MATHEMATICAL AMERICAN Exclusive Online Issue No

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty." So wrote British philosopher and logician Bertrand Russell nearly 100 years ago. He was not alone in this sentiment. French mathematician Henri Poincaré declared that "the mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful." Likewise, Einstein described pure mathematics as "the poetry of logical ideas." Indeed, many a scholar has remarked on the elegance of the science. It is in this spirit that we have put together a collection of Scientiﬁc American articles about math.

In this exclusive online issue, Martin Gardner, longtime editor of the magazine’s Mathematical Games column, reﬂects on 25 years of fun puzzles and serious discoveries; other scholars explore the concept of inﬁnity, the fate of mathematical proofs in the age of computers, and the thriving of native mathematics during Japan’s period of national seclusion. The anthology also includes articles that trace the long, hard roads to resolving Fermat’s Last Theorem and Zeno’s paradoxes, two problems that for centuries captivated--and tormented--some of the discipline’s most beautiful minds. —The Editors

TABLE OF CONTENTS

2 A Quarter-Century of Recreational Mathematics BY MARTIN GARDNER; SCIENTIFIC AMERICAN, AUGUST 1998 The author of Scientiﬁc American's column "Mathematical Games" from 1956 to 1981 recounts 25 years of amusing puzzles and serious discoveries

10 The Death of Proof BY JOHN HORGAN; SCIENTIFIC AMERICAN, OCTOBER 1993 Computers are transforming the way mathematicians discover, prove and communicate ideas, but is there a place for absolute certainty in this brave new world?

16 Resolving Zeno's Paradoxes BY WILLIAM I. MCLAUGHLIN; SCIENTIFIC AMERICAN, NOVEMBER 1994 For millennia, mathematicians and philosophers have tried to refute Zeno's paradoxes, a set of riddles suggesting that motion is inherently impossible. At last, a solution has been found

21 A Brief History of Infinity BY A. W. MOORE; SCIENTIFIC AMERICAN, APRIL 1995 The infinite has always been a slippery concept. Even the commonly accepted mathematical view, developed by Georg Cantor, may not have truly placed infinity on a rigorous foundation

25 Fermat's Last Stand BY SIMON SINGH AND KENNETH A. RIBET; SCIENTIFIC AMERICAN, NOVEMBER 1997 His most notorious theorem bafﬂed the greatest minds for more than three centuries. But after 10 years of work, one mathematician cracked it

29 Japanese Temple Geometry BY TONY ROTHMAN; SCIENTIFIC AMERICAN, MAY 1998 During Japan's period of national seclusion (1639-1854), native mathematics thrived, as evidenced in "sangaku"— wooden tablets engraved with geometry problems hung under the roofs of shrines and temples

The author of Scientiﬁc American’s column “Mathematical Games” from 1956 to 1981 recounts 25 years of amusing puzzles and serious discoveries

by Martin Gardner

“Amusement is one of the fields of ics were in print. The classic of the and serious math is a blurry one. Many applied math.” genre—Mathematical Recreations and professional mathematicians regard —William F. White, Essays, written by the eminent English their work as a form of play, in the A Scrapbook of mathematician W. W. Rouse Ball in same way professional golfers or bas- Elementary Mathematics 1892—was available in a version up- ketball stars might. In general, math is dated by another legendary figure, the considered recreational if it has a play- Canadian geometer H.S.M. Coxeter. ful aspect that can be understood and Dover Publications had put out a trans- appreciated by nonmathematicians. y “Mathematical Games” lation from the French of La Mathéma- Recreational math includes elementary column began in the De- tique des Jeux (Mathematical Recrea- problems with elegant, and at times Mcember 1956 issue of Sci- tions), by Belgian number theorist Mau- surprising, solutions. It also encompass- entific American with an article on rice Kraitchik. But aside from a few other es mind-bending paradoxes, ingenious hexaflexagons. These curious struc- puzzle collections, that was about it. games, bewildering magic tricks and tures, created by folding an ordinary Since then, there has been a remark- topological curiosities such as Möbius strip of paper into a hexagon and then able explosion of books on the subject, bands and Klein bottles. In fact, almost gluing the ends together, could be many written by distinguished mathe- every branch of mathematics simpler turned inside out repeatedly, revealing maticians. The authors include Ian Stew- than calculus has areas that can be con- one or more hidden faces. The struc- art, who now writes Scientific Ameri- sidered recreational. (Some amusing ex- tures were invented in 1939 by a group can’s “Mathematical Recreations” col- amples are shown on the following of Princeton University graduate stu- umn; John H. Conway of Princeton page.) dents. Hexaflexagons are fun to play University; Richard K. Guy of the Uni- with, but more important, they show versity of Calgary; and Elwyn R. Berle- Ticktacktoe in the Classroom the link between recreational puzzles kamp of the University of California at and “serious” mathematics: one of their Berkeley. Articles on recreational math- he monthly magazine published by inventors was Richard Feynman, who ematics also appear with increasing fre- Tthe National Council of Teachers went on to become one of the most fa- quency in mathematical periodicals. of Mathematics, Mathematics Teacher, mous theoretical physicists of the cen- The quarterly Journal of Recreational often carries articles on recreational top- tury. Mathematics began publication in ics. Most teachers, however, continue to At the time I started my column, only 1968. ignore such material. For 40 years I a few books on recreational mathemat- The line between entertaining math have done my best to convince educa-

1 2 ILLUSTRATIONS BY IAN WORPOLE ILLUSTRATIONS

1 1 1 3 3 3

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r. Jones, a cardsharp, puts three cards face down on a table. he matrix of numbers above is a curious type of magic square. MOne of the cards is an ace; the other two are face cards. You TCircle any number in the matrix, then cross out all the numbers place a finger on one of the cards, betting that this card is the ace. in the same column and row. Next, circle any number that has not 1 The probability that you’ve picked the ace is clearly /3. Jones now se- been crossed out and again cross out the row and column containing cretly peeks at each card. Because there is only one ace among the that number. Continue in this way until you have circled six numbers. three cards, at least one of the cards you didn’t choose must be a face Clearly, each number has been randomly selected. But no matter card. Jones turns over this card and shows it to you. What is the prob- which numbers you pick, they always add up to the same sum. What ability that your finger is now on the ace? is this sum? And, more important, why does this trick always work?

3 4 12 3 456

78910 CUT THE DECK

11 12 13 14 15 16

17 18 19 20 21 22 23

24 25 26 27 28 29 30 31 32

RIFFLE- 33 34 35 36 37 38 39 SHUFFLE

40 41 42 43 44 45 46

4847 5049 rinted above are the first three verses of Genesis in the King magician arranges a deck of cards so that the black and PJames Bible. Select any of the 10 words in the first verse: “In the A red cards alternate. She cuts the deck about in half, making sure beginning God created the heaven and the earth.” Count the num- that the bottom cards of each half are not the same color. Then she ber of letters in the chosen word and call this number x. Then go to allows you to riffle-shuffle the two halves together, as thoroughly or the word that is x words ahead. (For example, if you picked “in,” go to carelessly as you please. When you’re done, she picks the first two “beginning.”) Now count the number of letters in this word—call it cards from the top of the deck. They are a black card and a red card n—then jump ahead another n words. Continue in this manner until (not necessarily in that order). The next two are also a black card and your chain of words enters the third verse of Genesis. a red card. In fact, every succeeding pair of cards will include one of On what word does your count end? Is the answer happenstance each color. How does she do it? Why doesn’t shuffling the deck pro- or part of a divine plan? duce a random sequence?

2 1

√3

cd IAN WORPOLE

LOW-ORDER REP-TILES ﬁt together to make larger replicas of themselves. The isosceles right triangle (a) is a rep-2 ﬁgure: two such triangles form a larger triangle with the same shape. A rep-3 triangle (b) has angles of 30, 60 and 90 degrees. Other reptiles include a rep-4 quadrilateral (c) and a rep-4 hexagon (d). The sphinx (e) is the only known rep-4 pentagon.

tors that recreational math should be are some positive aspects of the new ed as I learned more, but the key to the incorporated into the standard curricu- new math, I was struck by the fact that column’s popularity was the fascinating lum. It should be regularly introduced the yearbook had nothing to say about material I was able to coax from some as a way to interest young students in the value of recreational mathematics, of the world’s best mathematicians. the wonders of mathematics. So far, which lends itself so well to cooperative Solomon W. Golomb of the Universi- though, movement in this direction has problem solving. ty of Southern California was one of been glacial. Let me propose to teachers the follow- the first to supply grist for the column. I have often told a story from my own ing experiment. Ask each group of stu- In the May 1957 issue I introduced his high school years that illustrates the di- dents to think of any three-digit num- studies of polyominoes, shapes formed lemma. One day during math study pe- ber—let’s call it ABC. Then ask the stu- by joining identical squares along their riod, after I’d finished my regular as- dents to enter the sequence of digits edges. The domino—created from two signment, I took out a fresh sheet of pa- twice into their calculators, forming the such squares—can take only one shape, per and tried to solve a problem that number ABCABC. For example, if the but the tromino, tetromino and pento- had intrigued me: whether the first play- students thought of the number 237, mino can assume a variety of forms: Ls, er in a game of ticktacktoe can always they’d punch in the number 237,237. Ts, squares and so forth. One of Gol- win, given the right strategy. When my Tell the students that you have the psy- omb’s early problems was to determine teacher saw me scribbling, she snatched chic power to predict that if they divide whether a specified set of polyominoes, the sheet away from me and said, “Mr. ABCABC by 13 there will be no remain- snugly fitted together, could cover a Gardner, when you’re in my class I ex- der. This will prove to be true. Now ask checkerboard without missing any pect you to work on mathematics and them to divide the result by 11. Again, squares. The study of polyominoes nothing else.” there will be no remainder. Finally, ask soon evolved into a flourishing branch The ticktacktoe problem would make them to divide by 7. Lo and behold, the of recreational mathematics. Arthur C. a wonderful classroom exercise. It is a original number ABC is now in the cal- Clarke, the science-fiction author, con- superb way to introduce students to culator’s readout. The secret to the trick fessed that he had become a “pentomi- combinatorial mathematics, game theo- is simple: ABCABC = ABC ≤ 1,001 = no addict” after he started playing with ry, symmetry and probability. More- ABC ≤ 7 ≤ 11 ≤ 13. (Like every other the deceptively simple figures. over, the game is part of every student’s integer, 1,001 can be factored into a Golomb also drew my attention to a experience: Who has not, as a child, unique set of prime numbers.) I know of class of figures he called “reptiles”— played ticktacktoe? Yet I know few no better introduction to number theory identical polygons that fit together to mathematics teachers who have includ- and the properties of primes than ask- form larger replicas of themselves. One ed such games in their lessons. ing students to explain why this trick of them is the sphinx, an irregular pen- According to the 1997 yearbook of always works. tagon whose shape is somewhat similar the mathematics teachers’ council, the to that of the ancient Egyptian monu- latest trend in math education is called Polyominoes and Penrose Tiles ment. When four identical sphinxes are “the new new math” to distinguish it joined in the right manner, they form a from “the new math,” which flopped ne of the great joys of writing the larger sphinx with the same shape as its so disastrously several decades ago. The OScientific American column over components. The pattern of rep-tiles newest teaching system involves divid- 25 years was getting to know so many can expand infinitely: they tile the plane ing classes into small groups of students authentic mathematicians. I myself am by making larger and larger replicas. and instructing the groups to solve prob- little more than a journalist who loves The late Piet Hein, Denmark’s illus- lems through cooperative reasoning. mathematics and can write about it glib- trious inventor and poet, became a “Interactive learning,” as it is called, is ly. I took no math courses in college. My good friend through his contributions substituted for lecturing. Although there columns grew increasingly sophisticat- to “Mathematical Games.” In the July

STAIRS

6 CHAIR 5 7 IAN WORPOLE

STEAMER SOMA PIECES are irregular shapes formed by joining unit cubes at their faces (above). The seven pieces can be arranged in 240 ways to build the 3-by-3-by-3 Soma cube. The pieces can also be assembled to form all but one of the structures pictured at the right. Can you determine which structure is impossible to build? The answer is on page 75.

1957 issue, I wrote about a topological ishing variety of forms can be created, CASTLE game he invented called Hex, which is including some that move across the played on a diamond-shaped board board like insects. I described Life in the made of hexagons. Players place their October 1970 column, and it became an markers on the hexagons and try to be instant hit among computer buffs. For the first to complete an unbroken chain many weeks afterward, business firms SKYSCRAPER from one side of the board to the other. and research laboratories were almost The game has often been called John be- shut down while Life enthusiasts exper- cause it can be played on the hexagonal imented with Life forms on their com- BATHTUB tiles of a bathroom floor. puter screens. Hein also invented the Soma cube, Conway later collaborated with fel- which was the subject of several columns low mathematicians Richard Guy and (September 1958, July 1969 and Sep- Elwyn Berlekamp on what I consider tember 1972). The Soma cube consists the greatest contribution to recreational of seven different polycubes, the three- mathematics in this century, a two-vol- dimensional analogues of polyominoes. ume work called Winning Ways (1982). TUNNEL They are created by joining identical One of its hundreds of gems is a two- cubes at their faces. The polycubes can person game called Phutball, which can be fitted together to form the Soma also be played on a go board. The Phut- cube—in 240 ways, no less—as well as ball is positioned at the center of the a whole panoply of Soma shapes: the board, and the players take turns plac- pyramid, the bathtub, the dog and so on. ing counters on the intersections of the In 1970 the mathematician John Con- grid lines. Players can move the Phutball SOFA way—one of the world’s undisputed ge- by jumping it over the counters, which niuses—came to see me and asked if I are removed from the board after they had a board for the ancient Oriental have been leapfrogged. The object of WELL game of go. I did. Conway then dem- the game is to get the Phutball past the onstrated his now famous simulation opposing side’s goal line by building a game called Life. He placed several chain of counters across the board. counters on the board’s grid, then re- What makes the game distinctive is that, moved or added new counters accord- unlike checkers, chess, go or Hex, Phut- ing to three simple rules: each counter ball does not assign different game piec- with two or three neighboring counters es to each side: the players use the same WALL is allowed to remain; each counter with counters to build their chains. Conse- one or no neighbors, or four or more quently, any move made by one Phut- neighbors, is removed; and a new coun- ball player can also be made by his or ter is added to each empty space adja- her opponent. cent to exactly three counters. By ap- Other mathematicians who contrib- plying these rules repeatedly, an aston- uted ideas for the column include Frank

5 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. Harary, now at New Mexico State Uni- April 1961 issue of Scientific American, a previously unknown type of molecu- versity, who generalized the game of as well as the nonperiodic tiling discov- lar structure called a quasicrystal. Since ticktacktoe. In Harary’s version of the ered by Roger Penrose, the British math- then, physicists have written hundreds game, presented in the April 1979 is- ematical physicist famous for his work of research papers on quasicrystals and sue, the goal was not to form a straight on relativity and black holes. their unique thermal and vibrational line of Xs or Os; instead players tried to Penrose tiles are a marvelous exam- properties. Although Penrose’s idea be the first to arrange their Xs or Os in ple of how a discovery made solely for started as a strictly recreational pursuit, a specified polyomino, such as an L or the fun of it can turn out to have an unit paved the way for an entirely new a square. Ronald L. Rivest of the Mas- expected practical use. Penrose devised branch of solid-state physics. sachusetts Institute of Technology al- two kinds of shapes, “kites” and lowed me to be the first to reveal—in “darts,” that cover the plane only in a Leonardo’s Flush Toilet the August 1977 column—the “public- nonperiodic way: no fundamental part key” cipher system that he co-invented. of the pattern repeats itself. I explained he two columns that generated the It was the first of a series of ciphers that the significance of the discovery in the Tgreatest number of letters were my revolutionized the field of cryptology. I January 1977 issue, which featured a April Fools’ Day column and the one also had the pleasure of presenting the pattern of Penrose tiles on its cover. A on Newcomb’s paradox. The hoax col- mathematical art of Maurits C. Escher, few years later a 3-D form of Penrose umn, which appeared in the April 1975 which appeared on the cover of the tiling became the basis for constructing issue, purported to cover great breakthroughs in science and math. The startling discoveries included a refutation of a IN THE GAME OF LIFE, forms evolve by following rules relativity theory and the disclosure that set by mathematician John H. Conway. If four “organ- isms” are initially arranged in a square block of cells (a), Leonardo da Vinci had invented the flush the Life form does not change. Three other initial patterns toilet. The column also announced that (b, c and d) evolve into the stable “beehive” form. The fifth the opening chess move of pawn to BLOCK pattern (e) evolves into the oscillating “traffic lights” figure, king’s rook 4 was a certain game win- which alternates between vertical and horizontal rows. ner and that e raised to the power of π ≤ √163 was exactly equal to the integer 262,537,412,640,768,744. To my b amazement, thousands of readers failed to recognize the column as a joke. Ac- companying the text was a complicated map that I said required five colors to ensure that no two neighboring regions BEEHIVE were colored the same. Hundreds of readers sent me copies of the map colored with only four colors, thus up- holding the four-color theorem. Many c readers said the task had taken days. Newcomb’s paradox is named after physicist William A. Newcomb, who originated the idea, but it was first described in a technical paper by Harvard BEEHIVE University philosopher Robert Nozick. The paradox involves two closed boxes, A and B. Box A contains $1,000. Box B contains either nothing or $1 d million. You have two choices: take only Box B or take both boxes. Taking both obviously seems to be the better choice, but there is a catch: a superbeing—God, if you like—has the power of BEEHIVE knowing in advance how you will choose. If he predicts that out of greed

e IAN WORPOLE

φ 1 φ ° 36° 36 36° 72° φ

1 + √5 φ = 2

PENROSE TILES can be constructed by dividing a rhombus into a “kite” and a “dart” such that the ratio of their diagonals is phi (φ), the golden ratio (above). Arranging five of the darts around a vertex creates a star. Placing 10 kites around the star and extending the tiling symmetrically generate the infinite star pattern (right). Other tilings around a vertex include the deuce, jack and queen, which can also generate infinite patterns of kites and darts (below right). you will take both boxes, he leaves B empty, and you will get only the $1,000 in A. But if he predicts you will take only Box B, he puts $1 million in it. You have watched this game played many times with others, and in every case when the player chose both boxes, he or she found DEUCE that B was empty. And every time a player chose only Box B, he or she became a millionaire. How should you choose? The prag- matic argument is that because of the previous games you have witnessed, JACK QUEEN you can assume that the superbeing does indeed have the power to make accurate predictions. You should therefore take only Box B to guarantee that B contains $1 million, you’ll get the mil- IAN WORPOLE you will get the $1 million. But wait! lion plus another thousand. So how can The superbeing makes his prediction you lose by choosing both boxes? contradiction, the impossibility of a su- before you play the game and has no Each argument seems unassailable. perbeing’s ability to predict decisions. I power to alter it. At the moment you Yet both cannot be the best strategy. No- wrote about the paradox in the July make your choice, Box B is either emp- zick concluded that the paradox, which 1973 column and received so many let- ty, or it contains $1 million. If it is emp- belongs to a branch of mathematics ters afterward that I packed them into a ty, you’ll get nothing if you choose only called decision theory, remains unre- carton and personally delivered them to Box B. But if you choose both boxes, at solved. My personal opinion is that the Nozick. He analyzed the letters in a least you’ll get the $1,000 in A. And if paradox proves, by leading to a logical guest column in the March 1974 issue. Magic squares have long been a popular part of recreational math. What makes these squares magical is the arrangement of numbers inside them: the numbers in every column, row and diagonal add up to the same sum. The numbers in the magic square are usually required to be distinct and run in TRAFFIC LIGHTS consecutive order, starting with one. There exists only one order-3 magic

7 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. any practical use. Why then are mathematicians trying to ﬁnd it? Because it The Vanishing Area Paradox might be there.

onsider the figures shown below. Each pattern is made with the same 16 The Amazing Dr. Matrix Cpieces: four large right triangles, four small right triangles, four eight-sided pieces and four small squares. In the pattern on the left, the pieces fit together very year or so during my tenure at snugly, but the pattern on the right has a square hole in its center! Where did this Scientific American, I would devote extra bit of area come from? And why does it vanish in the pattern on the left? E a column to an imaginary interview with a numerologist I called Dr. Irving Joshua Matrix (note the “666” provided by the number of letters in his first, middle and last names). The good doc- tor would expound on the unusual properties of numbers and on bizarre forms of wordplay. Many readers thought Dr. Matrix and his beautiful, half-Japanese daughter, Iva Toshiyori, were real. I recall a letter from a puzzled Japanese reader who told me that Toshiyori was a most peculiar surname in Japan. I had taken it from a map of Tokyo. My in- formant said that in Japanese the word means “street of old men.” The secret to this paradox—which I devised for the “Mathematical Games” col- I regret that I never asked Dr. Matrix umn in the May 1961 issue of Scientific American—will be revealed in the Letters for his opinion on the preposterous to the Editors section of next month’s issue. Impatient readers can find the an- 1997 best-seller The Bible Code, which swer at www.sciam.com on the World Wide Web. —M.G. claims to find predictions of the future

IAN WORPOLE in the arrangement of Hebrew letters in the Old Testament. The book employs a cipher system that would have made square, which arranges the digits one What if the numbers in a magic square Dr. Matrix proud. By selectively through nine in a three-by-three grid. are not required to run in consecutive applying this system to certain blocks (Variations made by rotating or reflect- order? If the only requirement is that of text, inquisitive readers can find hid- ing the square are considered trivial.) In the numbers be distinct, a wide variety den predictions not only in the Old Tes- contrast, there are 880 order-4 magic of order-3 magic squares can be con- tament but also in the New Testament, squares, and the number of arrange- structed. For example, there is an infin- the Koran, the Wall Street Journal— ments increases rapidly for higher orders. ite number of such squares that contain and even in the pages of The Bible Surprisingly, this is not the case with distinct prime numbers. Can an order-3 Code itself. magic hexagons. In 1963 I received in magic square be made with nine distinct The last time I heard from Dr. Matrix, the mail an order-3 magic hexagon de- square numbers? Two years ago in an he was in Hong Kong, investigating the vised by Clifford W. Adams, a retired article in Quantum, I offered $100 for accidental appearance of π in clerk for the Reading Railroad. I sent such a pattern. So far no one has come well-known works of fiction. He cited, the magic hexagon to Charles W. Trigg, forward with a “square of squares”— for example, the following sentence a mathematician at Los Angeles City but no one has proved its impossibility fragment in chapter nine of book two College, who proved that this elegant either. If it exists, its numbers would be of H. G. Wells’s The War of the pattern was the only possible order-3 huge, perhaps beyond the reach of to- Worlds: “For a time I stood regard- magic hexagon—and that no magic hex- day’s fastest supercomputers. Such a ing...” The letters in the words give π to agons of any other size are possible! magic square would probably not have six digits! SA

The Author Further Reading

MARTIN GARDNER wrote the “Mathematical Games” col- Recreations in the Theory of Numbers. Albert H. Beiler. Dover umn for Scientiﬁc American from 1956 to 1981 and continued to Publications, 1964. contribute columns on an occasional basis for several years after- Mathematics: Problem Solving through Recreational ward. These columns are collected in 15 books, ending with The Mathematics. Bonnie Averbach and Orin Chein. W. H. Freeman Last Recreations (Springer-Verlag, 1997). He is also the author of and Company, 1986. The Annotated Alice, The Whys of a Philosophical Scrivener, The Mathematical Recreations and Essays. 13th edition. W. W. Ambidextrous Universe, Relativity Simply Explained and The Rouse Ball and H.S.M. Coxeter. Dover Publications, 1987. Flight of Peter Fromm, the last a novel. His more than 70 other Penguin Edition of Curious and Interesting Geometry. books are about science, mathematics, philosophy, literature and David Wells. Penguin, 1991. his principal hobby, conjuring. Mazes of the Mind. Clifford Pickover. St. Martin’s Press, 1992.

1 I discussed Kruskal’s principle in my February 1978 column. 1. Most people guess that the probability has risen from /3 to 1 Mathematician John Allen Paulos applies the principle to word /2. After all, only two cards are face down, and one must be 1 chains in his upcoming book Once upon a Number. the ace. Actually, the probability remains /3. The probability 2 that you didn’t pick the ace remains /3, but Jones has elimi- nated some of the uncertainty by showing that one of the 4. For simplicity’s sake, imagine a deck of only 10 cards, with the 2 black and red cards alternating like so: BRBRBRBRBR. Cutting two unpicked cards is not the ace. So there is a /3 probability that the other unpicked card is the ace. If Jones gives you the this deck in half will produce two five-card decks: BRBRB and option to change your bet to that card, you should take it (un- RBRBR. At the start of the shuffle, the bottom card of one deck less he’s slipping cards up his sleeve, of course). is black, and the bottom card of the other deck is red. If the red card hits the table first, the bottom cards of both decks will then be black, so the next card to fall will create a black- red pair on the table. And if the black card drops first, the bottom cards of both decks will be red, so the next card to fall will create a red-black pair. After the first two cards drop—no mat- 2 1 3 3 ter which deck they came from—the situation will be the same as it was in the beginning: the bottom cards of the I introduced this problem in my October 1959 column in a decks will be different colors. The process then repeats, guar- slightly different form—instead of three cards, the problem anteeing a black and red card in each successive pair, even if involved three prisoners, one of whom had been pardoned some of the cards stick together (below). by the governor. In 1990 Marilyn vos Savant, the author of a popular column in Parade magazine, presented still another THOROUGH SHUFFLE STICKY SHUFFLE version of the same problem, involving three doors and a car behind one of them. She gave the correct answer but received thousands of angry letters—many from mathematicians—accusing her of ignorance of probability theory! The fracas generated a front-page story in the New York Times. OR This phenomenon is 2. The sum is 111. The trick always works because the matrix of known as the Gilbreath numbers is nothing more than an old-fashioned addition principle after its discover- table (below). The table is generated by two sets of numbers: er, Norman Gilbreath, a (3, 1, 5, 2, 4, 0) and (25, 31, 13, 1, 7, 19). Each number in the ma- California magician. I first trix is the sum of a pair of numbers in the two sets. When you explained it in my column choose the six circled in August 1960 and discussed it again in July 1972. Magicians 31 524 0 numbers, you are se- have invented more than 100 card tricks based on this princi- 25 lecting six pairs that ple and its generalizations. —M.G. together include all 31 12 of the generating numbers. So the sum 10 12 16 13 of the circled numbers is always equal to 13 4 2 19 1 the sum of the 12 generating numbers. 7 These special magic 15 8 573 squares were the sub- 19 ject of my January 14 6 1 17 1957 column. 91118 3. Each chain of words ends on “God.” This answer may seem providential, but it is actually the result of the Kruskal Count, a mathematical principle first noted by mathematician Martin MAGIC HEXAGON Kruskal in the 1970s. When the total number of words in a text has a unique property: every straight is significantly greater than the number of letters in the long- row of cells adds up to 38. est word, it is likely that any two arbitrarily started word chains will intersect at a keyword. Af- ter that point, of course, the SKYSCRAPER chains become identical. As the cannot be built from Soma pieces. text lengthens, the likelihood (The puzzle is on page 71.) of intersection increases. IAN WORPOLE

9 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. Originally published in October 1993 THE DEATH OF PROOF by John Horgan, senior writer Computers are transforming the way mathematicians discover, prove and communicate ideas, but is there a place for absolute certainty in this brave new world?

Legend has it that when Pythagoras and his followers discovered the theorem that bears his name in the sixth century B.C., they slaughtered an ox and feasted in celebration. And well they might. The relation they found between the sides of a right triangle held true not sometimes or most of the time but always—regardless of whether the triangle was a piece of silk or a plot of land or marks on papyrus. It seemed like magic, a gift from the gods. No wonder so many thinkers, from Plato to Kant, came to believe that mathematics offers the purest truths humans are permit- ted to know. That faith seemed reaffirmed this past June when Andrew J. Wiles of Princeton Uni- versity revealed during a meeting at the University of Cambridge that he had solved Fermat’s last theorem. This problem, one of the most famous in mathematics, was posed more than 350 years ago, and its roots extend back to Pythagoras himself. Since no oxen were available, Wiles’s listeners showed their appreciation by clapping their hands. But was the proof of Fermat’s last theorem the last gasp of a dying culture? Math- ematics, that most tradition-bound of intellectual enterprises, is undergoing profound changes. For millennia, mathematicians have measured progress in terms of what they can demonstrate through proofs—that is, a series of logical steps leading from a set of axioms to an irrefutable conclusion. Now the doubts riddling modern human thought have ﬁnally infected mathematics. Mathematicians may at last be forced to accept what many scientists and philosophers already have admitted: their assertions are, at best, only provisionally true, true until proved false.

10 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. This uncertainty stems, in part, from the more people doing mathematics without mathematical community by and large still growing complexity of mathematics. Proofs necessarily doing proofs.” regards computers as invaders, despoilers of are often so long and complicated that they Powerful institutional forces are promul- the sacred ground.” Last year Mumford pro- are difficult to evaluate. Wiles’s demonstra- gating these heresies. For several years, the posed a course in which instructors would tion runs to 200 pages—and experts estimate National Science Foundation has been urg- show students how to program a computer it could be five times longer if he spelled out ing mathematicians to become more in- to find solutions in advanced calculus. “I all its elements. One observer asserted that volved in computer science and other fields was vetoed,” he recalled, “and not on the only one tenth of 1 percent of the mathematics with potential applications. Some leading grounds—which I expected—that the stu- community was qualified to evaluate the lights, notably Phillip A. Griffiths, director dents would complain, but because half of proof. Wiles’s claim was accepted largely on of the Institute for Advanced Study in my fellow teachers couldn’t program!” the basis of his reputation and the reputations Princeton, N.J., and Michael Atiyah, who That situation is changing fast, if of those whose work he built on. Mathemati- won a Fields Medal (often called the Nobel the University of Minnesota’s Geometry cians who had not yet examined the argument Prize of mathematics) in 1966 and now Center is any indication. Founded two years in detail nonetheless commented that it “looks heads Cambridge’s Isaac Newton Institute ago, the Geometry Center occupies the fifth beautiful” and “has the ring of truth.” for Mathematical Sciences, have likewise en- floor of a gleaming, steel and glass polyhe- Another catalyst of change is the comput- couraged mathematicians to venture forth dron in Minneapolis. It receives $2 million a er, which is compelling mathematicians to from their ivory towers and mingle with the year from the National Science Foundation, reconsider the very nature of proof and, real world. At a time when funds and jobs the Department of Energy and the university. hence, of truth. In recent years, some proofs are scarce, young mathematicians cannot af- The center’s permanent faculty mem- have required enormous calculations by ford to ignore these exhortations. bers, most of whom hold positions elsewhere, computers. No mere human can verify these There are pockets of resistance, of course. include some of the most prominent mathe- so-called computer proofs, just other com- Some workers are complaining bitterly maticians in the world. puters. Recently investigators have proposed about the computerization of their field and On a recent day there, several young staff a computational proof that offers only the the growing emphasis on (oh, dirty word) members are editing a video demonstrating probability—not the certainty—of truth, a “applications.” One of the most vocal cham- how a sphere can be mashed, twisted, yanked statement that some mathematicians consid- pions of tradition is Steven G. Krantz of and finally turned inside out. In a conference er an oxymoron. Still others are generating Washington University. In speeches and arti- room, three computer scientists from major “video proofs” in the hopes that they will be cles, Krantz has urged students to choose universities are telling a score of high school more persuasive than page on page of for- mathematics over computer science, which teachers how to create computer graphics mal terminology. he warns could be a passing fad. Last year, he programs to teach mathematics. Other re- At the same time, some mathematicians recalls, a National Science Foundation repre- searchers sit at charcoal-colored NeXT ter- are challenging the notion that formal proofs sentative came to his university and an- minals, pondering luridly hued pictures of should be the supreme standard of truth. Al- nounced that the agency could no longer af- four-dimensional “hypercubes,” whirlpool- though no one advocates doing away with ford to support mathematics that was not ing fractals and lattices that plunge toward proofs altogether, some practitioners think “goal-oriented.” “We could stand up and infinity. No paper or pencils are in sight. the validity of certain propositions may be say this is wrong,” Krantz grumbles, “but At one terminal is David Ben-Zvi, a Har- better established by comparing them with mathematicians are spineless slobs, and they po Marx–haired junior at Princeton who is experiments run on computers or with real- don’t have a tradition of doing that.” spending six months here exploring nonlinear world phenomena. “Within the next 50 years David Mumford of Harvard University, dynamics. He dismisses the fears of some I think the importance of proof in mathe- who won a Fields Medal in 1974 for re- mathematicians that computers will lure matics will diminish,” says Keith Devlin of search in pure mathematics and is now them away from the methods that have Colby College, who writes a column on studying artificial vision, wrote recently that served them so well for so long. “They’re computers for Notices of the American “despite all the hype, the press, the pressure just afraid of change,” he says mildly. Mathematical Society. “You will see many from funding agencies, et cetera, the pure The Geometry Center is a hotbed of what

A Splendid Anachronism? Those who consider experimental mathematics and computer proofs to be tude for seven years. He shared his ideas with only a few colleagues toward abominations rather than innovations have a special reason to delight in the the end of his quest. conquest of Fermat’s last theorem by Andrew J. Wiles of Princeton University. Wiles’s proof has essentially the same classical, deductive form that Euclid’s Wiles’s achievement was a triumph of tradition, running against every current in geometric theorems did. It does not involve any computation, and it claims to modern mathematics. be absolutely—not probably—true. Nor did Wiles employ computers to repre- Wiles is a staunch believer in mathematics for its own sake. “I certainly sent ideas graphically, to perform calculations or even to compose his paper; a wouldn’t want to see mathematics just being a servant to applications, be- secretary typed his hand-written notes. cause it’s not even in the interests of the applications themselves,” he says. He concedes that testing conjectures with computers may be helpful. In the The problem he solved, first posed more than 350 years ago by the French 1970s computer tests suggested that a far-fetched proposal called the Taniya- polymath Pierre de Fermat, is a glorious example of a purely mathematical puz- ma conjecture might be true. The tests spurred work that laid the foundation zle. Fermat claimed to have found a proof of the following proposition: for the for Wiles’s own proof. equation X N + Y N = Z N, there are no integral solutions for any value of N Nevertheless, Wiles doubts he will take the trouble to learn how to perform greater than 2. The efforts of mathematicians to find the proof (which Fermat computer investigations. “It’s a separate skill,” he explains, “and if you’re in- never did disclose) helped to lay the foundation of modern number theory, vesting that much time on a separate skill, it’s quite likely it’s taking you away the study of whole numbers, which has recently become useful in cryptogra- from your real work on the problem.” phy. Yet Fermat’s last theorem itself “is very unlikely to have any applica- He rejects the possibility that there may be a finite number of truths accessi- tions,” Wiles says. ble to traditional forms of inquiry. “I disagree vehemently with the idea that Although funding agencies have been encouraging mathematicians to col- good theorems are running out,” he says. “I think we’ve barely scratched the laborate, both with each other and with scientists, Wiles worked in virtual soli- surface.”

11 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. is known as experimental mathematics, in They succeeded in representing these heli- tions you get here are the kinds you get in which investigators test their ideas by repre- coids—the first discovered since the 18th cen- many different sciences,” says John Milnor senting them graphically and doing calcula- tury—on a computer and went on to proof the State University of New York at Stony tions on computers. Last year some of the duce a formal proof of their existence. “Had Brook. Milnor is trying to fathom the prop- center’s faculty helped to found a journal, we not been able to see a picture that rough- erties of the four-dimensional set by examin- Experimental Mathematics, that showcases ly corresponded to what we believed, we ing two-dimensional slices of it generated by such work. “Experimental methods are not would never have been able to do it,” Hoff- a computer. His preliminary findings led off a new thing in mathematics,” observes the man says. the inaugural issue of Experimental Mathe- journal’s editor, David B. A. Epstein of the The area of experimental mathematics matics last year. Milnor, a 1962 Fields University of Warwick in England, noting that has received the lion’s share Medalist, says he occasionally performed that Carl Friedrich Gauss and other giants of attention over the past decade is known computer experiments in the days of punch often performed experimental calculations as nonlinear dynamics or, more popularly, cards, but “it was a miserable process. It has before constructing formal proofs. “What’s chaos. In general, nonlinear systems are gov- become much easier.” new is that it’s respectable.” Epstein acknowl- erned by a set of simple rules that, through The popularity of graphics-orient- edges that not all his co-workers are so ac- feedback and related effects, give rise to ed mathematics has provoked a backlash. cepting. “One of my colleagues said, ‘Your complicated phenomena. Nonlinear systems Krantz of Washington University charged journal should be called the Journal of Un- were investigated in the precomputer era, four years ago in the Mathematical Intelli- proved Theorems.’ ” but computers allow mathematicians to ex- gencer that “in some circles, it is easier to plore these systems and watch them evolve obtain funding to buy hardware to generate Bubbles and Tortellini in ways that Henri Poincaré and other pio- pictures of fractals than to obtain funding to neers of this branch of mathematics could study algebraic geometry.” A mathematician who epitomizes the new not. A broader warning about “speculative” style of mathematics is Jean E. Taylor of Cellular automata, which divide a com- mathematics was voiced this past July in the Rutgers University. “The idea that you don’t puter screen into a set of cells (equivalent to Bulletin of the American Mathematical Soci- use computers is going to be increasingly pixels), provide a particularly dramatic illus- ety by Arthur Jaffe of Harvard and Frank S. foreign to the next generation,” she says. tration of the principles of nonlinearity. In Quinn of the Virginia Polytechnic Institute. For two decades, Taylor has investigated general, the color, or “state,” of each cell is They suggested that computer experiments minimal surfaces, which represent the small- determined by the state of its neighbors. A and correspondence with natural phenome- est possible area or volume bounded by a change in the state of a single cell triggers a na are no substitute for proofs in establish- curve or surface. Perhaps the most elegant cascade of changes throughout the system. ing truth. “Groups and individuals within and simple minimal surfaces found in nature One of the most celebrated of cel- the mathematics community have from time are soap bubbles and films. Taylor has al- lular automata was invented by John H. to time tried being less compulsive about de- ways had an experimental bent. Early in her Conway of Princeton in the early 1970s. tails of arguments,” Jaffe and Quinn wrote. career she tested her handwritten models of Conway has proved that his automaton, “The results have been mixed, and they have minimal surfaces by dunking loops of wire which he calls “Life,” is “undecidable”: one occasionally been disastrous.” into a sink of soapy water. cannot determine wheth-er its patterns are Most mathematicians exploiting computer Now she is more likely to model endlessly variegated or eventually repeat graphics and other experimental techniques bubbles with a sophisticated computer themselves. Scientists have seized on cellular agree that seeing should not be believing and graphics program. She has also graduated automata as tools for studying the origin and that proofs are still needed to verify the con- from soap bubbles to crystals, which conform evolution of life. The computer scientist and jectures they arrive at through computation. to somewhat more complicated rules about physicist Edward Fredkin of Boston Uni- “I think mathematicians were contemplating minimal surfaces. Together with Frederick J. versity has even argued that the entire uni- their navels for too long, but that doesn’t Almgren of Princeton and Robert F. Alm- verse is a cellular automaton. mean I think proofs are irrelevant,” Taylor gren of the University of Chicago (her hus- More famous still is the Mandelbrot set, says. Hoffman offers an even stronger de- band and stepson, respectively) and Andrew whose image has become an icon for the en- fense of traditional proofs. “Proofs are the R. Roosen of the National Institute of Stan- tire field of chaos since it was popularized in only laboratory instrument mathematicians dards and Technology, Taylor is trying to the early 1980s by Benoit B. Mandelbrot of have,” he remarks, “and they are in danger mimic the growth of snowflakes and other the IBM Thomas J. Watson Research Center. of being thrown out.” Although computer crystals on a computer. Increasingly, she is The set stems from a simple equation con- graphics are “unbelievably wonderful,” he collaborating with materials scientists and taining a complex term (based on the square adds, “in the 1960s drugs were unbelievably physicists, swapping mathematical ideas and root of a negative number). The equation wonderful, and some people didn’t survive.” programming techniques in exchange for spits out solutions, which are then iterated, Indeed, veteran computer enthusiasts clues about how real crystals grow. or fed back, into the equation. know better than most that computational Another mathematician who has prowled The mathematics underlying the set had experiments—whether involving graphics or cyberspace in search of novel minimal sur- been invented more than 70 years ago by numerical calculations—can be deceiving. faces is David A. Hoffman of the University two Frenchmen, Gaston Julia and Pierre Fa- One cautionary tale involves the Riemann of Massachusetts at Amherst. Among his fa- tou, but computers laid bare their baroque hypothesis, a famous prediction about the vorite quarry are catenoids and helicoids, beauty for all to see. When plotted on a patterns displayed by prime numbers as they which resemble the pasta known as tortellini computer, the Mandelbrot set coalesces into march toward infinity. First posed more than and were first discovered in the 18th century. an image that has been likened to a tumor- 100 years ago by Bernhard Riemann, the hy- “We gain a tremendous amount of intuition ous heart, a badly burned chicken and a pothesis is considered to be one of the most by looking at images of these surfaces on warty snowman. The image is a fractal: its important unsolved problems in mathemat- computers,” he says. fuzzy borders are infinitely long, and it dis- ics. In 1992 Hoffman, Fusheng Wei of Am- plays patterns that recur at different scales. A contemporary of Riemann’s, Franz herst and Hermann Karcher of the Universi- Researchers are now studying sets that are Mertens, proposed a related conjecture in- ty of Bonn speculated on the existence of a similar to the Mandelbrot set but inhabit volving positive whole numbers; if true, the new class of helicoids, ones with handles. four dimensions. “The kinds of complica- conjecture would have provided strong evi-

12 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. dence that the Riemann hypothesis was also process real numbers rather than just inte- entific Revolutions, that scientific theories true. By the early 1980s computers had gers. are accepted for social reasons rather than shown that Mertens’s proposal did indeed Blum and Smale recently concluded that because they are in any objective sense hold for at least the first 10 billion integers. the Mandelbrot set is, in a technical sense, “true.” “That mathematics reduces in prin- In 1984, however, more extensive com-puta- uncomputable. That is, one cannot deter- ciple to formal proofs is a shaky idea” pecu- tions revealed that eventually—at numbers mine with certainty whether any given point liar to this century, Thurston asserts. “In as high as 10(10 70 )—the pattern predicted by on the complex plane resides within or out- practice, mathematicians prove theorems in Mertens vanishes. side the set’s hirsute border. These results a social context,” he says. “It is a socially One potential drawback of computers is suggest that “you have to be careful” in ex- conditioned body of knowledge and tech- that all their calculations are based on the trapolating from the results of computer ex- niques.” manipulation of discrete, whole numbers— periments, Smale says. The logician Kurt Gödel demonstrated in fact, ones and zeros. Computers can only These concerns are dismissed by Stephen more than 60 years ago through his incom- approximate real numbers, such as pi or the Wolfram, a mathematical physicist at the pleteness theorem that “it is impossible to square root of two. Someone knowledgeable University of Illinois. Wolfram is the creator codify mathematics,” Thurston notes. Any about the rounding-off functions of a simple of Mathematica, which has become the lead- set of axioms yields statements that are self- pocket calculator can easily induce it to gen- ing mathematics software since first being evidently true but cannot be demonstrated erate incorrect answers to calculations. marketed five years ago. He acknowledg- with those axioms. Bertrand Russell pointed More sophisticated programs can make es that “there are indeed pitfalls in ex- out even earlier that set theory, which is the more complicated and elusive errors. In perimental mathematics. As in all other basis of much of mathematics, is rife with 1991 David R. Stoutemyer, a software spe- kinds of experiments, you can do them logical contradictions related to the problem cialist at the University of Hawaii, presented wrong.” But he emphasizes that computa- of self-reference. (The self-contradicting 18 experiments in algebra that gave wrong tional experiments, intelligently performed statement “This sentence is false” illustrates answers when performed with standard and analyzed, can yield more results than the the problem.) “Set theory is based on polite mathematics software. old-fashioned conjecture-proof method. “In lies, things we agree on even though we Stephen Smale of the University of Cal- every other field of science there are a lot know they’re not true,” Thurston says. “In ifornia at Berkeley, a 1966 Fields Medalist, more experimentalists than theorists,” Wol- some ways, the foundation of mathematics has sought to place mathematical computa- fram says. “I suspect that will increasingly be has an air of unreality.” tion on a more secure foundation—or at the case with mathematics.” Thurston thinks highly formal proofs are least to point out the size and location of the “The obsession with proof,” Wolfram de- more likely to be flawed than those appeal- cracks running through the foundation. To- clares, has kept mathematicians from discov- ing to a more intuitive level of understand- gether with Lenore Blum of the Mathemati- ering the vast new realms of phenomena ac- ing. He is particularly enamored of the ability cal Sciences Research Institute at Berkeley cessible to computers. Even the most intrepid of computer graphics to communicate ab- and Michael Shub of IBM, he has created mathematical experimentalists are for the stract mathematical concepts to others both a theoretical model of a computer that can most part “not going far enough,” he says. within and outside the professional commu- “They’re taking existing questions in mathe- nity. Two years ago, at his urging, the Geom- matics and investigating those. They are etry Center produced a computer-generated HELICOID WITH A HOLE was discovered adding a few little curlicues to the top of a “video proof,” called Not Knot, that dra- last year by David A. Hoffman of the Universi- gigantic structure.” matizes a ground-breaking conjecture he ty of Massachusetts at Amherst and his col- Mathematicians may take this view with a proved a decade ago [see illustration on leagues, with the help of computer graphics. grain of salt. Although he shares Wolfram’s pages ?? and ??]. Thurston mentions proud- fascination with cellular automata, Conway ly that the rock band the Grateful Dead has contends that Wolfram’s career—as well as shown the Not Knot video at its concerts. his contempt for proofs—shows he is not a Whether Deadheads grok the substance of real mathematician. “Pure mathematicians the video—which concerns how mathemati- usually don’t found companies and deal with cal objects called three-manifolds behave in the world in an aggressive way,” Life’s cre- a non-Euclidean “hyperbolic” space—is an- ator says. “We sit in our ivory towers and other matter. Thurston concedes that the think about things.” video is difficult for nonmathematicians, and Purists may have a harder time ignoring even some professionals, to fathom, but he is William P. Thurston, who is also an enthusi- undaunted. The Geometry Center is now astic booster of experimental mathematics producing a video of yet another of his theo- and of computers in mathematics. Thurston, rems, which demonstrates how a sphere can who heads the Mathematical Sciences Re- be turned inside out. Last fall, moreover, search Institute at Berkeley and is a co-direc- Thurston organized a workshop at which tor of the Geometry Center (with Albert participants discussed how virtual reality Marden of the University of Minnesota), has and other advanced technologies could be impeccable credentials. In the mid-1970s he adapted for mathematical visualization. pointed out a deep potential connection be- Paradoxically, computers have catalyzed a tween two separate branches of mathemat- countertrend in which truth is obtained at ics—topology and geometry. Thurston won the expense of comprehensibility. In 1976 a Fields Medal for this work in 1982. Kenneth Appel and Wolfgang Haken of the Thurston emphasizes that he believes University of Illinois claimed they had proved mathematical truths are discovered and not the four-color conjecture, which stated that invented. But on the subject of proofs, he four hues are sufficient to construct even an sounds less like a disciple of Plato than of infinitely broad map so that no identical- Thomas S. Kuhn, the philosopher who ar- ly colored countries share a border. In some gued in his 1962 book, The Structure of Sci- respects, the proof of Appel and Haken was

13 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. conventional—that is, it consisted of a series chine. That may be a record, Radziszowski 15,000 pages and written by more than 100 of logical, traceable steps proceeding to a says, for a problem in pure mathematics. workers. It has been said that the only per- conclusion. The conclusion was that the con- The value of this work has been debated son who grasped the entire proof was its jecture could be reduced to a prediction in an unlikely forum—the newspaper col- general contractor, Daniel Gorenstein of about the behavior of some 2,000 different umn of advice-dispenser Ann Landers. In Rutgers. Gorenstein died last year. maps. June a correspondent complained to Landers Much shorter proofs can also raise Since checking this prediction by hand that resources spent on the party problem doubts. Three years ago Wu-Yi Hsiang of would be prohibitively time-consuming, Ap- should have been used to help “starving chil- Berkeley announced he had proved an old pel and Haken programmed a computer to dren in war-torn countries around the conjecture that one can pack the most do the job for them. Some 1,000 hours of world.” Some mathematicians raise another spheres in a given volume by stacking them computing time later, the machine conclud- objection to computer-assisted proofs. “I like cannonballs. Today some skeptics are ed that the 2,000 maps behave as expected: don’t believe in a proof done by a computer, convinced the 100-page proof is flawed; oth- the four-color conjecture was true. says Pierre Deligne of the Institute for Ad- ers are equally certain it is basically correct. vanced Study, an algebraic geometer and Indeed, the key to greater reliability, ac- The Party Problem 1978 Fields Medalist. “In a way, I am very cording to some computer scientists, is not egocentric. I believe in a proof if I understand less computerization but more. Robert S. Other computer-assisted proofs have fol- it, if it’s clear.” While recognizing that hu- Boyer of the University of Texas at Austin lowed. Just this year, a proof of the so-called mans can make mistakes, he adds: “A com- has led an effort to squeeze the entire party problem was announced by Stanislaw puter will also make mistakes, but they are sprawling corpus of modern mathematics P. Radziszowski of the Rochester Institute of much more difficult to find.” into a single data base whose consistency Technology and Brendan D. McKay of the Others take a more functional point of can be verified through automated “proof Australian National University in Canberra. view, arguing that establishing truth is more checkers.” The problem, which derives from work in set important than giving mathematicians an The manifesto of the so-called QED Pro- theory by the British mathematician Frank P. aesthetic glow, particularly if a result is ever ject states that such a data base will enable Ramsey in the 1920s, can be phrased as a to find an application. Defenders of this ap- users to “scan the entirety of mathematical question about relationships between people proach, who tend to be computer scientists, knowledge for relevant results and, using at a party. What is the minimum number of point out that conventional proofs are far tools of the QED system, build upon such guests that must be invited to guarantee that from immune to error. At the turn of the cen- results with reliability and confidence but at least X people are all mutual acquain- tury, most theorems were short enough to without the need for minute comprehension tances or at least Y are mutual strangers? read in one sitting and were produced by a of the details or even the ultimate founda- This number is known as a Ramsey number. single author. Now proofs often extend to tions.” The QED system, the manifesto pro- Previous proofs had established that 18 hundreds of pages or more and are so com- claims rather grandly, can even “provide guests are required to ensure that there are plicated that years may pass before they are some antidote to the degenerative effects of either four mutual acquaintances or four confirmed by others. cultural relativism and nihilism” and, pre- strangers. In their proof, Radziszowski and The current record holder of all conven- sumably, protect mathematics from the all- McKay showed that the Ramsey number for tional proofs was completed in the early too-human willingness to succumb to fash- four friends or five strangers is 25. Socialites 1980s and is called the classification of finite, ion. might think twice about trying to calculate simple groups. (A group is a set of elements, The debate over computer proofs has in- the Ramsey number for greater X’s and Y’s. such as integers, together with an operation, tensified recently with the advent of a tech- Radziszowski and McKay estimate that their such as ad-dition, that combines two ele- nique that offers not certainty but only a sta- proof consumed the equivalent of 11 years ments to get a third one.) The demonstration tistical probability of truth. Such proofs ex- of computation by a standard desktop ma- consists of some 500 articles totaling nearly ploit methods similar to those underlying

Silicon Mathematicians

The continuing penetration of computers into mathematics has revived an old have stumped people for years.” Another is Siemeon Fajtlowicz of the Univer- debate: Can mathematics be entirely automated? Will the great mathemati- sity of Houston, inventor of a program, called Graffiti, that has proposed “thou- cians of the next century be made of silicon? sands” of conjectures in graph theory. In fact, computer scientists have been working for decades on programs None of these achievements comes close to satisfying the “profound ef- that generate mathematical conjectures and proofs. In the late 1950s the ar- fect” criterion, according to David Mumford of Harvard University, a judge for tificial-intelligence guru Marvin Minsky showed how a computer could “redis- the prize. “Not now, not 100 years from now,” Mumford replies when asked to cover” some of Euclid’s basic theorems in geometry. In the 1970’s Douglas predict when the prize might be claimed. Lenat, a former student of Minsky’s, presented a program that devised even Some observers think computers will eventually surpass our mathematical more advanced geometry theorems. Skeptics contended that the results abilities. After all, notes Ronald L. Graham of AT&T Bell Laboratories, “we’re were, in effect, embedded in the original program. not very well adapted for thinking about the space-time continuum or the A decade ago the computer scientist and entrepreneur Edward Fredkin Riemann hypothesis. We’re designed for picking berries or avoiding being sought to revive the sagging interest in machine mathematics by creating eaten.” what came to be known as the Leibniz Prize. The prize, administered by Others side with the mathematical physicist Roger Penrose of the Univer- Carnegie Mellon University, offers $100,000 for the first computer program to sity of Oxford, who in his 1989 book, The Emperor’s New Mind, asserted that devise a theorem that has a “profound effect” on mathematics. computers can never replace mathematicians. Penrose’s argument drew on Some practitioners of what is known as automated reasoning think they quantum theory and Gödel’s incompleteness theorem, but he may have been may be ready to claim the prize. One is Larry Wos of Argonne National Labora- most convincing when discussing his personal experience. At its best, he sug- tory, editor of the Journal of Automated Reasoning. He claims to have gested, mathematics is an art, a creative act, that cannot be reduced to logic developed a program that has solved problems in mathematics and logic “that any more than King Lear or Beethoven’s Fifth can.

14 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. error-correction codes, which ensure that transmitted messages are not lost to noise and other ef-fects by making them highly re- dundant. The proof must ﬁrst be spelled out precisely in a rigorous form of mathematical logic. The logic then undergoes a further transformation called arithmetization, in which “and,” “or” and other functions are translated into arithmetic operations, such as addition and multiplication. Like a message transformed by an error- correction code, the “answer” of a probabilistic demonstration is distributed throughout its length—as are any errors. One checks the proof by querying it at different points and determining whether the answers are consistent; as the number of checks increases, so does the certainty that the argument is correct. Laszlo Babai of the University of Chicago, who developed the proofs two years ago (along with Lance Fortnow, Carsten Lund and Mario Szegedy of Chica- go and Leonid A. Levin of Boston Universi- ty), calls them “transparent.” Manuel Blum of Berkeley, whose work helped to pave the way for Babai’s group, suggests the term “holographic.”

The Uncertain Future

Whatever they are named, such proofs have practical drawbacks. Szegedy acknowl- edges that transforming a conventional demonstration into the probabilistic form is difficult, and the result can be a “much big- PARTY PROBLEM was solved after a vast computation by Stanislaw P. Radziszowski and ger and uglier animal.” A 1,000-line proof, Brendan D. McKay. They calculated that at least 25 people are required to ensure either that for example, could easily balloon to 1,0003 four people are all mutual acquaintances or that five are mutual strangers. This diagram, in (1,000,000,000) lines. Yet Szegedy contends which red lines connect friends and yellow lines link strangers, shows that a party of 24 that if he and his colleagues can simplify the violates the dictum. transformation process, probabilistic proofs might become a useful method for verifying thought alone,” he explains. Mathemati- into mathematics. This past January Lenore mathematical propositions and large compu- cians seeking to navigate uncharted waters Blum, the institute’s deputy director, orga- tations—such as those leading to the four- may become increasingly dependent on ex- nized a seminar devoted to the question color theorem. “The philosophical cost of periments, probabilistic proofs and other “Are Proofs in High School Geometry Ob- this efficient method is that we lose the abso- guides. “You may not be able to provide solete?” lute certainty of a Euclidean proof,” Babai proofs in a classical sense,” Graham says. The mathematicians insisted that proofs not-ed in a recent essay. “But if you do have Of course, mathematics may yield fewer are crucial to ensure that a result is true. The doubts, will you bet with me?” aesthetic satisfactions as inves-tigators be- high school teachers demurred, pointing out Such a bet would be ill advised, Levin become more dependent on computers. “It that students no longer considered tradition- lieves, since a relatively few checks can make would be very discouraging,” Graham real, axiomatic proofs to be as convincing as, the chance of error vanishingly small: one marks, “if somewhere down the line you say, visual arguments. “The high school divided by the number of particles in the uni- could ask a computer if the Riemann hy- teachers overwhelmingly declared that most verse. Even the most straightforward con- pothesis is correct and it said, ‘Yes, it is true, students now (Nintendo/joystick/MTV gen- ventional proofs, Levin points out, are sus- but you won’t be able to understand the eration) do not relate to or see the impor- ceptible to doubts of this scale. “At the mo- proof.’ ” tance of ‘proofs,’ ” the minutes of the meet- ment you find an error, your brain may Traditionalists no doubt shudder at the ing stated. Note the quotation marks around disappear because of the Heisenberg uncer- thought. For now, at least, they can rally be- the word “proofs.” SA tainty principle and be replaced by a new hind heros like Wiles, the conqueror of Fer- brain that thinks the proof is correct,” he mat’s last theorem, who eschews computers, says. applications and other abominations. But FURTHER READING Ronald L. Graham of AT&T Bell Labora- there may be fewer Wileses in the future if ISLANDS OF TRUTH: A MATHEMATI- tories suggests that the trend away from reports from the front of precollege educa- CAL MYSTERY CRUISE. Ivars Peterson. W. short, clear, conventional proofs that are be- tion are any guide. The Mathematical Scienc- H. Freeman and Company, 1990. yond reasonable doubt may be inevitable. es Research Institute at Berkeley, which is THE PROBLEMS OF MATHEMATICS. Ian “The things you can prove may be just tiny overseen by Thurston, has been holding an Stewart. Oxford University Press, 1992. PIINTHE SKY: COUNTING, THINKING, islands, exceptions, compared to the vast sea ongoing series of seminars with high school AND BEING. John D. Barrow. Oxford Uni- of results that cannot be proved by human teachers to find new ways to entice students versity Press, 1992.

For millennia, mathematicians and philosophers have tried to refute Zeno’s paradoxes, a set of riddles suggesting that motion is inherently impossible. At last, a solution has been found

by William I. McLaughlin

nce upon a time Achilles met a matter how fast he ran, a fraction of the dis- This portrayal of the world is contrary to tortoise in the road. The tor- tance remained. In fact, it appeared that the our everyday experience and relegates our O toise, whose mind was quicker hero could never overtake the plodding tor- most fundamental perceptions to the realm than his feet, challenged the swift hero to a toise. of illusion. Parmenides relied on Zeno’s race. Amused, Achilles accepted. The tor- Had Achilles spent less time in the gym powerful arguments, which were later toise asked if he might have a head start, as and more time studying philosophy, he recorded in the writings of Aristotle, to sup- he was truly much slower than the demigod. would have known that he was acting out port his case. For two and a half millennia, Achilles agreed happily, and so the tortoise the classic example used to illustrate one of Zeno’s paradoxes have provoked debates started off. After taking quite a bit of time to Zeno’s paradoxes, which argue against the and stimulated analyses. At last, using a for- fasten one of his sandal’s ankle straps, Achil- possibility of all motion. Zeno designed the mulation of calculus that was developed in les bolted from the starting line. In no time paradox of Achilles and the tortoise, and its just the past decade or so, it is possible to re- at all, he ran half the distance that separated companion conundra (more about them lat- solve Zeno’s paradoxes. The resolution de- him from the tortoise. Within another blink, er), to support the philosophical theories of pends on the concept of infinitesimals, he had covered three quarters of the stretch. his teacher, Parmenides. known since ancient times but until recently In another instant, he made up seven eighths Both men were citizens of the Greek viewed by many thinkers with skepticism. and in another, fifteen sixteenths. But no colony of Elea in southern Italy. In approxi- mately 445 B.C., Parmenides and Zeno met he tale of Achilles and the tortoise with Socrates in Athens to exchange ideas on depicts one of Zeno’s paradoxes, basic philosophical issues. The event, one of usually denoted “The Dichoto- WILLIAM I. MCLAUGHLIN is a technical T the greatest recorded intellectual encounters my”: any distance, such as that between the manager for advanced space astrophysics at (if it really took place), is commemorated in two contenders, over which an object must the Jet Propulsion Laboratory in Pasadena, 1 1 1 Plato’s dialogue Parmenides. Parmenides, a traverse can be halved ( ⁄ 2, ⁄ 4, ⁄ 8 and so Calif., where he has worked since 1971. He distinguished thinker nearly 65 years old, on) into an infinite number of spatial seg- has participated in many projects for the U.S. space program, including the Apollo lunar- presented to the young Socrates a startling ments, each representing some distance yet landing program, the Viking mission to Mars, thesis: “reality” is an unchanging single enti- to be traveled. As a result, Zeno asserts that the Infrared Astronomical Satellite (IRAS ) ty, seamless in its unity. The physical world, no motion can be completed because some and the Voyager project, about which he wrote he argued, is monolithic. In particular, mo- distance, no matter how small, always re- an article for Scientific American in No- tion is not possible. Although the rejection of mains. It is important to note that he does vember 1986. He received a B.S. in electrical plurality and change appears idiosyncratic, it not say that infinitely many stretches cannot engineering in 1963 and a Ph.D. has, in general outline, proved attractive to add up to a finite distance (glancing at the in mathematics in 1968, both from the University of California, Berkeley. Mc- numerous scholars. For example, the “abso- geometry of an infinitely partitioned line Laughlin conducts, in addition, research in lute idealism” of the Oxford philosopher F. shows immediately, without any sophisticat- epistemology. H. Bradley (1846–1924) has points in com- ed calculations, that an infinite number of mon with the Parmenidean outlook. pieces sum to a finite interval). Rather the

16 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. force of Zeno’s objection to the idea of mo- would be so very near zero as to be numeri- quantities, nor quantities infinitely small, nor tion comes from the obligation to explain cally impotent; such quantities would elude yet nothing. May we not call them ghosts of how an infinite number of acts—crossing all measurement, no matter how precise, like departed quantities?” He observed further: one interval—can be serially completed. sand through a sieve. “Whatever mathematicians may think of Zeno made a second attack on the con- Giovanni Benedetti (1530–1590), a pre- fluxions [rates of change], or the differential ceptual underpinnings of motion by viewing decessor of Galileo, postulated that when an calculus, and the like, a little reflexion will this first argument from a slightly different object appeared to be frozen in midair to shew them that, in working by those meth- perspective. His second paradox is as Zeno, he was in fact seeing only part of the ods, they do not conceive or imagine lines or follows: Before an object, say, an arrow, gets action, as though one were watching a slide surfaces less than what are perceivable to to the halfway mark of its supposed journey show instead of a movie. Between the static sense.”

Mathematicians found inﬁnitesimals hard to skirt in the course of their discoveries, no matter how distasteful they found them in theory.

(an achievement granted in the preceding images Zeno saw were infinitesimally small Indeed, mathematicians found infin- case), it must first travel a quarter of the dis- instants of time in which the object moved itesimals hard to skirt in the course of their tance. As in Zeno’s first objection, this rea- by equally small distances. discoveries, no matter how distasteful they soning can be continued indefinitely to yield Others sidestepped the issue by arguing found them in theory. Some historians be- an infinite regress, thus leading to his insis- that intervals in the physical world cannot lieve the great Archimedes (circa 287–212 tence that motion could never be initiated. simply be subdivided an infinite number of B.C.) achieved some of his mathematical re- Zeno’s third paradox takes a different tack times. Friedrich Adolf Trendelenburg sults using infinitesimals but employed more altogether. It asserts that the very concept of (1802–1872) of the University of Berlin built conventional modes for public presentations. motion is empty of content. Zeno invites us an entire philosophical system that explained Infinitesimals left their mark during the 17th to consider the arrow at any one instant of human perceptions in terms of motion. In and 18th centuries as well in the develop- its flight. At this point in time, the arrow oc- doing so, he freed himself from explaining ment of differential and integral calculus. El- cupies a region of space equal to its length, motion itself. ementary textbooks have long appealed to and no motion whatsoever is evident. Be- Similarly, in this century, the English “practical infinitesimals” to convey certain cause this observation is true at every instant, philosopher and mathematician Alfred ideas in calculus to students. the arrow is never in motion. This objection, North Whitehead (1861–1947) constructed When analysts thought about rigorously in a historical sense, proved the most trou- a system of metaphysics based on change, in justifying the existence of these small quanti- blesome for would-be explainers of Zeno’s which motion was a special case. Whitehead ties, innumerable difficulties arose. Eventual- paradoxes. responded to Zeno’s objections by insisting ly, mathematicians of the 19th century in- Many philosophers and mathematicians that events in the physical world had to have vented a technical substitute for infinitesi- have made various attempts to answer some extent; namely, they could not be mals: the so-called theory of limits. So Zeno’s objections. The most direct approach pointlike. Likewise, the Scottish philosopher complete was its triumph that some mathe- has simply been to deny that a problem ex- David Hume (1711–1776) wrote, “All the maticians spoke of the “banishment” of ists. For example, Johann Gottlieb Waldin, a ideas of quantity upon which mathemati- infinitesimals from their discipline. By the German professor of philosophy, wrote in cians reason, are nothing but particular, and 1960s, though, the ghostly tread of in- 1782 that the Eleatic, in arguing against mo- such as are suggested by the senses and finitesimals in the corridors of mathematics tion, assumed that motion exists. Evidently imagination and consequently, cannot be became quite real once more, thanks to the the good professor was not acquainted with infinitely divisible.” work of the logician Abraham Robinson of the form of argument known as reductio ad Either way, the subject of infinitesimals Yale University [see “Nonstandard Analy- absurdum: assume a state of affairs and then (and whether they exist or not) generated a sis,” by Martin Davis and Reuben Hersh; show that it leads to an illogical conclusion. long and acrimonious literature of its own. SCIENTIFIC AMERICAN, June 1972]. Nevertheless, other scholars made prog- Until recently, most mathematicians thought Since then, several methods in addition to ress by wrestling with how an infinite num- them to be a chimera. The Irish bishop Robinson’s approach have been devised that ber of actions might occur in the physical George Berkeley (1685–1753) is noted prin- make use of infinitesimals. world. Their explanations have continually cipally for his idealistic theory, which denied been intertwined with the idea of an the reality of matter, but he, too, wrestled hen my colleague Sylvia infinitesimal, an interval of space or time with infinitesimals. He believed them ill con- Miller and I started our work that embodies the quintessence of smallness. ceived by the mathematicians of the time, in- Won Zeno’s paradoxes, we had An infinitesimal quantity, some surmised, cluding Newton. “They are neither finite the advantage that infinitesimals had be-

17 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. difference between two concrete numbers must be concrete (and hence, standard). If this difference were infinitesimal, the definition of an infinitesimal as less than all standard numbers would be violated. The conse- quence of this fact is that both end points of an infinitesimal interval cannot be labeled using concrete numbers. Therefore, an infinitesimal interval can never be captured through measurement; infinitesimals remain forever beyond the range of observation.

o how can these phantom numbers be used to refute Zeno’s paradoxes? SFrom the above discussion it is clear that the points of space or time marked with concrete numbers are but isolated points. A trajectory and its associated time interval are in fact densely packed with inﬁnitesimal regions. As a result, we can grant Zeno’s third objection: the arrow’s tip is caught “strobo- scopically” at rest at concretely labeled points of time, but along the vast majority of the stretch, some kind of motion is taking place. This motion is immune from Zenoni- an criticism because it is postulated to occur

TRICIA J. WYNNE inside inﬁnitesimal segments. Their ineffabil-

PA ity provides a kind of screen or filter. RACE between Achilles and the tortoise illustrates one of Zeno’s paradoxes. Achilles gives the tor- Might the process of motion inside one of toise a head start. He must then make up half the distance between them, then three fourths, then these intervals be a uniform advance across seven eighths and so on, ad infinitum. In this way, it would seem he could never come abreast of the interval or an instantaneous jump from the sluggish animal. one end to the other? Or could motion com- prise a series of intermediate steps or else a process outside of time and space come mathematically respectable. We were numbers on the real line by adding three altogether? The possibilities are infinite, and intuitively drawn to these objects because rules, or axioms, to the set of 10 or so state- none can be verified or ruled out since an in- they seem to provide a microscopic view of ments supporting most mathematical sys- finitesimal interval can never be monitored. the details of motion. Edward Nelson of tems. (Zermelo-Fraenkel set theory is one Credit for this rebuttal is due to Benedetti, Princeton University created the tool we such foundation.) These additions introduce Trendelenburg and Whitehead for their earli- found most valuable in our attack, a brand a new term, standard, and help us to deter- er insights, which can now be formalized by of nonstandard analysis known by the rather mine which of our old friends in the number means of IST. arid name of internal set theory (IST). Nel- system are standard and which are nonstan- We can answer Zeno’s first two objections son’s method produces startling interpreta- dard. Not surprisingly, the infinitesimals fall more easily than we did the third, but we tions of seemingly familiar mathematical in the nonstandard category, along with need to use another mathematical fact from structures. The results are similar, in their some other numbers I will discuss later. IST. Every infinite set of numbers contains a strangeness, to the structures of quantum Nelson defines an infinitesimal as a num- nonstandard number. Before drawing out theory and general relativity in physics. Be- ber that lies between zero and every positive the Zenonian implications of this statement, cause these two theories have taken the bet- standard number. At first, this might not it is necessary to talk about the two other ter part of a century to gain widespread ac- seem to convey any particular notion of types of nonstandard numbers that are read- ceptance, we can only admire the power of smallness, but the standard numbers include ily manufactured from infinitesimal num- Nelson’s imagination. every concrete number (and a few others) bers. First, take all the infinitesimals, which Nelson adopted a novel means of defining you could write on a piece of paper or gener- by definition are wedged between zero and 1 infinitesimals. Mathematicians typically ex- ate in a computer: 10, pi, ⁄ 1000 and so on. all the positive, standard numbers, and put a pand existing number systems by tacking on Hence, an infinitesimal is greater than zero minus sign in front of each one. Now there is objects that have desirable properties, much but less than any number, however small, a symmetrical clustering of these small ob- in the same way that fractions were sprin- you could ever conceive of writing. It is not jects about zero. To create “mixed” nonstan- kled between the integers. Indeed, the num- immediately apparent that such infinitesi- dard numbers, take any standard number, ber system employed in modern mathemat- mals do indeed exist, but the conceptual va- say, one half, and add to it each of the non- ics, like a coral reef, grew by accretion onto a lidity of IST has been demonstrated to a de- standard infinitesimals in the grouping supporting base: “God made the integers, all gree commensurate with our justified belief around zero. This act of addition translates the rest is the work of man,” declared in other mathematical systems. the original cluster of infinitesimals to posi- Leopold Kronecker (1823–1891). Instead Still, infinitesimals are truly elusive enti- tions on either side of one half. Similarly, ev- the way of IST is to “stare” very hard at the ties. Their elusiveness rests on the mathemat- ery standard number can be viewed as hav- existing number system and note that it al- ical fact that two concrete numbers—those ing its own collection of nearby, nonstan- ready contains numbers that, quite reason- having numerical content—cannot differ by dard numbers, each one only an infinitesimal ably, can be considered infinitesimals. an infinitesimal amount. The proof, by re- distance from the standard number. Technically, Nelson finds nonstandard ductio ad absurdum, is easy: the arithmetic The third type of nonstandard number is

18 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. simply the inverse of an infinitesimal. Be- object is moot. Many descriptions of motion icates, such as standard, to define subsets; cause an infinitesimal is very small, its in- in the microrealm other than that containing the stricture is introduced to avoid contra- verse will be very large (in the standard the full series of checkpoints could apply, dictions. For example, imagine the set of all realm, the inverse of one millionth is one and just because his particular scenario caus- standard numbers in F. This set would be million). This type of nonstandard number es conceptual problems, there is no reason to finite because it is a subset of a finite set. It is called an unlimited number. The unlimited anathematize the idea of motion. His second would therefore have a least member, say, r. numbers, though large, are finite and hence argument, attempting to show that an object But then r–1 would be a standard number smaller than the truly infinite numbers creat- can never even start to move, suffers from less than r, when r was supposed to be the ed in mathematics. These unlimited numbers the same malady as the first, and we reject it smallest standard number. Thus, we cannot live in a kind of twilight zone between the fa- on like grounds. say the standard numbers are finite or miliar standard numbers, which are finite, infinite in extent, because we cannot form and the infinite ones. e have resolved Zeno’s three the set of them and count them. If, as demonstrated in IST, every infinite paradoxes using some techni- Nevertheless, the finite set F, though con- set contains a nonstandard number, then the W cal results from IST and the strained as to how it can be visualized, is infinite series of checkpoints Zeno used to principle that nonstandard numbers are not useful for constructing our theory of motion. gauge motion in his first argument must con- suitable for describing matters of fact, ob- This theory can be expressed quite simply as tain a mixed, nonstandard number. In fact, served or purported. Still, more can be said stepping through F, where each member of F as Zeno’s infinite series of numbers creeps regarding the matter than just the assurance represents a distinct moment. For conve- closer to one, a member of that series will that Zeno’s objections do not preclude mo- nience, consider only those members of F eventually be within an infinitesimal distance tion. Indeed, we can construct a theory of that fall between 0 and 1. Let time 0 be the from one. At that point, all succeeding mem- motion using a very powerful result from instant when we start tracking a moving ob- bers of the series will be nonstandard mem- IST. The theory yields the same results as do ject. The second instant when we might try bers of the cluster about one, and neither the tools of the calculus, and yet it is easier to to observe the object is at time f1, where f1 is Zeno nor anyone else will be able to chart visualize and does not fall prey to Zeno’s ob- the smallest member of F that is greater than the progress of a moving object in this inac- jections. 0. Ascending through F in this fashion, we cessible region. A theorem proved in IST states that there eventually reach time fn, where fn is the exists a finite set, call it F, that contains all largest member of F less than 1. In one more here is an element of irony in using the standard numbers! The corollary that step, we reach 1 itself, the destination in this infinity, Zeno’s putative weapon, there are only a finite number of standard example. In order to walk through a non- T to deflate his claims. To refute numbers would seem to be true, but surpris- infinitesimal distance, such as the span from Zeno’s first paradox, we need only state the ingly, it is not. In developing IST, Nelson 0 to 1 using infinitesimal steps, the subscript epistemological principle that we are not re- needed to finesse the conventional way n of fn must be an unlimited integer. The sponsible for explaining situations we can- mathematicians form objects. A statement in process of motion then is divided into n +1 not observe. Zeno’s infinite series of check- IST is called internal if it does not contain acts, and because n +1 is also finite, points contains nonstandard numbers, the label “standard.” Otherwise, the state- this number of acts can be completed which have no numerical meaning, and so ment is called external. Mathematicians fre- sequentially. we reject his argument based on these enti- quently create subsets from larger sets by Of the possible observing times identified ties. Because no one could ever, even in prin- predicating a quality that characterizes each earlier, the object’s progress could be reported ciple, observe the full domain of checkpoints of the objects in the subset—the balls that solely at those instants corresponding to cer- that his objection addresses, the objection- are red or the integers that are even. In IST, tain standard numbers in F. (By the way, f1 able behavior he postulates for the moving however, it is forbidden to use external pred- and fn would be nonstandard, as they are

Topology of the Real Line

—(N + 1) —N —3 —2 —1 0 1 2 5 NN + 1

he real numbers consist of the integers (positive and nega- Mixed nonstandard numbers, shown grouped around the inte- Ttive whole numbers), rational numbers (those that can be ger 5, result from adding and subtracting infinitesimal amounts expressed as a fraction) and irrational numbers (those that can- to standard numbers. In fact, every standard number is surround- not be expressed as a fraction). The real numbers can be repre- ed by such mixed, nonstandard neighbors. Unlimited nonstan- sented as points on a straight line known as the real line (above). dard numbers, represented as N and N +1, are the inverses of in- The mathematician Edward Nelson of Princeton University la- finitesimal nonstandard numbers. Each unlimited number is beled three types of numbers as nonstandard within this stan- greater than every standard number and yet less than the infinite dard number system. Infinitesimal nonstandard numbers are real numbers. The nonstandard real numbers prove useful in re- smaller than any positive standard number yet are greater than solving Zeno’s paradoxes. zero. JOHNNY JOHNSON

o see the relation between in- 32dt +16dt 2, divided by dt, is the de- Tfinitesimals and differential cal- 0 sired average velocity, 32 + 16dt. culus, consider the simple case of a Because 16dt is but an infinitesimal 50 falling stone. The distance the stone has amount, undetectable for all intents and traveled in feet can be calculated from purposes, it can be considered equal to 100 the formula s =16t 2, where t equals 0. Thus, after one second of travel, the the time elapsed in seconds. For exam- 150 formula yields the stone’s instanta-

ple, if a stone has fallen for two seconds, FEET neous velocity as 32 feet per second. it will have traveled 64 feet. 200 This manipulation, of course, resem- Suppose, however, one wishes to cal- bles those used in traditional, differen- culate the instantaneous velocity of the 250 tial calculus. There the small residue stone. The average speed of a moving 16dt cannot be dropped at the end of object equals the total distance it trav- 300 the calculation; it is a noninfinitesimal els divided by the total amount of time it 0 1342 quantity. Instead, in this calculus, it takes. By using this formula over an in- SECONDS must be argued away using the theory finitesimal change in the total distance of limits. In essence, the limit process and time, one can calculate a fair approximation of an object’s instantaneous renders the interval of length dt sufficiently small so that the average velocity velocity. is arbitrarily close to 32. As before, the instantaneous velocity of the stone af- Let dt represent an infinitesimal change in time and ds an infinitesimal ter one second of travel equals 32 feet per second. Similarly, judicious use of change in distance. The computation for the velocity of the stone after one infinitesimal regions facilitates the computation of the area of complicated re- second of travel, then, will be as follows: The time frame under consideration gions, a basic problem of integral calculus. Some think the newer calculus is ranges from t =1 to t =1+dt. The position of the stone during that time pedagogically superior to calculus without infinitesimals. Nevertheless, both changes from s =16(1)2 to s =16(1+ dt )2. The total change in distance, methods are equally rigorous and yield identical results.

infinitesimally close to 0 and 1, respectively.) nonstandard points of F are irrelevant given vation of certain events to discrete values. For example, although we can express a that they cannot be observed. Of course, this theory of motion is not a standard number to any finite (but not un- For many centuries, Zeno’s logic stood version of quantum mechanics (nor relativi- limited) number of decimal places and use mostly intact, proving the refractory nature ty theory, for that matter). Because the this approximation as a measurement label, of his arguments. A resolution was made theory resulted from a thought experiment we cannot access the unlimited tail of the ex- possible through two basic features of IST: on Zeno’s terms, it holds no direct pansion to alter a digit and thus define a first, the ability to partition an interval of connection to present physical theory. nonstandard, infinitesimally close neighbor. time or space into a finite number of ineffa- Moreover, the specific rules inherited from Only concrete standard numbers are effec- ble infinitesimals and, second, the fact that IST are probably not those best suited to tive as measurement labels; the utility of standardly labeled points—the only ones describe reality. Modern physics might their nonstandard neighbors for measure- that can be used for measurement—are iso- adapt the IST approach by modifying its rule ment is illusory. lated objects on the real line. Is our work system and introducing “physical con- merely the solution to an ancient puzzle? stants,” perhaps by assigning parameters to uch is superfluous in this the- Possibly, but there are several directions in the set F. ory of motion, and much is which it might prove extensible. But maybe not. Still, the simplicity and el- M left unsaid. It suffices, howev- Aside from its mathematical value, IST is egance of such thought experiments have er, in the sense that it can easily be translated ripe with epistemological import, as this catalyzed research throughout the ages. No- into the symbolic notation of the integral or analysis has shown. It might well be table examples include Heinrich W. M. Ol- differential calculus, commonly used to de- modified to constitute a general epistemic bers, questioning why the sky is dark at scribe the details of motion [see box on pre- logic. Also, infinitesimal intervals, or their night despite stars in every direction, or ceding page]. More important in the present generalization, would promise a technical re- James Clerk Maxwell, summoning a med- context, the finiteness of the set F enables us source to house Whitehead’s so-called actual dling, microscopic demon to batter the sec- to jump over the pitfalls in Zeno’s first two entities, the generative atoms of his philo- ond law of thermodynamics. Likewise, paradoxes. His third objection is dodged as sophical system. Zeno’s arguments have stimulated examina- before: motion in real time is an unknown Finally, the current theory of motion and tions of our ideas about motion, time and process that takes place in infinitesimal inter- the predictions of quantum physics are not space. The path to their resolution has been vals between the standard points of F; the dissimilar in that they both restrict the obser- eventful. SA

FURTHER READING A HISTORY OF GREEK PHILOSOPHY, Vol. 2: THE PRESOCRATIC TRADITION FROM PARMENIDES TO DEMOCRITUS. W. K. Guthrie. Cam- bridge University Press, 1965. ZENO OF ELEA. Gregory Vlastos in The Encyclopedia of Philosophy. Edited by Paul Edwards. Macmillan Publishing Company, 1967. NONSTANDARD ANALYSIS. Martin Davis and Reuben Hersh in Scientiﬁc American, Vol. 226, No. 6, pages 78–86; June 1972. INTERNAL SET THEORY: A NEW APPROACH TO NONSTANDARD ANALYSIS. Edward Nelson in Bulletin of the American Mathematical Society, Vol. 83, No. 6, pages 1165–1198; November 1977. AN EPISTEMOLOGICAL USE OF NONSTAN-DARD ANALYSIS TO ANSWER ZENO’S OB-JECTIONS AGAINST MOTION. William I. McLaugh- lin and Sylvia L. Miller in Synthese, Vol. 92, No. 3, pages 371–384; September 1992.

20 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. Originally published in April 1995 A Brief History of Infinity The infinite has always been a slippery concept. Even the commonly accepted mathematical view, developed by Georg Cantor, may not have truly placed infinity on a rigorous foundation

by A. W. Moore

or more than two millennia, math- so on, ad infinitum. It seems that Achilles the area of C. The same is not true of C*. ematicians, like most people, were can never overtake the tortoise. In like man- This fact, combined with a similar result for F unsure what to make of the infi- ner Zeno argued that it is impossible to com- circumscribed polygons and supplemented nite. Several paradoxes devised by Greek and plete a racecourse. To do so, it is necessary to with a refined version of the logic contained medieval thinkers had convinced them that reach the halfway point, then the three-quar- in that argument, finally enabled the infinite could not be pondered with im- ters point, then the seven-eighths point, and Archimedes to show, without ever invoking punity. Then, in the 1870s, the German so on. Zeno concluded not only that motion the infinite, that the area of a circle equals mathematician Georg Cantor unveiled is impossible but that we do best not to π r 2. transfinite mathematics, a branch of mathe- think in terms of the infinite. matics that seemingly resolved all the puzzles The mathematician Eudoxus, similarly The Actual and Potential Infinite the infinite had posed. In his work Cantor wary of the infinite, developed the so-called showed that infinite numbers existed, that method of exhaustion to circumvent it in cer- lthough Archimedes successfully they came in different sizes and that they tain geometric contexts. Archimedes exploit- A ducked the infinite in this particular could be used to measure the extent of ed that method some 100 years later to find exercise, the Pythagoreans (a religious soci- infinite sets. But did he really dispel all doubt the exact area of a circle. How did he pro- ety founded by Pythagoras) happened on a about mathematical dealings with infinity? ceed? In the box on page 23, I present not case in which the infinite was truly in- Most people now believe he did, but I shall his actual derivation but a corruption of it. escapable. This find shattered their belief in suggest that in fact he may have reinforced Part of Archimedes’ own procedure was to two fundamental cosmological principles: that doubt. consider the formula for the area of a poly- Peras (the limit), which subsumed all that The hostility of mathematicians toward gon with n equal sides—call it Pn—inscribed was good, and Apeiron (the unlimited or infinity began in the fifth century B.C., when inside a circle C. According to the distortion infinite), which encompassed all that was Zeno of Elea, a student of Parmenides, for- of his argument, this formula can be applied bad. They had insisted that the whole of cre- mulated the well-known paradox of Achilles to the circle itself, which is just a polygon ation could be understood in terms of, and and the tortoise [see “Resolving Zeno’s with infinitely many, infinitely small sides. indeed was ultimately constituted by, the Paradoxes,” by William I. McLaughlin; SCI- The perversion of Archimedes’ argument positive integers, each of which is finite. This ENTIFIC AMERICAN, November 1994]. has some intuitive appeal, but it would not reduction was made possible, they main- In this conundrum the swift demigod chal- have satisfied Archimedes. We cannot uncrit- tained, by the fact that Peras was ever subju- lenges the slow tortoise to a race and grants ically make use of the infinite as though it gating Apeiron. her a head start. Before he can overtake her, were just some unusually big integer. Part of Pythagoras had discovered, however, that he must reach the point at which she began, what is going on here is that the larger n is, the square of the hypotenuse (the longest by which time she will have advanced a lit- the more nearly Pn matches C. But it is also side) of a right-angled triangle is equal to the tle. Achilles must now make up the new dis- true that the larger n is, the more nearly Pn sum of the squares of the other two sides. tance separating them, but by the time he approximates a circle with a bulge—call it Given this theorem, the ratio of a square’s does so, she will have advanced again. And C*. The key point intuitively is that C, un- diagonal to each side is √ 2 to 1, since like its deformed counterpart C*, is the limit 12 +12 = (√ 2) 2. Were Peras impervious, this of the polygons—or what they are tending ratio should be expressible in the form p to A. W. MOORE is a tutorial fellow in phi- toward. q, where p and q are both positive integers. losophy at St. Hugh’s College of the Universi- Still, it is very hard to see any way of cap- Yet this is impossible. Imagine two positive ty of Oxford. He studied for his Ph.D. in the turing this intuition without, once again, integers, p and q, such that the ratio of p to philosophy of language at Balliol College at thinking of C as an “infinigon.” Archimedes q, or p divided by q, is equivalent to√ 2. We Oxford. His main academic interests are in provided a way. He pinpointed the crucial can assume that p and q have no common logic, metaphysics and the philosophies of Im- difference between C and C* by proving the factor greater than 1 (we could, if necessary, manuel Kant and Ludwig Wittgenstein, all of 2 2 which have informed his work on the infinite. following point: no matter how small an area divide by that factor). Now, p is twice q . ε 2 He is at work on a book about the meta- you consider, call it (the Greek letter ep- So p is even, which means that p itself is physics of objectivity and subjectivity. silon), there exists an integer n that is large even. Hence, q must be odd, otherwise 2 ε enough for the area of Pn to be within of would be a common factor. But consider: if p

21 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. is even, there must be a positive integer r that thing deeper and more abstract. Existing “in also include odd numbers. is exactly half of p. Therefore, (2r)2 equals time” or existing “all at once” assumed The medievals proffered many similar ex- 2q2, or 2r 2 equals q2, which means that q2 much broader meanings. To take exception amples, some of which were geometric. In is even, and so q itself is also even, contrary to the actual infinite was to object to the the 13th century the Scottish mathematician to what was proved above. very idea that some entity could have a John Duns Scotus puzzled over the case of For the Pythagoreans, this result was property that surpassed all finite measure. It two concentric circles: all the points on the nothing short of catastrophic. (According to was also to deny that the infinite was itself a shorter circumference of the smaller circle legend, one of them was shipwrecked at sea legitimate object of study. can be paired off with all the points on the for revealing the discovery to their enemies.) Some 2,000 years later the infinite, both longer circumference of the bigger circle. They had come on an “irrational” number. actual and potential, exercised mathemati- The same result applies to two spheres. In doing so, they had seen the limitations of cians once more as they developed the calcu- Some 350 years later Galileo discussed a the positive integers, and they had been lus. Early work on the calculus, ushered in variation of the pairing example of the even forced to acknowledge the presence of the by Isaac Newton and Gottfried Wilhelm integers, based instead on squared integers. infinite in their very midst. Indeed, a modern Leibniz in the late 17th century, fell far short Particularly striking is the fact that as in- mathematician would say that √2 of Greek standards of rigor. Indeed, mathe- creasingly larger segments of the sequence of Asking whether Cantor’s continuum hypothesis is true may be like asking whether Hamlet was left-handed. It may be that not enough is known to form an answer. is a kind of “infinite object.” Not only is its maticians had made extensive, uncritical use positive integers are considered, the propor- decimal expansion infinite, but this expan- of infinitesimals, items taken to be too small tion of these integers that are squares tends sion never adopts a recurring finite pattern. for measure. Sometimes these quantities toward zero. Nevertheless, the pairing still In the fourth century B.C. Aristotle recog- were considered equal to zero. For example, proceeds indefinitely. nized a more general problem. On the one when they were added to another number, It is certainly tempting, in view of these hand, we are under pressure to acknowledge the value of the original number remained difficulties, to eschew infinite sets entirely. the infinite. Quite apart from what we may the same. At other times, they were taken to More generally, it is tempting to deny, as did have to say about √ 2, time appears to con- be different from zero and used in division. Aristotle, that infinitely many things can be tinue indefinitely, numbers seem to go on Guillaume François Antoine de l’Hôpital gathered together all at once. Eventually, endlessly, and space, time and matter seem to wrote: “A curve may be regarded as the to- though, Cantor challenged the Aristotelian be forever divisible. On the other hand, we tality of an infinity of straight segments, each view. In work of great brilliance he took the are under pressure from various sources, in- infinitely small: or ... as a polygon with an paradoxes in his stride and formulated a co- cluding Zeno’s paradoxes, to repudiate the infinite number of sides.” Only in the 19th herent, systematic and precise theory of the infinite. century did French mathematician Augustin- actual infinite, ready for any skeptical gaze. Aristotle’s solution to this dilemma was Louis Cauchy and German mathematician Cantor accepted the “pairing off” principle masterful. He distinguished between two Karl Weierstrass resuscitate the method of and its converse, namely, that no two sets different kinds of infinity. The actual infinite exhaustion and give the calculus a secure are equinumerous unless their members can is that whose infinitude exists at some point foundation. be paired off. Accordingly, he accepted that in time. In contrast, the potential infinite is there are just as many even positive integers that whose infinitude is spread over time. All The Infinite and Equinumerosity as there are positive integers altogether (and the objections to the infinite, Aristotle insist- likewise in the other paradoxical cases). ed, are objections to the actual infinite. The s a result of Cauchy’s and Weier- Let us for the sake of argument, and con- potential infinite, on the other hand, is a fun- Astrass’s work, most mathematicians felt temporary mathematical convention for that damental feature of reality. It deserves recog- less threatened by Zeno’s paradoxes. Of matter, follow suit. If this principle means nition in any process that can never end, in- more concern by then was a family of para- that the whole is no greater than its parts, so cluding counting, the division of matter and doxes born in the Middle Ages dealing with be it. We can in fact use this idea to define the passage of time itself. This distinction be- equinumerosity. These puzzles derive from the infinite, at least in its application to sets: tween the two types of infinity provided a the principle that if it is possible to pair off a set is infinite if it is no bigger than one of solution to Zeno’s paradoxes. Traversing a all the members of one set with all those of its parts. More precisely, a set is infinite if it region of space does not involve moving another, the two sets must have equally many has as many members as does one of its across an actual infinity of subregions, which members. For example, in a nonpolygamous proper subsets. would be impossible. But it does mean cross- society there must be just as many husbands What remains an open question, once ing a potential infinity of subregions, in the as wives. This principle looks incontestable. things have been clarified in this way, is sense that there can be no end to the process Applied to infinite sets, however, it seems to whether all infinite sets are equinumerous. of dividing the space. This conclusion, fortu- flout a basic notion first articulated by Eu- Much of the impact of Cantor’s work came nately, is harmless. clid: the whole is always greater than any of in his demonstration that they are not. There Aristotle’s parting of the actual and the its parts. For instance, it is possible to pair are different infinite sizes. This proposition potential infinite long stood as orthodoxy. off all the positive integers with those that results from what is known as Cantor’s the- Nevertheless, scholars usually interpreted his are even: 1 with 2, 2 with 4, 3 with 6 and so orem: no set, and in particular no infinite set, reference to time as a metaphor for some- on—despite the fact that positive integers has as many members as it has subsets. In

ow did Archimedes use the method of exhaustion to find the original definition of Pn and allow n to be infinite. In this case, nbn is the area of a circle? Here is the corruption of his argument. Imag- circumference of C, which equals 2π r (which follows from the defini- H π π ine a circle C that has a radius r. For each integer n greater than 2, we tion of ), and hn is the radius r. So the area of C is 1/2(2 rr ), or sim- can construct a regular polygon with n sides and inscribe it inside C. ply πr2. This n-sided polygon—call it Pn— can be divided into n congruent triangles. Label the base of each trian- P4 P6 P8 P12 gle bn and its height hn. Then the area of each triangle is 1/2 bnhn. Thus, the area of Pn as a whole is n(1/2 bnhn ), or 1/2 nbnhn. But C itself is a polygon with infinitely many, infinitely small sides. In other words, C results when we extend the JARED SCHNEIDMAN DESIGN

other words, no set is as big as the set of its nowhere on this list of subsets, we make a gers. But how much smaller? Specifically, are subsets. Why not? Because if a set were, it new subset by moving down the “square’s there any sets of intermediate size? would be possible to pair off all its members diagonal,” replacing each yes with a no, and with all its subsets. Some members would vice versa. In the case illustrated, we write Cantor’s Continuum Hypothesis then be paired off with subsets that con- < yes, yes, no, no... >. What results repre- tained them, others not. So what of the set of sents the subset in question. For by construc- antor’s own hypothesis, his famous those members that were not included in the tion it differs from the first subset listed with C“continuum hypothesis,” was that set with which they had been paired? No respect to whether 1 belongs to it, from the there are not. But he never successfully member could be paired off with this subset second with respect to whether 2 belongs to proved this idea, nor did he disprove it. Sub- without contradiction. it, from the third with respect to whether 3 sequent work has shown that the situation is The argument can be recast in a diagram belongs to it, and so on. There is a pleasant far graver than he had imagined. Using all [see illustration above]. For convenience, I historical quirk here: just as studying a diag- the accepted methods of modern mathemat- will focus on the set of positive integers. I can onal had led the Pythagoreans to acknowl- ics, the issue cannot be settled. This problem represent any subset of the set of positive in- edge an infinitude beyond the grasp of the raises philosophical questions about the de- tegers by an infinite sequence of yeses and positive integers, the same was true in a dif- terminacy of Cantor’s conception. Asking noes, registering whether successive positive ferent way in Cantor’s case. whether the continuum hypothesis is true integers do or do not belong to the set. For Cantor later devised infinite cardinals— may be like asking whether Hamlet was left- example, the set of even integers can be rep- numbers that can be used to measure the size handed. It may be that not enough is known resented by the sequence , corresponding to 1, 2, 3, 4, 5, and so metic for them as well. Having defined his think how well Cantor’s work tames the ac- forth. We can do the same for the set of odd terms, he explored what happens when one tual infinite. integers , the set of infinite cardinal is added to another, when it Of even more significance are questions prime numbers is multiplied by another, when it is raised to surrounding the set of all sets. Given Can- and the set of squares . Generally, then, any assignment of mathematical craftsmanship of the highest than the set of sets of sets. But wait! Sets of different subsets to individual positive inte- caliber. But even in his own terms, difficul- sets are themselves sets, so it follows that the gers (such as the purely arbitrary example il- ties remained. The continuum problem is set of sets must be smaller than one of its lustrated) can be represented as an infinite perhaps the best known of these troubles. own proper subsets. That, however, is im- square of yeses and noes. The set of positive integers, we have seen, is possible. The whole can be the same size as To show that at least one subset is smaller than the set of sets of positive inte- the part, but it cannot be smaller. How did Cantor escape this trap? With wonderful pertinacity, he denied that there is any such thing as the set of sets. His reason lay in the 123 4 . . . n . . . following picture of what sets are like. There are things that are not sets, then there are sets of all these things, then there are sets of all those things, and so on, without end.

2 Each set belongs to some further set, but 1 4 9 16 . . . n . . . there never comes a set to which every set belongs. Cantor’s reasoning might seem somewhat ad hoc. But an argument of the sort is re- SETS ARE THE SAME SIZE if all their mem- quired, as revealed by Bertrand Russell’s bers can be paired with one another. But this principle seems to be violated in inﬁnite sets. memorable paradox, discovered in 1901. All the squared integers can be matched with This paradox concerns the set of all sets that every single positive integer (above), even do not belong to themselves. Call this set R. though the set of squares seems smaller. Simi- The set of mice, for example, is a member of larly, all the points on the smaller sphere can R; it does not belong to itself because it is

JARED SCHNEIDMAN DESIGN be paired o› with those on the larger one (left). a set, not a mouse. Russell’s paradox turns

23 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. on whether R can belong to itself. If it does, POSITIVE INTEGERS by deﬁnition it does not belong to R. If it SUBSETS 1 2345. . . does not, it satisﬁes the condition for mem- bership to R and so does belong to it. In any EVEN INTEGERS NO YES NO YES NO view of sets, there is something dubious about R. In Cantor’s view, according to ODD INTEGERS YES NO YES NO YES which no set belongs to itself, R, if it existed, would be the set of all sets. This argument PRIMES NO YES YES NO YES makes Cantor’s picture, and the rejection of R that goes with it, appear more reasonable. SQUARES YES NO NO YES NO But is the picture not strikingly Aris- totelian? Notice the temporal metaphor. Sets MULTIPLES OF 4 NO NO NO YES NO are depicted as coming into being “after” . . . .

their members—in such a way that there are JARED SCHNEIDMAN DESIGN always more to come. Their collective infini- DIAGONALIZED YES YES NO NO YES . . . tude, as opposed to the infinitude of any one SET of them, is potential, not actual. Moreover, is CANTOR’S THEOREM—that no set has as many members as it has subsets—is proved by diag- it not this collective infinitude that has best onalization, which creates an extra subset. Each subset of the set of positive integers is represent- claim to the title? People do ordinarily define ed as a series of yeses and noes. A yes indicates that the integer belongs to the subset; a no that it the infinite as that which is endless, unlimit- does not. Replacing each yes with a no, and vice versa, down the diagonal (shaded area) creates ed, unsurveyable and immeasurable. Few another subset. would admit that the technical definition of an infinite set expresses their intuitive under- be suggesting, with what most people would mathematical terminology. But I would urge standing of the concept. But given Cantor’s say. mathematicians and other scientists to use picture, endlessness, unlimitedness, unsur- Well, certainly most people would say the more caution than usual when assessing veyability and immeasurability more proper- set of positive integers is “really” infinite. how Cantor’s results bear on traditional con- ly apply to the entire hierarchy than to any But, then again, most people are unaware of ceptions of infinity. The truly infinite, it of the particular sets within it. Cantor’s results. They would also deny that seems, remains well beyond our grasp. SA In some ways, then, Cantor showed that one infinite set can be bigger than another. the set of positive integers, for example, is My point is not about what most people “really” finite and that what is “really” would say but rather about how they under- FURTHER READING infinite is something way beyond that. (He stand their terms—and how that under- INFINITY AND THE MIND: THE SCI- himself was not averse to talking in these standing is best able, for any given purpose, ENCE AND PHILOSOPHY OF THE INFI- terms.) Ironically, his work seems to have to absorb the shock of Cantor’s results. NITE. Rudy Rucker. Harvester, 1982. TO INFINITY AND BEYOND: A CULTUR- lent considerable substance to the Aris- Nothing here is forced on us. We could say AL HISTORY OF THE INFINITE. Eli Maor. totelian orthodoxy that “real” infinitude can some infinite sets are bigger than others. We Birkhauser, 1986. never be actual. could say the set of positive integers is only THE INFINITE. A. W. Moore. Routledge, Some scholars have objected to my sug- finite. We could hold back from saying either 1990. gestion that, in Cantor’s conception, the set and deny that the set of positive integers ex- INFINITY. Edited by A. W. Moore. Dart- of positive integers is “really” finite. They ists. mouth, 1993. complain that this assertion is at variance If the task at hand is to articulate certain UNDERSTANDING THE INFINITE. not only with standard mathematical termi- standard mathematical results, I would not Shaughan Lavine. Harvard University Press, 1994. nology but also, contrary to what I seem to advocate using anything other than standard

Diagonalization and Gödel’s Theorem

he diagonalization used in establishing Cantor’s theorem able if it can be defined using standard arithmetical terminology. Talso lies at the heart of Austrian mathematician Kurt Examples are the set of squares, the set of primes and the set of Gödel’s celebrated 1931 theorem. Seeing how offers a particular- positive integers less than, say, 821. ly perspicuous view of Gödel’s result. Definition 2: A set of positive integers is decidable if there is an Gödel’s theorem deals with formal systems of arithmetic. By algorithm for determining whether any given positive integer be- arithmetic I mean the theory of positive integers and the basic longs to the set. The same three sets above serve as examples. operations that apply to them, such as addition and multiplica- Lemma 1: There is an algorithmic way of pairing off positive tion. The theorem states that no single system of laws (axioms integers with arithmetically definable sets. and rules) can be strong enough to prove all true statements of Lemma 2: Every decidable set is arithmetically definable. arithmetic without at the same time being so strong that it Given lemma 1, diagonalization yields a set of positive inte- “proves” false ones, too. Equivalently, there is no single algo- gers that is not arithmetically definable. Call this set D. Now sup- rithm for distinguishing true arithmetical statements from false pose, contrary to Gödel’s theorem, there is an algorithm for dis- ones. Two definitions and two lemmas, or propositions, are need- tinguishing between true arithmetical statements and false ed to prove Gödel’s theorem. Proof of the lemmas is not possible ones. Then D, by virtue of its construction, is decidable. But given within these confines, although each is fairly plausible. lemma 2, this proposition contradicts the fact that D is not arith- Definition 1: A set of positive integers is arithmetically defin- metically definable. So Gödel’s theorem must hold after all. Q.E.D.

24 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. Originally published in November 1997 Fermat’s Last Stand His most notorious theorem bafﬂed the greatest minds for more than three centuries. But after 10 years of work, one mathematician cracked it

by Simon Singh and Kenneth A. Ribet

his past June, 500 mathemati- a small town in southwest France. He infinitely many sets of integer solutions, cians gathered in the Great pursued a career in local government such as a = 3, b = 4, c = 5, which are T Hall of Göttingen University and the judiciary. To ensure impartiali- known as Pythagorean triples. Fermat in Germany to watch Andrew J. Wiles ty, judges were discouraged from so- took the formula one step further and of Princeton University collect the pres- cializing, and so each evening Fermat concluded that there are no nontrivial tigious Wolfskehl Prize. The reward— would retreat to his study and concen- solutions for a whole family of similar established in 1908 for whoever proved trate on his hobby, mathematics. Al- equations, a n + b n = c n, where n repre- Pierre de Fermat’s famed last theorem— though an amateur, Fermat was highly sents any whole number greater than 2. was originally worth $2 million (in to- accomplished and was largely responsi- It seems remarkable that although day’s dollars). By the summer of 1997, ble for probability theory and the foun- there are infinitely many Pythagorean hyperinflation and the devaluation of the dations of calculus. Isaac Newton, the triples, there are no Fermat triples. Even mark had reduced it to a mere $50,000. father of modern calculus, stated that so, Fermat believed he could support But no one cared. For Wiles, proving he had based his work on “Monsieur his claim with a rigorous proof. In the Fermat’s 17th-century conundrum had Fermat’s method of drawing tangents.” margin of Arithmetica, the mischievous realized a childhood dream and ended Above all, Fermat was a master of genius jotted a comment that taunted a decade of intense effort. For the assem- number theory—the study of whole generations of mathematicians: “I have bled guests, Wiles’s proof promised to numbers and their relationships. He a truly marvelous demonstration of this revolutionize the future of mathematics. would often write to other mathemati- proposition, which this margin is too Indeed, to complete his 100-page cal- cians about his work on a particular narrow to contain.” Fermat made many culation, Wiles needed to draw on and problem and ask if they had the ingenu- such infuriating notes, and after his further develop many modern ideas in ity to match his solution. These chal- death his son published an edition of mathematics. In particular, he had to lenges, and the fact that he would never Arithmetica that included these teases. tackle the Shimura-Taniyama conjec- reveal his own calculations, caused oth- All the theorems were proved, one by ture, an important 20th-century insight ers a great deal of frustration. René Des- one, until only Fermat’s last remained. into both algebraic geometry and com- cartes, perhaps most noted for invent- Numerous mathematicians battled plex analysis. In doing so, Wiles forged ing coordinate geometry, called Fermat the last theorem and failed. In 1742 a link between these major branches of a braggart, and the English mathemati- Leonhard Euler, the greatest number mathematics. Henceforth, insights from cian John Wallis once referred to him as theorist of the 18th century, became so either field are certain to inspire new re- “that damned Frenchman.” frustrated by his inability to prove the sults in the other. Moreover, now that Fermat penned his most famous chal- last theorem that he asked a friend to this bridge has been built, other con- lenge, his so-called last theorem, while search Fermat’s house in case some vital nections between distant mathematical studying the ancient Greek mathemati- scrap of paper was left behind. In the realms may emerge. cal text Arithmetica, by Diophantus of 19th century Sophie Germain—who, be- Alexandria. The book discussed positive cause of prejudice against women math- The Prince of Amateurs whole-number solutions to the equation ematicians, pursued her studies under a2 + b2 = c2, Pythagoras’s formula de- the name of Monsieur Leblanc—made ierre de Fermat was born on August scribing the relation between the sides the first significant breakthrough. Ger- P20, 1601, in Beaumont-de-Lomagne, of a right triangle. This equation has main proved a general theorem that 25 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. went a long way toward solving Fer- ellipse. In general, cubic equations for that one can transform a complex num- mat’s equation for values of n that are elliptical curves take the form y2 = x3 + ber in many ways, and yet the function prime numbers greater than 2 and for ax2 + bx + c, where a, b and c are yields virtually the same result. In this which 2n + 1 is also prime. (Recall that whole numbers that satisfy some simple respect, modular forms are quite re- a prime number is divisible only by 1 conditions. Such equations are said to markable. Trigonometric functions are and itself.) But a complete proof for be of degree 3, because the highest ex- similar inasmuch as an angle, q, can be these exponents, or any others, re- ponent they contain is a cube. transformed by adding π, and yet the mained out of her reach. Number theorists regularly try to as- answer is constant: sin q = sin (q + π). At the start of the 20th century Paul certain the number of so-called rational This property is termed symmetry, and Wolfskehl, a German industrialist, be- solutions, those that are whole numbers trigonometric functions display it to a queathed 100,000 marks to whoever or fractions, for various equations. Lin- limited extent. In contrast, modular could meet Fermat’s challenge. Accord- ear or quadratic equations, of degree 1 forms exhibit an immense level of sym- ing to some historians, Wolfskehl was at and 2, respectively, have either no ratio- metry. So much so that when the French one time almost at the point of suicide, nal solutions or infinitely many, and it polymath Henri Poincaré discovered but he became so obsessed with trying is simple to decide which is the case. the first modular forms in the late 19th to prove the last theorem that his death For complicated equations, typically of century, he struggled to come to terms wish disappeared. In light of what had degree 4 or higher, the number of solu- with their symmetry. He described to happened, Wolfskehl rewrote his will. tions is always finite—a fact called Mor- his colleagues how every day for two The prize was his way of repaying a debt dell’s conjecture, which the German weeks he would wake up and search to the puzzle that saved his life. mathematician Gerd Faltings proved in for an error in his calculations. On the Ironically, just as the Wolfskehl Prize 1983. But elliptic curves present a unique 15th day he finally gave up, accepting was encouraging enthusiastic amateurs challenge. They may have a finite or in- that modular forms are symmetrical in to attempt a proof, professional mathe- finite number of solutions, and there is the extreme. maticians were losing hope. When the no easy way of telling. A decade or so before Wiles learned great German logician David Hilbert To simplify problems concerning el- about Fermat, two young Japanese was asked why he never attempted a liptic curves, mathematicians often re- mathematicians, Goro Shimura and Yu- proof of Fermat’s last theorem, he re- examine them using modular arithme- taka Taniyama, developed an idea in- plied, “Before beginning I should have tic. They divide x and y in the cubic volving modular forms that would ulti- to put in three years of intensive study, equation by a prime number p and keep mately serve as a cornerstone in Wiles’s and I haven’t that much time to only the remainder. This modified ver- proof. They believed that modular forms squander on a probable failure.” The sion of the equation is its “mod p” and elliptic curves were fundamentally problem still held a special place in the equivalent. Next, they repeat these divi- related—even though elliptic curves ap- hearts of number theorists, but they re- sions with another prime number, then parently belonged to a totally different garded Fermat’s last theorem in the another, and as they go, they note the area of mathematics. In particular, be- same way that chemists regarded alche- number of solutions for each prime cause modular forms have an L-series— my. It was a foolish romantic dream modulus. Eventually these calculations although derived by a different prescrip- from a past age. generate a series of simpler problems tion than that for elliptic curves—the that are analogous to the original. two men proposed that every elliptic The Childhood Dream The great advantage of modular curve could be paired with a modular arithmetic is that the maximum values form, such that the two L-series would hildren, of course, love romantic of x and y are effectively limited to p, match. Cdreams. And in 1963, at age 10, and so the problem is reduced to some- Shimura and Taniyama knew that if Wiles became enamored with Fermat’s thing finite. To grasp some understand- they were right, the consequences would last theorem. He read about it in his lo- ing of the original infinite problem, be extraordinary. First, mathematicians cal library in Cambridge, England, and mathematicians observe how the num- generally know more about the L-series promised himself that he would find a ber of solutions changes as p varies. of a modular form than that of an ellip- proof. His schoolteachers discouraged And using that information, they gener- tic curve. Hence, it would be unneces- him from wasting time on the impossi- ate a so-called L-series for the elliptic sary to compile the L-series for an ellip- ble. His college lecturers also tried to dis- curve. In essence, an L-series is an infin- tic curve, because it would be identical suade him. Eventually his graduate su- ite series in powers, where the value of to that of the corresponding modular pervisor at the University of Cambridge the coefficient for each pth power is de- form. More generally, building such a steered him toward more mainstream termined by the number of solutions in bridge between two hitherto unrelated mathematics, namely into the fruitful modulo p. branches of mathematics could benefit research area surrounding objects called In fact, other mathematical objects, both: potentially each discipline could elliptic curves. The ancient Greeks orig- called modular forms, also have L-se- become enriched by knowledge already inally studied elliptic curves, and they ries. Modular forms should not be con- gathered in the other. appear in Arithmetica. Little did Wiles fused with modular arithmetic. They The Shimura-Taniyama conjecture, as know that this training would lead him are a certain kind of function that deals it was formulated by Shimura in the back to Fermat’s last theorem. with complex numbers of the form (x + early 1960s, states that every elliptic Elliptic curves are not ellipses. Instead iy), where x and y are real numbers, curve can be paired with a modular they are named as such because they are and i is the imaginary number (equal to form; in other words, all elliptic curves described by cubic equations, like those the square root of –1). are modular. Even though no one could used for calculating the perimeter of an What makes modular forms special is find a way to prove it, as the decades

26 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. passed the hypothesis became increas- power. So a proof that the discriminant This special form of addition can be ingly influential. By the 1970s, for in- of an elliptic curve can never be an nth applied to any pair of points within the stance, mathematicians would often as- power would contain, implicitly, a infinite set of all points on an elliptic sume that the Shimura-Taniyama con- proof of Fermat’s last theorem. Frey curve, but this operation is particularly jecture was true and then derive some saw no way to construct that proof. He interesting because there are finite sets new result from it. In due course, many did, however, suspect that an elliptic of points having the crucial property major findings came to rely on the con- curve whose discriminant was a perfect that the sum of any two points in the jecture, although few scholars expected nth power—if it existed—could not be set is again in the set. These finite sets of it would be proved in this century. Trag- modular. In other words, such an elliptic points form a group: a set of points that For seven years, Wiles worked in complete secrecy. Not only did he want to avoid the pressure of public attention, but he hoped to keep others from copying his ideas.

ically, one of the men who inspired it did curve would defy the Shimura-Tani- obeys a handful of simple axioms. It not live to see its ultimate importance. yama conjecture. Running the argu- turns out that if the elliptic curve is On November 17, 1958, Yutaka Tani- ment backwards, Frey pointed out that modular, so are the points in each finite yama committed suicide. if someone proved that the Shimura- group of the elliptic curve. What Ribet Taniyama conjecture is true and that proved is that a specific finite group of The Missing Link the elliptic equation y2 = x(x – A)(x + Frey’s curve cannot be modular, ruling B) is not modular, then they would out the modularity of the whole curve. n the fall of 1984, at a symposium in have shown that the elliptic equation For three and half centuries, the last IOberwolfach, Germany, Gerhard Frey cannot exist. In that case, the solution theorem had been an isolated problem, of the University of Saarland gave a lec- to Fermat’s equation cannot exist, and a curious and impossible riddle on the ture that hinted at a new strategy for at- Fermat’s last theorem is proved true. edge of mathematics. In 1986 Ribet, tacking Fermat’s last theorem. The theo- Many mathematicians explored this building on Frey’s work, had brought it rem asserts that Fermat’s equation has link between Fermat and Shimura-Tani- center stage. It was possible to prove no positive whole-number solutions. To yama. Their first goal was to show that Fermat’s last theorem by proving the test a statement of this type, mathema- the Frey elliptic curve, y2 = x(x – A)(x + Shimura-Taniyama conjecture. Wiles, ticians frequently assume that it is false B), was in fact not modular. Jean-Pierre who was by now a professor at Prince- and then explore the consequences. To Serre of the College of France and Bar- ton, wasted no time. For seven years, he say that Fermat’s last theorem is false is ry Mazur of Harvard University made worked in complete secrecy. Not only to say that there are two perfect nth important contributions in this direc- did he want to avoid the pressure of powers whose sum is a third nth power. tion. And in June 1986 one of us (Ri- public attention, but he hoped to keep Frey’s idea proceeded as follows: Sup- bet) at last constructed a complete others from copying his ideas. During pose that A and B are perfect nth pow- proof of the assertion. It is not possible this period, only his wife learned of his ers of two numbers such that A + B is to describe the full argument in this ar- obsession—on their honeymoon. again an nth power—that is, they are a ticle, but we will give a few hints. solution to Fermat’s equation. A and B To begin, Ribet’s proof depends on Seven Years of Secrecy can then be used as coefficients in a spe- a geometric method for “adding” two cial elliptic curve: y2 = x(x – A)(x + B). points on an elliptic curve. Visually, the iles had to pull together many of A quantity that is routinely calculated idea is that if you project a line through Wthe major findings of 20th-centu- whenever one studies elliptic curves is a pair of distinct solutions, P1 and P2, ry number theory. When those ideas the “discriminant” of the elliptic curve, the line cuts the curve at a third point, were inadequate, he was forced to cre- A2B2(A + B)2. Because A and B are so- which we might provisionally call the ate other tools and techniques. He de- lutions to the Fermat equation, the dis- sum of P1 and P2. A slightly more com- scribes his experience of doing mathe- criminant is a perfect nth power. plicated but more valuable version of matics as a journey through a dark, un- The crucial point in Frey’s tactic is that this addition is as follows: first add two explored mansion: “You enter the first if Fermat’s last theorem is false, then points and derive a new point, P3, as al- room of the mansion, and it’s complete- whole-number solutions such as A and ready described, and then reflect this ly dark. You stumble around bumping B can be used to construct an elliptic point through the x axis to get the final into the furniture, but gradually you curve whose discriminant is a perfect nth sum, Q. learn where each piece of furniture is.

27 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. Finally, after six months or so, you find 81, then to 93, or 729, and so on. If he Taylor. Together they wrestled with the the light switch. You turn it on, and could reach an infinitely large group problem, trying to patch up the method suddenly it’s all illuminated. You can and prove that it, too, is modular, that Wiles had already used and applying see exactly where you were. Then you would be equivalent to proving that the other tools that he had previously reject- move into the next room and spend an- entire curve is modular. ed. They were at the point of admitting other six months in the dark. So each of Wiles accomplished this task via a defeat and releasing the flawed proof so these breakthroughs, while sometimes process loosely based on induction. He that others could try to correct it, when, they’re momentary, sometimes over a had to show that if one group was mod- on September 19, 1994, they found the period of a day or two, they are the cul- ular, then so must be the next larger vital fix. Many years earlier Wiles had mination of, and couldn’t exist with- group. This approach is similar to top- considered using an alternative approach out, the many months of stumbling pling dominoes: to knock down an in- based on so-called Iwasawa theory, but around in the dark that precede them.” finite number of dominoes, one merely it floundered, and he abandoned it. As it turned out, Wiles did not have to has to ensure that knocking down any Now he realized that what was causing prove the full Shimura-Taniyama con- one domino will always topple the next. the Kolyvagin-Flach method to fail was jecture. Instead he had to show only that Eventually Wiles felt confident that his exactly what would make the Iwasawa a particular subset of elliptic curves— proof was complete, and on June 23, theory approach succeed. one that would include the hypothetical 1993, he announced his result at a con- Wiles recalls his reaction to the dis- elliptic curve Frey proposed, should it ference at the Isaac Newton Mathemat- covery: “It was so indescribably beauti- exist—is modular. It wasn’t really much ical Sciences Institute in Cambridge. ful; it was so simple and so elegant. The of a simplification. This subset is still His secret research program had been a first night I went back home and slept infinite in size and includes the majority success, and the mathematical commu- on it. I checked through it again the of interesting cases. Wiles’s strategy used nity and the world’s press were sur- next morning, and I went down and told the same techniques employed by Ribet, prised and delighted by his proof. The my wife, ‘I’ve got it. I think I’ve found plus many more. And as with Ribet’s front page of the New York Times ex- it.’ And it was so unexpected that she argument, it is possible to give only a claimed, “At Last, Shout of ‘Eureka!’ in thought I was talking about a children’s hint of the main points involved. Age-Old Math Mystery.” toy or something, and she said, ‘Got The difficulty was to show that every As the media circus intensified, the what?’ I said, ‘I’ve fixed my proof. I’ve elliptic curve in Wiles’s subset is modu- official peer-review process began. Al- got it.’” lar. To do so, Wiles exploited the group most immediately, Nicholas M. Katz of For Wiles, the award of the Wolfskehl property of points on the elliptic curves Princeton uncovered a fundamental and Prize marks the end of an obsession that and applied a theorem of Robert P. devastating flaw in one stage of Wiles’s lasted more than 30 years: “Having Langlands of the Institute for Advanced argument. In his induction process, solved this problem, there’s certainly a Study in Princeton, N.J., and Jerrold Wiles had borrowed a method from sense of freedom. I was so obsessed by Tunnell of Rutgers University. The the- Victor A. Kolyvagin of Johns Hopkins this problem that for eight years I was orem shows, for each elliptic curve in University and Matthias Flach of the thinking about it all of the time—when Wiles’s set, that a specific group of points California Institute of Technology to I woke up in the morning to when I inside the elliptic curve is modular. This show that the group is modular. But it went to sleep at night. That particular requirement is necessary but not suffi- now seemed that this method could not odyssey is now over. My mind is at rest.” cient to demonstrate that the elliptic be relied on in this particular instance. For other mathematicians, though, ma- curve as a whole is modular. Wiles’s childhood dream had turned jor questions remain. In particular, all The group in question has only nine into a nightmare. agree that Wiles’s proof is far too com- elements, so one might imagine that its plicated and modern to be the one that modularity represents an extremely Finding the Fix Fermat had in mind when he wrote his small first step toward complete modu- marginal note. Either Fermat was mis- larity. To close this gap, Wiles wanted or the next 14 months, Wiles hid taken, and his proof, if it existed, was to examine increasingly larger groups, Fhimself away, discussing the error flawed, or a simple and cunning proof 2 stepping from groups of size 9 to 9 , or only with his former student Richard awaits discovery. SA

The Authors Further Reading

SIMON SINGH and KENNETH A. RIBET Yutaka Taniyama and His Time: Very Personal Recollections from Shimura. share a keen interest in Fermat’s last theorem. Goro Shimura in Bulletin of the London Mathematical Society, Vol. 21, pages 186–196; Singh is a particle physicist turned television 1989. science journalist, who wrote Fermat’s Enig- From the Taniyama-Shimura Conjecture to Fermat’s Last Theorem. Kenneth A. ma and co-produced a documentary on the Ribet in Annales de la Faculté des Sciences de L’Université de Toulouse, Vol. 11, No. 1, subject. Ribet is a professor of mathematics at pages 115–139; 1990. the University of California, Berkeley, where Modular Elliptic Curves and Fermat’s Last Theorem. Andrew Wiles in Annals of his work focuses on number theory and arith- Mathematics, Vol. 141, No. 3, pages 443–551; May 1995. metic algebraic geometry. For his proof that Ring Theoretic Properties of Certain Hecke Algebras. Richard Taylor and An- the Shimura-Taniyama conjecture implies Fer- drew Wiles in Annals of Mathematics, Vol. 141, No. 3, pages 553–572; May 1995. mat’s last theorem, Ribet and his colleague Notes on Fermat’s Last Theorem. A. J. van der Poorten. Wiley Interscience, 1996. Abbas Bahri won the ﬁrst Prix Fermat. Fermat’s Enigma. Simon Singh. Walker and Company, 1997.

28 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. Originally published in May 1998 Japanese Temple Geometry During Japan’s period of national seclusion (1639–1854), native mathematics thrived, as evidenced in sangaku—wooden tablets engraved with geometry problems hung under the roofs of shrines and temples

by Tony Rothman, with the cooperation of Hidetoshi Fukagawa

f the world’s countless cus- impossible, and modern geometers in- Another tablet, from Kyoto, is dated toms and traditions, perhaps variably tackle them with advanced 1686, and a third is from 1691. The O none is as elegant, nor as methods, including calculus and affine 19th-century travel diary of the mathe- beautiful, as the tradition of sangaku, transformations. Although most of the matician Kazu Yamaguchi refers to an Japanese temple geometry. From 1639 problems would be classified today as even earlier tablet—now lost—dated to 1854, Japan lived in strict, self-im- recreational or educational mathemat- 1668. So historians guess that the cus- posed isolation from the West. Access ics, a few predate known Western re- tom first arose in the second half of the to all forms of occidental culture was sults, such as the Malfatti theorem, the 17th century. In 1789 the first collec- suppressed, and the influx of Western Casey theorem and the Soddy hexlet tion containing typical sangaku prob- scientific ideas was effectively curtailed. theorem. One problem reproduces the lems was published. Other collections During this period of seclusion, a kind Descartes circle theorem. Many of the followed throughout the 18th and 19th of native mathematics flourished. tablets are exceptionally beautiful and centuries. These books were either hand- Devotees of math, evidently samurai, can be regarded as works of art. written or printed with wooden blocks merchants and farmers, would solve a and are remarkably beautiful. Today wide variety of geometry problems, in- Pleasing the Kami more than 880 tablets survive, with ref- scribe their efforts in delicately colored erences to hundreds of others in the wooden tablets and hang the works un- t is natural to wonder who created various collections. From a survey of der the roofs of religious buildings. These Ithe sangaku and when, but it is easier the extant sangaku, the tablets seem to sangaku, a word that literally means to ask such questions than to answer have been distributed fairly uniformly mathematical tablet, may have been acts them. The custom of hanging tablets at throughout Japan, in both rural and ur- of homage—a thanks to a guiding spirit— shrines was established in Japan cen- ban districts, with about twice as many or they may have been brazen challeng- turies before sangaku came into exis- found in Shinto shrines as in Buddhist es to other worshipers: Solve this one if tence. Shintoism, Japan’s native religion, temples. you can! is populated by “eight hundred myriads Most of the surviving sangaku con- For the most part, sangaku deal with of gods,” the kami. Because the kami, it tain more than one theorem and are fre- ordinary Euclidean geometry. But the was said, love horses, those worshipers quently brightly colored. The proof of problems are strikingly different from who could not present a living horse as the theorem is usually not given, only those found in a typical high school ge- an offering to the shrine might instead the result. Other information typically ometry course. Circles and ellipses play give a likeness drawn on wood. As a re- includes the name of the presenter and a far more prominent role than in West- sult, many tablets dating from the 15th the date. Not all the problems deal sole- ern problems: circles within ellipses, el- century and earlier depict horses. ly with geometry. Some ask for the vol- lipses within circles. Some of the exercis- Of the sangaku themselves, the oldest umes of various solids and thus require es are quite simple and could be solved surviving tablet has been found in To- calculus. (This point raises the interest-

BRYAN CHRISTIE, AFTER TOSHIHISA IWASAKI AFTER TOSHIHISA CHRISTIE, BRYAN by first-year students. Others are nearly chigi Prefecture and dates from 1683. ing question of what techniques the practitioners brought into play; some speculations will be offered in the fol- SANGAKU PROBLEMS typically involve multitudes of circles within circles or of lowing discussion.) Other tablets con- spheres within other figures. This problem is from a sangaku, or mathematical wooden tablet, dated 1788 in Tokyo Prefecture. It asks for the radius of the nth largest blue tain Diophantine problems—that is, al- circle in terms of r, the radius of the green circle. Note that the red circles are identical, gebraic equations requiring solutions in each with radius r/2. (Hint: The radius of the fifth blue circle is r/95.) The original so- integers. lution to this problem deploys the Japanese equivalent of the Descartes circle theorem. In modern times the sangaku have (The answer can be found on page 35.) been largely forgotten but for a few

Here is a simple problem that has survived on an 1824 tablet in Gumma Prefecture. The orange and blue circles touch each other at one point and are tangent to the same line. The small red circle touches both of the larger circles and is also tangent to the same line. How are the radii of the three circles related?

P This striking problem was written in 1912 on a tablet extant in Miyagi Prefecture; the date of the problem itself is unknown. At a point P on an ellipse, draw the normal PQ such that it intersects the other side. Find the least value of PQ. At first glance, the problem appears to be trivial: the minimum PQ is the minor axis of the ellipse. Indeed, this is the solution if b < a ≤√2b, where a and b are the major and minor axes, respectively; however, the tablet does not give this solution but another, if 2b2 < a2.

This beautiful problem, which requires no more than high school geometry to solve, is written on a tablet dated 1913 in Miyagi Prefecture. Three orange squares are drawn as shown in the large, green right triangle. How are the radii of the three blue circles related? BRYAN CHRISTIE BRYAN

31 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. devotees of traditional Japanese mathe- these is the Chou-pei Suan-ching, which matics. Among them is Hidetoshi Fu- contains an example of the Pythagore- kagawa, a high school teacher in Aichi an theorem and the diagram commonly Prefecture, roughly halfway between used to prove it. This part of the tome Tokyo and Osaka. About 30 years ago is at least as old as the sixth century B.C. Fukagawa decided to study the history A more advanced state of knowledge of Japanese mathematics in hopes of is represented in the Chiu-chang Suan- finding better ways to teach his courses. shu, considered the most influential of A mention of the math tablets in an old Chinese books on mathematics. The library book greatly astonished him, for Chiu-chang describes methods for find- he had never heard of such a thing. Since ing the areas of triangles, quadrilater- then, Fukagawa, who holds a Ph.D. in als, circles and other figures. It also con- mathematics, has traveled widely in Ja- tains simple word problems of the type In this problem, from an 1803 san- pan to study the tablets and has amassed that torment many high school students gaku found in Gumma Prefecture, the a collection of books dealing not only today: “If five oxen and two sheep cost base of an isosceles triangle sits on a diameter of the large green circle. with sangaku but with the general field eight taels of gold, and two oxen and This diameter also bisects the red cir- of traditional Japanese mathematics. eight sheep cost eight taels, what is the cle, which is inscribed so that it just To carry out his research, Fukagawa price of each animal?” The dates of the touches the inside of the green circle had to teach himself Kambun, an archa- Chiu-chang are also uncertain, but most and one vertex of the triangle, as ic form of Japanese that is closely relat- of it was probably composed by the shown. The blue circle is inscribed so ed to Chinese. Kambun is the Japanese third century B.C. If this information is that it touches the outsides of both equivalent of Latin; during the Edo pe- correct, the Chiu-chang contains per- the red circle and the triangle, as well riod (1603–1867), scientific works were haps the first known mention of nega- as the inside of the green circle. A line written in this language, and only a few tive numbers and an early statement of segment connects the center of the people in modern Japan are able to read the quadratic equation. (According to blue circle and the intersection point it fluently. As new tablets have been dis- some historians, the ancient Egyptians between the red circle and the trian- covered, Fukagawa has been called in had begun studying quadratic equations gle. Show that this line segment is to decipher them. In 1989 Fukagawa, centuries before, prior to 2000 B.C.) perpendicular to the drawn diameter along with Daniel Pedoe, published the Despite the influx of Chinese learn- of the green circle. first collection of sangaku in English. ing, mathematics did not then take root Most of the geometry problems accom- in Japan. Instead the country entered a panying this article were drawn from dark age, roughly contemporaneous that collection. with that of Western Europe. In the West, church and monastery became Wasan versus Yosan the centers of learning; in Japan, Bud- dhist temples served the same function, lthough the origin of the sangaku although mathematics does not seem to A cannot be pinpointed, it can be lo- have played much of a role. By some calized. There is a word in Japanese, accounts, during the Ashikaga shogun- wasan, that is used to refer to native ate (1338–1573) there could hardly be Japanese mathematics. Wasan is meant found in all Japan a person versed in to stand in opposition to yosan, or the art of division. Western mathematics. To understand It is not until the opening of the 17th This problem comes from an 1874 how wasan came into existence—and century that definite historical records tablet in Gumma Prefecture. A large with it the unusual sangaku problems— exist of any Japanese mathematicians. blue circle lies within a square. Four one must first appreciate the peculiar The first of these is Kambei Mori, who smaller orange circles, each with a history of Japanese mathematics. prospered around the year 1600. Al- different radius, touch the blue circle Of the earliest times, very little is defi- though only one of Mori’s works—a as well as the adjacent sides of the nitely known about mathematics in Ja- booklet—survives, he is known to have square. What is the relation between pan, except that a system of exponen- been instrumental in developing arith- the radii of the four small circles and tial notation, similar to that employed metical calculations on the soroban, or the length of the side of the square? by Archimedes in the Sand Reckoner, Japanese abacus, and in popularizing it (Hint: The problem can be solved by had been developed. More concrete in- throughout the country. applying the Casey theorem, which formation dates only from the mid-sixth The oldest substantial Japanese work describes the relation between four century A.D.,when Buddhism—and, on mathematics actually extant belongs circles that are tangent to a fifth circle with it, Chinese mathematics—made its to Mori’s pupil Koyu Yoshida (1598– or to a straight line.) way to Japan. Judging from the works 1672). The book, entitled Jinko-ki (lit- that were taught at official schools at erally, “small and large numbers”), was the start of the eighth century, histori- published in 1627 and also concerns op- ans infer that Japan had imported the erations on the soroban. Jinko-ki was great Chinese classics on arithmetic, also influential that the name of the work *Answers are on page 35. gebra and geometry. often was synonymous with arithmetic. According to tradition, the earliest of Because of the book’s influence, compu-

32 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. From a sangaku dated 1825, this problem was probably solved by using the enri, or the Japanese circle principle. A cylinder intersects a sphere so that the outside of the cylinder is tangent to the inside of the sphere. What is the surface area of the part of the cylinder contained inside the sphere? (The inset shows a three-dimensional view of the problem.)

This problem is from an 1822 tablet in Kanagawa Prefecture. It pre- dates by more than a century a theorem of Frederick Soddy, the famous British chemist who, along with Ernest Rutherford, discovered transmutation of the elements. Two red spheres touch each other and also touch the inside of the large green sphere. A loop of smaller, different-size blue spheres circle the “neck” between the red spheres. Each blue sphere in the “necklace” touches its nearest neighbors, and they all touch both the red spheres and the green sphere. How many blue spheres must there be? Also, how are the radii of the blue spheres related? (The inset shows a three-dimensional view of the problem.)

Hidetoshi Fukagawa was so fascinated with this problem, which dates from 1798, that he built a wooden model of it. Let a large sphere be surrounded by 30 small, identical spheres, each of which touches its four small-sphere neighbors as well as the large sphere. How is the radius of the large sphere related to that of the small spheres? (The inset shows a three-dimensional view of the problem.)

*Answers are on page 35. BRYAN CHRISTIE BRYAN

33 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. tation—as opposed to logic—became the a third accomplishment sometimes at- tion was not complete, then it was most most important concept in traditional tributed to Seki, and one that might also nearly so, and any foreign influence on Japanese mathematics. To the extent bear on sangaku, is the development of Japanese mathematics would have been that it makes sense to credit anyone the enri, or circle principle. minimal. with the founding of wasan, that honor The enri was quite similar to the meth- The situation began to change in the probably goes to Mori and Yoshida. od of exhaustion developed by Eudox- 19th century, when the wasan gradual- us and Archimedes in ancient Greece ly became supplanted by yosan, a pro- A Brilliant Flowering for computing the area of circles. The cess that produced hybrid manuscripts main difference was that Eudoxus and written in Kambun with Western math- asan, though, was created not so Archimedes used n-sided polygons to ematical notations. And, after the open- Wmuch by a few individuals but approximate the circle, whereas the enri ing of Japan by Commodore Perry and by something much larger. In 1639 the divided the circle into n rectangles. Thus, the subsequent collapse of the Toku- ruling Tokugawa shogunate (during the the limiting procedure was somewhat gawa shogunate in 1867, the new gov- Edo period), to strengthen its power different. Nevertheless, the enri repre- ernment abandoned the study of native and diminish challenges to its reign, de- sented a crude form of integral calculus mathematics in favor of yosan. Some creed the official closing of Japan. Dur- that was later extended to other figures, practitioners, however, continued to ing this time of sakoku, or national se- including spheres and ellipses. A type of hang tablets well into the 20th century. clusion, the government banned foreign differential calculus was also developed A few sangaku even date from the cur- books and travel, persecuted Christians around the same period. It is conceiv- rent decade. But almost all the prob- and forbade Portuguese and Spanish able that the enri and similar techniques lems from this century are plagiarisms. ships from coming ashore. Many of were brought to bear on sangaku. To- The final and most intriguing ques- these strictures would remain for more day’s mathematicians would use mod- tion is, Who produced the sangaku? than two centuries, until Commodore ern calculus to solve these problems. Were the theorems so beautifully drawn Matthew C. Perry, backed by a fleet of on wooden tablets the works of profes- U.S. warships, forced the end of sakoku Spheres within Ellipsoids sional mathematicians or amateurs? in 1854. The evidence is meager. Yet the isolationist policy was not en- uring Seki’s lifetime, the first books Only a handful of sangaku are men- tirely negative. Indeed, during the late Demploying the enri were published, tioned in the standard A History of Jap- 17th century, Japanese art and culture and the first sangaku evidently made anese Mathematics, by David E. Smith flowered so brilliantly that those years their appearance. The dates are almost and Yoshio Mikami. They cite the 1789 go by the name of Genroku, for “re- certainly not coincidental; the followers collection Shimpeki Sampo, or Mathe- naissance.” In that era, haiku developed of Yoshida and Seki must have influ- matical Problems Suspended before the into a fine art form; No and Kabuki enced the development of wasan, and, in Temple, which was published by Kagen theater reached the pinnacle of their de- turn, wasan may have influenced them. Fujita, a professional mathematician. velopment; ukiyo-e, or “floating world” Fukagawa believes that Seki encoun- Smith and Mikami mention a tablet on pictures, originated; and tea ceremonies tered sangaku on his way to the shogun- which the following was appended af- and flower arranging reached new ate castle, where he was officially em- ter the solution: “Feudal district of Ka- heights. Neither was mathematics left ployed as court mathematician, and kegawa in Enshu Province, third month behind, for Genroku was also the age that the tablets pushed him to further of 1795, Sonobei Keichi Miyajima, pu- of Kowa Seki. researches. A legend? Perhaps. But by pil of Sadasuke Fujita of the School of By popular accounts, Seki (1642– the next century, books were being pub- Seki.” Mikami, in his Development of 1708) was Japan’s Isaac Newton or lished that contained typical native Jap- Mathematics in China and Japan, men- Gottfried Wilhelm Leibniz, although anese problems: circles within triangles, tions the “Gion Temple Problem,” this reputation is difficult to substanti- spheres within pyramids, ellipsoids sur- which was suspended at the Gion Tem- ate. If the numbers of manuscripts at- rounding spheres. The problems found ple in Kyoto by Enkyu Tsuda, pupil of tributed to him are correct, then most in these books do not differ in any im- Enri Nishimura. Furthermore, the tab- of his work has been lost. Still, there is portant way from those found on the lets were written in the specialized lan- no question that Seki left many disci- tablets, and it is difficult to avoid the guage of Kambun, signifying the mark ples who were influential in the further conclusion that the peculiar flavor of all of an educated class of practitioners. development of Japanese mathematics. wasan problems—including the sanga- From such scraps of information, it is The first—and incontestable—achieve- ku—is a direct result of the policy of na- tempting to conclude that the tablets ment of Seki was his theory of determi- tional seclusion. were the work primarily of professional nants, which is more powerful than that But the question immediately arises: mathematicians and their students. Yet of Leibniz and which antedates the Ger- Was Japan’s isolation complete? It is there are reasons to believe otherwise. man mathematician’s work by at least a certain that apart from the Dutch who Many of the problems are elementary decade. Another accomplishment, more were allowed to remain in Nagasaki and can be solved in a few lines; they relevant to temple geometry but of de- Harbor on Kyushu, the southernmost are not the kind of work a professional batable origin, is the development of island, all Western traders were banned. mathematician would publish. Fuka- methods for solving high-degree equa- Equally clear is that the Japanese them- gawa has found a tablet from Mie Pre- tions. (Much traditional Japanese math- selves were severely restricted. The mere fecture inscribed with the name of a ematics from that era involves equa- act of traveling abroad was considered merchant. Others have names of wom- tions to hundreds of degrees; one such high treason, punishable by death. It en and children—12 to 14 years of age. equation is of the 1,458th degree.) Yet appears safe to assume that if the isola- Most, according to Fukagawa, were

34 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE DECEMBER 2003 COPYRIGHT 2003 SCIENTIFIC AMERICAN, INC. created by the members of the highly with such artistic care speciﬁcally to at- spend days working on it in peaceful educated samurai class. A few were tract nonmathematicians. contemplation. After ﬁnally arriving at probably done by farmers; Fukagawa The best answer, then, to the question a solution, he might allow himself a recalls how about 10 years ago he visit- of who created temple geometry seems short rest to savor the result of his hard ed the former cottage of mathematician to be: everybody. On learning of the san- labor. Convinced the proof was a wor- Sen Sakuma (1819–1896), who taught gaku, Fukagawa came to understand thy offering to his guiding spirits, he wasan to the farmers in nearby villages that, in those days, many of the Japan- would have the theorem inscribed in in Fukushima Prefecture. Sakuma had ese loved and enjoyed math, as well as wood, hang it in his local temple and about 2,000 students. poetry and other art forms. begin to consider the next challenge. Such instruction recalls the Edo peri- It is pleasant to realize that some san- Visitors would notice the colorful tablet od itself, when there were no colleges gaku were the works of ordinary math- and admire its beauty. Many people or universities in Japan. During that ematics devotees, carried away by the would leave wondering how the author time, teaching was carried out at pri- beauty of geometry. Perhaps a village arrived at such a miraculous solution. vate schools or temples, where ordinary teacher, after spending the day with stu- Some might decide to give the problem people would go to study reading, writ- dents, or a samurai warrior, after sharp- a try or to study geometry so that the ing and the abacus. Because laypeople ening his sword, would retire to his attempt could be made. A few might are more often drawn to problems of study, light an oil lamp and lose the leave asking, “What if the problem geometry than of algebra, it would not world to an intricate problem involving were changed just so....” be surprising if the tablets were painted spheres and ellipsoids. Perhaps he would Something for us all to consider. SA

Answers to Sangaku Problems Answer: If a is the length of the square’s side, and r1, r2, r3 and r4 nfortunately, because of space limitations the complete solutions to the prob- are the radii of the upper right, up- Ulems could not be given here. per left, lower left and lower right orange circles, respectively, then 2(r r – r r ) + √2(r – r )(r – r )(r – r )(r – r ) a = 1 3 2 4 1 2 1 4 3 2 3 4 2 2 r – r + r – r Answer: r/[(2n – 1) + 14]. The original Answer: r2 = r1r3, where r1, r2 1 2 3 4 solution to this problem applies the and r3 are the radii of the large, Japanese version of the Descartes cir- medium and small blue circles, Answer: 16t √t(r–t), where r and t cle theorem several times. The answer respectively. (In other words, r2 are the radii of the sphere and given here was obtained by using the in- is the geometric mean of r1 and r3.) The problem cylinder, respectively. version method, which was unknown to the can be solved by first realizing that all the interior Japanese mathematicians of that era. green triangles formed by the orange squares are similar. The original solution then looks at Answer: Six spheres. The Soddy hex- Answer: 1/√r3 = 1/√r1 + 1/√r2, how the three squares are related. let theorem states that there must where r1, r2 and r3 are the radii be six and only six blue spheres (thus of the orange, blue and red the word “hexlet”). Interestingly, the circles, respectively. The prob- Answer: In the original solution to theorem is true regardless of the position of the lem can be solved by applying the Pythagorean this problem, the author draws a ﬁrst blue sphere around the neck. Another in- theorem. line segment that goes through triguing result is that the radii of the different the center of the blue circle and is blue spheres in the “necklace” (t1 through t6) are perpendicular to the drawn diameter related by 1/t + 1/t = 1/t + 1/t = 1/t + 1/t . √27 a2b2 1 4 2 5 3 6 Answer: PQ = of the green circle. The author assumes that this 3/ (a2+ b2) 2 line segment is different from the line segment Answer: R = √5r, where R and r described in the statement of the problem on are the radii of the large and The problem can be solved by using analytic ge- page 87. Thus, the two line segments should in- small spheres, respectively. The ometry to derive an equation for PQ and then tersect the drawn diameter at different locations. problem can be solved by real- taking the first derivative of the equation and The author then shows that the distance be- izing that the center of each setting it to zero to obtain the minimum value tween those locations must necessarily be equal small sphere lies on the midpoint for PQ. It is not known whether the original au- to zero—that is, that the two line segments are of the edge of a regular dodecahedron, a 12-sid-

BRYAN CHRISTIE BRYAN thors resorted to calculus to solve this problem. identical, thereby proving the perpendicularity. ed solid with pentagonal faces.

The Author Further Reading

TONY ROTHMAN received his Ph.D. in 1981 from the Center A History of Japanese Mathematics. David E. Smith and Yoshio for Relativity at the University of Texas at Austin. He did postdoc- Mikami. Open Court Publishing Company, Chicago, 1914. (Also toral work at Oxford, Moscow and Cape Town, and he has taught available on microﬁlm.) at Harvard University. Rothman has published six books, most re- The Development of Mathematics in China and Japan. Sec- cently Instant Physics. His next book is Doubt and Certainty, with ond edition (reprint). Yoshio Mikami. Chelsea Publishing Compa- E.C.G. Sudarshan, to be published this fall by Helix Books/Addi- ny, New York, 1974. son-Wesley. He has also recently written a novel about nuclear fu- Japanese Temple Geometry Problems. H. Fukagawa and D. Pedoe. sion. Scientiﬁc American wishes to acknowledge the help of Charles Babbage Research Foundation, Winnipeg, Canada, 1989. Hidetoshi Fukagawa in preparing this manuscript. Fukagawa re- Traditional Japanese Mathematics Problems from the 18th ceived a Ph.D. in mathematics from the Bulgarian Academy of Sci- and 19th Centuries. H. Fukagawa and D. Sokolowsky. Science ence. He is a high school teacher in Aichi Prefecture, Japan. Culture Technology Publishing, Singapore (in press).