Genius (6 page)

Even when he was young, absorbing such wisdom, Feynman sometimes glimpsed the limits of his father’s understanding of science. As he was going to bed one night, he asked his father what algebra was.

“It’s a way of doing problems that you can’t do in arithmetic,” his father said.

“Like what?”

“Like a house and a garage rents for $15,000. How much does the garage rent for?”

Richard could see the trouble with that. And when he started high school, he came home upset by the apparent triviality of Algebra 1. He went into his sister’s room and asked, “Joanie, if 2
^x
is equal to 4 and
x
is an unknown number, can you tell me what
x
is?” Of course she could, and Richard wanted to know why he should have to learn anything so obvious in high school. The same year, he could see just as easily what
x
must be if 2
^x
was 32. The school quickly switched him into Algebra 2, taught by Miss Moore, a plump woman with an exquisite sense of discipline. Her class ran as a roundelay of problem solving, the students making a continual stream to and from the blackboard. Feynman was slightly ill at ease among the older students, but he already let friends know that he thought he was smarter. Still, his score on the school IQ test was a merely respectable 125.

At School

The New York City public schools of that era gained a reputation later for high quality, partly because of the nostalgic reminiscences of famous alumni. Feynman himself thought that his grammar school, Public School 39, had been stultifyingly barren: “an intellectual desert.” At first he learned more at home, often from the encyclopedia. Having trained himself in rudimentary algebra, he once concocted a set of four equations with four unknowns and showed it off to his arithmetic teacher, along with his methodical solution. She was impressed but mystified; she had to take it to the principal to find out whether it was correct. The school had one course in general science, for boys only, taught by a blustering, heavyset man called Major Connolly—evidently his World War I rank. All Feynman remembered from the course was the length of a meter in inches, 39.37, and a futile argument with the teacher over whether rays of light from a single source come out radially, as seemed logical to Richard, or in parallel, as in the conventional textbook diagrams of lens behavior. Even in grade school he had no doubt that he was right about such things. It was just obvious, physically—not the sort of argument that could be settled by an appeal to authority. At home, meanwhile, he boiled water by running 110-volt house current through it and watched the lines of blue and yellow sparks that flow when the current breaks. His father sometimes described the beauty of the flow of energy through the everyday world, from sunlight to plants to muscles to the mechanical work stored in the spring of a windup toy. Assigned at school to write verse, Richard applied this idea to a fancifully bucolic scene with a farmer plowing his field to make food, grass, and hay:

… Energy plays an important part
And it’s used in all this work;
Energy, yes, energy with power so great,
A kind that cannot shirk.
If the farmer had not this energy,
He would be at a loss,
But it’s sad to think, this energy
Belongs to a little brown horse.

Then he wrote another poem, brooding self-consciously about his own obsession with science and with the idea of science. Amid some borrowed apocalyptic imagery he expressed a feeling that science meant skepticism about God—at least about the standardized God to whom he had been exposed at school. Over the Feynmans’ rational and humanistic household God had never held much sway. “Science is making us wonder,” he began—then on second thought he scratched out the word
wonder
.

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 recreational 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:

T
HE
M
OST
R
EMARKABLE
F
ORMULA
I
N
M
ATH
.
e
^iπ
+ 1 = 0

(F
ROM
S
CIENCE
H
ISTORY OF
T
HE
U
NIVERSE
)

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.