_RE The Science of Patterns LYNNARTHUR STEEN forcesoutside mathematics while contributingto humancivilization The rapid growth of computing and applicationshas a rich and ever-changingvariety of intellectualflora and fauna. helpedcross-fertilize the mathematicalsciences, yielding These differencesin perceptionare due primarilyto the steep and an unprecedentedabundance of new methods,theories, harshterrain of abstractlanguage that separatesthe mathematical andmodels. Examples from statisiicalscienceX core math- rainforest from the domainof ordinaryhuman activity. ematics,and applied mathematics illustrate these changes, The densejungle of mathematicshas beennourished for millennia which have both broadenedand enrichedthe relation by challengesof practicalapplications. In recentyears, computers between mathematicsand science. No longer just the have amplifiedthe impact of applications;together, computation study of number and space, mathematicalscience has and applicationshave swept like a cyclone across the terrainof become the science of patterlls, with theory built on mathematics.Forces -unleashed by the interactionof theseintellectu- relations among patterns and on applicationsderived al stormshave changed forever and for the better the morpholo- from the fit betweenpattern and observation. gy of mathematics.In their wake have emergednew openingsthat link diverseparts of the mathematicalforest, making possible cross- fertilizationof isolatedparts that has immeasurablystrengthened the whole. MODERN MATHEMATICSJUST MARKED ITS 300TH BIRTH- Throughoutthe 20th century,mathematics has grown rapidlyon day. The publicationin 1687 of Newton'sPrincipia manyfronts. The classicalcore has remainedrooted in the Newtoni- Mathematicaestablished mathematics asthe methodolog- an mathematicsof analysis,a synthesis of algebraand geometry icalparadigm of theoreticalscience. Newton perceived patterns in appliedto the studyof how things change.But even as this core has the accumulatedastronomical data of his time;he abstractedfrom expandedunder explosivepost-World War II growth, it has been thesepatterns certain general principles (whence Principia); then he supplementedby major developmentsin other mathematicalsci- usedthese principles to deducepatterns both known and unknown ences in number theory, logic, statistics, operations research, in the behaviorof planetarybodies. His wasa scienceof patterns- probability,computation, topology, and combinatorics,in addition rootedin data,supported by deduction,confirmed by observation. to algebra,geometry, and analysis. By the endof the 19thcentury, Newton's creation had flowered In each of these subdisciplines,applications parallel theory. Even magnificently,producing unprecedented intellectual blossoms. Eu- the most esotericand abstractparts of mathematics numbertheory ropeangiants such as Euler, Lagrange, and Weierstrass had elaborat- and logic, for example arenow used routinelyin applications(for ed andrefined the calculus, establishing the foundations for modern example,in computerscience and cryptography).Fifty years ago G. analysis.James Clerk Maxwell used Newton's derivatives to write H. Hardycould boast of numbertheory as the most pure and least the lawsof electromagnetism,and GeorgBernhard Riemann ap- usefulpart of mathematics(1); todaynumber theory is studiedas an plieddifferentials to geometryin apt(albeit unintentional) prepara- essentialprerequisite to manyapplications of coding, includingdata tion forAlbert Einstein, who soonwould discover in Riemanniantransmission from remote satellites,protection of financialrecords, geometrythe keyto a generaltheory of gravitation. and eflicientalgorithms for computation. At the sametime, on a separatecontinent, the peopleof the In 1960, at a time when theoreticalphysics was still the central UnitedStates were beginning their second century without mathe- jewel in the crown of appliedmathematics, Eugene Wignerwrote maticalor scientificgiants in theirmidst. Yet in 1888,two centuries aboutthe "unreasonableeffectiveness" of mathematicsin the natural afterPrincipia,a fewfar-sighted individuals foundedwhat is nowthe sciences:"The miracleof the appropriatenessof the languageof AmericanMathematical Society, thereby setting in motiona process mathematicsfor the formulationof the laws of physicsis a wonder- that createdthe world'sstrongest environment for mathematicsful gift whichwe neitherunderstand nor deserve"(2, p. 14). Indeed, research.In recognitionof thisanniversary, American mathematics theoretical physics has continuedto adopt (andoccasionally invent) is now celebratingits 100thbirthday. increasinglyabstract mathematical models as the logicalfoundation for currenttheories: Lie groups and gauge theories,exotic expres- sions of symmetry,have joined fermionsand baryonsas fundamen- tal tools in the physicist'ssearch for a unified theory of both Forcesfor Change microscopicand macroscopicforces of nature. Manyeducated persons, especially scientists and engineers, harbor Duringthis sameperiod, however, striking applications of mathe- an imageof mathematicsas akinto a treeof knowledge:formulas, matics emergedacross the entire landscapeof natural,behavioral, theorems,and results hang like ripe fruits to be pluckedby passing and socialscience. Moreover, applications of one partof mathemat- scientiststo nourishtheir theories. Mathematicians, in contrast, see ics to another-of geometryto analysis,of probabilityto number theirfield as a rapidlygrowing rain forest, nourished and shaped by theory provide renewed evidence of the fundamentalunity of mathematics.Despite the ubiquityof connectionsamong problems The authoris professorof mathematicsat St. Olaf Collegein Northfield,MN 55057 in science and mathematics,the discoveryof new links retains a and chairmanof the ConferenceBoard of the MathematicalSciences. surprisingdegree of unpredictabilityand serendipity.Whether 29 APRIL I988 ARTICLES 6II This content downloaded on Thu, 14 Mar 2013 16:48:35 PM All use subject to JSTOR Terms and Conditions plannedor unplanned,the cross-fertilizationbetween science and ideasrooted in antiquity.Its tools areabstraction and deduction; its mathematicsin problems, theories, concepts, and paradigmshas edificesinclude functions, equations, operators, and infinite-dimen- never been greaterthan it is now, in the last quarterof the 20th sional spaces.Within core mathematicsare found the traditional century. In 1988 one can say with some justificationthat the subjectsof numbertheory, algebra, geometry, analysis, and topolo- effectivenessof mathematicsis even more "unreasonable"than ever gy. After a half-centuryof explosive specialized growth, core before. mathematicsis experiencinga renaissanceof renewedintegrity based Parallelingthe growingpower of applicationsof mathematicshas on the unexpectedbut welcome discoveryof deep links among its been the extraordinaryimpact of computing. It is ironic but variouscomponents. inuisputablethat computerswere made possible by applicationof Appliedmathematics fits mathematicalmethods to the observa- abstracttheories of mathematicianssuch as Boole, Cantor,Turing, tions and theoriesof science.It is a principalconduit for scientific andvon Neumann,theories that just a few decadesago werewidely ideas to stimulatemathematical innovation and for mathematical deridedby criticsof the "new math"as wild abstractionsirrelevant tools to solve scientificproblems. Traditional methods of applied to practicalpurposes. It is doubly ironic that the computeris now mathematicsinclude differential equations, numerical computation, the most powerfulforce changingthe natureof mathematics.Even controltheory, and dynamical systems; such traditional methods are mathematicianswho never use computersmay frequentlydevote today being appliedin majornew areasof applications,including their entireresearch careers to problemsgenerated by the presence combustion,turbulence, optimization, physiology, and epidemiol- of computers.Across all parts of mathematics,computers have ogy. In addition,new tools fromgame theory, decision science, and posed new problemsfor research,provided new tools to solve old discretemathematics are being appliedto the humansciences where problems,and introducednew researchstrategies. choices,decisions, and coalitionsrather than continuous change are Althoughthe publicoften views computersas a replacementfor the apt metaphorsfor descriptionand prediction. mathematics,each is in realitya power tool for the other. Indeed, All attempts to divide mathematicsinto parts are necessarily just as computersprovide new opportunitiesfor mathematics,so artificialand perpetuallyin flux: statisticalscience, core mathemat- also mathematicsmakes computers so incrediblyeffiective. Mathe- ics, and appliedmathematics represent just one of many possible maticsprovides abstract models for naturalphenomena, as well as structuresthat may help one understandthe whole. Thesedivisions algorithmsfor implementingthese models in computerlanguages. do not representintrinsic differences in the natureof the discipline Applications,computers, and mathematicsform a tightly coupled so muchas differencesin style,purpose, and history; they may more systemyielding results never before possible and ideasnever before aptlydescribe types of mathematiciansthan types of mathematics. * . lmagmea. Othershave attempted to portraythe natureof modernmathematics in somewhat differentterms [see, for example, (3-5)]. What is importantabout these labels is that they help focus attentionon certaincharacteristics