The Heart of Mathematics

The Heart of Mathematics Author(s): P. R. Halmos Source: The American Mathematical Monthly, Vol. 87, No. 7 (Aug. - Sep., 1980), pp. 519-524 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2321415 Accessed: 03/06/2010 05:31 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=maa. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

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THE HEART OF MATHEMATICS P. R. HALMOS really consistof? Axioms(such as the parallelposWhat does mathematics Introduction. theoremof algebra)?Proofs(suchas Godel's proof tulate)?Theorems(suchas thefundamental Concepts (such as sets and classes)? Definitions(such as the Menger of undecidability)? definitionof dimension)?Theories(such as categorytheory)?Formulas(such as Cauchy's integralformula)?Methods(such as themethodof successiveapproximations)? theyare all essential.It is Mathematicscould surelynot existwithouttheseingredients; a tenablepointof viewthatnone of themis at theheartof the subject,thatthe nevertheless what main reason for existenceis to solve problems,and that,therefore, mathematician's reallyconsistsof is problemsand solutions. mathematics but "problem"is "Theorem"is a respectedwordin thevocabularyof mostmathematicians, use theword,are lowlyexercisesthat sometimes notalwaysso. "Problems,"as theprofessionals Theseemotionalovertones are assignedto studentswhowilllaterlearnhowto provetheorems. are,however,notalwaystherightones. of addition for naturalnumbersand the solvabilityof polynomial The commutativity butone of themis regardedas trivial(near equationsoverthecomplexfieldare boththeorems, is easy to prove),and theotheras deep (thestatement easy to understand, thebasic definitions, not obvious,the proofcomes via seeminglydistantconcepts,the resulthas manysurprising and to locate all thezeroesof the fortic-tac-toe applications).To findan unbeatablestrategy Riemannzeta functionare both problems,but one of themis trivial(anybodywho can effort and no can findtheanswerquickly,withalmostno intellectual thedefinitions understand and the otheris and the answerhas no consequencesof interest), feelingof accomplishment, deep (no one has foundtheansweralthoughmanyhave soughtit,theknownpartialsolutions answerwould implymany requiregreateffortand providegreatinsight,and an affirmative corollaries).Moral: theoremscan be trivialand problemscan be profound.Those non-trivial wrong. consistsof problemsare not necessarily who believethattheheartof mathematics

an articleor to mathematics bywriting ProblemBooks. If youwantedto makea contribution a book on mathematicalproblems,how should you go about it? Should the problemsbe or or graduatestudents, shouldtheybe at thelevelofundergraduates (pre-calculus), elementary should theybe researchproblemsto whichno one knowsthe answer?If the solutionsare known,shouldyour workcontainthemor not? Should the problemsbe arrangedin some order(in whichcase theverylocationof theproblemis somehintto itssolution),or systematic shouldtheybe arrangedin some"random"way?Whatshouldyouexpectthereaderto getfrom or facts(or someof each)? yourwork:fun,techniques, problemshave All possibleanswersto thesequestionshave alreadybeengiven.Mathematical A visitto thepartof thestacks and flowering. whichis stillgrowing literature, quitean extensive can be an excitingand memorablerevelation, labeledQA43 (Libraryof Congressclassification) and thereare richsourcesof problemsscatteredthroughotherpartsof the stackstoo. What followsis a quick reviewof some not quite randomlyselectedbut probablytypicalproblem collectionsthatevena casual librarysearchcould uncover. The authorreceivedhis Ph.D. fromthe Universityof Illinois; he has held positionsat (consecutively)Illinois, Syracuse,Chicago, Michigan,Hawaii, Indiana, Santa Barbara, and Indiana, withvisitingpositionsfor various periods at the Institutefor Advanced Study,Montevideo,Miami, Harvard, Tulane, Universityof Washington (Seattle),Berkeley,Edinburgh,and Perth.He has publishedeightor morebooks and manyarticles;he has held a GuggenheimFellowshipand is a memberof the Royal Society of Edinburghand the HungarianAcademy of Sciences.The MAA has givenhim a ChauvenetPrize and two Ford awards. He has been active in the affairsof both the AMS and the MAA, and will become editorof thisMoNrHLY on Jan. 1, 1982. His mathematicalinterestsare in measureand ergodictheory,algebraiclogic,and operatorson Hilbertspace, withexcursionsto probability,statistics,topologicalgroups,and Boolean algebras.-Editors

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kindof problemcollection Hilbert'sProblems.The mostriskyand possiblyleastrewarding publicis theone thatconsistsofresearchproblems.Your problems to offerto themathematical be could becomesolvedin a fewweeks,or months,or years,and yourworkwould,therefore, If you are notof thestature exposition. out of date muchmorequicklythanmostmathematical of Hilbert,you can neverbe surethatyourproblemswon'tturnout to be trivial,or impossible, or,perhapsworseyet,just orthogonalto thetruththatwe all seek-wronglyphrased,leading nowhere,and havingno lastingvalue. researchof the A listof researchproblemsthathas had a greateffecton themathematical centuryat the twentieth centurywas offeredby Hilbertin the last year of the nineteenth in Paris[3]. The firstof Hilbert's23 problemsis the International Congressof Mathematicians is everyuncountablesubsetof theset R of real numbersin one-to-one continuumhypothesis: with R? Even in 1900 the questionwas no longernew, and althoughgreat correspondence progresshas been made sincethenand somethinkthattheproblemis solved,thereare others who feelthatthefactsare farfromfullyknownyet. Some are Hilbert'sproblemsare of varyingdepthsand touchmanypartsof mathematics. intothesame have thesamevolume,can theyalwaysbe partitioned geometric (iftwotetrahedra piecesare congruent?-theansweris so thatcorresponding finitenumberof smallertetrahedra answeris yes). Severalof the (is 2'2 transcendental?-the no), and someare number-theoretic accumulatedup to 1974was broughtup to problemsare stillunsolved.Much oftheinformation did curiosity community's date and collectedin one volumein 1976[5],but themathematical contributions has and substantive not stop there-a considerablenumberof both expository been made sincethen. P6lya-Szego.Perhapsthe mostfamousand stillrichestproblembook is thatof Polya and in 1972and (in Englishtranslation) Szego [6],whichfirstappearedin 1925and was republished of vigorouslife(so far)it has been themainstayof uncountably 1976.In itsoverhalfa century sourceof examination book,and an almostinexhaustible manyseminars, a standardreference fromhighschoolto thefrontiers and doable. Its levelstretches questionsthatare bothinspiring of research.The firstproblemasks about thenumberof waysto makechangefora dollar,the of theavailablecoinsbeing1,5, 10,25, and 50,of course;in theoriginaledition denominations were 1, 2, 5, 10,20, and 50. From thequestionwas about Swissfrancs,and thedenominations theproblemsproceed,in gentlebutchallenging steps,to theHadamard thisinnocentbeginning latticepoints,determinants, and Eisenstein's threecirclestheorem,Tchebychevpolynomials, theoremaboutpowerserieswithrationalcoefficients. of mathematics" is theoriginaltitle(in German)of Dorrie'sbook [1]. "The triumph Dormre. This is a book thatdeservesto be muchbetterknownthanit seemsto be. It is eclectic,it is fromelementary to arithmetic spread over 2000 years of history,and it rangesin difficulty thesubjectof graduatecourses. materialthatis frequently attributed to Newton(Arithmetica UniversaIt contains,forinstance,thefollowing curiosity lis, 1707).If "a cowsgrazeb fieldsbarein c days,a' cowsgrazeb' fieldsbarein c' days,a" cows a to c"? It is grazeb" fieldsbare in c" days,whatrelationexistsbetweentheninemagnitudes assumedthatall fieldsprovidethe same amountof grass,thatthe dailygrowthof the fields remainsconstant,and thatall thecows eat thesame amounteach day." Answer:

(

b

det b' \b"

bc

b'c' b"c"

ac

a'c' a"c"

= 0.

This is Problem3, out of a hundred. The problemslean moretowardgeometry thananything else,buttheyincludealso Catalan's question about the numberof ways of forminga productof n prescribedfactorsin a ("how manydifferent and non-associative multiplicative systemthatis totallynon-commutative

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factorsbe calculatedby pairs?,"Problem7), and the ways can a productof n different theorem("the sum of two cubic numberscannot be a cubic Fermat-Gaussimpossibility 21). Problem number," Two moreexamplesshouldgivea fairidea of theflavorof thecollectionas a whole:"every imageof a square"(Problem72), and "at what can be consideredas a perspective quadrilateral suspendedrod appearthelongest?"(Problem pointof theearth'ssurfacedoes a perpendicularly but manyof theproblemsare of theeternally 94). The styleand theattitudeare old-fashioned, kind;thisis an excellentbook to browsein. interesting Steinhaus[71, which (like is of a Polish contribution, Steinhaus.My next mini-review and good solid fun. Dorrie's) has exactly100 problems,and theyare genuinelyelementary likethisone,and,indeed, Whensomeonesays"problembook" mostpeoplethinkof something exemplarof thespecies.The problemsare,however,notequallyinteresting it is an outstanding anotheraspectof problemsolving:itis sometimes moreover, Theyillustrate, or equallydifficult. itis,till howinteresting a problemis,or,forthatmatter, almostimpossibleto guesshowdifficult afterthesolutionis known. Considerthreeexamples.(1) Does thereexista sequence{x1,x2,...,x10}of tennumberssuch halves that(a) xl is containedin theclosedinterval[0,1],(b) xl and X2are containedin different on up so and of the interval, third in a different of [0,1],(c) each of X1,X2, and X3is contained line, a on lie three straight no that are such in the plane If 3000 points (2) through x1,x2,...,x10? and boundary)withthesepointsas verticessuch (meaninginterior do thereexist1000triangles have anypointsin common?(3) Does thereexista discin theplane thatno twoof thetriangles and boundaryofa circle)thatcontainsexactly71 latticepoints(pointsbothof (meaninginterior whosecoordinatesare integers)? are subjective,so all I can do is recordmy and interests Of coursejudgmentsof difficulty easy -andmildlyinterest(2) is astonishingly and uninteresting, own evaluations.(1) is difficult In defenseof interesting. facie even quite and looks prima it harder than ing,and (3) is a little theseopinions,I mentionone criterionthatI used: if the numbers(10,1000,71)cannotbe replaced by arbitrarypositiveintegers,I am inclinedto conclude that the corresponding problemis specialenoughto be dull. It turnsout thattheanswerto (1) is yes,and Steinhaus a solution(quiteconcretely: provesit by exhibiting x, =.95, x2=.05, x3=.34, x4=.74, etc.). He proves(the same way) that the answeris yes for 14 insteadof 10, and, by threepages of unpleasantlookingcalculation,thatthe answeris no for75. He mentionsthat,in fact,the answeris yesfor17 and no foreveryintegergreaterthan17.I say that'sdull.For (2) and (3) the answersare yes(forall n in place of 1000,or in place of 71). The book of Glazmanand Ljubic[2]is an unusualone (I don'tknowofany Glazman-Ljubic. to the othersof its kind),and, despitesome faults,it is a beautifuland excitingcontribution linear a newkindof textbookof (finite-dimensional) The book is, in effect, problemliterature. of (complex)vectorspaces and the algebraand linearanalysis.It beginswiththe definitions conceptsof lineardependenceand indepedence;thefirstproblemin thebook is to provethata if and only if x #0. The chapters set consistingof just one vectorx is linearlyindependent kind: of theconventional just as theydo in textbooks followone anotherin logicaldependence, Normedspaces,etc. Bilinearfunctionals, Linearoperators, poetry.It prose,however;perhapsit could be called expository The book is not expository backgroundmaterialwithsome care. The mainbody and relatedexplanatory givesdefinitions and theproblemis to as assertions, of thebook consistsof problems;theyare all formulated but thereaderis toldthathe provethem.The proofsare notin thebook. Thereare references, willnotneed to consultthem. analysis, The reallynewidea in thebook is itssharpfocus:thisis reallya book on functional writtenfor an audience who is initiallynot even assumedto know what a matrixis. The studenttheeasycase, thetransparent ingeniousidea of theauthorsis to presentto a beginning

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case, the motivating case, thefinite-dimensional case, thepurelyalgebraiccase of some of the deepestanalyticfactsthatfunctional analystshave discovered.The subjectsdiscussedinclude spectraltheory,the Toeplitz-Hausdorff the Hahn-Banachtheorem, theorem, partiallyordered vectorspaces,momentproblems,dissipativeoperators, and manyothersuchanalyticsounding results.A beautiful coursecould be givenfromthisbook (I wouldloveto giveit),and a student broughtup in sucha coursecould becomean infantprodigyfunctional analystin no time. (A regrettable featureof thebook, at leastin its Englishversion,is thewillfully unorthodox terminology. Example: the (canonical)projectionfroma vectorspace to a quotientspace is called a "contraction", and what most people call a contractionis called a "compression". Fortunately theconceptwhosestandardtechnicalnameis compression is notdiscussed.) to be reviewedhereis Klambauer's Klambauer.The last additionto theproblemliterature itslevelis problems, [4]. Its subjectis real analysis,and, althoughit does have someelementary relatively advanced.It is an excellentand excitingbook. It does have some faults,of course, includingsome misprints and some pointlessrepetitions, and the absence of an index is an exasperating featurethatmakesthebook muchharderto use thanit oughtto be. It is,however, a greatsourceof stimulating questions,of well knownand not so well knownexamplesand and of standardand notso standardproofs.It shouldbe on thebookshelfof counterexamples, of everyproblemlover,of everyteacherof analysis(fromcalculuson up), and,forthatmatter, everyseriousstudentof thesubject. and The table of contentsrevealsthatthebook is dividedinto fourchapters:Arithmetic Here are someexamples combinatorics, Inequalities,Sequencesand series,and Real functions. fromeach thatshould serveto illustratethe rangeof the work,perhapsto communicate its flavor,and, I hope,stimulate theappetiteformore. The combinatorics chapterasks for a proofof the "rule forcastingout nines" (is that of an integerby 9 via the sum of its decimaldigitstoo expressionfortestingthe divisibility it asks how manyzeroesthereare at theend of thedecimal old-fashioned to be recognized?), ofxk in (1 +x+x2+ **_*+ Xn-)2. Alongwith expansionof 1000!,and it asksforthecoefficient suchproblemsthereare also unmotivated formulasthatprobablyonlytheirfathercould love, and thereare a fewcuriosities (such as theproblemthatsuggeststhe use of thewell ordering of V ). A simplebutstriking ifm principleto provetheirrationality oddityis thisstatement: then and n are distinct positiveintegers, mn' nmn.

The chapteron inequalitiescontainsmanyof thefamousones (Holder,Minkowski, Jensen), and manyothersthatare analytically valuablebut somewhatmorespecializedand therefore theanswerto whichveryfewpeopleare likelyto guessis this somewhatlessfamous.A curiosity one: foreach positiveintegern, whichis bigger Vr

f

or Vin+

n?

The chapteron sequenceshas theonlydetailedand completediscussionthatI haveeverseen of the fascinating(and non-trivial) problemabout the convergencecsfthe infiniteprocess indicatedby thesymbol xxx

to learnthattheresultis due to Euler;thereference Studentsmightbe interested givenis to the articleDe formulisexponentialibus replicatis, Acta Academica ScientiarumImperialisPetropolitanae,1777.One moreteaser:whatis theclosureof theset of all real numbersof theform /n- Vm (wheren and m are positiveintegers)? The chapteron real functions is richtoo. It includesthetranscendentality of e, someof the basic properties of theCantorset,Lebesgue'sexampleof a continuousbutnowheredifferentiable function, and F. Riesz'sproof(via the"risingsunlemma")thateverycontinuousmonotone

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There is a discussionof thatvestigialcuriosity almosteverywhere. functionis differentiable theoremforcontinuous whichis theLebesgueboundedconvergence called Osgood's theorem, theoremis polynomialapproximation functions on a closed boundedinterval.The Weierstrass brokendown intobite-sizelemmas),and so is one of Gauss's proofsof the here(intelligently theoremofalgebra.For a finalexampleI mentiona questionthatshouldbe asked fundamental continuouson muchmoreoftenthatit probablyis: is therean exampleof a seriesof functions, but for which the a closed bounded interval,that convergesabsolutelyand uniformly, M-testfails? Weierstrass Our ProblemCourses.How can we, the teachersof today,use the problemliterature? engineers, knowledgeto the technicians, assignedtaskis to pass on thetorchof mathematical of tomorrow:do probteachers,and, not least,researchmathematicians humanists, scientists, lemshelp? Yes, theydo. The major part of everymeaningfullife is the solutionof problems;a etc.,is thesolution scientists, engineers, lifeof technicians, considerablepartof theprofessional in of mathematical problems.It is the dutyof all teachers,and of teachersof mathematics to exposetheirstudentsto problemsmuchmorethanto facts.It is, perhaps,more particular, M-testthanto to strideintoa classroomand givea polishedlectureon theWeierstrass satisfying sessionthatendsin thequestion:"Is theboundednessassumpconducta fumble-and-blunder tionof thetestnecessaryforits conclusion?"I maintain,however,thatsuch a fumblesession, morevaluable. is infinitely intendedto motivatethestudentto searchfora counterexample, students (and then I have taughtcourseswhose entirecontentwas problemssolved by were in such a course students that the exposed theorems The number of presentedto theclass). half the numberthattheycould have been exposed to in a seriesof to was approximately questionlectures.In a problemcourse,however,exposuremeanstheacquiringof an intelligent ing attitudeand of some techniqueforpluggingtheleaks thatproofsare likelyto spring;in a meansnotmuchmorethanlearningthenameof a theorem, lecturecourse,exposuresometimes aboutwhether it wouldappearon the by itscomplicatedproof,and worrying beingintimidated examination. CoveringMaterial.Many teachersare concernedabout the amountof materialtheymust cover in a course.One cynic suggesteda formula:since,he said, studentson the average remember onlyabout 40% of whatyou tellthem,the thingto do is to cramintoeach course 250%of whatyou hope willstick.Glib as thatis, it probablywouldnotwork. Problemcoursesdo work.Studentswho have takenmyproblemcourseswereoftencompliwere on theiralertattitude,on their mentedby theirsubsequentteachers.The compliments searchingquestions abilityto get to the heartof the matterquickly,and on theirintelligently whatwas happeningin class.All thishappenedon morethan thatshowedthattheyunderstood one level,in calculus,in linearalgebra,in set theory,and, of course,in graduatecourseson measuretheoryand functional analysis. learn?Even if (to stay thatwe hope studentswillultimately Whymustwe covereverything with an example already mentioned)we thinkthat the WeierstrassM-test is supremely studentmustknowthatit existsand mustunderstand and thateverymathematics important, branchof analysismightbe betterfor how to applyit-even thena courseon thepertinent it.Supposethatthereare 40 suchimportant topicsthata studentmustbe exposedto in omitting a term.Does itfollowthatwe mustgive40 completelecturesand hope thattheywillall sinkin? mention(the name, the Mightit not be betterto give 20 of the topicsjust a ten-minute in whichit can be applied),and to treatthe and an indicationof one of thedirections statement, and counterexamples, other 20 in depth,by student-solved problems,student-constructed student-discovered applications?I firmlybelieve that the lattermethodteaches more and (a telling teachesbetter.Some of thematerialdoesn'tgetcoveredbuta lot of it getsdiscovered

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old pun that deservesto be kept alive), and the methodtherebyopens doors whose very ofsettledfacts.As for existencemightneverhavebeen suspectedbehinda solidlybuiltstructure in class-well, booksandjournals else was givenshortshrift M-test,or whatever theWeierstrass do exist,and studentshave been knownto read themin a pinch. ProblemSeminars.Whilea problemcoursemightbe devotedto a sharplyfocusedsubject,it the questioningattitudeand improving also mightnot-it mightjust be devotedto fostering techniqueby discussingproblemswidelyscatteredoverseveralfields.Such techniquecourses, forPh.D. candidates, sometimes called ProblemSeminars,can existat all levels(forbeginners, or foranyintermediate group). butit isjust as The bestwayto conducta problemseminaris,of course,to presentproblems, teacherto do all theaskingin a problemseminaras itis foran omniscient bad foran omniscient in a problem thatstudents recommend teacherto do all thetalkingin a lecturecourse.I strongly modifying seminarbe encouragedto discoverproblemson theirown(at firstperhapsby slightly problemsthattheyhave learnedfromothers),and thattheyshouldbe givenpublicpraise(and Justas you shouldnottellyourstudentsall theanswers,you gradecredit)forsuchdiscoveries. shouldalso notask themall thequestions.One of thehardestpartsof problemsolvingis to ask the rightquestion,and the only way to learnto do so is to practice.On the researchlevel, thesisproblemto a candidate,I am notdoingmyjob of teaching especially,ifI pose a definite him? himto do research.How willhe findhis nextproblem,whenI am no longersupervising just as thereis no easywayto Thereis no easyway to teachsomeoneto ask good questions, teachsomeoneto swimor to playthecello,butthat'sno excuseto giveup. You cannotswimfor therightkindof and reinforce someoneelse; thebestyou can do is to supervisewithsympathy helpsto makegood questionsout of fumbleby approval.You can give advicethatsometimes forrepeatedtrialand practice. bad ones,butthereis no substitute less obviousone is: specialize;a moderately is: generalize;a slightly An obvioussuggestion Anotherwellspecializationof a generalization. one is: look for a non-trivial sophisticated knownpieceofadviceis due to Polya: makeit easier.(P6lya'sdictumdeservesto be propagated overand overagain.In slightly greaterdetailit says:ifyoucannotsolvea problem,thenthereis an easierproblemthatyoucannotsolve,and yourfirst job is to findit!) The adviceI am fondest on askingthenaturalquestion of is: makeit sharp.By thatI mean: do not insistimmediately ("what is .. .?", "when is ... ?", "how much is ... ?"), but focus firston an easy (but nontrivial) yes-or-noquestion ("is it ... ?").

and I hope that as Epilogue.I do believe that problemsare the heartof mathematics, in theclassroom,in seminars, and in thebooksand articleswe write,we willemphasize teachers, and themmore and more,and thatwe will trainour studentsto be betterproblem-posers thanwe are. problem-solvers References ofElementary 1. H. Dorrie,100GreatProblems Mathematics, Dover,NewYork,1965. 2. I. M. Glazmanand Ju. I. Ljubic,Finite-Dimensional LinearAnalysis:A Systematic Presentation in ProblemForm,MIT, Cambridge, 1974. 3. D. Hilbert, Mathematical problems, Bull.Amer.Math.Soc.,8 (1902)437-479. 4. G. Klambauer, Problems in Analysis, and Propositions NewYork,1979. Dekker, 5. Mathematical Developments Arising fromHilbertProblems, AMS, Providence, 1976. 6. G. PolyaandG. Szego,Problems inAnalysis, Springer, Berlin,1972,1976. and.Theorems 7. H. Steinhaus, One HundredProblems in Elementary Mathematics, BasicBooks,NewYork,1964. DEPARTMENTOF MATHEMATICS,INDLANAUNIVERSITY, BLOOMINGTON,IN

47405.