On the world's numerical recipe

All too often, we ignore goals, genres, or values, or we assume that they are so apparent that we do not bother to highlight them. Yet judgments about whether Note by Frank Wilczek an exercise — a paper, a project, an essay response on an examination - has been done intelligently or stupidly are often difficult for students to fathom. And since these evaluations are not well understood, few if any lessons can be drawn from them. Laying out the criteria by which judgments of quality are made may not suffice in itself to improve quality, but in the absence of such clarification, we have little reason to expect our students to go about their work ~ Frank Wilczek | on the world's numerical recipe intelligently. Twentieth-century physics began around 600 B.C. when Pythagoras of Samos proclaimed 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 + 254 MCxs"Z)E PE Frank Wilczek, Herman Feshbach Professor of Physics at MIT, is known, among other things, for the discovery of asymptoticfreedom, the development of quantum chromodynamics, the invention of axions, and the discovery and exploitation of newforms of quantum statistics (anyons). When only twenty-one years old and a graduate student at Princeton University, he and David Gross defined the properties of gluons, which hold atomic nuclei together. He has been a Fellow of the American Academy since 1993.

2:1 sounds a musical octave, 3:2 a musical fifth, and 4:3 a musical fourth. only to unravel completely. Students today still learn about 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 the Pythagorean Brotherhood, a mixedlaws, this great speculative thinker had announced another Jaw — 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 scientific academy. The Brotherhood was responsible for it is entirely wrong. Yet it was his discovery of the zeroth law that fired Kepler’s many fine 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 remains a staple of introductory geome- Kepler’s zeroth law concerns the relative try courses. It is also the point of departure for the Riemann-Einstein theories formulate it, we must imagine that the of curved space and gravity. spheres around the Sun. His law states Unfortunately, this very theorem undermined the Brotherhood’s credo. that the successive planetary spheres are Using the Pythagorean Theorem, it is not hard to prove that the ratio of the inscribed within and circumscribed hypotenuse of an isosceles right triangle to either of its two shorter sides cannot be expressed in whole numbers. A member of the Brotherhood who revealed this dreadful secret drowned shortly afterwards, in suspicious circumstances. remarkable solids - tetrahedron, cube, size of the orbits of different planets. To planets are carried about on concentric of such proportions that they can be about the five Platonic solids. These five octahedron, dodecahedron, icosahedron - have faces that are congruent equilateral polygons. The Pythagoreans studied them, Plato employed them Today, when we say V2 is irrational, our in the speculative cosmology of the Timaeus, and Euclid climaxed his Elelanguage still reflects these ancient anximents with the first known proof that eties. only five such regular polyhedra exist. Still, the Pythagorean vision, broadly understood — and stripped of cultic, if not entirely of mystical, trappings — remained for centuries a touchstone for pioneers of mathematical science. Those working within this tradition did not insist on whole numbers, but continued to postulate that the deep structure of the physical world could be captured in purely conceptual constructions. Considerations of symmetry and abstract geometry were allowed to supplement simple numerics. In the work of the German astronomer Johannes Kepler (1570-1630), this program reached a remarkable apotheosis - The world’s numerical recipe Kepler was enraptured by his discovery. He imagined that the spheres emitted music as they rotated, and he even speculated on the tunes. (This is the source of the phrase “music of the spheres.”) It was a beautiful realization of the Pythagorean ideal. Purely conceptual, yet sensually appealing, the zeroth law seemed a production worthy of a mathematically sophisticated Creator. To his great credit as an honest man and - though the concept is anachronistic-as a scientist, Kepler did not wallow in mystic rapture, but actively strove to see whether his law accurately matched reality. He discovered that it does not. In

Note by Frank Wilczek wrestling with the precise observations Sun, and the size is a factor 10"2? smallof Tycho Brahe, Kepler was forced to er; but the leitmotif of Bohr's model is give up circular in favor of elliptical orbits. He couldn’t salvage the ideas that first inspired him. After this, the Pythagorean vision went into a long, deep eclipse. In Newunmistakably “Things are Number.” Through Bohr’s model, Kepler's idea ton’s classical synthesis of motion and gravitation, there is no sense in which al ideal emerged from its embers, reborn like a phoenix, after three hundred years’ quiescence. If anything, Bohr’s structure is governed by numerical or model conforms more closely to the conceptual constructs. All is dynamics. Newton's laws inform us, given the posi- Pythagorean ideal than Kepler’s, since its preferred orbits are defined by whole tions, velocities, and masses of a system numbers rather than geometric conof gravitating bodies at one time, how they will move in the future. They do not fix a unique size or structure for the solar system. Indeed, recent discoveries of planetary systems around distant stars structions. Einstein responded with great empathy and enthusiasm, referring have revealed quite different patterns. Schrédinger, which defined modern The great developments of nineteenthcentury physics, epitomized in Maxquantum mechanics, superseded Bohr’s well’s equations of electrodynamics, to Bohr’s work as “the highest form of musicality in the sphere of thought.” Later work by Heisenberg and model. This account of subatomic matter is less tangible than Bohr’s, but ultibrought many new phenomena with the mately much richer. In the Heisenbergscope of physics, but they did not alter Schrédinger theory, electrons are no this situation essentially. There is nothlonger particles moving in space, elements of reality that at a given time are ing in the equations of classical physics that can fix a definite scale of size, whether for planetary systems, atoms, “just there and not anywhere else.” Rather, they define oscillatory, spaceor anything else. The world-system of classical physics is divided between inifilling wave patterns always “here, there, tial conditions that can be assigned arbitrarily, and dynamical equations. In attracted to a positively charged nucleus and can form localized standing wave those equations, neither whole numbers patterns around it. The mathematics and everywhere.” Electron waves are nor any other purely conceptual eledescribing the vibratory patterns that ments play a distinguished role. define the states of atoms in quantum Quantum mechanics changed everything. mechanics is identical to that which 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 Einuncanny 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 protons rather than planets orbiting the 144 that the orbits that occur in nature are precisely those that embody a conceptu- The big difference between nature’s instruments and those of human con-

struction is that her designs depend not on craftsmanship refined by experience, masses or their other properties; these were simply taken from experiment. but rather on the ruthlessly precise ap- That pragmatic approach was extremely plication of simple rules. Now if you browse through a textbook on atomic fruitful and to this day provides the quantum mechanics, or look at atomic of physics in chemistry, materials scivibration patterns using modern visualience, and biology. But it failed to proworking basis for practical applications zation tools, “simple” might not be the vide a theory that was in our sense simword that leaps to mind. But it has a precise, objective meaning in this context. ple, and so it left the ultimate ambitions of a Pythagorean physics unfulfilled. Atheory is simpler the fewer noncon- Starting in the early 1930s, with elecceptual elements, which must be taken from observation, enter into its controns under control, the frontier of fundamental physics moved inward, to the struction. In this sense, Kepler’s zeroth law provided a simpler (as it turns out, nuclei. This is not the occasion to recount the complex history of the heroic too simple) theory of the solar system than Newton’s, because in Newton’s theory the relative sizes of planetary constructions and ingenious deductions international effort, fully exposed the orbits must be taken from observation, secrets of this inaccessible domain. Forthat at last, after fifty years of strenuous whereas in Kepler’s they are determined tunately, the answer is easier to describe, conceptually. From this perspective, modern atomic theory is extraordinarily simple. The and it advances and consummates our Schrédinger equation, which governs is quantum chromodynamics, or QCD. electrons in atoms, contains just two As its name hints, QCD is firmly based nonconceptual quantities. These are the mass of the electron and the so-called fine-structure constant, denoted a, that on quantum mechanics. Its mathematispecifies the overall strength of the electromagnetic interaction. By solving this structure supporting enhanced symmeone equation, finding the vibrations it supports, we make a concept-world that reproduces a tremendous wealth of realas an icosahedron stands to a triangle. The basic players in QCD are quarks and world data, notably the accurately measured spectral lines of atoms that encode The world’s numerical recipe theme. The theory that governs atomic nuclei cal basis is a direct generalization of QED, incorporating a more intricate try. Metaphorically, QCD stands to QED gluons. For constructing an accurate model of ordinary matter just two kinds of quarks, called up and down or simply their inner structure. The marvelous theory of electrons and their interactions with light is called quantum elecu and d, need to be considered. (There are four other kinds, at least, but they trodynamics, or QED. for ordinary matter.) Protons, neutrons, are highly unstable and not important In the initial modeling of atoms, the © mesons, and a vast zoo of very shortfocus was on their accessible, outlying parts, the electron clouds. The nuclei of lived particles called resonances are conatoms, which contain most of their mass and all of their positive charge, were treated as so many tiny (but very heavy!) particles and resonances observed in black boxes, buried in the core. There concept-world of QCD, much as states was no theory for the values of nuclear of atoms match the resonant wave patstructed from these building blocks. The the real word match the resonant wave patterns of quarks and gluons in the

terns of electrons. You can predict their masses and properties directly by solv- Note by Frank Wilczek ing the equations. A peculiar feature of QCD, and a major reason why it was hard to discover, is that the quarks and gluons are never found in isolation, but always in complex associations. QCD actually predicts this “confinement” property, but that’s behavior of real-world matter. These not easy to prove. pleased. Considering how much it accounts for, QCD is an amazingly simple theory, in our objective sense. Its equations contain just three nonconceptual ingredients: the masses of the u and d quarks and the strong coupling constant AX, analogous to the fine structure constant of QED, which specifies how powerfully quarks couple to gluons. The gluons are automatically massless. Actually even three is an overestimate. objects are vibratory wave patterns. Stable elements of reality - protons, atomic nuclei, atoms - correspond, not just metaphorically but with mathematical precision, to pure tones. Kepler would be This tale continues in several directions. Given two more ingredients, Newton’s constant G, and Fermi’s constant Gy, which parametrize the strength of gravity and of the weak interaction, respectively, we can expand our conceptworld beyond ordinary matter to describe virtually all of astrophysics. There is a brilliant series of ideas involving unified field theories and supersymmetry that might allow us to get by with just The quark-gluon coupling varies with five ingredients. (Once you're down to distance, so we can trade it in for a unit of distance. In other words, mutant so few, each further reduction marks an QCDs with different values of a, genersively in coming years, especially as the ate concept-worlds that behave identi- Large Hadron Collider (LHC) at CERN, cally, but use different-sized metersticks. Also, the masses of the u and d quarks around 2007. turn out not to be very important, quantitatively. Most of the mass of strongly justice to the properties of many exotic, interacting particles is due to the pure short-lived particles discovered at highenergy of the moving quarks and gluons energy accelerators, things get much epoch.) These ideas will be tested decinear Geneva, swings into operation On the other hand, if we attempt to do they contain, according to the converse more complicated and unsatisfactory. of Einstein’s equation, m = E/c?. The We have to add pinches of many new inmasses of the u and d quarks are much gredients to our recipe, until it may seem smaller than the masses of the protons and other particles that contain them. that rather than deriving a wealth of insight from a small investment of facts, Putting all this together, we arrive at a we are doing just the opposite. That’s the most remarkable conclusion. To the state of our knowledge of fundamental extent that we are willing to use the prophysics today - simultaneously triton itself as a meterstick, and ignore the umphant, exciting, and a mess. small corrections due to the u and d quark masses, QCD becomes a theory with no nonconceptual elements whatsoever. Let me summarize. Starting with precisely four numerical ingredients, which must be taken from experiment, QED 146 and QCD cook up a concept-world of mathematical objects whose behavior matches, with remarkable accuracy, the Dedalus Winter 2002 The last word [ leave to Einstein: I would like to state a theorem which at present can not be based upon anything more than upon a faith in the simplicity, ¡.e., intelligibility, of nature: there are no

arbitrary constants... that is to say, nature is so constituted that it is possible logically to lay down such strongly determined laws that within these laws only rationally completely determined constants occur (not constants, therefore, whose numerical value could be changed without destroying the theory). David G. Nathan on clinical research & thefuture of medicine Biomedical inquiry as it is practiced in America today is an amalgam of three different kinds of research: basic research, population research, and clinical research. While all three are of critical importance, it is clinical research that underpins our national medical efforts. Only clinical researchers are able to apply the knowledge of the cell and organ systems developed by basic researchers, and the population data gathered by epidemiologists and biostatisticians, to patients, making this knowledge and data relevant to medical practice by “translating” it into novel David G. Nathan has been a Fellow of the American Academy since 1983. The Robert A. Stranahan Distinguished Professor of Pediatrics at Harvard Medical School and president emeritus of the Dana-Farber Cancer Institute, he was also physician in chief at the Children’s Hospital in Bostonfrom 1985 to 1995. A recipient of the National Medal of Sciencefor his research on blood disorders, he is also well known for his mentorship of young physicians during theformative years of their research careers.