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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- 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 qual- ity, but in the absence of such clari½- cation, we have little reason to expect our students to go about their work intelligently.

Twentieth-century 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, Professor of Physics at MIT, is known, among other things, for the discovery of , the develop- ment of , the invention of , and the discovery and exploitation of new forms of quantum statistics (). When only twenty-one years old and a graduate student at , he and de½ned the properties of , which hold atom- ic 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 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- , 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 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 ’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. A theory is simpler the fewer noncon- Starting in the early 1930s, with elec- ceptual elements, which must be taken trons under control, the frontier of fun- from observation, enter into its con- damental physics moved inward, to the struction. In this sense, Kepler’s zeroth nuclei. This is not the occasion to re- law provided a simpler (as it turns out, count the complex history of the heroic too simple) theory of the solar system constructions and ingenious deductions than Newton’s, because in Newton’s that at last, after ½fty years of strenuous theory the relative sizes of planetary international effort, fully exposed the orbits must be taken from observation, secrets of this inaccessible domain. For- whereas in Kepler’s they are determined tunately, the answer is easier to describe, conceptually. and it advances and consummates our From this perspective, modern atomic theme. theory is extraordinarily simple. The The theory that governs atomic nuclei Schrödinger equation, which governs is quantum chromodynamics, or qcd. electrons in atoms, contains just two As its name hints, qcd is ½rmly based nonconceptual quantities. These are the on quantum mechanics. Its mathemati- of the electron and the so-called cal basis is a direct generalization of ½ne-structure constant, denoted a, that qed, incorporating a more intricate speci½es the overall strength of the elec- structure supporting enhanced symme- tromagnetic interaction. By solving this try. Metaphorically, qcd stands to qed one equation, ½nding the vibrations it as an icosahedron stands to a triangle. supports, we make a concept-world that The basic players in qcd are and reproduces a tremendous wealth of real- gluons. For constructing an accurate world data, notably the accurately meas- model of ordinary matter just two kinds ured spectral lines of atoms that encode of quarks, called up and down or simply their inner structure. The marvelous u and d, need to be considered. (There theory of electrons and their interac- are four other kinds, at least, but they tions with light is called quantum elec- are highly unstable and not important trodynamics, or qed. for ordinary matter.) Protons, neutrons, In the initial modeling of atoms, the p mesons, and a vast zoo of very short- focus was on their accessible, outlying lived particles called resonances are con- parts, the electron clouds. The nuclei of structed from these building blocks. The atoms, which contain most of their mass particles and resonances observed in and all of their positive charge, were the real word match the resonant wave treated as so many tiny (but very heavy!) patterns of quarks and gluons in the 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 pat-

Dædalus Winter 2002 145 Note by terns of electrons. You can predict their and qcd cook up a concept-world of Frank Wilczek masses and properties directly by solv- mathematical objects whose behavior ing the equations. matches, with remarkable accuracy, the A peculiar feature of qcd, and a major behavior of real-world matter. These reason why it was hard to discover, is objects are vibratory wave patterns. Sta- that the quarks and gluons are never ble elements of reality–protons, atomic found in isolation, but always in com- nuclei, atoms–correspond, not just plex associations. qcd actually predicts metaphorically but with mathematical this “con½nement” property, but that’s precision, to pure tones. Kepler would be not easy to prove. pleased. Considering how much it accounts for, This tale continues in several direc- qcd is an amazingly simple theory, in tions. Given two more ingredients, New- our objective sense. Its equations con- ton’s constant gn and Fermi’s constant tain just three nonconceptual ingredi- gf, which parametrize the strength of ents: the masses of the u and d quarks gravity and of the , a and the strong s, respectively, we can expand our concept- analogous to the ½ne structure constant world beyond ordinary matter to de- of qed, which speci½es how powerfully scribe virtually all of . There quarks couple to gluons. The gluons are is a brilliant series of ideas involving uni- automatically massless. ½ed ½eld theories and supersymmetry Actually even three is an overestimate. that might allow us to get by with just The - coupling varies with ½ve ingredients. (Once you’re down to distance, so we can trade it in for a unit so few, each further reduction marks an of distance. In other words, mutant epoch.) These ideas will be tested deci- a qcds with different values of s gener- sively in coming years, especially as the ate concept-worlds that behave identi- Large Hadron Collider (lhc) at cern, cally, but use different-sized metersticks. near Geneva, swings into operation Also, the masses of the u and d quarks around 2007. turn out not to be very important, quan- On the other hand, if we attempt to do titatively. 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 high- energy of the moving quarks and gluons energy accelerators, things get much they contain, according to the converse more complicated and unsatisfactory. of Einstein’s equation, m = E/c2. The We have to add pinches of many new in- masses of the u and d quarks are much gredients to our recipe, until it may seem smaller than the masses of the protons that rather than deriving a wealth of and other particles that contain them. 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 pro- physics today–simultaneously tri- ton itself as a meterstick, and ignore the umphant, exciting, and a mess. small corrections due to the u and d The last word I leave to Einstein: quark masses, qcd becomes a theory I would like to state a theorem which at with no nonconceptual elements whatsoever. present can not be based upon anything Let me summarize. Starting with pre- more than upon a faith in the simplicity, cisely four numerical ingredients, which i.e., intelligibility, of nature: there are no must be taken from experiment, qed

146 Dædalus Winter 2002 arbitrary constants . . . that is to say, nature is so constituted that it is possible logical- ly to lay down such strongly determined laws that within these laws only rationally completely determined constants occur David G. Nathan (not constants, therefore, whose numeri- cal value could be changed without on clinical destroying the theory). research & the future 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 clini- cal research. While all three are of criti- cal 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 biosta- tisticians, 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 Ameri- can Academy since 1983. The Robert A. Strana- han Distinguished Professor of Pediatrics at Har- vard Medical School and president emeritus of the Dana-Farber Cancer Institute, he was also physician in chief at the Children’s Hospital in Boston from 1985 to 1995. A recipient of the National Medal of Science for his research on blood disorders, he is also well known for his men- torship of young physicians during the formative years of their research careers.

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