Джеймс Глик - Genius - The Life and Science of Richard Feynman

omniscience. He believed—in the face of the increasing esotericism of his own subject—that true understanding implied a kind of clarity. A physicist once asked him to explain in simple terms a standard item of the dogma, why spin-one-half particles obey Fermi-Dirac statistics.

Feynman promised to prepare a freshman lecture on it. For once, he failed. “I couldn’t reduce it to the freshman level,”

he said a few days later, and added, “That means we real y don’t understand it.”

It was his own children, however, who crystal ized many of his attitudes toward teaching. In 1964 he had made the rare decision to serve on a public commission, responsible for choosing mathematics textbooks for California’s grade schools. Traditional y this commissionership was a sinecure that brought various smal perquisites under the table from textbook publishers. Few commissioners—as Feynman discovered—read many textbooks, but he determined to read them al , and had scores of them delivered to his house. This was the era of the so-cal ed new mathematics in children’s education: the much-debated effort to modernize the teaching of mathematics by introducing such high-level concepts as set theory and nondecimal number systems. New math swept the nation’s schools startlingly fast, in the face of parental nervousness that was captured in a New Yorker cartoon: “You see, Daddy,” a little girl explains, “this set equals al the dol ars you earned; your expenses are a sub-set within it. A sub-set of that is your deductions.”

Feynman did not take the side of the modernizers.

Instead, he poked a blade into the new-math bubble. He argued to his fel ow commissioners that sets, as presented in the reformers’ textbooks, were an example of the most insidious pedantry: new definitions for the sake of definition, a perfect case of introducing words without introducing ideas. A proposed primer instructed first-graders: “Find out if the set of the lol ipops is equal in number to the set of the girls.” Feynman described this as a disease. It removed clarity without adding any precision to the normal sentence: “Find out if there are just enough lol ipops for the girls.” Specialized language should wait until it is needed, he said, and the peculiar language of set theory never is needed. He found that the new textbooks did not reach the areas in which set theory does begin to contribute content beyond the definitions: the understanding of different degrees of infinity, for example.

It is an example of the use of words, new definitions of new words, but in this particular case a most extreme example because no facts whatever are given… . It wil perhaps surprise most people who have studied this textbook to discover that the symbol

? or ? representing union and intersection of sets … al the elaborate notation for sets that is given in these books, almost never appear in any writings in theoretical

physics,

in

engineering,

business,

arithmetic, computer design, or other places where mathematics is being used.

Feynman could not make his real point without drifting into philosophy. It was crucial, he argued, to distinguish clear language from precise language. The textbooks placed a new emphasis on precise language: distinguishing

“number” from “numeral,” for example, and separating the symbol from the real object in the modern critical fashion—

pilpul for schoolchildren, it seemed to Feynman. He objected to a book that tried to teach a distinction between a bal and a picture of a bal —the book insisting on such language as “color the picture of the bal red.”

“I doubt that any child would make an error in this particular direction,” Feynman said dryly.

As a matter of fact, it is impossible to be precise

… whereas before there was no difficulty. The picture of a bal includes a circle and includes a background.

Should we color the entire square area in which the bal image appears al red? … Precision has only been pedantical y increased in one particular corner when there was original y no doubt and no difficulty in the idea.

In the real world, he pointed out once again, absolute precision is an ideal that can never be reached. Nice distinctions should be reserved for the times when doubt arises.

Feynman had his own ideas for reforming the teaching of mathematics to children. He proposed that first-graders learn to add and subtract more or less the way he worked

out complicated integrals—free to select any method that seems suitable for the problem at hand. A modern-sounding notion was, The answer isn’t what matters, so long as you use the right method. To Feynman no educational philosophy could have been more wrong. The answer is al that does matter, he said. He listed some of the techniques available to a child making the transition from being able to count to being able to add. A child can combine two groups into one and simply count the combined group: to add 5 ducks and 3 ducks, one counts 8

ducks. The child can use fingers or count mental y: 6, 7, 8.

One can memorize the standard combinations. Larger numbers can be handled by making piles—one groups pennies into fives, for example—and counting the piles.

One can mark numbers on a line and count off the spaces

—a method that becomes useful, Feynman noted, in understanding measurement and fractions. One can write larger numbers in columns and carry sums larger than 10.

To Feynman the standard texts seemed too rigid. The problem 29 + 3 was considered a third-grade problem, because it required the advanced technique of carrying; yet Feynman pointed out that a first-grader could handle it by thinking 30, 31, 32. Why should children not be given simple algebra problems (2 times what plus 3 is 7?) and encouraged to solve them by trial and error? That is how real scientists work.

We must remove the rigidity of thought… . We must leave freedom for the mind to wander about in trying to

solve the problems… . The successful user of mathematics is practical y an inventor of new ways of obtaining answers in given situations. Even if the ways are wel known, it is usual y much easier for him to invent his own way—a new way or an old way—than it is to try to find it by looking it up.

Better to have a jumbled bag of tricks than any one orthodox method. That was how he taught his own children at homework time. Michel e learned that he had a thousand shortcuts; also that they tended to get her into trouble with her arithmetic teachers.

Do You Think You Can Last On

Forever?

Although he had never liked athletic activity, he tried to stay fit. After he broke a kneecap fal ing over a Chicago curb, he took up jogging. He ran almost daily up and down the steep paths above his house in the Altadena hil s. He owned a wet suit and swam often at the beachfront house in Mexico that he had bought with his Nobel Prize money. (It had been a shambles when he and Gweneth first saw it. He told her that they did not want it. She looked at the glass wal facing the warm currents sweeping up from the Tropic of Cancer and replied, “Oh yes, we do.”)

Traveling in the Swiss Alps in the summer of 1977, he frightened Gweneth by suddenly running to the bathroom of

their cabin and vomiting—something he never did as an adult. Later that day he passed out in the téléphérique.

Twice that year his physician diagnosed “fever of undetermined origin.” It was not until October 1978 that cancer was discovered: a tumor that had grown to the size of a melon, weighing six pounds, in the back of his abdomen. A bulge was visible at his waistline when he stood straight. He had ignored the symptoms for too long.