Sunday, November 30, 2014
being a zoo (or algebraic gadfly)
When we look at a photograph, we project a third dimension (depth, “going back” “in” the imagespace). 3-space is mapped into the 2-space photo. We see this because we live in 3-space—or rather, we see within the 3-space (4-space actually, of course) in which we move and live.
But someone who was congenitally blind, then gains sight has to learn to see 3-space. You take his hand (which is so familiar to him having done to him, as he has always lived by touch and embodied spatiality) and move it toward the glass on the table. His fingers touch the glass, and he takes hold of it immediately, as he’s always done—yet now amazed that there is the glass seen at the distance that his reach knows well. (Seeing in 3-space is learned by infants, as they also learn to focus).
Suppose an ant on a plane (flat surface, not a jet). Dependent on everything being in his plane, there is no such thing as something being above or below the plane. Above and below don’t exist. (This little fiction is stipulating absurd cognitive ability with severe constraints, but what the heck...). If a circle that is positioned above the plane (to we beings who know an x, y, z coordinate system) passes through the plane, the ant sees a point suddenly emerge, which becomes two points which move away from each other, each stopping, then moving back to each other, finally becoming a point that disappears.
The ant’s experience of the third dimension is like a shadow of the circle’s movement in 3-space. Better perhaps, the movement of points is a mapping of the circle’s movement into 2-space. The movement figures or models (within its available dimensions) that which is really the movement of a circle through the plane. ("Real" is relative to a given dimensionality.)
Likewise, someone living in a pure, eternal present sees something suddenly appear then disapplear, because it’s in time; but appearing where someone lives only the 3-dimensional aspect of time (as if there is an eternity without eternity, since the notion itself is relative to there being time).
Depictions of hyperspaces I've seen look very weird (like Escher on steroids). Only the mathematical architecturing (or hyper-“space”ing) is fair to its world.
The recent “proof” of gravity waves was depicted as a pattern of little line segments atop an image of measured cosmic background radiation because the claimed discovery was about a plasticity in space-time itself (a graphical point that is valid for dust in the data). The patterns of little line segments were like the weather maps we see where thousands of local wind vectors become Big Data for images of weather patterns. Gravity distorts space-time “like” a heavy weight (a planet) depresses a plane (space-time itself), creating a trough which causes an object to be drawn to the planet like a ball-bearing rolling “down” into a trough. The weighty planet-ball distorts the plane for the ball-bearing “like” gravity distorts space-time. These depictions, these tropes are “like” shadows of dimensions that are beyond the given dimensional paradigm.
It’s said that, relative to a 10-dimensionality of “strings” in string theory (or is it 11 dimensions?), each 4+n dimension is “wrapped around” its “lower” dimension. This is like referring to the little line segments in the gravitational wave “evidence” (recently recanted, but the trOpical claim remains what it is), as if gravitational waves are shaping space-time itself like heat waves shape the winds.
This introduces a way to understand a comment by Edward Frenkel, in the NYTimes last week, in an article that honored a recently-deceased genius of algebraic geometry. For me, Frenkel calls to mind a retort to the great physicist Richard Feynman, who had a wicked sense of humor: “Surely you’re joking Mr. Feynman!” He wasn’t joking.
Evidently, neither is Mr. Frenkel, a leading mathematical researcher in number theory (which has kinship with algebric geometry through “schemes”) when he refers to algebraic “spaces” as “shadows” of schemes.
In the Times article, Frenkel is referring to different solution spaces resulting from solving equations with real numbers vs. solving with imaginary numbers. “Thus, for a given equation we get a whole zoo of spaces.” That’s quite a lot! In string theory, there “is” a whole zoo of dimensions. What does it mean that there is a whole here?
“How are they related to one another and to the equation itself? Which came first, the equation or the space?” In other words, does the mathematics create the space or is there modeling of “existing” space? The notion of “came first” is mysterious. What if neither “came first”?
“These questions had perplexed mathematicians for centuries.” Are the objects of mathematics “real,” “ideal,” or nominal?
Apparently, Plato gets the prize. “Grothendieck’s genius was to recognize that”—he kids us not—”to recognize that there is a ‘being’ hiding behind a given algebraic equation (or a system of equations) called a scheme.” Presumably, “being” (keep the quote marks) is about a way of going on—a be-ing—rather than something, an Origin, a be-ing. Earlier in the article, Frenkel says (trOping how it goes with a zoo of solution spaces): “Visualize a living and breathing circle evolving in time.” This is apparently no mere changing through time, rather “evolving in time.” The “in” is an in-ing that would itself be outside of time. The way of going on, of scheme generativity, would be “hiding” (concealed, awaiting unconcealment or disclosure) relative to a system of equations.
However, who generates the zoo? Does the zoo exist without the mathematician? Is the mathematizability a function of there being mathematicality to mathematize? What is the mirrorplay of discovery and stipulation?
“The spaces of solutions,” Frenckel continues, “are mere projections, or shadows of this scheme”—mere projections, mere shadows—shadows that are projected, by the inanimate scheme? Or only by there being the projecter?
“Moreover, he realized that these schemes inhabit a rich world.” There is a world containing multiple schemes! This world would have to be a trans-schemic dimensionality (a domain ranging over schemicness) in which “They ‘interact’ with one another, can be ‘glued’ together and so on.” And so on? Does “so on” include refusing each other or abandoning each other? Do gods have sex?
O, well. So it goes with “the obsessive, sustained search for truth in its most universal and abstract form.”
Don’t you love it? (Frenkel's well-selling book is titled Love and Math.) Why does one search for a “most universal and abstract form”? Because one surmises, hopes, expects? that It “is” “there” to be—to be found? to be created?
When the Big Bang happened, space-time as such resulted. So, there is no point in our Universe at which It began, for Beginning became possible at that Timing, so to speak. The desire (need?) to retroject a First Point doesn’t arise from physics itself. Relative to an “anthropic” model of “our” Universe, It’s one that permits the physics that leads to the intelligent form of life that “discovers” zoos of spaces (and spins string theories and so on). The TOE—the Theory of Everything—appealing to us from prospected Higgs fields and so on (that too is now questioned by further examination of the data) could pertain to a multiverse in which the physics of ours is evolving—not only that the “life of the cosmos” is an evolving one, not only that Our Universe is evolving as such (i.e., Our Universe as such is evolving). More!: that the “laws” of physics are evolving.
But my narrative point is a simple recursive fold into the notion of evolving within the physical universe, as if there is evolving of physicality as such. The recursivity is conceptual—and fanciful. Or maybe there are multiple universes. After all—After All—The Universe is “expanding” in where? A Where that is beyond space-time as such?
Nonetheless, it’s also plausible that there are multiple universes, but that their evidence is intrinsically undecidable, “given” that—at this Point in physical research—given that the energy of the Higgs field is midway between, on the one hand, that which corroborates there being multiple universes (because the energy has a propensity for instability that implies multiple earlier, now-retrojected states, thus alternative universes?) and, on the other hand, there being The One (because the energy shows tendency toward more and more stability?). Therefore, we are a Goldilocks universe where both hands hold together Ultimate ambivalence of intelligibility?
Anyway, it’s delightful to entertain the resort to linguistic language to portray phenomena disclosed through mathematical “language.” This event of appropriation yields easy absurdity.
“...Grothendieck ‘had to understand things”—matters of mind—“from the most general possible point of view’,” as if the most general possible is a point of view: The Archimedean Point that there can be. “...[A]nd once he achieved that, everything ‘became so clear that proofs seemed almost trivial’.” This would be because the formalism is Complete. The proofs “emerge” from the softened dimensionality. (Grothendieck’s trope is that the nut shell has been softened so that the nut emerges as if from an easily-peeled avocado.) There is Completeness! Gödel was wrong? There is proof—though given a prover? (Formalism can't represent its own representability. It has no intentionality—though that wasn't Gödel's kind of proof that there is truth about a formalism as such in terms of the formalism, but that can't be proved in or by the formalism. So, what's the deal: There's transcendental logic? No. But there's There here.)
“Perhaps that’s why Grothendieck’s ideas ‘penetrated the unconscious of mathematicians’.” Mathematical unconscious! More to love, as if there’s no difference between the repressed (unconscious) and deep, deep implicity (e.g., a grammar relative to its “shadow” articulations; or a neuroelectrical field stability relative to phenomenalities “there” like shadows on a cave wall).
Oddly, Grothendieck abandoned mathematics, allegedly “at the height of his powers.” He moved into extended meditations and mysticism—and admirable political causes.
It’s a story of intelligence in Our Universe evolving beyond the shadows.