Proc. Nati. Acad. Sci. USAVol. 85, pp. 8371-8375, November 1988Symposium Paper

This paper was presented at a symposium organized by Felix M. Browder to discuss "Mathematics and the Sciences"during the 124th Annual Meeting of the National Academy of Sciences, April 25, 1987, Washington, DC.

Physics and mathematics at the frontierDAVID J. GROSSJoseph Henry Laboratories, Princeton University, Princeton, NJ 08544

ABSTRACT The relation between elementary particlephysics and mathematics is explored. The beauty and effec-tiveness of the mathematical structures that appear in funda-mental physics are discussed.

We celebrate today 100 years of the American MathematicalSociety, and thereby American mathematics, by exploring therelations between mathematics and the natural sciences, fromelementary particle physics to computer science. I shall bediscussing the interaction of physics and mathematics in thefield of elementary particle physics. Here, in the exploration ofthe fundamental laws of nature, mathematics and physics havehad the longest union and the most fruitful exchanges.

I should, however, qualify my use of "fundamental," a buzzword that might raise the hackles of my colleagues who objectto statements that imply that one field is more fundamental thananother. By fundamental I do not mean supreme, preeminent,or dominant but rather basic, elementary, and underlying. Inthat sense, the fundamental laws of physics are those that wewould start with ifwe were to teach physics in a logical fashion,as opposed to the traditional historical method. In this approachthe laws of fluid dynamics would be seen as a consequence ofthe microscopic laws of classical dynamics, themselves anexcellent approximation to the nonrelativistic laws of thequantum mechanics ofatoms. The atoms would be understood,in an excellent approximation, by nonrelativistic quantummechanics that describes electrons interacting with nuclei; thenuclei would be understood as bound states of quarks andgluons-all these ingredients being part of the standard theoryof elementary particle physics, which itself (together with thelaw of gravity) is part of who knows what. It is the business ofelementary particle physics to search for the next rung in thisladder, to discover the "who knows what" from which wecould deduce our current, somewhat incomplete, description ofmatter and its interactions. It is this realm of fundamentalphysics that is intimately intertwined with mathematical re-search at the frontiers of mathematical study, where newpatterns are being discovered and new edifices are beingconstructed.

This has been true from the beginning of modem physics,when Galileo first enunciated the proposition that the naturallanguage of physics was mathematics. Newton, one of thegreatest mathematicians of his day, invented the calculus ofinfinitesimals to calculate planetary orbits as well as to solvepure mathematical problems. In the following centuries therewas little distinction between theoretical physics and mathe-matics, with many of the greatest contributors-Laplace, Le-gendre, Hamilton, Gauss, Fourier-being regarded as physi-cists by physicists and as mathematicians by mathematicians.The 20th century has witnessed two revolutions in physics

and the completion of a theory of ordinary matter and itsinteractions. Once again we have called on mathematics tosupply the tools and framework for this task. When Einsteincreated general relativity, the dynamical theory of space andtime, in 1915, the necessary tools of differential geometry wereavailable. They had been created by Gauss and by Riemann in

the previous century. The effect of general relativity on math-ematics was electrifying; Riemannian geometry became a cen-tral topic ofgeometry. The development ofquantum mechanicsbuilt on the understanding of Hilbert spaces and influenced thedevelopment offunctional analysis. Early particle physics drewheavily on the theory of continuous groups, which itself waspartly motivated by the desire to understand the spatial sym-metry of crystalline structure.

Nonetheless, during the middle part of this century math-ematics and fundamental physics have developed in verydifferent directions with little significant interaction betweenthem. This was due, in part, to an atmosphere of increasedabstraction in the mathematics community, as well as aninsistence on rigid formal rigor as exemplified by the famousBourbaki school. (This school, by the way, has had adisastrous effect on the style of mathematics writing,whereby authors are encouraged to remove from the descrip-tion of their work all traces of intuitive reasoning or any hintat how they arrived at their ideas. This style, which lately isbeginning to change, has made it difficult for nonspecialiststo follow the progress of modem mathematics.) However,much of the reason for this separation was due to develop-ments in physics. First, the early development of quantummechanics and the early applications of quantum mechanicsto elucidating the structure of matter required little mathe-matical sophistication. It has been said that "In the 1930s,under the demoralizing influence of quantum theoretic per-turbation theory, the mathematics required of a theoreticalphysicist was reduced to a rudimentary knowledge of theLatin and Greek alphabets" (R. Jost, quoted in ref. 1). Thesetools largely sufficed for the first applications of quantummechanics to the study of matter. During the first decadesafter World War II the vistas of particle physics rapidlyexpanded. These times were dominated by experimentalsurprises and theoretical model building required little morethan traditional mathematical tools.

This situation changed dramatically 10 years ago when,prompted by decades of experimental exploration, we arrivedat the nonabelian gauge theories of the strong, weak, andelectromagnetic interactions. These theories are now univer-sally accepted as yielding a complete description of all theinteractions of matter at energies and distances that are exper-imentally accessible at present. This development is surely oneof the most remarkable accomplishments of 20th centuryscience. Attention has more recently turned to the explorationof the structure of these theories and to even more ambitiousattempts to construct unified theories of all the interactions ofmatter together with gravity. In the development ofthese gaugetheories-the so-called "standard model"-it has happenedthat many significant physical problems have led to significantconcepts in modem mathematics. Many of these concepts infact were invented independently by physicists and by mathe-maticians. Thus, for example, in 1931 Paul Dirac discussed, inone of the most beautiful papers in theoretical physics, thepossible existence of elementary magnetic charges-magneticmonopoles. He showed that in quantum mechanics such

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Proc. Natl. Acad. Sci. USA 85 (1988)

magnetic monopoles made sense if and only if the product oftheir charge, g, with the electric charge of the electron, e, wasan integer multiple of Planck's constant A: ge = nA. This wasvery exciting since it meant that as long as there existed onemagnetic monopole in the universe all charges had to bequantized in units of tug. In mathematical terms Dirac haddiscovered an integer that characterized the topological clas-sification of vector bundles, mathematical constructs thatwere being invented at about the same time by mathemati-cians. These concepts have come to play a role of increasingimportance in modern gauge theories.We have borrowed much from modern mathematics but

now the debt is being paid back. Methods developed inquantum gauge theories, using so called "instantons," wereborrowed a few years ago by Donaldson, Taubes, and Floerto deduce some deep and astounding properties of thegeometry of three- and four-dimensional spaces (9, 10). Inwhat is unlikely to be the final chapter in this saga, Witten hasrecently reinterpreted Donaldson's theory in physical terms,using it to speculate on a new phase of quantum gravity (11).Finally, recent developments in superstring theory, an am-bitious theory that attempts to construct a unified quantumtheory of matter and gravity, have begun to meet realmathematical frontiers. These theories have attracted muchattention from mathematicians since they give strong hints ofconnections between hitherto separate parts of mathematics.Many physicists believe that the final understanding of thestructure of string theory will involve fundamental general-izations of geometry. Perhaps we are entering a golden era inthe long history of cooperation between fundamental math-ematics and physics. More on this later.

On the Unreasonable Effectiveness of Mathematics inPhysics

Eugene Wigner wondered, almost 30 years ago, about the"unreasonable effectiveness of mathematics in the naturalsciences." The effectiveness of mathematics in physics isindeed impressive and, all too often, taken for granted.Wigner argued that it is not at all obvious that mathematicalconcepts are appropriate for the description of natural phe-nomenon. These concepts are certainly not conceptuallysimple, conceptual simplicity is not one of the primary goalsof mathematics, nor are they necessarily inevitable. How-ever, they are certainly useful. The mathematical formulationofphysics often leads to a remarkably accurate description ofmany phenomena. This record of agreement provides con-vincing evidence that mathematics is the correct language forphysics. Wigner pointed out that "The enormous usefulnessofmathematics in the natural sciences is something borderingon the mysterious and there is no rational explanation for it.It is not at all natural that 'laws of nature' exist, much less thatman is able to discover them. The miracle of the appropri-ateness of the language of mathematics for the formulation ofthe laws of physics is a wonderful gift which we neitherunderstand nor deserve" (2). I suppose he meant by the lastremark that if we do not understand it we do not deserve it.

Indeed it is something of a miracle that we are able todevise theories that allow us to make incredibly precisepredictions regarding physical phenomena and that we cancarry out controlled experiments that allow us to measurethese quantities to incredible precision. To give one of themost astounding examples, consider the magnetic moment ofthe electron, ou = g(eh/2mc)S. Naively, the gyromagneticratio of the electron, g, should equal 2, and the deviation ofg from 2 was one of the anomalies that stimulated thedevelopment of the relativistic quantum theory of the elec-tromagnetic field. After long, long calculations quantumelectrodynamicists can predict and after careful, carefulmeasurements atomic scientists can measure this parameter

to one part in hundred billion! The result is [using a-'137.035963(15)]

gtheory = 2.[1 + - 0.328478445 (-)

+ 1.183(11) (

- 2'[1.000,159,652,459 ± 0.000,000,000,123]

gexperiment = 2-[1.000,159,652,193 ± 0.000,000,000,004].

I am not sure which is more impressive, the theoreticalaccuracy or the experimental accuracy. Experimentally, thisaccuracy is achieved by the tour de force of trapping a singleelectron for a very long time in a magnetic-electric bottle (aPenning trap) (3). Theoretically, one must calculate up tofourth order in the perturbative expansion of quantum elec-trodynamics, with the fourth-order term requiring the eval-uation of 891 Feynman diagrams, an equivalent theoreticaltour de force (4). The error is totally dominated by theuncertainties in the determination of the fine-structure con-stant a.The ability to achieve this kind of precision is dependent on

many fortunate circumstances-on our ability to isolate thestudied physical phenomena from the environment and on theinvariance of basic physics under time and spatial transla-tions so that one can repeat experiments elsewhere at someother time-features lacking in the discussion, say, of socialphenomena. However, they surely also depend on -thismiraculous overlap (or isomorphism, to use the mathematicalterm) between the pure mathematical structures that under-lay quantum field theory and the real, material physicalworld. Perhaps we can gain some insight into this mystery ifwe deepen it by examining yet another miracle in the con-nection between mathematics and physics.

On the Unreasonable Beauty of Mathematics in Physics

The mystery of the effectiveness of mathematics in funda-mental physics is much deeper than just the miracle of itsastonishing utility. After all, it is no surprise that we needmathematics to deal with complicated situations involvingsystems composed of many parts, all of which are inthemselves simple. We have also learned that even simplesystems whose microscopic laws of evolution are easy todescribe can exhibit extremely complex behavior. However,we might expect to be able to describe the microscopic lawsin terms of simple mathematics. The strangest thing is that forthe fundamental laws of physics we still need deep mathe-matics and that as we probe deeper to reveal the ultimatemicroscopic simplicity, we require deeper and deeper math-ematical structures. Even more, these mathematical struc-tures are not just deep but they are also interesting, beautiful,and powerful. As D)irac put it "It seems to be one of thefundamental features of nature that fundamental physicallaws are described in terms of great beauty and power" and"As time goes on it becomes increasingly evident that therules that the mathematician finds interesting are the same asthose that Nature has chosen" (5).

This hyperbole is full of ill-defined terminology: interest-ing, beautiful, powerful. What do we mean when we say thatan equation is beautiful or that a physical concept is power-ful? Consider the mathematical formulation of the "standardmodel," the aforementioned theory of the strong, weak, andelectromagnetic interactions, that we believe describes all theconstituents of matter and their interactions down to dis-tances of 10- " cm. The action that describes this theory,from which we believe that, in a frenzy of reductionism, we

8372 Symposium Paper: Gross

Proc. Natl. Acad. Sci. USA 85 (1988) 8373

could describe all of low energy physics, is given by thefollowing mathematical expression:

S = d4xV[{g, pgya{Aa9AI + TrBc7B93 + TrCaVC9-}

+32r2

TrB +32ir2

3

+ E gv(QiyD Qi + LieyD"Li)

+ gSATr(D1-I3)t(D/-I3) - V(¢I)

+ (ofQiQri a + LinI7aLj4a) + R].

Is this beautiful? Perhaps, but only in the eye of aninformed beholder. Clearly the notion of beauty in mathe-matics, as in art, is an acquired taste. To appreciate mathe-matical beauty requires long education and training and isalways a subjective judgment. Nonetheless, there tends to bea large degree of consensus among mathematicians andphysicists as to what is beautiful and what is not. In the abovetheory there is much that most of us feel is beautiful and thereis much that is not. The beautiful parts are those that explainthe forces of nature as arising from the powerful symmetryprinciples that are the essence of these "gauge theories."These are beautiful to physicists since from a simple principleof symmetry we deduce in an almost unique fashion thenature of the forces of nature and the existence of the carriersof these forces-the graviton that underlies gravitationalforces, the photon of light, the gluons that hold nucleitogether, and the W and Z mesons that are responsible fortheir radioactive decay. This part of the theory is alsobeautiful to mathematicians since these gauge theories pro-vide interesting, very interesting as it turns out, mathematicalstructures-the fiber bundles I referred to above.The ugly parts are those that describe the strange spectrum

of matter. These do not follow from any symmetry principleand must be put in by hand, with many, much too many,parameters to yield agreement with observation. In fact it islargely due to this lack of beauty, as well as the large numberof unexplained parameters, 19 all in all, that we believe thatthis theory is not the end of the story-it is simply not prettyenough. Of course these two defects are correlated. In bothmathematics and physics the beauty and power of a conceptare strongly correlated. We find concepts and structuresbeautiful if they enable us to derive new results, understandnew phenomena-if they are powerful.The most beautiful part of the standard model is the idea of

local, or gauge, symmetries. These are unlike the more familiarglobal symmetries of the world, according to which the laws ofphysics are unchanged if we perform a symmetry transforma-tion on the whole world at once, say by rotating it about someaxis. If the world possesses a gauge symmetry one can makelocal rotations by an amount that might differ from place toplace. This symmetry first appeared in Maxwell's formulationof the laws of electromagnetism, although its full significancewas not realized until the development of quantum mechanics.A theory that possesses a local gauge symmetry necessarilyrequires a special field (the gauge field, or mathematically theconnection) with which one can connect objects that areseparated in space. Associated with the gauge field is a particleand a force that is mediated by this particle. In the case ofelectromagnetism, the gauge field is simply the electromagneticfield and the associated gauge particle is the photon of light thatmediates the electromagnetic forces between charged particles.In the case of the nonabelian generalizations of gauge theory

that appear in the standard model, the gauge particles are thegluons and the Wand Zparticles that mediate the strong and theweak interactions. Mathematically, it is convenient to think ofattaching to each point ofordinary space an internal space uponwhich the local symmetry acts. This combined object is calleda fiber bundle and is of central concern to modem mathematics.

C. N. Yang, one of the inventors of nonabelian gaugetheories, tells the story of his meeting with the mathematicianChern, who had done pioneering work on the differentialgeometry of fiber bundles. He relates that when he learnedthat mathematicians had been talking for years about theidentical structure that physicists had discovered he was verysurprised. He remarked to Chern that "it is both thrilling andpuzzling, since you mathematicians dreamed up these con-cepts out of nowhere." Chern replied "No, no, theseconcepts were not dreamed up. They were natural and real"(6). This is a fascinating reply. Chern was expressing a pointof view that, from my experience, is not uncommon amongcreative mathematicians-namely, that the mathematicalstructures that they arrive at are not artificial creations of thehuman mind but rather have a naturalness to them as if theywere as real as the structures created by physicists todescribe the so-called real world. Mathematicians, in otherwords, are not inventing new mathematics, they are discov-ering it.

If this is the case then perhaps some of the mysteries thatwe have been exploring are rendered slightly less mysterious.If mathematics is about structures that are a real part of thenatural world, as real as the concepts of theoretical physics,then it is not so surprising that it is an effective tool inanalyzing the real world. Similarly, we might expect thatphysical and mathematical structures would share the char-acteristics that we call beauty. Our minds have surelyevolved to find natural patterns pleasing.There is an obvious objection to this point of view.

Theoretical physicists are constrained by experiment. Theirconstructions must not only be beautiful and powerful theymust also be correct. They must agree with experiment andgo beyond mere explanation to successful prediction. Math-ematicians seem not to be constrained by these shackles. Ifphysicists are searching for the one logical structure thatdescribes the real world, mathematicians are exploring thespace of all conceivable logical structures, only a portion ofwhich overlaps the real, unique world. This is quite correct,nonetheless it does not contradict the idea of a commonunderlying structure that is a real feature of nature.

If it is the case that the concepts and structures thatunderlie fundamental mathematics and physics are common,then it might be advantageous for workers in both fields tosearch for new ideas and structures in each other's back yard.This strategy was promoted and followed by Dirac who said"The research worker in his efforts to express the funda-mental laws of nature in mathematical form should strivemainly for mathematical beauty" and "It may well be that thenext advance in physics will come along these lines: peoplefirst discovering the equations and needing a few years ofdevelopment in order to find the physical ideas behind theequation" (7). Conversely, mathematicians should study thestructures that physicists discover for possible hints of newmathematics. The absorption of structures from physics wasof enormous importance in the early development of math-ematics and this cross fertilization has recently been revital-ized, notjust in particle physics but also in the study of chaosin simple dynamical systems, in the discovery of fractalgeometry, and in many other examples.The revitalization ofthe connections between mathematics

and physics is especially true in the realm of elementaryparticle physics. Recent attempts to construct unified theo-ries of matter and gravity have led to a radically new kind oftheory-string theory, which gives hints of essential connec-

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Proc. Natl. Acad. Sci. USA 85 (1988)

tions to many frontier areas of modem mathematics. Stringtheory, which was originally discovered accidentally in anattempt to understand nuclear forces, has emerged in recentyears as a promising realistic theory of all the interactionsand, for the first time, a consistent theory ofquantum gravity.To some extent string theory is a simple generalization of theordinary framework of quantum field theory, in which thebasic constituents ofnature are not pointlike but are extendedone-dimensional objects-strings. Remarkably, this seem-ingly minor extension from pointlike particles to extendedstrings, without modifying in any other way the fundamentalprinciples of physics, leads to an incredible structure. Thisstructure implies that the only forces that can exist are just ofthe kind we see-gravitational and gauge interactions. It canalso produce the matter content of the world as we know itas well as the specific pattern of forces that we observe. Italso has bizarre implications, requiring that space-time be10-dimensional. To agree with the crudest of observations, itmust be the case that 6 of the spatial dimensions are curledup into a little closed space so that we do not notice them.This can be achieved since, as a generalization of Einstein'stheory of general relativity, the theory incorporates thedynamics of space-time and possesses solutions with 6compact, curled up, directions of space.

String theory has already provided many interesting math-ematical connections. The theory makes use of deep struc-tures in differential geometry and in algebraic geometry,connects to the theory of modular functions and finite groups.It even appears to have a place for branches of mathematicsthat I thought would never play a role in physics-such asnumber theory and knot theory. I once described thisdevelopment to a famous mathematician who was intriguedby the theory and the mathematical ideas it drew upon.However, his first question to me was "But is it physics?"The original, highly optimistic expectation that this theory,

which in principle has the power to allow us to calculate allthe parameters of the standard model as well as understandthe reason behind many of its features, would lead rapidly tonew predictions and tests, has undergone sober reevaluation.It is not that there are any experimental contradictions, norare there any indications of internal inconsistency, rather itis clear that we do not yet know enough about the structureof the theory to control its dynamics sufficiently to makecontact with experiment. Part of the problem is that we havestumbled on this theory by accident, without knowing whatthe basic logical setting for the theory is or will be (it has beensaid that string theory is of the 21st century, discovered byaccident in the 20th century).A more immediate problem is that in trying to discover the

principles of this theory and applying it to the real world totest its validity, we are faced with the fact that the basicdistance scale of the theory is very, very small. The funda-mental length scale of string theory, or indeed of any unifiedtheory of gravity and matter, is the Planck scale, the lengththat can be formed from the three-dimensional fundamentalconstants of nature: Newton's constant of gravitation G,Planck's quantum constant h, and the speed of light c:

'planck - -c 33 cm.

(Alternatively, we can express this scale in units of time:tplanck 10 s, or in units of mass: M 19 nucicoplanck-10 ~planck 2z101 Mnced

Unfortunately, this length scale is smaller by 17 orders ofmagnitude than the smallest distances that we can see withour most powerful microscopes, our most energetic particleaccelerators. The fact that this number is so small is respon-sible for some of the most striking features of our universe.For example, the reason stars are so big is that at the scale

of the radius of ordinary atoms and nuclei, gravity is veryweak (because this scale is 17 orders of magnitude below thePlanck scale). Thus, gravitationally bound aggregates ofnuclei can contain approximately (Mplanck/Mnucleon)3 105nuclei before collapsing.The value of this number presents us with one of the major

problems of theoretical physics, since it is very difficult togive a natural explanation for a number that is so small. In anycase it implies that string theory is an attempt to extrapolatefar, far beyond present day experiment. Even if we have anidea of the physics at these incredibly small planckiandistances, it is very hard to make our way up to the distancesat which measurements are done at present. There is a lot ofphysics that occurs along the way that we must understandif we are to make contact with experiment.An extrapolation of this enormity is unprecedented in the

history of physics. One has every right to express skepticismas to the chances of success of such a risky venture, as someof my colleagues have recently done in a vocal and publicfashion. It does little good to remind these critics that at highenergy we have learned that the correct scale of distances islogarithmic (i.e., physics changes as the logarithm of thedistance scale for very short distances), so that an extrapo-lation by a factor of 1017 is really only a factor of log(1017)40. It is equally useless to point out that we have no choicebut to attempt such an extrapolation if we wish to addressfundamental questions.What is clear is that new strategies are required in today's

climate, different from those that we used in previousdecades when our field was led by the nose by experimentaldiscovery. Not that we do not expect new experimentaldiscoveries. All unified theories, string theory included,predict much new physics that could be seen at the super-conducting supercollider (SSC), which we hope and trust willbe built. Experiments with the SSC, although they will nottake us to the energy scale at which gravity becomes as strongas the nuclear force, will be of crucial importance in providingclues to the connection between Planck-scale physics and ourlow-energy world. Without the SSC particle physics will die.At this moment, however, when we are faced with no

experimental surprises or paradoxes and when string theoryhints at deep mathematical structures, Dirac's strategy hasbecome more and more appealing. Many string theorists areexploring the mathematical structures that have been thrownup by string theory in the hope that they will provide theunderlying framework for the theory and give clues as to itsdynamics.Our critical colleagues denounce these efforts, indeed all of

string theory, and call it by the dirtiest name they can comeup with-recreational mathematics. Although I resent beingcalled a recreational mathematician, I admit that there is avalid (albeit small) point to these criticisms. They remind usof the danger, in following the diracian dictum, of turning intomathematicians. This for some theorists is an ever-presenttemptation. This would not be a good outcome for physicsnor I suspect for mathematics. Let us remember some of thedifferences between mathematics and physics.The bottom line for mathematicians is the proof of their

theorems, the logical consistency of their results. The finaljudge of theoretical physicists is experiment. Personally I feelthat experiment is a harsher mistress than consistency. Dirac,motivated as he was by mathematical ideas, neverthelessstated "I am not interested in proofs but only in what naturedoes." Indeed, when faced with the astounding prediction ofhis relativistic electron equation that there should exist apositively charged particle with precisely the same mass as

the electron and no evidence for such a particle (the positronwas discovered by Anderson 5 years later), he contemplatedabandoning some of the beautiful symmetry of his theory toidentify the positron with the proton, which is 1836 times

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Proc. Natl. Acad. Sci. USA 85 (1988) 8375

heavier than the electron. Weyl, who more than any othermathematician in this century saw mathematics and physicsas an organic whole said, in contrast "My work always triedto unite the true with the beautiful, but when I had to chooseone over the other, I usually chose the beautiful." On theother hand, in the case of the positron, it was Weyl whorecognized the charge conjugation symmetry of the Diracequation and who objected strenuously to the identificationof the positron with the proton. In the end the positron wasdiscovered with the same mass as the electron in accord withthe symmetry of the Dirac equation. Weyl was proved rightand beauty prevailed. In this case truth and beauty were thesame, consistent with my message that most ofthe time thereneed not be conflict between these two principles. Never-theless, if a choice between beauty and truth arises each ofus must retreat to our individual corners of security.

Mathematicians also think differently and have differenthabits of work than physicists, even when they are exploringsimilar structures. Mathematicians love to generalize, toextend their concepts to the most general possible case, toconstruct the most inclusive possible theory. Physicists are ofcourse interested not in the most general case but in thespecial case of the real world. They also work by simplifi-cation, idealization, and by the construction of specificexamples. We might say that mathematicians labor to con-struct interesting and useful definitions from which goodtheorems flow, physicists to construct interesting and usefulmodels from which good predictions flow.Mathematicians and physicists also have different

strengths. I find the most remarkable attribute of greatmathematicians is their power of abstraction. They arecapable of feats of abstraction that leave me breathless. Isuspect that mathematicians similarly admire physicists fortheir intuition, by which they are able to use mathematicalformalism much as a poet uses language. Unlike mathema-ticians they are allowed to neglect the constraints of rigor, toguess what is true without proving it, proceeding as rapidlyas possible to confront one's ideas with experiment. TheSoviet mathematician Manin agrees: "The choice of a La-grangian on the unified theory of weak and electromagneticinteractions . . . the introduction of Higgs fields, the sub-traction of vacuum expectation values and other sorcery,

which leads, say, to the prediction of neutral currents-allthis leaves the mathematician dumbfounded" (8).

Finally, physicists and mathematicians are taught to thinkdifferently. Even the chronology of the standard curriculumis different for the two fields. Physics is always taughthistorically, from the bottom up. We start with classicalmechanics, then proceed to teach nonrelativistic quantummechanics, and only at the last stages teach modem, rela-tivistic physics. This allows us to teach our students physicalintuition by allowing them to practice on concrete everydayphenomena. Modem mathematics is often taught from thetop down. This teaches the power of abstraction. Maninwrites that "it would be wonderful to master both types ofthinking, just as we master the use of a right and a left hand"(8). This is probably impossible, it must violate some kind ofuncertainty principle:

AMathematics x APhysics - C.

In any case both approaches are necessary. We need eachother's special talents and insights. Let us continue thecollaboration and extend it.But vive la difference!

This research was supported in part by National Science Foun-dation Grant PHY80-19754.

1. Streater, R. F. & Wightman, A. S. (1964) PCT, Spin andStatistics and All That (Benjamin, New York).

2. Wigner, E. P. (1960) Commun. Pure Appl. Math. 13, 1.3. Van Dyck, R. S., Schwinberg, P. & Dehmelt, H. (1984) in

Atomic Physics 9, eds. Van Dyck, R. S. & Fortson, E. N.(World Sci., Singapore).

4. Kinoshita, T. & Sapirstein, J. (1984) in Atomic Physics 9, eds.Van Dyck, R. S. & Fortson, E. N. (World Sci., Singapore).

5. Dirac, P. A. M. (1939) Proc. R. Soc. Edinburgh Sect. A 59,122.6. Yang, C. N. (1977) Ann. N.Y. Acad. Sci. 294, 86.7. Dirac, P. A. M. (1963) Sci. Am. 208, 45.8. Manin, Yu. I. (1979) Mathematics and Physics (Birkhaeuser,

Boston).9. Donaldson, S. (1983) J. Diff. Geom. 18, 269.

10. Floer, A. (1987) Bull. Am. Math. Soc. 16, 279.11. Witten, E. (1988) Comm. Math. Phys. 117, in press.

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