Death of Proof

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TRENDS IN MATHEMATICS

THE DEATH OF PROOFby John Horgan, senior writer

Copyright 1993 Scientific American, Inc.


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Computers are transforming the waymathematicians discover, prove and

communicate ideas, but is there a placefor absolute certainty in this brave new world?

Legend has it that when Pythagoras and his followers dis-

covered the theorem that bears his name in the sixthcentury B.C., they slaughtered an ox and feasted in cel-

ebration. And well they might. The relation they found be-tween the sides of a right triangle held true not sometimes ormost of the time but alwaysregardless of whether the trian-gle was a piece of silk or a plot of land or marks on papyrus. Itseemed like magic, a gift from the gods. No wonder so manythinkers, from Plato to Kant, came to believe that mathematicsoers the purest truths humans are permitted to know.

That faith seemed rearmed this past June when Andrew J.Wiles of Princeton University revealed during a meeting at the

University of Cambridge that he had solved Fermats last the-orem. This problem, one of the most famous in mathematics,was posed more than 350 years ago, and its roots extend backto Pythagoras himself. Since no oxen were available, Wilesslisteners showed their appreciation by clapping their hands.

But was the proof of Fermats last theorem the last gasp of adying culture? Mathematics, that most tradition-bound of in-tellectual enterprises, is undergoing profound changes. Formillennia, mathematicians have measured progress in termsof what they can demonstrate through proofsthat is, a se-ries of logical steps leading from a set of axioms to an irre-futable conclusion. Now the doubts riddling modern human

thought have nally infected mathematics. Mathematiciansmay at last be forced to accept what many scientists and phi-losophers already have admitted: their assertions are, at best,only provisionally true, true until proved false.

This uncertainty stems, in part, from the growing complexityof mathematics. Proofs are often so long and complicated that

they are dicult to evaluate.Wiless demonstration runs to200 pagesand experts esti-mate it could be ve timeslonger if he spelled out allits elements. One observer as-

VIDEO PROOF dramatizes a theo-rem, proved by William P. Thurstonof the Mathematical Sciences Re-search Institute (left), that establish-es a profound connection betweentopology and geometry. The theo-rem shows how the space surround-ing a complex knot (represented bythe lattice in this scene) yields a hy-perbolic geometry, in which par-allel lines diverge and the sides ofpentagons form right angles. Thecomputer-generated video, calledNot Knot, was produced at the Ge-

ometry Center in Minnesota.



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serted that only one tenth of 1 percentof the mathematics community wasqualied to evaluate the proof. Wilessclaim was accepted largely on the ba-sis of his reputation and the reputa-tions of those whose work he built on.Mathematicians who had not yet exam-ined the argument in detail nonethe-less commented that it looks beauti-ful and has the ring of truth.

Another catalyst of change is thecomputer, which is compelling mathe-maticians to reconsider the very natureof proof and, hence, of truth. In recentyears, some proofs have required enor-mous calculations by computers. Nomere human can verify these so-calledcomputer proofs, just other comput-ers. Recently investigators have pro-posed a computational proof that of-fers only the probabilitynot the cer-taintyof truth, a statement that somemathematicians consider an oxymoron.Still others are generating video proofs

in the hopes that they will be more per-suasive than page on page of formalterminology.

At the same time, some mathemati-cians are challenging the notion thatformal proofs should be the supremestandard of truth. Although no one ad-

vocates doing away with proofs alto-gether, some practitioners think the va-lidity of certain propositions may be

better established by comparing themwith experiments run on computers orwith real-world phenomena. Within thenext 50 years I think the importance ofproof in mathematics will diminish,says Keith Devlin of Colby College, whowrites a column on computers for No-

tices of the American Mathematical So-ciety. You will see many more peopledoing mathematics without necessarilydoing proofs.

Powerful institutional forces are pro-mulgating these heresies. For severalyears, the National Science Foundationhas been urging mathematicians to be-come more involved in computer sci-ence and other elds with potential ap-plications. Some leading lights, notablyPhillip A. Griths, director of the Insti-tute for Advanced Study in Princeton,N.J., and Michael Atiyah, who won a

Fields Medal (often called the NobelPrize of mathematics) in 1966 and nowheads Cambridges Isaac Newton Insti-tute for Mathematical Sciences, havelikewise encouraged mathematicians toventure forth from their ivory towersand mingle with the real world. At a

time when funds and jobs are scarce,young mathematicians cannot aord toignore these exhortations.

There are pockets of resistance, ofcourse. Some workers are complaining

bitterly about the computerization oftheir eld and the growing emphasison (oh, dirty word) applications. Oneof the most vocal champions of tra-dition is Steven G. Krantz of Washing-

ton University. In speeches and articles,Krantz has urged students to choosemathematics over computer science,which he warns could be a passing fad.Last year, he recalls, a National ScienceFoundation representative came to his

university and announced that the agen-cy could no longer aord to supportmathematics that was not goal-orient-ed. We could stand up and say this iswrong, Krantz grumbles, but mathe-maticians are spineless slobs, and theydont have a tradition of doing that.

David Mumford of Harvard Universi-

ty, who won a Fields Medal in 1974 forresearch in pure mathematics and isnow studying articial vision, wrote re-cently that despite all the hype, thepress, the pressure from funding agen-cies, et cetera, the pure mathematicalcommunity by and large still regards

94 SCIENTIFIC AMERICAN October 1993

A Splendid Anachronism?

hose who consider experimental mathematics andcomputer proofs to be abominations rather than inno-

vations have a special reason to delight in the conquest of

Fermats last theorem by Andrew J. Wiles of Princeton Uni-versity. Wiless achievement was a triumph of tradition, run-ning against every current in modern mathematics.

Wiles is a staunch believer in mathematics for its ownsake. I certainly wouldnt want to see mathematics justbeing a servant to applications, because its not even inthe interests of the applications themselves, he says.

The problem he solved, first posedmore than 350 years ago by theFrench polymath Pierre de Fermat, isa glorious example of a purely math-ematical puzzle. Fermat claimed tohave found a proof of the followingproposition: for the equation XN+ YN

= ZN

, there are no integral solutionsfor any value of N greater than 2.The efforts of mathematicians to findthe proof (which Fermat never diddisclose) helped to lay the founda-tion of modern number theory, thestudy of whole numbers, which hasrecently become useful in cryptogra-phy. Yet Fermats last theorem itselfis very unlikely to have any applica-tions, Wiles says.

Although funding agencies havebeen encouraging mathematicians tocollaborate, both with each other and

with scientists, Wiles worked in virtual solitude for sevenyears. He shared his ideas with only a few colleagues to-ward the end of his quest.

Wiless proof has essentially the same classical, deductiveform that Euclids geometric theorems did. It does not in-volve any computation, and it claims to be absolutelynotprobablytrue. Nor did Wiles employ computers to repre-sent ideas graphically, to perform calculations or even tocompose his paper; a secretary typed his hand-written notes.

He concedes that testing conjectures with computersmay be helpful. In the 1970s comput-er tests suggested that a far-fetchedproposal called the Taniyama conjec-ture might be true. The tests spurredwork that laid the foundation forWiless own proof.

Nevertheless, Wiles doubts he will

take the trouble to learn how to per-form computer investigations. Itsa separate skill, he explains, and ifyoure investing that much time on aseparate skill, its quite likely its tak-ing you away from your real workon the problem.

He rejects the possibility that theremay be a finite number of truths ac-cessible to traditional forms of in-quiry. I disagree vehemently withthe idea that good theorems are run-ning out, he says. I think wevebarely scratched the surface.

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computers as invaders, despoilers of thesacred ground. Last year Mumford pro-posed a course in which instructorswould show students how to program acomputer to nd solutions in advancedcalculus. I was vetoed, he recalled,and not on the groundswhich I ex-

pectedthat the students would com-plain, but because half of my fellowteachers couldnt program!

That situation is changing fast, ifthe University of Minnesotas GeometryCenter is any indication. Founded twoyears ago, the Geometry Center occu-pies the fth oor of a gleaming, steeland glass polyhedron in Minneapolis.It receives $2 million a year from theNational Science Foundation, the De-partment of Energy and the university.The centers permanent faculty mem-

bers, most of whom hold positions else-where, include some of the most promi-

nent mathematicians in the world.On a recent day there, several young

sta members are editing a video dem-onstrating how a sphere can be mashed,twisted, yanked and nally turned in-side out. In a conference room, threecomputer scientists from major univer-sities are telling a score of high schoolteachers how to create computer graph-ics programs to teach mathematics.Other researchers sit at charcoal-col-ored NeXT terminals, pondering luridlyhued pictures of four-dimensional hy-percubes, whirlpooling fractals and lat-

tices that plunge toward innity. Nopaper or pencils are in sight.

At one terminal is David Ben-Zvi, aHarpo Marxhaired junior at Princetonwho is spending six months here explor-ing nonlinear dynamics. He dismissesthe fears of some mathematicians thatcomputers will lure them away fromthe methods that have served them sowell for so long. Theyre just afraid ofchange, he says mildly.

The Geometry Center is a hotbed ofwhat is known as experimental mathe-matics, in which investigators test their

ideas by representing them graphical-

ly and doing calculations on comput-ers. Last year some of the centers facul-ty helped to found a journal, Experimen-tal Mathematics, that showcases suchwork. Experimental methods are nota new thing in mathematics, observesthe journals editor, David B. A. Epsteinof the University of Warwick in England,noting that Carl Friedrich Gauss and

other giants often performed experi-mental calculations before constructingformal proofs. Whats new is that itsrespectable. Epstein acknowledges thatnot all his co-workers are so accepting.One of my colleagues said, Your jour-nal should be called theJournal of Un-proved Theorems.

Bubbles and Tortellini

A mathematician who epitomizes thenew style of mathematics is Jean E.Taylor of Rutgers University. The idea

that you dont use computers is goingto be increasingly foreign to the nextgeneration, she says. For two decades,Taylor has investigated minimal surfac-es, which represent the smallest possi-

ble area or volume bounded by a curveor surface. Perhaps the most elegantand simple minimal surfaces found innature are soap bubbles and lms. Tay-lor has always had an experimental

bent. Early in her career she tested herhandwritten models of minimal sur-faces by dunking loops of wire into asink of soapy water.

Now she is more likely to model

bubbles with a sophisticated computergraphics program. She has also graduat-ed from soap bubbles to crystals, whichconform to somewhat more complicat-ed rules about minimal surfaces. To-gether with Frederick J. Almgren ofPrinceton and Robert F. Almgren of theUniversity of Chicago (her husband andstepson, respectively) and Andrew R.

Roosen of the National Institute of Stan-dards and Technology, Taylor is tryingto mimic the growth of snowakes andother crystals on a computer. Increas-ingly, she is collaborating with materi-als scientists and physicists, swappingmathematical ideas and programmingtechniques in exchange for clues abouthow real crystals grow.

Another mathematician who hasprowled cyberspace in search of novelminimal surfaces is David A. Homanof the University of Massachusetts atAmherst. Among his favorite quarry are

catenoids and helicoids, which resemblethe pasta known as tortellini and wererst discovered in the 18th century. Wegain a tremendous amount of intuition

by looking at images of these surfaceson computers, he says.

In 1992 Homan, Fusheng Wei of Am-herst and Hermann Karcher of the Uni-versity of Bonn speculated on the exis-tence of a new class of helicoids, oneswith handles. They succeeded in repre-senting these helicoidsthe rst discov-ered since the 18th centuryon a com-puter and went on to produce a formal

proof of their existence. Had we not

SCIENTIFIC AMERICAN October 1993 95

EXPERIMENTAL MATHEMATICIAN Jean E . Taylor of Rutgers University seeks therules governing minimal surfaces by studying real phenomena, such as soap bub-

bles, and computer-generated ones, such as idealized crystals (left).



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been able to see a picture that roughlycorresponded to what we believed, wewould never have been able to do it,Homan says.

The area of experimental mathemat-ics that has received the lions shareof attention over the past decade isknown as nonlinear dynamics or, morepopularly, chaos. In general, nonlinearsystems are governed by a set of sim-

ple rules that, through feedback andrelated eects, give rise to complicatedphenomena. Nonlinear systems were in-vestigated in the precomputer era, butcomputers allow mathematicians to ex-plore these systems and watch themevolve in ways that Henri Poincar andother pioneers of this branch of mathe-matics could not.

Cellular automata, which divide acomputer screen into a set of cells(equivalent to pixels), provide a partic-

ularly dramatic illustration of the prin-

ciples of nonlinearity. In general, thecolor, or state, of each cell is deter-mined by the state of its neighbors. Achange in the state of a single cell trig-gers a cascade of changes throughoutthe system.

One of the most celebrated of cel-lular automata was invented by JohnH. Conway of Princeton in the early1970s. Conway has proved that his au-

tomaton, which he calls Life, is unde-cidable: one cannot determine wheth-er its patterns are endlessly variegatedor eventually repeat themselves. Scien-tists have seized on cellular automata astools for studying the origin and evolu-tion of life. The computer scientist andphysicist Edward Fredkin of Boston Uni-versity has even argued that the entire

universe is a cellular automaton.More famous still is the Mandelbrot

set, whose image has become an iconfor the entire eld of chaos since it waspopularized in the early 1980s by Be-

noit B. Mandelbrot of the IBM Thomas J.Watson Research Center. The set stemsfrom a simple equation containing acomplex term (based on the square rootof a negative number). The equation

spits out solutions, which are then iter-ated, or fed back, into the equation.

The mathematics underlying the sethad been invented more than 70 yearsago by two Frenchmen, Gaston Juliaand Pierre Fatou, but computers laid

bare their baroque beauty for all tosee. When plotted on a computer, theMandelbrot set coalesces into an imagethat has been likened to a tumorous

heart, a badly burned chicken and awarty snowman. The image is a fractal:its fuzzy borders are innitely long,and it displays patterns that recur atdierent scales.

Researchers are now studying setsthat are similar to the Mandelbrot set

but inhabit four dimensions. The kindsof complications you get here are thekinds you get in many dierent scienc-es, says John Milnor of the State Uni-versity of New York at Stony Brook.Milnor is trying to fathom the proper-ties of the four-dimensional set by ex-

amining two-dimensional slices of itgenerated by a computer. His prelimi-nary ndings led o the inaugural issueof Experimental Mathematicslast year.Milnor, a 1962 Fields Medalist, says he


HELICOID WITH A HOLE (bottom left) was discovered last year by David A. Ho-man of the University of Massachusetts at Amherst and his colleagues, with thehelp of computer graphics. Edward C. Thayer, one of Homans graduate students,recently found a structure (below) that mimics the pattern of certain polymers.



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occasionally performed computer ex-periments in the days of punch cards,

but it was a miserable process. It hasbecome much easier.

The popularity of graphics-orient-ed mathematics has provoked a back-lash. Krantz of Washington Universitycharged four years ago in the Mathe-matical Intelligencerthat in some cir-cles, it is easier to obtain funding to

buy hardware to generate pictures offractals than to obtain funding to studyalgebraic geometry.

A broader warning about specula-tive mathematics was voiced this pastJuly in the Bulletin of the AmericanMathematical Societyby Arthur Jae ofHarvard and Frank S. Quinn of the Vir-ginia Polytechnic Institute. They sug-

gested that computer experiments andcorrespondence with natural phenome-na are no substitute for proofs in es-tablishing truth. Groups and individu-als within the mathematics communityhave from time to time tried being lesscompulsive about details of arguments,Jae and Quinn wrote. The resultshave been mixed, and they have occa-sionally been disastrous.

Most mathematicians exploiting com-puter graphics and other experimentaltechniques agree that seeing shouldnot be believing and that proofs are

still needed to verify the conjecturesthey arrive at through computation. Ithink mathematicians were contem-plating their navels for too long, butthat doesnt mean I think proofs are ir-relevant, Taylor says. Homan oersan even stronger defense of traditionalproofs. Proofs are the only laboratoryinstrument mathematicians have, heremarks, and they are in danger of be-ing thrown out. Although computergraphics are unbelievably wonderful,he adds, in the 1960s drugs were un-

believably wonderful, and some people

didnt survive.

Indeed, veteran computer enthusiastsknow better than most that computa-tional experimentswhether involvinggraphics or numerical calculationscan be deceiving. One cautionary taleinvolves the Riemann hypothesis, a fa-

mous prediction about the patterns dis-played by prime numbers as they marchtoward innity. First posed more than100 years ago by Bernhard Riemann,the hypothesis is considered to be oneof the most important unsolved prob-lems in mathematics.

A contemporary of Riemanns, FranzMertens, proposed a related conjectureinvolving positive whole numbers; iftrue, the conjecture would have provid-ed strong evidence that the Riemannhypothesis was also true. By the early1980s computers had shown that Mer-

tenss proposal did indeed hold for

at least the rst 10 billion integers. In1984, however, more extensive com-putations revealed that eventuallyatnumbers as high as 10 (10

70)the pat-tern predicted by Mertens vanishes.

One potential drawback of comput-

ers is that all their calculations arebased on the manipulation of discrete,whole numbersin fact, ones and ze-ros. Computers can only approximatereal numbers, such as pior the squareroot of two. Someone knowledgeableabout the rounding-o functions of asimple pocket calculator can easily in-duce it to generate incorrect answersto calculations. More sophisticated pro-grams can make more complicated andelusive errors. In 1991 David R. Stoute-myer, a software specialist at the Uni-versity of Hawaii, presented 18 experi-

ments in algebra that gave wrong an-


UNEARTHLY LANDSCAPES emerge when a computer generates slices of a four-dimensional map similar to the well-known Mandelbrot set. John Milnor of the StateUniversity of New York at Stony Brook studies similar two-dimensional images inorder to understand the properties of the complex mathematical object.



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swers when performed with standardmathematics software.

Stephen Smale of the University ofCalifornia at Berkeley, a 1966 FieldsMedalist, has sought to place mathe-matical computation on a more securefoundationor at least to point out thesize and location of the cracks runningthrough the foundation. Together withLenore Blum of the Mathematical Sci-

ences Research Institute at Berkeley andMichael Shub of IBM, he has createda theoretical model of a computer thatcan process real numbers rather than

just integers.Blum and Smale recently concluded

that the Mandelbrot set is, in a technicalsense, uncomputable. That is, one can-not determine with certainty whetherany given point on the complex planeresides within or outside the sets hir-sute border. These results suggest thatyou have to be careful in extrapolat-ing from the results of computer ex-

periments, Smale says.These concerns are dismissed by Ste-phen Wolfram, a mathematical phys-icist at the University of Illinois. Wol-fram is the creator of Mathematica,which has become the leading mathe-matics software since rst being mar-keted ve years ago. He acknowledg-es that there are indeed pitfalls in ex-

perimental mathematics. As in all otherkinds of experiments, you can do themwrong. But he emphasizes that compu-tational experiments, intelligently per-formed and analyzed, can yield more re-sults than the old-fashioned conjecture-proof method. In every other eld ofscience there are a lot more experimen-talists than theorists, Wolfram says. Isuspect that will increasingly be the

case with mathematics.The obsession with proof, Wolfram

declares, has kept mathematicians fromdiscovering the vast new realms of phe-nomena accessible to computers. Eventhe most intrepid mathematical exper-imentalists are for the most part notgoing far enough, he says. Theyretaking existing questions in mathemat-ics and investigating those. They areadding a few little curlicues to the topof a gigantic structure.

Mathematicians may take this viewwith a grain of salt. Although he shares

Wolframs fascination with cellular auto-mata, Conway contends that Wolframscareeras well as his contempt forproofsshows he is not a real mathe-matician. Pure mathematicians usuallydont found companies and deal withthe world in an aggressive way, Lifescreator says. We sit in our ivory tow-ers and think about things.

Purists may have a harder time ignor-ing William P. Thurston, who is also anenthusiastic booster of experimentalmathematics and of computers in math-ematics. Thurston, who heads the Math-ematical Sciences Research Institute atBerkeley and is a co-director of the Ge-ometry Center (with Albert Marden ofthe University of Minnesota), has im-peccable credentials. In the mid-1970s

he pointed out a deep potential con-nection between two separate branchesof mathematicstopology and geome-try. Thurston won a Fields Medal forthis work in 1982.

Thurston emphasizes that he be-lieves mathematical truths are discov-ered and not invented. But on the sub-

ject of proofs, he sounds less like a dis-ciple of Plato than of Thomas S. Kuhn,the philosopher who argued in his 1962

book, The Structure of Scientific Revo-lutions, that scientic theories are ac-cepted for social reasons rather than

because they are in any objective sensetrue. That mathematics reduces inprinciple to formal proofs is a shakyidea peculiar to this century, Thurstonasserts. In practice, mathematiciansprove theorems in a social context, hesays. It is a socially conditioned bodyof knowledge and techniques.

The logician Kurt Gdel demonstrat-ed more than 60 years ago through hisincompleteness theorem that it is im-possible to codify mathematics, Thur-ston notes. Any set of axioms yieldsstatements that are self-evidently true

but cannot be demonstrated with those

axioms. Bertrand Russell pointed outeven earlier that set theory, which is the

basis of much of mathematics, is rifewith logical contradictions related tothe problem of self-reference. (The self-contradicting statement This sentenceis false illustrates the problem.) Settheory is based on polite lies, things weagree on even though we know theyrenot true, Thurston says. In some ways,the foundation of mathematics has anair of unreality.

Thurston thinks highly formal proofsare more likely to be awed than those

appealing to a more intuitive level ofunderstanding. He is particularly enam-ored of the ability of computer graphics


PARTY PROBLEM was solved after a vastcomputation by Stanislaw P. Radziszow-ski and Brendan D. McKay. They calcu-lated that at least 25 people are requiredto ensure either that four people are allmutual acquaintances or that ve aremutual strangers. This diagram, in whichred lines connect friends and yellowlines link strangers, shows that a party

of 24 violates the dictum.



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to communicate abstract mathematicalconcepts to others both within and out-side the professional community. Twoyears ago, at his urging, the GeometryCenter produced a computer-generatedvideo proof, called Not Knot, that dra-matizes a ground-breaking conjecture

he proved a decade ago [see illustrationon pages 92 and 93]. Thurston men-tions proudly that the rock band theGrateful Dead has shown the Not Knotvideo at its concerts.

Whether Deadheads grok the sub-stance of the videowhich concernshow mathematical objects called three-manifolds behave in a non-Euclideanhyperbolic spaceis another matter.Thurston concedes that the video is dif-cult for nonmathematicians, and evensome professionals, to fathom, but heis undaunted. The Geometry Center is

now producing a video of yet anotherof his theorems, which demonstrateshow a sphere can be turned inside out[see cover illustration]. Last fall, more-over, Thurston organized a workshopat which participants discussed howvirtual reality and other advanced tech-nologies could be adapted for mathe-matical visualization.

Paradoxically, computers have cat-alyzed a countertrend in which truth isobtained at the expense of comprehensi-

bility. In 1976 Kenneth Appel and Wolf-gang Haken of the University of Illinois

claimed they had proved the four-color

conjecture, which stated that four huesare sucient to construct even an in-nitely broad map so that no identical-ly colored countries share a border. Insome respects, the proof of Appel andHaken was conventionalthat is, it con-sisted of a series of logical, traceable

steps proceeding to a conclusion. Theconclusion was that the conjecture could

be reduced to a prediction about thebehavior of some 2,000 dierent maps.

Since checking this prediction byhand would be prohibitively time-con-suming, Appel and Haken programmeda computer to do the job for them.Some 1,000 hours of computing timelater, the machine concluded that the2,000 maps behave as expected: thefour-color conjecture was true.

The Party Problem

Other computer-assisted proofs havefollowed. Just this year, a proof of theso-called party problem was announced

by Stanislaw P. Radziszowski of theRochester Institute of Technology andBrendan D. McKay of the Australian Na-tional University in Canberra. The prob-lem, which derives from work in set the-ory by the British mathematician FrankP. Ramsey in the 1920s, can be phrasedas a question about relationships be-tween people at a party. What is theminimum number of guests that must

be invited to guarantee that at leastX

people are all mutual acquaintances orat least Yare mutual strangers? Thisnumber is known as a Ramsey number.

Previous proofs had established that18 guests are required to ensure thatthere are either four mutual acquain-tances or four strangers. In their proof,

Radziszowski and McKay showed thatthe Ramsey number for four friends orve strangers is 25. Socialites mightthink twice about trying to calculate theRamsey number for greaterXs and Ys.Radziszowski and McKay estimate thattheir proof consumed the equivalent of11 years of computation by a standarddesktop machine. That may be a rec-ord, Radziszowski says, for a problemin pure mathematics.

The value of this work has been de-bated in an unlikely forumthe news-paper column of advice-dispenser Ann

Landers. In June a correspondent com-plained to Landers that resources spenton the party problem should have been

used to help starving children in war-torn countries around the world. Somemathematicians raise another objectionto computer-assisted proofs. I dont be-lieve in a proof done by a computer,says Pierre Deligne of the Institute forAdvanced Study, an algebraic geometerand 1978 Fields Medalist. In a way, Iam very egocentric. I believe in a proofif I understand it, if its clear. While rec-ognizing that humans can make mis-

takes, he adds: A computer will also


he continuing penetration of computers into mathe-matics has revived an old debate: Can mathematics

be entirely automated? Will the great mathematicians ofthe next century be made of silicon?

In fact, computer scientists have been working for de-cades on programs that generate mathematical conjec-tures and proofs. In the late 1950s the artificial-intelli-gence guru Marvin Minsky showed how a computer couldrediscover some of Euclids basic theorems in geometry.In the 1970s Douglas Lenat, a former student of Minskys,presented a program that devised even more advancedgeometry theorems. Skeptics contended that the resultswere, in effect, embedded in the original program.

A decade ago the computer scientist and entrepreneurEdward Fredkin sought to revive the sagging interest inmachine mathematics by creating what came to be knownas the Leibniz Prize. The prize, administered by CarnegieMellon University, offers $100,000 for the first computerprogram to devise a theorem that has a profound effecton mathematics.

Some practitioners of what is known as automated rea-

soning think they may be ready to claim the prize. One isLarry Wos of Argonne National Laboratory, editor of the

Journal of Automated Reasoning. He claims to have devel-oped a program that has solved problems in mathematics

and logic that have stumped people for years. Anotheris Siemeon Fajtlowicz of the University of Houston, inven-tor of a program, called Graffiti, that has proposed thou-sands of conjectures in graph theory.

None of these achievements comes close to satisfyingthe profound effect criterion, according to David Mum-ford of Harvard University, a judge for the prize. Not now,not 100 years from now, Mumford replies when asked topredict when the prize might be claimed.

Some observers think computers will eventually surpassour mathematical abilities. After all, notes Ronald L. Gra-ham of AT&T Bell Laboratories, were not very well adapt-ed for thinking about the space-time continuum or the Rie-mann hypothesis. Were designed for picking berries oravoiding being eaten.

Others side with the mathematical physicist RogerPenrose of the University of Oxford, who in his 1989book, The Emperors New Mind, asserted that computerscan never replace mathematicians. Penroses argumentdrew on quantum theory and Gdels incompleteness the-orem, but he may have been most convincing when dis-

cussing his personal experience. At its best, he suggest-ed, mathematics is an art, a creative act, that cannot bereduced to logic any more than King Learor BeethovensFifth can.

Silicon Mathematicians

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make mistakes, but they are much moredicult to nd.

Others take a more functional pointof view, arguing that establishing truthis more important than giving mathe-maticians an aesthetic glow, particularlyif a result is ever to nd an application.Defenders of this approach, who tendto be computer scientists, point out thatconventional proofs are far from im-

mune to error. At the turn of the cen-tury, most theorems were short enoughto read in one sitting and were pro-duced by a single author. Now proofsoften extend to hundreds of pages ormore and are so complicated that yearsmay pass before they are conrmed

by others.The current record holder of all con-

ventional proofs was completed in theearly 1980s and is called the classica-tion of nite, simple groups. (A groupis a set of elements, such as integers,together with an operation, such as ad-

dition, that combines two elements toget a third one.) The demonstration con-sists of some 500 articles totaling near-ly 15,000 pages and written by morethan 100 workers. It has been said thatthe only person who grasped the entireproof was its general contractor, DanielGorenstein of Rutgers. Gorenstein diedlast year.

Much shorter proofs can also raisedoubts. Three years ago Wu-Yi Hsiangof Berkeley announced he had provedan old conjecture that one can packthe most spheres in a given volume bystacking them like cannonballs. Today

some skeptics are convinced the 100-page proof is awed; others are equal-ly certain it is basically correct.

Indeed, the key to greater reliability,according to some computer scientists,is not less computerization but more.Robert S. Boyer of the University ofTexas at Austin has led an eort tosqueeze the entire sprawling corpus ofmodern mathematics into a single data

base whose consistency can be veriedthrough automated proof checkers.

The manifesto of the so-called QEDProject states that such a data base will

enable users to scan the entirety ofmathematical knowledge for relevantresults and, using tools of the QED sys-tem, build upon such results with re-liability and condence but withoutthe need for minute comprehension ofthe details or even the ultimate founda-tions. The QED system, the manifestoproclaims rather grandly, can even pro-vide some antidote to the degenerativeeects of cultural relativism and nihil-ism and, presumably, protect mathe-matics from the all-too-human willing-ness to succumb to fashion.

The debate over computer proofs has

intensied recently with the advent ofa technique that oers not certainty butonly a statistical probability of truth.Such proofs exploit methods similar tothose underlying error-correction codes,which ensure that transmitted messag-es are not lost to noise and other ef-fects by making them highly redundant.The proof must rst be spelled out pre-cisely in a rigorous form of mathemat-

ical logic. The logic then undergoes afurther transformation called arithmet-ization, in which and, or and oth-er functions are translated into arith-metic operations, such as addition andmultiplication.

Like a message transformed by anerror-correction code, the answer of aprobabilistic demonstration is distribut-ed throughout its lengthas are anyerrors. One checks the proof by query-ing it at dierent points and determin-ing whether the answers are consistent;as the number of checks increases, so

does the certainty that the argument iscorrect. Laszlo Babai of the Universityof Chicago, who developed the proofstwo years ago (along with Lance Fort-now, Carsten Lund and Mario Szegedyof Chicago and Leonid A. Levin of Bos-ton University), calls them transpar-ent. Manuel Blum of Berkeley, whosework helped to pave the way for Babaisgroup, suggests the term holographic.

The Uncertain Future

Whatever they are named, such proofshave practical drawbacks. Szegedy ac-

knowledges that transforming a con-ventional demonstration into the prob-abilistic form is dicult, and the resultcan be a much bigger and uglier ani-mal. A 1,000-line proof, for example,could easily balloon to 1,0003 (1,000,-000,000) lines. Yet Szegedy contendsthat if he and his colleagues can simplifythe transformation process, probabilis-tic proofs might become a useful meth-od for verifying mathematical proposi-tions and large computationssuch asthose leading to the four-color theorem.The philosophical cost of this ecient

method is that we lose the absolute cer-tainty of a Euclidean proof, Babai not-ed in a recent essay. But if you do havedoubts, will you bet with me?

Such a bet would be ill advised, Levinbelieves, since a relatively few checkscan make the chance of error vanish-ingly small: one divided by the num-

ber of particles in the universe. Eventhe most straightforward conventionalproofs, Levin points out, are suscepti-

ble to doubts of this scale. At the mo-ment you nd an error, your brain maydisappear because of the Heisenberg

uncertainty principle and be replaced

102 SCIENTIFIC AMERICAN October 1993 Copyright 1993 Scientific American, Inc.


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by a new brain that thinks the proof iscorrect, he says.

Ronald L. Graham of AT&T Bell Lab-oratories suggests that the trend awayfrom short, clear, conventional proofsthat are beyond reasonable doubtmay be inevitable. The things you canprove may be just tiny islands, excep-tions, compared to the vast sea of re-sults that cannot be proved by human

thought alone, he explains. Mathema-ticians seeking to navigate unchartedwaters may become increasingly de-pendent on experiments, probabilisticproofs and other guides. You may not

be able to provide proofs in a classicalsense, Graham says.

Of course, mathematics may yieldfewer aesthetic satisfactions as inves-tigators become more dependent oncomputers. It would be very discour-aging, Graham remarks, if somewheredown the line you could ask a comput-er if the Riemann hypothesis is correct

and it said, Yes, it is true, but you wontbe able to understand the proof. Traditionalists no doubt shudder at

the thought. For now, at least, they canrally behind heros like Wiles, the con-queror of Fermats last theorem, whoeschews computers, applications andother abominations. But there may befewer Wileses in the future if reportsfrom the front of precollege educationare any guide. The Mathematical Scienc-es Research Institute at Berkeley, whichis overseen by Thurston, has been hold-ing an ongoing series of seminars withhigh school teachers to nd newways to

entice students into mathematics. Thispast January Lenore Blum, the institutesdeputy director, organized a seminardevoted to the question Are Proofs inHigh School Geometry Obsolete?

The mathematicians insisted thatproofs are crucial to ensure that a resultis true. The high school teachers de-murred, pointing out that students nolonger considered traditional, axiomaticproofs to be as convincing as, say, visu-al arguments. The high school teach-ers overwhelmingly declared that moststudents now (Nintendo/joystick/MTV

generation) do not relate to or see theimportance of proofs, the minutes ofthe meeting stated. Note the quotationmarks around the word proofs.


FURTHER READING

ISLANDS OF TRUTH: A MATHEMATICALMYSTERY CRUISE. Ivars Peterson. W. H.Freeman and Company, 1990.

THE PROBLEMS OF MATHEMATICS. IanStewart. Oxford University Press, 1992.

PI IN THE SKY: COUNTING, THINKING,ANDBEING. John D. Barrow. Oxford Universi-ty Press, 1992.


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Death of Proof