by James Gleick
Science is making us wander,
Wander, far and wide;
And know, by this time,
Our face we ought to hide.
Some day, the mountain shall wither,
While the valleys get flooded with fire;
Or men shall be driven like horses,
And stamper, like beasts, in the mire.
And we say, “The earth was thrown from the sun,”
Or, “Evolution made us come to be
And we come from lowest of beasts,
Or one step back, the ape and monkey.”
Our minds are thinking of science,
And science is in our ears;
Our eyes are seeing science,
And science is in our fears.
Yes, we’re wandering from the Lord our God,
Away from the Holy One;
But now we cannot help it,
For it is already done.
But poetry was (Richard thought) “sissy-like.” This was no small problem. He suffered grievously from the standard curse of boy intellectuals, the fear of being thought, or of being, a sissy. He thought he was weak and physically awkward. In baseball he was inept. The sight of a ball rolling toward him across a street filled him with dread. Piano lessons dismayed him, too, not just because he played so poorly, but because he kept playing an exercise called “Dance of the Daisies.” For a while this verged on obsession. Anxiety would strike when his mother sent him to the store for “peppermint patties.”
As a natural corollary he was shy about girls. He worried about getting in fights with stronger boys. He tried to ingratiate himself with them by solving their school problems or showing how much he knew. He endured the canonical humiliations: for example, watching helplessly while some neighborhood children turned his first chemistry set into a brown, useless, sodden mass on the sidewalk in front of his house. He tried to be a good boy and then worried, as good boys do, about being too good—“goody-good.” He could hardly retreat from intellect to athleticism, but he could hold off the taint of sissiness by staying with the more practical side of the mental world, or so he thought. The practical man—that was how he saw himself. At Far Rockaway High School he came upon a series of mathematics primers with that magical phrase in the title—Arithmetic for the Practical Man; Algebra for the Practical Man—and he devoured them. He did not want to let himself be too “delicate,” and poetry, literature, drawing, and music were too delicate. Carpentry and machining were activities for real men.
For students whose competitive instincts could not be satisfied on the baseball field, New York’s high schools had the Interscholastic Algebra League: in other words, math team. In physics club Feynman and his friends studied the wave motions of light and the odd vortex phenomenon of smoke rings, and they re-created the already classic experiment of the California physicist Robert Millikan, using suspended oil drops to measure the charge of a single electron. But nothing gave Ritty the thrill of math team. Squads of five students from each school met in a classroom, the two teams sitting in a line, and a teacher would present a series of problems. These were designed with special cleverness. By agreement they could require no calculus—nothing more than standard algebra—yet the routines of algebra as taught in class would never suffice within the specified time. There was always some trick, or shortcut, without which the problem would just take too long. Or else there was no built-in shortcut; a student had to invent one that the designer had not foreseen.
According to the fashion of educators, students were often taught that using the proper methods mattered more than getting the correct answer. Here only the answer mattered. Students could fill the scratch pads with gibberish as long as they reached a solution and drew a circle around it. The mind had to learn indirection and flexibility. Head-on attacks were second best. Feynman lived for these competitions. Other boys were president and vice president, but Ritty was team captain, and the team always won. The team’s number-two student, sitting directly behind Feynman, would calculate furiously with his pencil, often beating the clock, and meanwhile he had a sensation that Feynman, in his peripheral vision, was not writing—never wrote, until the answer came to him. You are rowing a boat upstream. The river flows at three miles per hour; your speed against the current is four and one-quarter. You lose your hat on the water. Forty-five minutes later you realize it is missing and execute the instantaneous, acceleration-free about-face that such puzzles depend on. How long does it take to row back to your floating hat?
A simpler problem than most. Given a few minutes, the algebra is routine. But a student whose head starts filling with 3s and 4¼s, adding them or subtracting them, has already lost. This is a problem about reference frames. The river’s motion is irrelevant—as irrelevant as the earth’s motion through the solar system or the solar system’s motion through the galaxy. In fact all the velocities are just so much foliage. Ignore them, place your point of reference at the floating hat—think of yourself floating like the hat, the water motionless about you, the banks an irrelevant blur—now watch the boat, and you see at once, as Feynman did, that it will return in the same forty-five minutes it spent rowing away. For all the best competitors, the goal was a mental flash, achieved somewhere below consciousness. In these ideal instants one did not strain toward the answer so much as relax toward it. Often enough Feynman would get this unstudied insight while the problem was still being read out, and his opponents, before they could begin to compute, would see him ostentatiously write a single number and draw a circle around it. Then he would let out a loud sigh. In his senior year, when all the city’s public and private schools competed in the annual championship at New York University, Feynman placed first.
For most people it was clear enough what mathematics was—a cool body of facts and rote algorithms, under the established headings of arithmetic, algebra, geometry, trigonometry, and calculus. A few, though, always managed to find an entry into a freer and more colorful world, later called “recreational” mathematics. It was a world where rowboats had to ferry foxes and rabbits across imaginary streams in nonlethal combinations; where certain tribespeople always lied and others always told the truth; where gold coins had to be sorted from false-gold in just three weighings on a balance scale; where painters had to squeeze twelve-foot ladders around inconveniently sized corners. Some problems never went away. When an eight-quart jug of wine needed to be divided evenly, the only measures available were five quarts and three. When a monkey climbed a rope, the end was always tied to a balancing weight on the other side of a pulley (a physics problem in disguise). Numbers were prime or square or perfect. Probability theory suffused games and paradoxes, where coins were flipped and cards dealt until the head spun. Infinities multiplied: the infinity of counting numbers turned out to be demonstrably smaller than the infinity of points on a line. A boy plumbed geometry exactly as Euclid had, with compass and straightedge, making triangles and pentagons, inscribing polyhedra in circles, folding paper into the five Platonic solids. In Feynman’s case, the boy dreamed of glory. He and his friend Leonard Mautner thought they had found a solution to the problem of trisecting an angle with the Euclidean tools—a classic impossibility. Actually they had misunderstood the problem: they could trisect one side of an equilateral triangle, producing three equal segments, and they mistakenly assumed that the lines joining those segments to the far corner mark off equal angles. Riding around the neighborhood on their bicycles, Ritty and Len excitedly imagined the newspaper headlines: “Two Children in High School First Learning Geometry Solve the Age-Old Problem of the Trisection of the Angle.”
This cornucopian world was a place for play, not work. Yet unlike its stolid high-school counterpart it actually connected here and there to real, adult mathematics. Illusory though the feeling was at first, Feynman had the sense of conducting research, solving unsolved problems, actively exploring a live frontier instead of passively receiving the wisdom of a dead era. In school every problem had an answer. In recr
eational mathematics one could quickly understand and investigate problems that were open. Mathematical game playing also brought a release from authority. Recognizing some illogic in the customary notation for trigonometric functions, Feynman invented a new notation of his own: √x for sin √x for cos (x), √x for tan (x). He was free, but he was also extremely methodical. He memorized tables of logarithms and practiced mentally deriving values in between. He began to fill notebooks with formulas, continued fractions whose sums produced the constants π and e.
A page from one of Feynrnan's teenage notebooks.
A month before he turned fifteen he covered a page with an elated inch-high scrawl:
THE MOST REMARKABLE
FORMULA
IN MATH.
eiπ + 1 = 0
(FROM SCIENCE HISTORY OF THE UNIVERSE)
By the end of this year he had mastered trigonometry and calculus, both differential and integral. His teachers could see where he was heading. After three days of Mr. Augsbury’s geometry class, Mr. Augsbury abdicated, putting his feet up on his desk and asking Richard to take charge. In algebra Richard had now taught himself conic sections and complex numbers, domains where the business of equation solving acquired a geometrical tinge, the solver having to associate symbols with curves in the plane or in space. He made sure the knowledge was practical. His notebooks contained not just the principles of these subjects but also extensive tables of trigonometric functions and integrals—not copied but calculated, often by original techniques that he devised for the purpose. For his calculus notebook he borrowed a title from the primers he had studied so avidly, Calculus for the Practical Man. When his classmates handed out yearbook sobriquets, Feynman was not in contention for the genuinely desirable Most Likely to Succeed and Most Intellectual. The consensus was Mad Genius.
All Things Are Made of Atoms
The first quantum idea—the notion that indivisible building blocks lay at the core of things—occurred to someone at least twenty-five hundred years ago, and with it physics began its slow birth, for otherwise not much can be understood about earth or water, fire or air. The idea must have seemed dubious at first. Nothing in the blunt appearance of dirt, marble, leaves, water, flesh, or bone suggests that it is so. But a few Greek philosophers in the fifth century B.C. found themselves hard pressed to produce any other satisfactory possibilities. Things change—crumble, fade, wither, or grow—yet they remain the same. The notion of immutability seemed to require some fundamental immutable parts. Their motion and recombination might give the appearance of change. On reflection, it seemed worthwhile to regard the basic constituents of matter as unchanging and indivisible: atomos—uncuttable. Whether they were also uniform was disputed. Plato thought of atoms as rigid blocks of pure geometry: cubes, octahedrons, tetrahedrons, and icosahedrons for the four pure elements, earth, air, fire, and water. Others imagined little hooks holding the atoms together (of what, though, could these hooks be made?).
Experiment was not the Greek way, but some observations supported the notion of atoms. Water evaporated; vapor condensed. Animals sent forth invisible messengers, their scents on the wind. A jar packed with ashes could still accept water; the volumes did not sum properly, suggesting interstices within matter. The mechanics were troubling and remained so. How did these grains move? How did they bind? “Cloudy, cloudy is the stuff of stones,” wrote the poet Richard Wilbur, and even in the atomic era it was hard to see how the physicist’s swarming clouds of particles could give rise to the hard-edged world of everyday sight and touch.
Someone who trusts science to explain the everyday must continually make connections between textbook knowledge and real knowledge, the knowledge we receive and the knowledge we truly own. We are told when we are young that the earth is round, that it circles the sun, that it spins on a tilted axis. We may accept the knowledge on faith, the frail teaching of a modern secular religion. Or we may solder these strands to a frame of understanding from which it may not so easily be disengaged. We watch the sun’s arc fall in the sky as winter approaches. We guess the time from the shadow of a lamppost. We walk across a merry-go-round and strain against the sideways Coriolis force, and we try to connect the sensation to our received knowledge of the habits of earthly cyclones: northern hemisphere, low pressure, counterclockwise. We time the vanishing point of a tall-masted ship below the horizon. The sun, the winds, the waves all join in preventing our return to a flat-earth world, where we could watch the tides follow the moon without understanding.
All things are made of atoms—how much harder it is to reconcile this received fact with the daily experience of solid tables and chairs. Glancing at the smooth depressions worn in the stone steps of an office building, we seldom recognize the cumulative loss of invisibly small particles struck off by ten million footfalls. Nor do we connect the geometrical facets of a jewel to a mental picture of atoms stacked like cannonballs, favoring a particular crystalline orientation and so forcing regular angles visible to the naked eye. If we do think about the atoms in us and around us, the persistence of solid stone remains a mystery. Richard Feynman asked a high-school teacher (and never heard a satisfactory reply), “How do sharp things stay sharp all this time if the atoms are always jiggling?”
The adult Feynman asked: If all scientific knowledge were lost in a cataclysm, what single statement would preserve the most information for the next generations of creatures? How could we best pass on our understanding of the world? He proposed, “All things are made of atoms—little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another,” and he added, “In that one sentence, you will see, there is an enormous amount of information about the world, if just a little imagination and thinking are applied.” Although millennia had passed since natural philosophers broached the atomic idea, Feynman’s lifetime saw the first generations of scientists who truly and universally believed in it, not just as a mental convenience but as a hard physical reality. As late as 1922 Bohr, delivering his Nobel Prize address, felt compelled to remind his listeners that scientists “believe the existence of atoms to be proved beyond a doubt.” Richard nevertheless read and reread in the Feynmans’ Encyclopaedia Britannica that “pure chemistry, even to-day, has no very conclusive arguments for the settlement of this controversy.” Stronger evidence was at hand from the newer science, physics: the phenomenon called radioactivity seemed to involve the actual disintegration of matter, so discretely as to produce audible pings or visible blips. Not until the eighties could people say that they had finally seen atoms. Even then the seeing was indirect, but it stirred the imagination to see shadowy globules arrayed in electron-microscope photographs or to see glowing points of orange light in the laser crossfire of “atom traps.”
Not solids but gases began to persuade seventeenth- and eighteenth-century scientists of matter’s fundamental granularity. In the heady aftermath of Newton’s revolution scientists made measurements, found constant quantities, and forged mathematical relationships that a philosophy without numbers had left hidden. Investigators made and unmade water, ammonia, carbonic acid, potash, and dozens of other compounds. When they carefully weighed the ingredients and end products, they discovered regularities. Volumes of hydrogen and oxygen vanished in a neat two-to-one ratio in the making of water. Robert Boyle found in England that, although one could vary both the pressure and the volume of air trapped at a given temperature in a piston, one could not vary their product. Pressure multiplied by volume was a constant. These measures were joined by an invisible rod—why? Heating a gas increased its volume or its pressure. Why?
Heat had seemed to flow from one place to another as an invisible fluid—“phlogiston” or “caloric.” But a succession of natural philosophers hit on a less intuitive idea—that heat was motion. It was a brave thought, because no one could see the things in motion. A scientist had to imagine uncountable corpuscles banging invisibly this way and
that in the soft pressure of wind against his face. The arithmetic bore out the guess. In Switzerland Daniel Bernoulli derived Boyle’s law by supposing that pressure was precisely the force of repeated impacts of spherical corpuscles, and in the same way, assuming that heat was an intensification of the motion hither and thither, he derived a link between temperature and density. The corpuscularians advanced again when Antoine-Laurent Lavoisier, again with painstaking care, demonstrated that one could keep reliable account books of the molecules entering and exiting any chemical reaction, even when gases joined with solids, as in rusting iron.
“Matter is unchangeable, and consists of points that are perfectly simple, indivisible, of no extent”—that the atom could itself contain a crowded and measurable universe remained for a later century to guess—“& separated from one another.” Ruggiero Boscovich, an eighteenth-century mathematician and director of optics for the French navy, developed a view of atoms with a strikingly prescient bearing, a view that Feynman’s single-sentence credo echoed two centuries later. Boscovich’s atoms stood not so much for substance as for forces. There was so much to explain: how matter compresses elastically or inelastically, like rubber or wax; how objects bounce or recoil; how solids hold together while liquids congeal or release vapors; “effervescences & fermentations of many different kinds, in which the particles go & return with as many different velocities, & now approach towards & now recede from one another.”
The quest to understand the corpuscles translated itself into a need to understand the invisible attractions and repulsions that gave matter its visible qualities. Attracting each other when they are a little distance apart, but repelling upon being squeezed into one another, Feynman would say simply. That mental picture was already available to a bright high-school student in 1933. Two centuries had brought more and more precise inquiry into the chemical behavior of substances. The elements had proliferated. Even a high-school laboratory could run an electric current through a beaker of water to separate it into its explosive constituents, hydrogen and oxygen. Chemistry as packaged in educational chemistry sets seemed to have reduced itself to a mechanical collection of rules and recipes. But the fundamental questions remained for those curious enough to ask, How do solids stay solid, with atoms always “jiggling”? What forces control the fluid motions of air and water, and what agitation of atoms engenders fire?
