under discussion. And yes, last edition closed with a promise to return with the prime motivator of the entirety: the Higgs Boson. However, in order to get there, we have to sum up the journey, to gather all the far-flung assumptions under one roof (or page) because even I am still a little loose as to what’s going on…and what I’m trying to pull off.
So, because everybody should have their say, after all the scientists here’s a couple philosophers.
“Every idea, extended to infinity, becomes its own opposite.”—Hegel
“I should point out how Beginning and End meet together and how closely and intimately Eros and Death--?”—Schopenauer
And now, the one authentic mystic who was swirling around my consciousness at the outset. P.D. Ouspensky was, over the course of his life, a Russian radical, mathematician, Theosophist, one of the original followers of Gurdjieff, refugee, teacher, and cat fancier. The following is just one bit from his early, ground-breaking work “Tertium Organum” (1919).
“The mystery of infinity is that the visible universe has no dimension in comparison; that they are both equal to a point—a mathematical point which has no dimension whatever, and at the same time points which are not measureable as far as they may have different extensions and different dimensions.”
See? You have to love the fact that way back when, there were guys who were already anticipating the 10-dimensional universe of Calabi-Yau spaces. (I know. Well, maybe we’ll get to that and maybe you’ll just have to read the Green book.) What is also striking is that the second Einstein quote above so much resembles the entirety of “Tertium Organum” that it is almost as karmic as the other is comic.
As these notes bring out salient points, so do they wave. Each argument builds a sand castle; each wave turns it back to entropic mush. 'So why build them’, you ask? Because it wastes time, and it does so oh so wonderfully. ‘So why not build your sand castles further back from the ocean’, you ask? (Which is also my favorite restaurant on Fire Island—the Sand Castle in Cherry Grove, that is. Very good lobster bisque, friendly drag queen hostess, and nice place to view bikini bottoms, beach bums, and 2nd degree burns on the boardwalk above.) Because it is much too hard to carry water that far with the surface temperature of granulated silica is +/- a few degrees around pizza oven. And sand without water is merely dust. And the threat of imminent annihilation also makes the construction that much more precious. ‘So why’s it so wonderful and precious’, you ask?
Because it takes time.
Way back when this began, there was a concurrent free showing, at Lincoln Center, of Christian Marclay’s magnum opus, “The Clock”. Briefly, imagine you walk into a gallery, say Paula Cooper’s in Chelsea, on your average Saturday afternoon art-crawl, find a bunch of comfortable couches and a video playing. So you settle in and settle down, and notice that, like Mr. M’s previous masterpiece—“Quartet”—it is all made up of movie clips. If you came in without noting the title, is isn’t long before you see that every cut gets made around some form of time-keeping device (there’s even a sundial) or people mentioning the hour. And then comes the next revelation: it is—to the minute—the exact same one on your wrist chronometer! Once these two pieces of information come together…the world is your oyster. Or Rolex.
This was part and parcel of the original inspiration for this investigation. And why should it lead off the second part of “Infinity” (already a contradiction: how can something that is infinite be divided into two parts?)? Because that this installation also runs, in its complete version, 24 hours. I have no data on how many cuts there are, nor how many times certain movies show up, but still the mind reels, and unreels. Unreal. It is a loop of one day…and you can put it right up there next to James Joyce’s “Ulysseus” if you like for the sheer tour-de force that encompassing one sidereal Terran cycle can be, and this is.
There are few who have seen it in its entirety due to the fact that it is free admission (some people never leave…which also, oddly enough, brings up the central conceit of David Foster Wallace’s “Infinite Jest”—but if I were to go there you would truly learn the meaning of endlessness in detail), and yet, having seen enough of a few sections, it is easy to state in all confidence that one does not tire of free association either. Much like a Rorschach test, or the aforementioned “The Lady or The Tiger”, what you get from it is pretty much what you bring to it. Some may bring a wealth of knowledge of world cinema, some nothing more than curiosity. Others, a box lunch.
However, some may see it even more as a meditation on the nature of Time itself. Sure, it is all the same: one second after another, minutes, hours…and so it goes, and so it flows. Is it the same ticking towards the appointed hour in the execution chamber of Sing-Sing, awaiting a call from the governor? As a child in a schoolroom, staring from the open window to summer and back up to the magic hands approaching 3PM? As Bruno Antony going to plant Guy Haines’s watch at the amusement park while the tennis player, on the courts at Forest Hills, fights to end a furious match and catch the evening local out of old Pennsylvania Station to stop him in Hitchcock’s “Strangers on a Train”? As the Hilltown clocktower or the digital readout in the Doc’s Delorean in any of Marty McFly’s several choices of alternate realities available in “Back to the Future”? (…which coincidentally also started this mishegoss…) The closer you examine it, even psycho-socially, the more it resembles “Ulysseus”, to represent the human scale in infinity--or at least all we can comprehend of it in our lives: the time that doesn't stop.
And the more it resembles quantum mechanics.
Which brings us back to the Higgs Boson. (Thought we’d never get there, didn’t you?)
If you spent any time at the press conference [see link in pt. 3] you could not help but be charmed by Rolf Heuer, CERN’s director-general (how great is that title? Like he should have a uniform with epaulets, gold braid, a cocked hat with plumes!), who sounds like a second cousin to the Roamin’ Gnome and looks nearly as cute with same neatly-trimmed goatee. For someone whose focus is on nano-seconds of existence, he also has surprisingly broad, and articulate, views of the grander scale of things.
When, toward the end, he was asked about the merits of such things as the hunt for the Higgs, in a world where famine and disease are rampant, and the everyday necessities of life are still in contest, he put it into a succinct perspective. (Note: the attempt to capture the man’s style is, admittedly, caricature. But it is a marvelous accent. The rest is some paraphrase.) ‘Yah, vell, zhere is applied research and blue-sky science, and while it may be better to have more problems solved now… It is like you have a bag of rice, yah? Zso, you may eat all the rice today, but you will starve tomorrow. But if you plant it all today, you will starve before you can harvest, yah? …Zhen vhat you vant is a Virtuous Circle, yah? Not a Vicious Circle. One that feeds the other and increases the benefits to each, yah? Zso ze answer is somewhere in between.’
Between: a negotiation of present needs vs. future plans is the way the world works, and an answer given by someone who sounds like he has had to deal with it often in meetings with boards and committees. Allocations of resources. Projected schedules and realistic expectations. This is all the most practical aspect of scaling, as well. And the Virtuous Circle? A positive feedback loop; geometry that even a 4th-grader can get behind.
And when it comes to loops, there is one in particular which has been at the forefront of Western Civ. 101 since Athens opened the Academy: Zeno’s Paradox. Briefest: Achilles, the world’s fastest runner, challenges a tortoise to a race, but gives himself a handicap—he’ll run 100 meters and the tortoise will do ten centimeters. So—they’re off…but here’s the trick: for every half-length Achilles covers, so does the tortoise: and that’s what makes it impossible. As the closer they get to the finish line the more they slow down and Achilles can’t get ahead of the tortoise because the remaining length keeps breaking down into millimeters, micrometers, nanometers, etc. and it NEVER ENDS. (For those of you old enough to remember, this was exactly what your slide rule was telling you too. It’s called a logorhythmic progression, and one of the earliest examples of infinity.) This is the most human scale of all because the universe does not understand futility…only utility.
So you say, ‘Well, Greek philosophy or not. That’s impossible, in reality’, and you’d be right. Sure, we know about angular momentum and the mass of Achillies’ body in motion would give him additional kinetic velocity, etc...buuuut Newton hadn’t shown up yet. Nope, it was another thing entirely that Zeno didn’t factor in: Time. You see, the early Hellenes were mostly farmers, herders, and the agrarian lifestyle didn't need clocks, only seasons. The USA didn't institute a standardized national time until the advent of trains that had to run on schedules. (Yes. That WAS long ago.)
The reason the words “human scale” keep cropping up is that we don’t want to forget that philosophers from there to here have always emphasized that if you can’t find some way to make the abstract semi-concrete—put flesh on the bones, so to speak—then you’re not constructing a world, even sand castles; you’re living a fantasy.
Which is how mathematicians and physicists sometimes find themselves at odds, even. A fascinating presentation on this can be found in “The Character of Physical Law”—the collected Messenger Lectures (no: just a co-incidence. The guy who endowed them was named Messenger…but fun to play with, ain’t it?) given at Cornell University by the late, lamented physicist and all-round-fun-guy, Richard Feynman, in 1964.
Yup. Pretty long ago warn’t it? Uh-yeh. And that’s what’s so amazing: everything he says is coming is what shows up in “The Elegant Universe” and then some. And what doesn’t is as relevant today as it was nearly fifty years earlier. You get bonus insights everywhere, but some of the goodies include: when you find one real, solid physical law of the universe, it becomes a can-opener to others, many just via Einstein’s Gedankenexperiment (“thought experiment”). Like Archimedes said: Give me a fulcrum, and I shall move the world. But even the simple explanation of Newton’s Law of Gravity is a boon. F=G mm’/r(squared): force is proportional to the product (multiple) of the masses of two objects and inversely as the square of the distance between them. It may sound too technical as a formula but written out is makes perfect sense. (The closer you get the more you need to get closer, and faster. Simplest.)
And that’s another Feynman confirmation of what we already know: “The burden of this lecture is just to emphasize the fact that it is impossible to explain honestly the beauties of the laws of nature in a way that people can feel, without their having some deep understanding of mathematics. …You might say, ‘Alright, then if there is no explanation of the law, at least tell me what the law IS. Why not tell me in words instead of symbols? Mathematics is just a language, and I want to be able to translate that language’. …But I do not think that is possible, because mathematics is NOT just another language. Mathematics is language plus reasoning; it is like language plus logic. …In mathematics, it is possible to connect one statement with another.”
I could go on but what seems like a good place to segue is right here because that's how we, in proseland, connect OUR statements. See, even this series of arguments, propositions and pronouncements is only made up of language. And it is considered artful if the transitions between paragraphs have some, if only a token, of reasoning. (As to whether this works for yourself, well, your narrator isn’t even too sure if he likes a lot of them.) But the one thing to get at this junction is that the purpose of bringing up Garrett Lisi before was to show how he was able to bring together so many different mathematical and geometrical languages to reach his conclusions. What did was in math, yes, but there was also English language and, despite what Feynman says above, some things can be understood the same way that we read anything: you just have to figure out what some of the technical terms are. Whether Lisi's right or just a waverider—can’t say. But just trying to follow this reasoning set me off on a voyage of discovery not seen since map borders read ‘Here Be Dragons’. Whether or not they will prove to be valid is beyond the scope of this work; they are offered as the path I followed to get here.
And, boy!—does that take a lot of time!
Was that supposed to be funny? Nope. Human scale, again. I paid attention to the words and found answers that made sense. See, that’s what we forget about math: in the same way the Bible talks about the Word-of-God-made-Man (in Jesus, I gather) so is geometry just mathematics made flesh…and conversely, mathematics is geometry made thought. Thus, we can see postulate/proof/theorem in our everyday lives, if we only try to find them.
(Oho? Postulate or apostate? Who’s delivering the message here? Hang in there—we are making the turn; don’t fall out on the curves!)
Which is all very well and good to SAY, but if it has no use, turn it loose! Then, like all vacuous and empty statements (outside of those made with high television ratings among target demographics), it should and will fade out of the universe. “Thus the yeoman work in any science, and especially physics, is done by the experimentalist, who must keep the theoreticians honest.”— Michio Kaku. This is the same as saying (pretty much the same as in Greene’s book): without some empirical evidence, no matter how many theories a 10-D, superstring/superpartner particle satisfies, until it gets off the page and into CERN, we can’t use the term “existence”.
Which brings us back to? The Higgs? Messenger particles? Signals? Or, the other way around—Information Theory, Media Theory, Brane Theory? Either way , we end up at odds with the other. But that’s part of the fun too. “The test of a first-rate intelligence is the ability to hold two opposing ideas in mind at the same time and still retain the ability to function.”—F. Scott Fitzgerald
If you note a preponderance towards quotes in this one it is 1) to give credit where credit is due, and 2) to show that these premises, however intangible, were previously occupied. The one I can’t give attribution for is due to the fact that I can’t remember where I heard it, but, being so common, I figure I can just throw it down without: “A mark of genius is the ability to create analogies.” As true as that is, it gets better when you change analogies to metaphors, because metaphors are being able to take a lesson from one situation and apply it to another…sorta like Scotty says above, but with a twist. The most popular metaphorical reasoner of the present day is/was Gregory House, MD, who could take a crime scenario and apply it to liver failure, or such. What we enjoyed so much about this was the immediate connection we could make to the story being told by House, and see the way this also lit up the faces of his associates.
“In every work of genius we recognize our own rejected thoughts: they come back to us with a certain alienated majesty.”—Ralph Waldo Emerson.
Ok. I think its time to stop with this summary of judgments, lest you think all these citation of “genius” reference are meant to imply your not-so-humble narrator.
There is, then, one more bit of intelligence to gather before the last part. What follows is something of a “vocabulary lesson”. It is not meant for the ardent fan of inside field jargon, but neither is it more than a just a bit arcane for the amateur ponderer. It is more to remind me of the journey to get here from there. It may not even be germane to the next portion…but probably will. Otherwise, I wouldn’t feel compelled to explain it to myself.
* * *
- The first thing to establish is point = particle. This is the basis of math-to-geometry and geometry-to-math: physics talks about particles, but the only measure (to “get the message”, which, beyond the photon-return-of-data thang that everyone agrees is so barely acceptable they had to create quantum physics to do it) they can make is in points. (Now, it DOES show up as a 14-tetravolt spike, true, but...*) And it is the same in an x-y-z coordinate graph as it is in a Cantor set of transfinite numbers. (And yes, I’d love to go into that too, but it would be a needless expenditure in blood & treasure and other valuable resources so unless you enjoyed the quagmire of Vietnam, or Afghanistan, I feel it best to drop the threat of quotient equations and walk away.) As for a wave? Math doesn’t make that distinction; only physics. (*...but, everything else between the first asterisk and this one IS speaking of a wave, only as a graphic element, which is--again!--geometrical. In physics, their only interest besides frequency is whether it is enough to cook your tv dinner, or show if your tooth has an absess.)
- Starting at the primal terms, a topological space is composed of a homogenous (smooth) set of points, which is the same as metric space—meaning measurable via distances between points as vectors and so, in laws of motion, velocities. When talking about Klein geometry or Riemannian geometry or Cartan geometry or even pseudo-Riemannian geometry, this is all that they are talking about: the stuff that matter is (more or less) made of. (In a pseudo-Riemannian inner product of the metric doesn’t even have to be positive-definite—meaning: exactly exact copy—as long as it is still isomorphic: two sets that are indistinguishable given only a selection of their features.)
- Klein stepped away from Euclid by saying that each geometric language had its own appropriate concepts: meaning—you could talk conic sections (as in ice cream cones, street hazards, party hats, etc.) in projective geometry but not in plane geometry. Q.E.D., yah? Stuff that projects OFF a surface? But you could bring them back together by way of subgroups of related symmetry. (Again: no brainer. If certain parts behave alike, or look alike, those parts are probably the same and can be treated via the same equations. Like angles with curves on a plane may have similar measurements to projected cones.)
- Klein geometry is then a space and a law of motion within that space. Nothing fancy there either. So when a velocity is included in this set of points, the space takes on the values of a vector space. (A vector is no different than the Euclidan “ray”—like a line with an arrowhead, yah?, where something can be measured between point A and point B. A ray like in a "radius", but without a circle.)
- It was Riemann, however, who threw the curve, literally. WIKI sez his language is one: “concerning the geometry of surfaces and the behavior of geodesics on them…applied to the study of differentiable manifolds of higher dimensions.” This may sound daunting but it ain’t; you just have to use creative visualization, remember? There’re two things here to make pictures of, in your mind. One is a “geodesic”. Those are the half-spheres of Buckminster Fuller’s 1960s projects, Expo ’67, some of our favorite sci-fi movies from “Silent Running” to “Logan’s Run”, and even the Biosphere experiment from a few years back. For our purpose, the best definition is “curves whose tangent vectors remain parallel if they are transported along it”—“it” being the curves. Which probably should have read as “them”, but hey? These are scientists, not grammarians. The other is “manifolds”. Riemann’s “tangent vectors are orthogonal (just what you think it is: visualize an architectural drawing of a house where it shows a corner and two sides; the base line has the 90-degree perpendicular and twin angles of 30-degrees to represent the bottom of the two walls, which make up parallelograms with angles of 60-degrees each. See it? That's what they call an "orthogonal view".) and orthonormal (same thing as isomorphic: see above) frames”…and that’s all smooth manifolds are! “MANY-FOLDS”, so to speak. Think of an accordion, or better, a bandoneon—one of those biiig Argentinian tango squeezeboxes. Or a slinky. They all maintain an integrity of form even when moving their orthonormal frames/manifolds around.
- So what does this have to do with “points”? They create an “inner product space”, which is STILL point-as-particles, far as we’re concerned. It may have bee Klein who put up the space-w/law-of-motion idea, but it was Riemann who made the expressions HOW stuff EXISTS and MOVES in curved space. And that’s what got Einstein stroking his mustache and thinking, “…hmmmm…ach zso…”
- So, the whole reason for this trip through obscure technical terms was to follow the thorny path to see how Lisi got from there to here. Remember “spinors”? The one thing to take away from the mindset of the surfin’ physicist is a knowledge that spinors on a manifold require a spin structure. So think of those orthogonal frames—manifolds—in rotation and suddenly it begins to look like Quarkville. Or Boson Square. The Clifford Bundles are what he figured would act on the manifolds through an affine connection—which isn’t that different from “affinity”, like all the manifolds (ortho-frames) move in parallel transport (equidistant from one another in their spatial relationships, right?—again: think accordion pleats, steady, regular, and even).
- Now, where Lisi ended up with was the McDowelll-Mansori Approach to Gravity. So what makes this so interesting? It is Cartan geometry. That’s where the affine connection comes in: the geometry of the principal bundle (say Clifford, for one) is tied to the base manifold (remember? The ¾ view of the house? That’s the same base, even if it isn’t flat) via an affine connection. In the McDowelll-Mansori Approach it is called a “solder connection”. (And why is it called a “solder conncetion”? See for yourself. That's the thing connecting the two wireframe wastebaskets above.) What makes it even better is that flat Cartan geometry is the same as flat Klein geometry. And a Riemann is just a flat Klein with an inner product space (where all the ortho-frames/manifolds--easier to see now how each one of those "wires" IS an OF/manifold, isn't it?) are all made up of similar points) and because the LAW OF MOTION STILL APPLIES.
- This isn’t opaque if I can see it; trust me. Go all the way back to Cantor and you’ll see that, like the Triadic Koch Island cited in the Cesaro quote above—a rumination of a fractal series of space-filling replications via many scale transformations—the self-similarity of the orthonormal bases means that points can exist in equal, or even slightly unequal, neighborhoods and still behave in manifolds… Like every atom can be slightly dissimilar and still follow Klein's Law of Motion.
- …and that’s all Cartan connections promise: a method of moving frames. Pretty simple.
- All the previous was just to get to this premise. And why? Because Cartan Geometry appears to be the final language to regard the whole, as described here—and in doing so, the infinite.
- Now take another look at that solder connection again. It connects two Riemannian manifold spaces, ok? And what does it DO? It relates the movements of one to the other, sorta like two bandoneons in orbit: one flowing concave, the other convex, tied into on seamless motions.
- And that’s very much like gravity.
- …which brings us back to… (No. For real. It HAS to…)