Frank Wilczek on the World's Numerical Recipe

Note by All too often, we ignore goals, genres, or Frank Wilczek values, or we assume that they are so apparent that we do not bother to high- light them. Yet judgments about whether an exercise–a paper, a project, an essay Frank Wilczek response on an examination–has been done intelligently or stupidly are often on the dif½cult for students to fathom. And since these evaluations are not well world’s understood, few if any lessons can be numerical drawn from them. Laying out the criteria recipe by which judgments of quality are made may not suf½ce in itself to improve quality, but in the absence of such clari½- cation, we have little reason to expect our students to go about their work intelligently. Twentieth-century physics began around 600 b.c. when Pythagoras of Samos pro- claimed an awesome vision. By studying the notes sounded by plucked strings, Pythagoras discovered that the human perception of harmony is connected to numerical ratios. He examined strings made of the same material, having the same thickness, and under the same tension, but of different lengths. Under these conditions, he found that the notes sound harmonious precisely when the ratio of the lengths of string can be expressed in small whole numbers. For example, the length ratio Frank Wilczek, Herman Feshbach Professor of Physics at MIT, is known, among other things, for the discovery of asymptotic freedom, the develop- ment of quantum chromodynamics, the invention of axions, and the discovery and exploitation of new forms of quantum statistics (anyons). When only twenty-one years old and a graduate student at Princeton University, he and David Gross de½ned the properties of gluons, which hold atomic nuclei together. He has been a Fellow of the American Academy since 1993. 142 Dædalus Winter 2002 2:1 sounds a musical octave, 3:2 a musi- only to unravel completely. The world’s numerical cal ½fth, and 4:3 a musical fourth. Students today still learn about recipe The vision inspired by this discovery Kepler’s three laws of planetary motion. is summed up in the maxim “All Things But before formulating these celebrated are Number.” This became the credo of laws, this great speculative thinker had the Pythagorean Brotherhood, a mixed- announced another law–we can call it sex society that combined elements of Kepler’s zeroth law–of which we hear an archaic religious cult and a modern much less, for the very good reason that scienti½c academy. it is entirely wrong. Yet it was his discov- The Brotherhood was responsible for ery of the zeroth law that ½red Kepler’s many ½ne discoveries, all of which it enthusiasm for planetary astronomy, in attributed to Pythagoras. Perhaps the particular for the Copernican system, most celebrated and profound is the and launched his extraordinary career. Pythagorean Theorem. This theorem Kepler’s zeroth law concerns the relative remains a staple of introductory geome- size of the orbits of different planets. To try courses. It is also the point of depar- formulate it, we must imagine that the ture for the Riemann-Einstein theories planets are carried about on concentric of curved space and gravity. spheres around the Sun. His law states Unfortunately, this very theorem that the successive planetary spheres are undermined the Brotherhood’s credo. of such proportions that they can be Using the Pythagorean Theorem, it is inscribed within and circumscribed not hard to prove that the ratio of the about the ½ve Platonic solids. These ½ve hypotenuse of an isosceles right triangle remarkable solids–tetrahedron, cube, to either of its two shorter sides cannot octahedron, dodecahedron, icosahe- be expressed in whole numbers. A mem- dron–have faces that are congruent ber of the Brotherhood who revealed equilateral polygons. The Pythagoreans this dreadful secret drowned shortly studied them, Plato employed them afterwards, in suspicious circumstances. in the speculative cosmology of the Today, when we say Ö2 is irrational, our Timaeus, and Euclid climaxed his Ele- language still reflects these ancient anxi- ments with the ½rst known proof that eties. only ½ve such regular polyhedra exist. Still, the Pythagorean vision, broadly Kepler was enraptured by his discov- understood–and stripped of cultic, if ery. He imagined that the spheres emit- not entirely of mystical, trappings– ted music as they rotated, and he even remained for centuries a touchstone for speculated on the tunes. (This is the pioneers of mathematical science. Those source of the phrase “music of the working within this tradition did not spheres.”) It was a beautiful realization insist on whole numbers, but continued of the Pythagorean ideal. Purely concep- to postulate that the deep structure of tual, yet sensually appealing, the zeroth the physical world could be captured in law seemed a production worthy of a purely conceptual constructions. Con- mathematically sophisticated Creator. siderations of symmetry and abstract To his great credit as an honest man geometry were allowed to supplement and–though the concept is anachronis- simple numerics. tic–as a scientist, Kepler did not wallow In the work of the German astronomer in mystic rapture, but actively strove to Johannes Kepler (1570–1630), this pro- see whether his law accurately matched gram reached a remarkable apotheosis– reality. He discovered that it does not. In Dædalus Winter 2002 143 Note by wrestling with the precise observations -22 Frank Sun, and the size is a factor 10 small- Wilczek of Tycho Brahe, Kepler was forced to er; but the leitmotif of Bohr’s model is give up circular in favor of elliptical unmistakably “Things are Number.” orbits. He couldn’t salvage the ideas that Through Bohr’s model, Kepler’s idea ½rst inspired him. that the orbits that occur in nature are After this, the Pythagorean vision precisely those that embody a conceptu- went into a long, deep eclipse. In New- al ideal emerged from its embers, reborn ton’s classical synthesis of motion and like a phoenix, after three hundred gravitation, there is no sense in which years’ quiescence. If anything, Bohr’s structure is governed by numerical or model conforms more closely to the conceptual constructs. All is dynamics. Pythagorean ideal than Kepler’s, since Newton’s laws inform us, given the posi- its preferred orbits are de½ned by whole tions, velocities, and masses of a system numbers rather than geometric con- of gravitating bodies at one time, how structions. Einstein responded with they will move in the future. They do not great empathy and enthusiasm, referring ½x a unique size or structure for the solar to Bohr’s work as “the highest form of system. Indeed, recent discoveries of musicality in the sphere of thought.” planetary systems around distant stars Later work by Heisenberg and have revealed quite different patterns. Schrödinger, which de½ned modern The great developments of nineteenth- quantum mechanics, superseded Bohr’s century physics, epitomized in Max- model. This account of subatomic mat- well’s equations of electrodynamics, ter is less tangible than Bohr’s, but ulti- brought many new phenomena with the mately much richer. In the Heisenberg- scope of physics, but they did not alter Schrödinger theory, electrons are no this situation essentially. There is noth- longer particles moving in space, ele- ing in the equations of classical physics ments of reality that at a given time are that can ½x a de½nite scale of size, “just there and not anywhere else.” whether for planetary systems, atoms, Rather, they de½ne oscillatory, space- or anything else. The world-system of ½lling wave patterns always “here, there, classical physics is divided between ini- and everywhere.” Electron waves are tial conditions that can be assigned arbi- attracted to a positively charged nucleus trarily, and dynamical equations. In and can form localized standing wave those equations, neither whole numbers patterns around it. The mathematics nor any other purely conceptual ele- describing the vibratory patterns that ments play a distinguished role. de½ne the states of atoms in quantum Quantum mechanics changed every- mechanics is identical to that which thing. describes the resonance of musical Emblematic of the new physics, and instruments. The stable states of atoms decisive historically, was Niels Bohr’s correspond to pure tones. I think it’s fair atomic model of 1913. Though it applies to say that the musicality Einstein in a vastly different domain, Bohr’s praised in Bohr’s model is, if anything, model of the hydrogen atom bears an heightened in its progeny (though Ein- uncanny resemblance to Kepler’s system stein himself, notoriously, withheld his of planetary spheres. The binding force approval from the new quantum is electrical rather than gravitational, the mechanics). players are electrons orbiting around The big difference between nature’s protons rather than planets orbiting the instruments and those of human con- 144 Dædalus Winter 2002 struction is that her designs depend not masses or their other properties; these The world’s numerical on craftsmanship re½ned by experience, were simply taken from experiment. recipe but rather on the ruthlessly precise ap- That pragmatic approach was extremely plication of simple rules. Now if you fruitful and to this day provides the browse through a textbook on atomic working basis for practical applications quantum mechanics, or look at atomic of physics in chemistry, materials sci- vibration patterns using modern visuali- ence, and biology. But it failed to pro- zation tools, “simple” might not be the vide a theory that was in our sense sim- word that leaps to mind. But it has a pre- ple, and so it left the ultimate ambitions cise, objective meaning in this context. of a Pythagorean physics unful½lled.

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