ten misconceptions about mathematics

8/2/2019 Ten Misconceptions About Mathematics

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Michael J. Crowe

Ten Misconceptions about Mathematicsand Its History

For over two decades , one of my major interes ts has been reading,teaching, and w riting history of m athem atics. D uring those decades, I havebecome convinced that ten claims I form erly accepted concerning m ath-ematics and i ts development are both ser iously wrong and a hindranceto the historical study of mathemat ics . In analyzing these claims, I shalla t tempt to establish their in itial plausibility by show ing tha t one or moreem inent scholars have endorsed each of the m ; in fact, all seem to be heldby many persons not fully inform ed abo ut recent s tudies in history andphiloso phy of m athem atics. This paper is in one sense a case study ; it has,however , th e peculiar feature that in it I serve both as dissector and f rog.In candidly recounting my changes of view, I hope to help newcomersto history of mathematics to formulate a satisfactory historiography andto encou rage other pract i t ioners to present their ow n ref lect ions . M y a t-tempt to counter these ten claims should be prefaced by two qualifica-tions. First , in advocating their aban don m ent, I am not in most cases urg-in g their inverses; to deny that all swans are white does not imply thatone b elieves no swa ns are wh ite. Second, I realize that the evidenc e I ad-vance in opposition to these claims is scarcely adequate; m y argumentsare presented prim arily to suggest approaches that co uld be taken in m orefully formulated analyses.

1. The Methodology of Mathematics Is Deduct ionIn a widely republished 1945 essay, Carl G. Hempel stated that the

method employed in mathematics "is the method of mathematical dem-onstration, which consists in the logical deduction of the propositionto be proved from other propositions, previously established." Hempeladded the qua lif icat ion that m athema tical sys tems res t ul tim ately on ax-ioms and postulates, which cannot themselves be secured by deduct ion .1Hempel 's claim concerning the method of mathemat ics is widely shared;

260


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TE N M I S C O N C E P T IO N S A B O U T M A T H E M A T I C S 261I accepted i t as a young historian of mathematics, but was uneasy withtwo aspects of it. First, it seemed to make mathematic ians unnecessaryby implying that a machine program m ed w ith approp ria te rules of in-ference and, say, Euclid's definitions, axioms, and postulates could deduceall 465 propo sitions presented in his Elements. Second, it reduce d the roleof historians of mathematics to reconstructing the deductive chains at-tained in the development of mathematics.

I came to realize that Hempel's claim could not be correct by readinga later pub lication, also by Hem pel; in his Philosophy of Natural Science(1966), he presented an elementary proof that leads to the conclusion thatdeduction cannot be the sole method of mathematics. In part icular, hedemonstra ted tha t from even a single tru e statem ent, an infinity of othertrue statements can be validly deduced. If we take "or" in the nonexclusivesense and are given a t rue proposit ion p, Hempel asserted that we candeduce an infinity of statements o f the form "/? or q" where q is any p rop-osit ion whatsoever. Note that all these propo sitions are t rue because withthe nonexclusive meaning of "or," all proposit ions of the form "p orq" are true if p is true. As Hempel stated, this example shows that th erules of logical inference p rovide only tests of the validity of arguments ,not methods of discovery. 2 Nor, i t is important to note, do they provideguidance as to whether the deduced proposit ions are in any way signifi-cant . Thus we see that an enti ty, be i t man or machine, possessing thedeductive rules of infe rence and a set of axiom s from w hich to start , couldgenerate an infinite num be r of t rue conclusions, non e of wh ich would besignificant. W e wo uld not call such resul ts mathem atics . C onsequ ent ly,mathematics as we know it cannot arise solely from deduct ive m ethods .A m achine given Eu clid's d efinitions, axiom s, and postulates m ight deducethousands of valid propositions without deriving any Euclidean theorem s.Moreover , Hempel ' s analysis show s that even if definitions, axioms, andpostulates could be produced deductively, st i l l mathematics cannot relysolely on deduction. Furthermore, we see from this that historians ofmathemat ics must not confine their efforts to reconstruct ing deduct ivechains from the past of mathematics. This is not to deny that deduct ionplays a ma jor ro le in m athem atical metho dolog y; al l I have attem pted toshow is that i t cannot be the sole method of mathematics.

2. Mathematics Provides Certain KnowledgeIn the same 1945 essay cited previously, H em pel stated: "The m ost


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262 Michael J. Crowedistinctive characteristic wh ich differentiates m athematics from the variousbranches of empirical s c i e n c e . . . is no doubt th e peculiar certainty andnecessity of its results." And, he added: "a mathematical theorem, onceproved, is established once and for all. . . ."3 In noting the certainty ofm athema tics, Hem pel was merely reasserting a view proclaimed for cen-turies by dozens of au thors wh o freq uen tly cited E uclid 's Elements as theprim e exemp lification of that certainty. W riting in 1843, Philip Kellandremarked: "It is certain that from its completeness, uniformity and fault-l e s s n es s , . . . and from the universal adoption of the completest and bestl ine of argument, Euclid 's 'Elements ' stand preeminently at the head ofall h u m a n product ions ."4 A careful reading of Hempel's essay reveals astriking feature; immediately after noting the certainty of mathematics,he devoted a section to "The Inadequacy of Euclid's Postulates." Herein Hilbertian fashion, Hempel showed that Euclid's geometry is marredby the fact that it does not contain a number of postulates necessary forproving m any of i ts propositions. Hem pel was of course correct; as earlyas 1892, C. S. Peirce had dramatically summarized a conclusion reachedby most late-nineteenth-century mathematicians: "The truth is , that ele-m entary [Euclidean] geom etry, instead of being the perfection of h um anreasoning, is riddled with fa l l ac ies . . . ."5W hat is striking in H em pel's essay is that he seems not to hav e realizedthe tension between his claim for the certainty of mathematics and hisdemonstration that perhaps the most famous exemplar of that certaintycontains numerous faulty arguments. Hempel 's claim may be construedas containing the implicit assertion that a m athematical system emb odiescertainty only after all defects have been removed from it . What is prob-lematic is whether we can ever be certain that this has been done. Surelyth e fact that th e inadequacy of some of Eu clid's argum ents escaped detec-tion for over two millennia suggests that certainty is more elusive thanusually assumed. Moreover, in opposition to the belief that certainty canbe secured for formalized mathematical systems, Re uben Hersh has stated:"It is jus t not the case that a doubtful proof would become certain bybeing formalized. On the contrary, the doubtfulness of the proof wouldthen be replaced b y the d ou btfuln ess of the coding and programming ." 6Morris Kline has recently presented a powerful demonstration that thecertainty purpo rtedly present thro ugh ou t the developm ent of m athem aticsis an illusion; I refer to his Mathematics: The Loss of Certainty, in whichhe states: "The hope of f inding objective, infallible laws and standards


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T E N M ISC O NC EPT IO NS A BO U T M A T HEM A T IC S 263has faded. The Age of Reason is gone."7 Much in wha t follows shedsfurther l ight on the purported certainty of mathematics, but let us nowproceed to two related claims.3 . Mathematics Is Cumula t ive

A n elegant formulation of the claim for the cum ulative character ofmathematics is due to Hermann Hankel , w ho wrote : "In most sciencesone generation tears down what another has built and what one hasestablished another undo es. In Mathematics alone each generation buildsa new story to the old structure."8 Pierre Duhem made a similar claim:"Physics does not progress as does geometry, w hich adds new final andindisputable propositions to the final and indisputable propositions italready p o s s e s s e d . . . ."9 T he most frequently cited illustration of thecum ulative character of mathematics is non-Euclidean geometry. ConsiderW illiam K ingdon Clifford's statement: "What Vesalius was to Galen, whatCopernicus w as to Ptolemy, that w as Lobatchewsky to Euclid."10 Clif-ford ' s claim cannot, however, be quite correct; whereas acceptance ofVesalius entailed rejection of Galen, whereas adoption of C opernicus ledto abandonment of Ptolemy, Lobachevsky did not refute Euclid; ratherhe revealed tha t another geometry is possible. Although this instance il-lustrates the rem ark able degree to w hich m athem atics is cum ulative, othercases exhibit opposing patterns of development. As I wrote my Historyof Vector Analysis,11 I realized tha t I was also, in effect, writing TheDecline of the Quaternion System. Massive areas of mathematics have,for all practical purposes, been abandoned . The nineteenth-centurymathematic ians w ho extended tw o millennia of research on conic sectiontheory h ave now been forg otten; invariant theory, so pop ular in the nine-teenth century, fell from favor.1 2 Of the hundreds of proofs of the Py-thagorean theorem, nearly all are now nothing more than curiosities.1 3In short , al though many previous areas, proofs, and concepts in mathe-matics have persisted, others are now abandoned. Scattered over th e land-scape of the past of mathematics are numerous ci tadels, once proudlyerected, but w hich, a lthough never a ttacked, are now left unoccupied byactive mathematic ians.

4. Mathematical Statements A re Invariably CorrectThe m ost challenging aspect of the question of the cum ulative character

of mathematics concerns whether mathematical assertions are ever refuted.


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264 Michael J. CroweT he previously cited quotations from H ankel and Duhem typify thewidespread belief that Joseph Fourier expressed in 1822 by stat ing thatm athem atics "is f orm ed slo w ly, but it preserves every principle it has o ncea c q u i r e d . . . . "uAlthough mathematicians m ay lose interest in a particularprinciple, proof , or problem solution, al though m ore elegant ways of for-mulating them may be found, nonetheless they purportedly remain. In-fluenced by this belief, I stated in a 1975 paper that "Revolutions neveroccur in mathemat ics ."1 5 In making this claim, I added tw o importantqualifications: the first of these was the "minimal st ipulation that anecessary characteristic of a revolution is that some previously existingentity (be it king, const i tut ion, or theory) must be ove r th rown and ir-revocably discarded"; second, I stressed th e significance of the phrase "inm athem atics," urging that al though "revolutions m ay occur in mathe-matical nomenclature, symbolism, metamathematics, [and] methodolo-gy. . . , " they do not occur within mathematics itself.1 6 In making thatclaim concerning revolutions, I was influenced by the widespread beliefthat mathematical statements and proofs have invariably been correct.I w as first led to question this belief by reading Imre Lakatos 's bri l l iantProofs and Refutations, which contains a history of Euler's claim thatfor polyhedra V-E + F = 2, where K is the nu m ber of vert ices, E the num -ber of edges, and Fthe num be r of faces.1 7 Lakatos showed not only thatEuler's claim w as repeatedly falsified, but also that publ ished proofs forit were on many occasions found to be flawed. Lakatos 's history alsodisplayed the rich repertoire of techniques mathematicians possess forrescuing theorems from refutat ions.Whereas Lakatos had focused on a single area, Philip J. Davis tooka broader view when in 1972 he listed an array of errors in mathematicsthat he had encountered.1 8 Philip Kitcher, in his recent Nature of Math-ematical Knowledge, has also discussed this issue, noting numerous er-rors, especially from the history of analysis.1 9 Morris Kline called atten-tion to many faulty mathematical claims and proofs in his Mathematics:The Loss of Certainty. For exam ple, he noted that A m pere in 1806 provedthat every function is different iable at every point w here it is cont inuous ,and that Lacroix, Bertrand, and others also prov ided pro ofs u nti l W eier-strass dramatically demonstrated th e existence of functions that are every-where cont inuous but nowhere differentiable. 2 0 In s tudying the historyof complex numbers, Ernest Nagel foun d that such mathematicians as Car-dan, Simson, Playfair, and Frend denied their existence.21 Moreover,


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TEN M I S C O N C E P T I O N S A B O U T M A T H E M A T I C S 255M aurice Lecat in a 1935 bo ok l isted nearly 50 0 errors pu blished by over300 mathematicians. 22 On the other hand, Rene Tho rn has asserted: "Thereis no case in the history of mathematics where th e mistake of one manhas thrown the ent i re field on the wrong t r a c k . . . . Never has a signifi-cant error slipped into a conclusion without almost immediately beingdiscovered."23 Even if Thorn's claim is correct, the quotations from Duhemand Fourier seem difficult to reconcile with th e inform at ion cited aboveconcern ing cases in w hich concepts an d conjectures, principles and p roo fswithin mathematics have been rejected.5. The Structure of Mathematics Accurately Reflects Its History

In recent years, I have been teaching a course for hum anit ies studentsthat begins with a careful reading of Book I of Euclid's Elements. Thatexperience has convinced m e that the most crucial misconception thatstudents have about mathematics is that i ts structure accurately reflectsits history. Almost invariably, th e students read this text in light of theassumption that the deductive progression from i ts opening definit ions,postulates, and comm on not ions through its forty-eight proposit ions ac-curately reflects th e development of Euclid's thought. Their conviction inthis regard is reinforced by the fact that most of them have earlier read Aris-totle's Posterior Analytics, in which that great philosopher specified thatfor a valid dem on stration "the p r e m i s e s . . . mus t be . . . bet ter kno w n thanand prior to the conclusion.... "24My ownconception isthat the develop-ment of Euclid's thought was drastically different . Isn't it plausible thatin composing Book I of the Elements, Euclid began not with his defini-tions, postulates, and common notions but rather either with his extremelypowerfu l 45th proposi tion, w hich shows how to reduce areas bound ed b ystraight lines to a cluster of m easurab le triangular areas, o r w ith his magnifi-cent 47th proposition, th e Pythagorean theorem, fo r which he forged a proofthat h as been adm ired for cen turies. W ere not these two propo sitions theones he knew best and of which he was most deeply convinced? Isn't itreasonable to assume that i t was only after Euclid had decided on thesepropositions as the culmination for his first book that he set out to con-struct the deductive chains that support them? Is i t probable that Euclidbegan his efforts with his sometim es abstruse and arbi t rary definit ions"a point is that which has no parts"and somehow arrived forty-sevenproposit ions later at a result known to the Babylonians fifteen centuriesearlier? An ex aminat ion of Eucl id 's 4 5th and 47 th proposi t ions shows that


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266 Michael J. Crowethey depend u po n the proposit ion that if two coplanar straight lines meetat a point and make an angle with each other equal to two right angles,then those lines are collinear. S hould it be seen as a rem arkab le coincidencethat thirty-o ne propositions earlier Euclid had p roved precisely this result,but had not used it a single time in the intervening propositions? It seemsto me that accepting the claim that the history and deduct ive s t ructuresof mathematical systems are identical is comparable to believing that Sac-cheri w as surprised when after proving dozens of propositions, he finallyconcluded that he had established the parallel postulate.

Is not the axiomatization of a field frequently one of the last stages,rather than the first, in its developm ent? Recall that it took W hitehead andRussell 362 pages of their Principia Mathematica to prove that 1 + 1 = 2 .Calculus texts open with a formulation of the limit concept, which tooktw o centuries to develop. G eom etry book s begin with prim ary notions anddefinit ions with which Hilbert cl imaxed tw o millennia of searching.Second-grade students encounter sets as well as the associative and com-mutat ive lawsall hard-won at ta inments of the nineteenth century. Ifthese students are gifted and diligent, they m ay years later be able tocomprehend some of the esoteric theorem s advanced by Archimedes orApollonius. When Cauchy established the fundamental theorem of thecalculus, tha t subject w as nearly tw o centuries old; when Gauss provedthe fundamental theorem of algebra, he cl imaxed more than two millen-nia of advancement in that area.2 5 In teaching complex numbers, we firstjust i fy them in terms of ordered couples of real numbers, a creation ofth e 1830s. After they have magically appeared from this process, w edevelop them to the point of attaining, say, Demoive's theorem, whichcame a century before the Hamilton-Bolyai ordered-couple just ificationof t hem. In presenting a theorem, first w e name it and state it preciselyso as to exclude th e exceptions it has encountered in the years since itsfirst formulat ion; then we prove i t ; and, finally, we employ i t to proveresults that were probably kn ow n long befo re i ts discovery. In sho rt , wereverse history. Hamilton created quaternions in 1843 and simultaneous-ly supplied a form al justification for them, this being th e first case in whicha number system w as discovered and justified at the same time; half acentury later G ibbs and Heaviside, viewing the quaternion method of spaceanalysis as un satisfac tory , proposed a simp ler system derived from quater-nions by a process now largely forgotten.

Do not misunders tand: I am not claiming that the structure of math-


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TEN MISCONCEPTIONS ABOUT MATH EMATICS 267ematics, as a whole or in its parts, is in every case the opposite of its history.Rather I am suggesting that the view that students frequen t ly have, im-plicitly or explicitly, that th e st ructure in which they encounter areas ofmathematics is an adequate approximation of its history, is seriously defec-tive. Mathem atics is ofte n com pared to a tree, ever attaining new heights.The lat ter feature is certainly present, but m athem atics also grow s in rootand t r u n k ; it develops as a whole. T o take another metaphor, the math-ematical research frontier is frequently found to l ie not at some remoteand u nexplored region, but in the very m idst of the m athematical dom ain.M athem atics is often com pared to art ; yet reflect for a m om ent . H om er 'sOdyssey, Da Vinc i ' s Mona Lisa, and Beethoven's Fifth Symphony arecompleted w ork s, wh ich no later artist dare alter. Nonetheless, the latestexpert on analysis works alongside Leibniz and Newton in ordering thearea they created; a new Ph.D. in number theory joins Eucl id, Fermat ,and Gau ss in perfecting know ledge of the prim es. Kelvin called Fo urier 'sTheorie analytique de la chaleur a "mathematical poem,"26 but manyauth ors shaped its verses. W hy did some m athematicians oppose introduc -tion of com plex or transfinite nu m bers, charging tha t they co nflicted w iththe foundat ions of mathematics? Part of the reason is that , lacking ahistorical sense, they failed to see that foun dat ions are themselves opento a l teration, that not only premises but results dictate w hat is desirablein mathematics.

6. Mathematical Proof in Unproblemat icPierre Duhem in his Aim and Structure of Physical Theory reiteratedth e widely held view that there is nothing problematic in mathematical

proof by stat ing that geometry "grows by the cont inual con tr ibut ion ofa new theorem demonstrated once and for all and added to theoremsalready d e m o n s t r a t e d . . . . "27 In short , Duhem w as claim ing that once aproposition has been demonstrated, it remains true for all time. Variousauthors , both before and after D uh em , have taken a less absolutist viewof the nature and conclusiveness of proof. In 1739, David Hu m e observed:

There is no.. . Mathematician so e x p e r t . . . as to place en tire confidencein any truth immediately upon his discovery of it, or regard it as anyth ing , but a m ere prob abi l ity. Every t im e he runs over his pro ofs , hisconfidence encreases; but still m ore by the approbation of his friends;and is rais'd to its utmost perfect ion by the universal assent and ap-plauses of the learned world."28


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268 Michael J. CroweG. H . Hardy concluded in a 1929 paper entitled "M athematical Proof"that "If we were to push it to its extreme, w e should be led to rather aparadoxical conclusion: that there is , strictly, no such thing as mathemati-cal pro of; th at w e can, in the last analysis, do nothin g but point; that proofsare what Litt lewood and I call gas, rhetorical flourishes designed to af-fect p s y c h o l o g y . . . . "29 E. T. Bell in a number of his writings developedthe point that standards of proof have changed dramatically throughouthistory. For example, in his Development of Mathematics (1940), hechallenged th e assertion of an unnamed "eminent scholar of Greek math-ematics" that the Greeks, by their " 'uner r ing logic, ' had attained suchperfect m athematical results that ' there has been no need to reconstruct ,still less to reject as unsound, any essential part of their doctrine....'"Bell responded that among, for example, Euclid 's proofs, "many havebeen demolished in detail, and it would be easy to destroy m ore were itworth th e trouble."30 Raymond Wilder, w ho also discussed th e processof proof in variou s writings , asserted in 1944 tha t "w e don ' t possess, andprobably will never possess, any standard of proof that is independentof th e t ime, th e thing to be proved, or the person or school of thoughtusing it." Over thre e decades later, he put this point most succinctly:"'proof in mathematics is a culturally determined, relative matter."31That research in history and philosophy of m athem atics has con tributedfar mo re tow ard u nderstanding the nature of proof than simply showingthat standards of proof have repeatedly changed can be illustrated by brief-ly examining the relevant writ ings of Imre Lakatos. In his Proofs andRefutations (1963-64), Lakatos, proceeding from his conviction (derivedfrom Karl Popper) that conjectures play a vital role in the developmentof mathematics and his hope (derived from George Polya) that heuristicmethods for mathematics can be formulated, reconstructed the historyof Euler 's conjecture concerning polyhedra so as to show that its historyil l accords with the traditional accumulationist historiography of math-ematics. Whereas some had seen its history as encompassing little morethan Euler's formulation of the conjecture and Poincare's later proof forit, Lakatos showed that numerous "proofs" had been advanced in theinterim, each being falsified by counterexamples. Fundamental to this es-say is Lakatos's definition of proof as "a thought-experiment or 'quasi-experiment' which suggests a decomposition of the original conjectureinto subconjectures or lemmas, t hus embedding it in a po ssibly qu ite dis-tant body of knowledge."32 O n this basis, Lakatos, in opposition to the


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TE N MI SCONCE PTI ONS AB OUT MATHE MATI CS 269belief that th e proof or refutation of a mathematical claim is final , arguedforceful ly that on the one hand mathematicians should seek counterex-amples to proved theorems (pp. 50ff . ) and on the other hand be cautiousin abandoning refuted theorems (pp. 13ff .) . Moreover, he warned of thedangers involved in recourse, if counterexamples are found, to the tech-niques he called "monster-barring," "monster-adjustment," and "excep-tion-barring" (pp. 14-33). Lakatos also wrote other papers relevant to thenature of mathematical proof; fo r example, in his "Infinite Regress andthe Foundations of Mathematics" (1962), he provided insightful critiquesof the "Euclidean programme" as well as of the formalist conception ofmathematical method. In his "A Renaissance of Empiricism in RecentPhilosophy of Mathematics" (1967), he stressed the importance of em-pirical considerations in mathematical proof, while in his "Cauchy andthe Continuum...," he urged that Abraham Robinson's methods of non-standard analysis could be used to provide a radically new interpretationof the role of infinitesimals in the creation of the calculus.33 T he some-t imes enigmatic character of Lakatos's writings and the fact that his in-terests shifted in the late 1960s toward the history and philosophy ofsc ienceto which he contributed a "methodology of scientific researchp r o g r a m m e s " l e f t , after his death in 1974, many unanswered questionsabout h is views on mathematics. Various authors have attempted to sys-tematize his thought in this regard,34 and Michael Hallett has advancedand historically illustrated the thesis that "mathematical theories can beappraised by criteria like those of [Lakatos's] methodology of scientificresearch programmes...."35

7. Standards of Rigor A re UnchangingWri t ing in 1873, the Oxford mathematician H. J. S. Smith repeated

a conclusion often voiced in earlier centuries; Smith stated: "The methodsof Euclid are, by almost universal consent, unexceptionable in point ofrigour."36 By the beginning of the present century, Smith's claim concern-ing Euclid's "perfect rigorousness" could no longer be sustained. In hisValue of Science (1905), Henri Poincare asked: "Have w e finally attainedabsolute rigor? A t each stage of the evolution our f a t h e r s . . . thought theyhad reached it. If they deceived themselves, do we not likewise cheatourselves?" Surprisingly, Poincare went on to assert that "in the analysisof today, when one cares to take the trouble to be rigorous, there canb e nothing bu t syllogisms o r appeals to this intuition o f pure number, th


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270 Michael J. Croweonly intuition which can not deceive us. It m ay be said that tod ay abso luterigor is attained."37 More recently, Morris Kline remarked: "N o proofis final. New counterexamples underm ine old proofs. The p roofs are thenrevised and mistakenly considered proven for all t ime. But history tellsus that this merely means that the t ime has not yet come for a crit icalexamination of the proof."38

Not only do standards of rigor intensify, they also change in nature;w hereas in 1700 geometry w as viewed as providing th e paradigm for suchstandards, by the late nineteenth cen tury arithm etic-algebraic considera-tions had assumed primacy, w ith these eve ntually giving way to standardsformulated in term s of set theo ry. Both these poin ts, as well as a num berof others relating to rigor, have been discussed with unusual sensitivityby Philip K itcher. For example, in opposition to the traditional view thatrigor should always be given primacy, K itcher has suggested in his essay"Mathematical Ri gor Who Needs It?" the following answer: "Somem athem aticians at some tim es, but by no m eans all m athem aticians at alltimes."39 W hat has st ruck m e most forcefully about the position Kitcherdeveloped concerning rigor in that paper and in his Nature of MathematicalKnowledge are its implications for the historiography of mathematics. Irecall be ing puzzled some years ago wh ile studying the history of com-plex numbers by the terms that practitioners of that most rational dis-cipline, mathematics, used for these numbers. Whereas their inventorCardan called them "sophistic," Napier, Girard, Descartes, Huygens, andEuler respectively branded them "nonsense," "inexplicable," "imagi-nary," "incomprehensible," and "impossible." Even m ore m ysteriously,it seemed, m ost of these m athem aticians, despite the invective impliedin thus naming these numbers, did not hesitate to use them. A s ErnestNagel observed, "for a long time no one could defend the ' imaginarynumbers ' with any plausibility, except on the logically inadeq uate grou ndof their mathematical usefulness." H e added: "Nonetheless, mathema-ticians who refused to banish t h e m . . . were no t f o o l s . . . a s subsequen tevents showed."40What I understand Kitcher to be suggesting is tha t the apparent irra-tionality of the disregard for rigor found in the pre-1830 history of bothcomplex numbers and the calculus is largely a product of unhistorical,present-centered conceptions of mathematics. In particular, if onerecognizes that need for rigor is a relative value that may be and has attim es been rationally set aside in favor of such other values as usefulness,


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T E N M I S C O N C E P T I O N S A B O U T M A T H E M A T IC S 277then one will be less ready to describe variou s p eriods of mathematics asages of unreason and m ore prone to u nd ertake the properly historical taskof und erstanding w hy mathematicians adopted such entities as "impossi-ble numbers" or infinitesimals. As Kitcher suggests, i t may be wise forhistorians of mathematics to follow the lead of historians of science w holong ago became suspicious of philosophic and historiographic systemsthat entail the reconstruction of scientific controversies in t e rms of suchcategories as irrationality, illogicality, and s tubbornness .4 18. The Methodology of Mathematics Is Radically Different from

th e Methodology of ScienceThe quotations previously ci ted from Duhem's Aim and Structure ofPhysical Theory illustrate a subtheme running through tha t book: tha t

the methodology of mathematics differs greatly from that of physics. Inother passages, D uhem lam ented that physics had not achieved "a growthas calm and as regular as that of mathematics" (p. 10 ) and that physics ,unlike mathematics, possesses few ideas that appear "clear, pure , andsimple" (p. 266). M oreo ver, largely because Du hem believed that "thesetw o m ethods reveal them selves to b e pro fou nd ly differen t" (p. 265), heconcluded that, w hereas history of p hysics contributes im po rtantly to un-derstanding physics, "The history of mathematics is, [although] alegitimate object of curiosity, not essential to the und erstan ding of m ath-ematics" (p . 269). The position developed in this and the next section isthat important parallels exist between the methods employed inmathematics and in physics .T he first author w ho explicitly described th e method that, accordingto most contemporary phi losophers of science, chara cterizes phy sics w asCh rist iaan H uyg ens. He prefaced his Treatise on Light by stat ing that inpresenting his theory of light he had relied upon "demonstrations of thosekinds which do not produce as great a cert i tude as those of Geometry,and which even differ much therefrom, since whereas th e Geom eters pro vetheir Propositions by fixed and incontestable Principles, here th e Prin-ciples are verified by the conclusions to be drawn from t h e m . . . . "42 Wha tI wish to suggest is that , to a far greater extent than is commonly real-ized, m athematicians have employed precisely th e same methodthe so-called hypo thetico-deduct ive m ethod . W hereas the p retense is that m ath-ematical axioms jus t i fy the conclusions drawn from t hem, the reality isthat to a large extent mathematicians have accepted axiom systems on the


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272 Michael J. Crowebasis of the ability of those axioms to bring order and intelligibility toa field and/or to generate interesting and fruitful conclusions. In an im -po rtant sense, w hat legit im ized the calculus in the eyes of its creators w asthat by means of its m etho ds they attained conclusions that w ere recognizedas correct and m eaningfu l . Al though H am ilton, Grassm ann, and Can-tor, to name but a few, presented the new systems for which they arenow famous in the context of particular ph ilosophies of mathematics (nowlargely discarded), what above all just ified their new creations, both intheir own eyes and among their contemporaries, were the conclusionsdrawn from them. This should not be misunderstood; I am not urgingthat only uti l i tarian cri teria have determined the acceptabil i ty of math-matical systems, al though usefulness has undoubtedly been impor tant .R ather I am claiming that characteristics of the results at ta inedfor ex-ample, their intelligibilityhave played a major role in determining theacceptability of the source from which th e results w ere deduced. To putit differently, calculus, complex numbers, non-Euclidean geometries, etc.,were in a sense hypo theses that m athematicians sub jected to test in wayscomparable in logical form to those used by physicists.My claim that mathematicians have repeatedly employed th e hypo-thetico-deductive method is not original; a num be r of recent authorshave m ade essentially th e same suggestion. Hilary Putnam began a 1975paper b y asking how we would react to find ing that M artian m athem ati-cians employ a methodology that , although using full-blown proofs whenpossible, also relies upon quasi-empirical tests; for example, his Mart iansaccept the four-color conjecture because much empirical evidence sup-ports and none con tradicts it. Putnam proceeded to claim th at w e shouldnot see this as resulting fro m some bizarre m isunderstanding of the natureof mathematics; in fact, he asserted that "w e have been using quasi-empirical and even empirical methods in mathematics all a l o n g . . . ,"43The first example he used to illustrate this claim is Descartes 's creationof analytical geometry, which depends upon th e possibility of a one-to-one correspondence between the real numbers and the points on a line.The fact that no just i f icat ion for this correspondence, let alone for thereal numbers, was available in Descartes 's day did not deter him or hiscontem poraries; th ey proceeded con fidently ahead. A s Pu tnam com m entedon his Descartes illustration: "This is as much an example of the use ofhypothetico-deductive methods as anything in physics is" (p. 65). PhilipKitcher , who has stressed the parallels between the evolut ion of ma th-


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T E N M I S C O N C E P T I O N S A B O U T M A T H E M A T IC S 27 3em atics and of science, has adv ocated a similar view . In 1981, he stated

Although we can sometimes present parts of mathematics in axiomaticf o r m , . . . th e statements taken as axioms usually lack th e epistemologicalfeatures w hich [deductivists] attribute to first principles. Our knowledgeof th e axioms is frequently less certain than our knowledge of thestatements w e derive from t h e m . . . . In fact, our knowledge of the ax-ioms is sometimes obtained by nondeductive inference from knowledgeof the theorems they are used to systematize.44

Finally, statem ents urging that mathem atical system s are, like scien tificsystems, tested by their results occur in the writ ings of Haskell Cu rry ,Willard V an Orman Quine, and Kurt Godel.45

9. Mathematical Claims Admit of Decisive FalsificationIn the most widely acclaimed section of his Aim and Structure of

Physical Theory, Duhem attacked the view that crucial experiments arepossible in physics. H e stated: "Unlike the reduction to absurdity [method]employed by geometers, experimental contradiction does not have thepower to t rans fo rm a physical hypothesis into an indisputable truth" (p .190). The chief reason he cited fo r this inability is that a supposed crucialexperiment can at m ost decide "between two sets of theories each of w hichhas to b e taken as a whole, i.e., between tw o entire s y s t e m s . . . " (p. 189).Because physical theories can be tested only in clusters, the physicist, whe nfaced with a contradiction, can, according to Duhem, save a particulartheory by modifying one or more elements in the cluster, leaving the par-ticular theory of m ost concern (for exam ple, the w ave or particle theory)intact . In effect, Du hem was stating that individual physical theories canalways be rescued from apparent refutations. Having criticized a numberof Duhemian claims, I wish now to pay t r ibu te to him by urging that acomparable analysis be applied to mathematics. In particular, I suggestthat in history of mathematics one frequently encou nters cases in whicha mathematical claim, faced with an apparent logical falsification, hasbeen rescued by m odifying some other aspect of the system. In other words,m athem atical assertions are usually not tested in isolation b ut in con junc -tion with other elements in the system.

Let us consider som e examples. Euclid brough t his Elements to acon-clusion with his celebrated theorem that "no other figure, besides [thefive regular solids], can be constructed which is contained by equilateraland equiangular figures equal to one another." How would Eucl id re-


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274 Michael J. Crowespond if presented with a contradiction to this t h eo remfo r example, witha hexahedron formed by placing two regular tetrahedra face to face? Itseems indisputable that rather than rejecting his theorem, he would rescueit by revising his definition of regular solid so as to exclude polyhedrapossessing noncongruent vertices. F or centuries, complex numbers werebeset with contradictions; some charged that they were contradicted bythe rules that every number must be less than, greater than, or equal tozero and that th e square of any number be positive. Moreover, others urgedthat no geometric interpretation of them is possible.46 Complex numberssurvived such attacks, whereas the cited rules and the traditional defini-tion of number did not. Many additional cases can be found; in fact,Lakatos's Proofs and Refutations is rich in examples of refutations thatwere themselves rejected. Of course, mathematicians do at times chooseto declare apparent logical contradictions to be actual refutations;nonetheless , an element of choice seems present in many such cases.10. In Specifying the Methodology Used in Mathematics, the Choices

Are Empiricism, Formalism, Intuitionism, and PlatonismFor decades, mathematicians, philosophers, and historians have de-scribed the alternative positions concerning the methodology of math-ematics as empiricism, formalism, intuitionism, and Platonism. Thisdelineation of the options seems ill-conceived in at least tw o ways. First,it tends to blur the distinction between the epistemology and methodologyof mathematics. Although related in a number of complex ways, the tw oareas can and should be dist ingu ished. Epistemology of mathematics dealswith how mathematical knowledge is possible, whereas methodology ofmathematics focuses on what methods are used in mathematics. The ap-propriateness of distinguishing between these two areas is supported bythe fact that the history of the philosophy of mathematics reveals thatdifferent epistemological positions have frequently incorporated many ofthe same methodological claims. Second, this fourfold characterizationtends to blur the distinction between normative and descriptive claims.To ask what methods mathematicians should use is certainly different fromasking what methods they have in fact used. Failure to recognize thisdistinction leads not only to the so-called naturalistic fa l lacythe prac-tice of inferring from "is" to "ought"but also to the unnamed oppositefallacy of inferring from "ought" to "is" (or to "was").

It has been my experience that both mathematicians and historians of


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TE N M I S C O N C E P T I O N S ABOUT MATHEMATICS 275mathematics are primarily interested in issues of the methodology ratherthan of the epis tem ology of mathematics . W hen they turn to w ri t ings inthe philosophy o f m athematics , they usually find these composed largelyin terms of one or more of the four prim arily epistemological positionsmentioned previously. Unfortunately, these categories seem relativelyunil luminating in exploring methodological issues. Reuben Hersh veryeffectively discussed this problem in a 1979 paper in which he asked:"Do we really h ave to choose between a formalism tha t is falsified by oureveryday experience, and a Platonism that postulates a m ythical fairylandwhere the uncountable and the inaccessible lie waiting to be o b s e r v e d . . . ?"Hersh proposed a different and more modest program for those of us in-terested in investigating the na ture o f m athem atics; he suggested that w eattempt "to give an account of mathematical knowledge as it really isfallible, corrigible, tentative and e v o l v i n g . . . . That is , reflect honestlyon what we do when we use, teach, invent or discover mathemat i c sbystudying history, by introspection, and by observing ourselves and eacho t h e r . . . . "47 If Hersh 's proposal is taken seriously, new categories willprobably emerge in the philosophy and his tor iography of mathematics ,and these categories should prov e mo re interesting and i l lum inating tha nthe traditional ones. Moreover, increased study of the descriptive meth-odology of mathematics should itself shed light on epistemological issues.This paper not only concludes with an endorsement of Hersh 's proposedprogram of research, but has been designed to serve as an exemplifica-tion of it.

Notes1. Carl G . Hempel , "Geometry and Empir ica l Science," in Readings in the Philosophyof Science, ed. Philip P. Wiener (New York: Appleton-Century-Crofts, 1953), p. 41; reprintedfrom American Mathematical Monthly 52 (1945): 7-17.2. Carl G . Hempel , Philosophy of Natural Science (Englewood Cliffs, N . J . : Prentice-Hall, 1966) pp. 16-17.3 . Hempel, "Geometry," pp. 40-41.4. As quoted in On Mathematics and Mathematicians, ed. Rob ert Edouard M oritz (NewYork: Dover, 1942), p. 295.5. Charles S. Peirce, "The Non-Euclidean Geometry," in Collected Papers of CharlesSandersPeirce, vol . 8, ed . A r t h u r W . Burk s (Cambr idge , Mass . : H a r va r d Univers i ty Press ,1966), p. 72.6. Reuben Hersh , "Some Proposals fo r Reviving the Philosophy of Mathematics," Ad-vances in Mathematics 31 (1979): 43 .7. Morris Kline, Mathematics: The Loss of Certainty (New York: Oxford University Press,1980), p. 7.8 . A s quoted in Mor i t z , Mathematics, p. 14.9 . Pierre D u h e m , The Aim and Structure of Physical Theory, t rans. Phil ip P. Wie ne r


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276 Michael J . Crowe(Princeton, N .J.: Princeton Unive rsity Press, 1954), p.177.1 0. As quoted in Moritz , Mathematics, p. 354.11. Michael J . Crowe, A History of Vector Analysis: The Evolution of the Idea of aVectorial System (Notre D ame, Ind . : Un iversity of No tre Dame P ress, 1967); reprinted NewYork: Dover Publications, 1985.12. Charles S. Fisher, "The Death of a Mathem atical Theory: A Study in the Sociologyof Knowledge," Archive for the History of the Exact Sciences 3 (1966-67): 137-59.13. For a com pilation of 362 pro ofs, see Elisha Scott Loom is, The Pythagorean Prop-osition (Ann Arbor, Mich.: National Council of Teachers of Mathematics, 1940).14. Joseph F ourier, Analytical Theory of Heat, t rans. Alexander Freeman (N ew Y ork:Dover, 1955), p. 7.15. Michael J. Crowe, "Ten 'Laws ' Concerning Patterns of Change in the History ofMathematics," Historia Mathematica 2 (1975): 161-66. For discussions of my claim concern-ing revolut ions, see I. Bernard C ohen, Rev olution in Science (Cambridge, Mass.: HarvardUniversity Press, 1985) pp. 489-91, 505-7; Joseph Dauben, "Conceptual Revolutions andthe History of Mathemat ics ," in Transformation and Tradition in the Sciences, ed. EverettMendelsohn (Cambridge: Cambridge University Press, 1984), pp. 81-103; Herbert Mehrtens,"T. S. K uh n's Theories and Mathematics: A Discussion Paper on the 'New Historiography'of Mathematics," Historia Mathematica 3 (1976): 297-320; and Raymond L. Wilder,Mathematics as a Cultural System (Oxford: Pergamon, 1981), pp. 142-44.16. Crowe, "Patterns," pp. 165-66.17. Imre Lakatos, Proofs and Refutations: The Logic of Mathematical Discovery, ed.John Worrall and Elie Zahar (Cambridge: Cambridge University Press, 1976). This studyfirst appeared in 1963-64 in British Journal for the Philosophy of Science.

18. Philip J. Davis, "Fidelity in Mathem atical Discourse: Is One and One Really Two?"in American Mathematical Monthly 79 (1972): 252-63; see esp. pp. 260-62.19. Philip Kitcher, The Nature of Mathematical Knowledge (New York: Oxford Univer-sity Press, 1983), pp.155-61, 178-85, 236-6820. Kline, Mathematics, pp. 161, 177.21 . Ernest Nagel, " Impossible Nu mb er s ' : A Chapter in the History of Modern Logic,"Studies in the History of Ideas 3 (1935): 427-74; reprinted in Nagel's Teleology Revisited(New Y ork : Colum bia Un iversity Press, 1979), pp. 166-94.22. Maurice Lecat, Erreurs de mathematiciens des origines a nos jours (Brussels: Castaigne,1935), p. viii.23. Rene" Thorn, "'Modern Mathematics ' : An Educational and Philosophic Error?"American Scientist 59 (1971): 695-99.24. Aristotle, Posterior Analytics, in The Basic W orks of Aristotle, ed. Richard McKeon(New Yo rk: Random House, 1941), book 1, chap. 2, lines 20-22. In the same chapter, Aristotleadmitted that "prior" can be understood in two senses, but he apparently saw no significancein this fact.25. Morris Kline, Mathematical Thought from Ancient to Modern Times (New York:Oxfo rd Un iversity Press, 1972), pp. 958, 595.26. As quoted in Silvanus P. Thompson , The Life of William Thomson, Baron KelvinofLargs, vol. 2 (London: Macmillan, 1910), p. 1139.27. D u h e m , Aim, p. 204.28. David Hume, Treatise of Human Nature, ed. Ernest C. M ossner (Baltimore: Penguin,1969), p. 231.29. G. H. Hardy, "Mathematical Proof," Mind 38 (1929): 1 8.30 . Eric Temple Bell, The Development of Mathematics, 2d ed. (New Yo rk: McGraw-Hill, 1945), p. 10. See also Bell's "The Place of Rigor in Mathematics," AmericanMathematical Monthly 41 (1934): 599-607.31. Raymond L. Wilder, "The Nature of Mathematical Proof," American Ma thematicalMonthly 51 (1944): 319. See Also Wilder, Mathematics, e.g. pp. 39-41, and his "Relativityof Standards of Mathematical Rigor," Dictionary of the History of Ideas, ed. Philip P.Wiener, vol. 3 (New York: Charles Scribner 's, 1973), pp. 170-77.


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TE N M I S C O N C E P T I O N S A B O U T M A T H E M A T IC S 27732. Lakatos, Proofs, p. 9.33. Imre Lakatos , "Infinite Regress and the Foun dations of Mathem atics," "ARenaissance of Empiricism in Recent Philosophy of Mathematics," and "Cauchy and the

Cont inuum: The Significance of the Non-Standard Analysis for the History and Philosophyof Mathematics," in Lakatos's Mathematics, Science and Epistemology. Philosophical Papers,vol. 2, ed. John Wo rrall and Gregory Currie (Cambridge: Cam bridge University Press, 1978),pp. 3-23, 24-42, 43-60.34. See, fo r example, Hugh Lehman, "A n Examination of Imre Lakatos' Philosophyof Mathematics ," Philosophical Fo rum 12 (1980): 33-48; Peggy Marchi, "Mathematics asa Critical En terprise," in Essays in Memory of Imre Lakatos, ed. R. S. Coh en, P. K . Feyera-bend, and M. W. Wartofsky (Dordrecht and Boston: Reidel, 1976), pp. 379-93.35. Michael Hallett, "Towards a Theory of Mathematical R esearch Program m es," BritishJournal for the Philosophy of Science 30 (1979): 1-25, 135-59.36. H. J. S. Sm ith, "Opening Address by the President...," Nature 8 (25 Sept. 1873):450. Smith in that year was president of Section A of the British Association for the Ad-vancement of Science.37. Henri Poincar, The Value of Science, trans. George Bruce Halsted (New Y ork: D over,1958), pp. 19-20.38. Kline, Mathematics, p. 313; see also his Why Johnny Can't Add (New Y ork: Vin-tage, 1974), p. 69.39. Philip Kitcher, "Mathematical R igorWho Needs It?" Nous 15 (1980): 490.40. Nagel, "Numbers," pp. 435, 437.41. Kitcher , Mathematical Knowledge, pp. 155-58. For his "rational reco nstruc tion"of the history of calculus, see chap. 10.42. Chris t iaan Huygens, Treatise on Light, t rans . Silvanus P. Thom pson (New Y ork :Dover , n.d.) , p. vi .43. Hilary Putnam , "What Is Mathematical T r u t h?" in Putnam , Mathematics, Matterand M ethod. Philosophical Papers, vol. 1, 2d ed. (Cambridge: C am bridge U niversity Press,1979), p. 64.44. Kitcher, "Rigor," p. 471; se e also K itcher,Mathematical Knowledge, pp. 217-24, 271.45. See quotations in Kline, Mathematics, pp. 330-31.46. Nagel, "Numbers," pp. 434, 437, 441.47. Hersh , "Proposals," p. 43 . See also Philip J. Davis and Re uben H ersh, TheMathematical Experience (Boston: Birkhauser, 1981), p. 406.

ten misconceptions about mathematics

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