Lakatos - Proofs and Refutations (I)

The British Society for the Philosophy of Science

Proofs and Refutations (I) Author(s): I. Lakatos Source: The British Journal for the Philosophy of Science, Vol. 14, No. 53 (May, 1963), pp. 125 Published by: Oxford University Press on behalf of The British Society for the Philosophy of Science Stable URL: http://www.jstor.org/stable/685347 Accessed: 04/11/2008 18:09 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=oup. 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 organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected].

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The

British

Journal

Philosophy VOLUME

XIV

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Science No.

May, I963

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53

(I)*t

I. LAKATOS For GeorgePdlya's 75th and Karl Popper's6oth birthday Introduction

? I.

A Problem and a Conjecture.

?2.

A Proof.

? 3. Criticism of the Proof by Counterexamples which are Local but not Global. ? 4. Criticism of the Conjecture by Global Counterexamples. (a) Rejection of the conjecture. The method of surrender. (b) Rejection of the counterexample. The method of monsterbarring. (c) Improving the conjecture by exception-barring methods. Piecemeal exclusions. Strategic withdrawal. (d) The method of monster-adjustment. (e) Improving the conjecture by the method of lemma-incorp oration. Proof-generated theorem versus naive conjecture. * Received 3.x.60 t This paper was written in I958-59 at King's College, Cambridge. It was first read in Karl Popper's seminar in London in March 1959. The paper was then mimeographed and widely circulated. An improved version has been included in the author's Cambridge Ph.D. thesis prepared under Professor R. B. Braithwaite's supervision(Essaysin the Logic of Mathematical Discovery,1961). The author also received much help, encouragement and valuable criticism from Dr T. J. Smiley. The thesiswould not have been written but for the generous help of the Rockefeller Foundation. When preparingthis latest version at the London School of Economics the author tried to take note especially of the criticismsand suggestions of Dr J. Agassi, Dr I. Hacking, ProfessorsW. C. Kneale and R. Montague, A. Musgrave, Professor M. Polanyi andJ. W. N. Watkins. The treatmentof the exception-barringmethod was improved under the stimulusof the criticalremarksof ProfessorsG. Polya and B. L. van der Waerden. The distinction between the methods of monster-barringand monster-adjustmentwas suggested by B. MacLennan. The paper should be seen againstthe backgroundof Polya's revival of mathematical heuristic,and of Popper'scriticalphilosophy. A I

I. LAKATOS

whichareGlobalbut not ? 5. Criticismof the Proofby Counterexamples Local. and theorem(a) The role of refutationsin proof-elaboration formation. The allianceof proofandrefutations. (b) Reductivestructures.

?6.

(c) Fallibilism. Concept-formation.

Introduction IT frequentlyhappensin the history of thought that when a powerful new method emerges the study of those problemswhich can be dealt with by the new method advancesrapidly and attractsthe limelight, while the rest tends to be ignored or even forgotten, its study despised. This situationseemsto have arisenin our centuryin the Philosophy of Mathematics as a result of the dynamic development of metamathematics. The subjectmatterof metamathematicsis an abstractionof mathematicsin which mathematicaltheoriesare replacedby formal systems, proofs by certain sequencesof well-formed formulae, definitions by 'abbreviatory devices' which are 'theoretically dispensable' but 'typographically convenient '. This abstraction was devised by Hilbert to provide a powerful technique for approachingsome of the problemsof the methodology of mathematics. At the sametime there are problems which fall outside the range of metamathematical abstractions. Among these are all problems relating to informal mathematicsand to its growth, and all problemsrelatingto (inhaltliche) the situationallogic of mathematicalproblem-solving. I shallreferto the school of mathematicalphilosophywhich tendsto identify mathematicswith its metamathematicalabstraction(and the philosophy of mathematicswith metamathematics)as the ' formalist' school. One of the cleareststatementsof the formalistposition is to be found in Carnap[I937].2 Camap demandsthat (a) 'philosophy is to be replaced by the logic of science . . . ', (b) ' the logic of science is

nothing other than the logical syntax of the languageof science . . . 1

Church[1956]I, pp. 76-77. Also cf. Peano [1894],p. 49 and Whitehead-Russell [I9I0-I3], I, p. 12. This is an integralpart of the Euclideanprogrammeas formulated by Pascal[I657-58]: cf. Lakatos[I962], p. I58. 2 For full detailsof this and similarreferencessee the list of works at the end of Part III of this article. 2

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is the syntaxof mathematical language'(pp.xiii (c)' metamathematics is to be replacedby metaand 9). Or: philosophyof mathematics mathematics. from the philothe historyof mathematics Formalismdisconnects sophy of mathematics,since, accordingto the formalistconceptof thereis no historyof mathematics mathematics, proper. Anyformalist would basicallyagreewith Russell's'romantically'put but seriously meantremark,accordingto whichBoole'sLawsof Thought (1854)was 'the firstbook everwrittenon mathematics'.1Formalismdeniesthe to mostof whathasbeencommonlyunderstood statusof mathematics and can say nothingaboutits growth. None of to be mathematics, the 'creative' periodsand hardlyany of the 'critical' periodsof theorieswould be admittedinto the formalistheaven, mathematical theoriesdwelllikethe seraphim,purgedof all the wheremathematical of impurities earthlyuncertainty. Formalists,though, usuallyleave opena smallbackdoor for fallenangels: if it turnsout thatfor some 'mixturesof mathematicsand somethingelse' we can find formal systems' whichincludethemin a certainsense', thentheytoo maybe admitted(Curry[1951],pp. 56-57). On thosetermsNewton hadto wait four centuriesuntil Peano,Russell,and Quinehelpedhim into heaven by formalisingthe Calculus. Dirac is more fortunate: Schwartzsaved his soul during his lifetime. Perhapswe should mentionhere the paradoxicalplight of the metamathematician: by formalist,or even by deductivist,standards,he is not an honest mathematician.Dieudonnetalksabout'theabsolutenecessityimposed whocares on any mathematician for intellectual integrity'[my italics]to present his reasonings in axiomatic form ([I939], p. 225).

Under the present dominance of formalism, one is tempted to paraphraseKant: the history of mathematics,lacking the guidanceof philosophy, has become blind,while the philosophy of mathematics, turning its back on the most intriguing phenomena in the history of mathematics,has become empty. ' Formalism' is a bulwarkof logical positivistphilosophy. According to logical positivism, a statement is meaningful only if it is 'tautological' or empirical. Since informal mathematicsis neither 1 B. Russell[190o]. The essaywas republished as ChapterV of Russell's[I9I8], underthe title'Mathematicsandthe Metaphysicians '. In the I953 PenguinEdition thequotationcanbe foundon p. 74. Intheprefaceof [9I8] Russellsaysof theessay: 'Its toneis partlyexplainedby the factthatthe editorbeggedme to makethe article " as romantic as possible".'

3

I. LAKATOS

sheernonsense.1 'tautological'nor empirical,it mustbe meaningless, The dogmasof logicalpositivismhavebeendetrimental to the history andphilosophy mathematics. of The purposeof theseessaysis to approachsome problemsof the I use the word 'methodology' in a sense of mathematics. methodology akin to Polya's and Bernays''heuristic'2 and Popper's'logic of discovery' or ' situationallogic '.3

The recent expropriationof the

term 'methodology of mathematics'to serve as a synonym for 'metamathematics'has undoubtedlya formalisttouch. It indicates that in formalistphilosophyof mathematics thereis no properplace for methodologyqua logic of discovery.4 Accordingto formalists, 1

According to Turquette, G6delian sentences are meaningless ([1950], p. 129). Turquette argues against Copi who claims that since they are a prioritruthsbut not analytic, they refute the analytic theory of a priori([1949] and [I950]). Neither of them notices that the peculiarstatusof G6delian sentencesfrom this point of view is that these theorems are theorems of informal mathematics, and that in fact they discussthe statusof informalmathematicsin a particularcase. Neither do they notice that theorems of informal mathematics are surely guesses, which one can hardly classify dogmatist-wise as ' a priori' and ' a posteriori' guesses. 2 P6lya [I945], especially p. I02, and also [I954], [1962a]; Bemays [I947], esp. p. I87 3

Popper [1934], then [I945], especiallyp. 90 (or the fourth edition (1962) p. 97);

and also [1957], pp. 147 ff.

4 One canillustratethis, e.g. by Tarski[I930a] and Tarski[193ob]. In the first paperTarskiusesthe term ' deductivesciences' explicitlyas a shorthandfor ' formalised deductive sciences'. He says: ' Formaliseddeductive disciplinesform the field of research of metamathematicsroughly in the same sense in which spatial entities form the field of research in geometry.' This sensible formulation is given an intriguingimperialisttwist in the secondpaper: ' The deductivedisciplinesconstitute the subject-matterof the methodology of the deductive sciencesin much the same sense in which spatialentities constitute the subject-matterof geometry and animals that of zoology. Naturally not all deductive disciplines are presented in a form suitablefor objects of scientific investigation. Those, for example, are not suitable which do not rest on a definitelogical basis,have no preciserulesof inference,and the theorems of which are formulated in the usually ambiguous and inexact terms of colloquial language-in a word those which are not formalised. Metamathematical investigationsare confined in consequenceto the discussionof formalised deductive disciplines.' The innovation is that while the first formulation stated that the subject-matterof metamathematicsis the formaliseddeductive disciplines,the second formulationstatesthatthe subject-matterof metamathematicsis confinedto formalised deductive disciplinesonly becausenon-formaliseddeductive sciencesare not suitable objectsfor scientificinvestigationat all. This implies that the pre-historyof a formalised discipline cannot be the subject-matterof a scientificinvestigation-unlike the pre-historyof a zoological species,which can be the subject-matterof a very scientific 4

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mathematics is identical with formalised mathematics. But what can one discoverin a formalised theory? Two sorts of things. First, one can discover the solution to problems which a suitably programmed Turing machine could solve in a finite time (such as: is a certain alleged proof a proof or not?). No mathematician is interested in following out the dreary mechanical 'method' prescribed by such decision procedures. Secondly, one can discover the solutions to problems (such as: is a certain formula in a non-decidable theory a theorem or not?), where one can be guided only by the 'method' of 'unregimented insight and good fortune'. Now this bleak alternative between the rationalism of a machine and the irrationalism of blind guessing does not hold for live mathematics:l an investigation of informal mathematics will yield a rich situational logic for working mathematicians, a situational logic which is neither mechanical nor irrational, but which cannot be recognised and still less, stimulated, by the formalist philosophy. The history of mathematics and the logic of mathematical discovery, theoryof evolution. Nobodywill doubtthatsomeproblemsabouta mathematical afterit has been formalised, theorycan only be approached just as some problems abouthumanbeings(say concerningtheiranatomy)can only be approached after theirdeath. Butfew will inferfromthisthathumanbeingsare' suitableforscientific investigation'only whenthey are'presentedin " dead" form', andthatbiological areconfinedin consequence to the discussion of deadhumanbeingsinvestigations although,I shouldnot be surprisedif some enthusiastic pupil of Vesaliusin those gloriousdaysof earlyanatomy,whenthepowerfulnewmethodof dissection emerged, hadidentifiedbiologywith the analysisof deadbodies. In the prefaceof his [1941]Tarskienlargeson his negativeattitudetowardsthe possibilityof anysortof methodologyotherthanformalsystems:' A coursein the and methodologyof empiricalsciences. . . mustbe largelyconfinedto evaluations criticisms of tentativegropingsandunsuccessful efforts.' Thereasonis thatempirical sciencesareunscientific:for Tarskidefinesa scientifictheory' asa systemof asserted statements arrangedaccordingto certainrules' (Ibid.). 1 One of the most dangerousvagariesof formalistphilosophyis the habitof (I) statingsomething-rightly-aboutformalsystems;(2) then sayingthatthisapplies to ' mathematics '-this is againrightif we acceptthe identification of mathematics andformalsystems;(3) subsequently, shiftin meaning,usingthe with a surreptitious term ' mathematics' in the ordinarysense. So Quine says ([I951], p. 87), that ' this

reflectsthe characteristic situation:the mathematician mathematical hits upon his

proof by unregimented insight and good fortune, but afterwardsother mathematicians can check his proof'. But often the checking of an ordinaryproof is a very delicate enterprise,and to hit on a ' mistake' requiresas much insight and luck as to hit on a proof: the discovery of' mistakes' in informal proofs may sometimes take decades-if not centuries. 5

I. LAKATOS

i.e. the phylogenesisand the ontogenesisof mathematical thought,l cannotbe developedwithoutthe criticismand ultimaterejectionof formalism. But formalistphilosophyof mathematics hasverydeeproots. It is the latestlink in the long chainof dogmatist philosophiesof mathematics. For more thantwo thousandyearstherehas been an argument betweendogmatists and sceptics.The dogmatistshold that-by the power of our humanintellectand/orsenses-we can attaintruth and know that we have attainedit. The scepticson the otherhand eitherhold thatwe cannotattainthe truthat all (unlesswith the help of mysticalexperience),or thatwe cannotknow if we canattainit or thatwe haveattainedit. In thisgreatdebate,in whichargumentsare time and againbroughtup-to-date,mathematics has been the proud fortressof dogmatism. Wheneverthe mathematical dogmatismof the day got into a 'crisis', a new versiononceagainprovidedgenuine rigour and ultimatefoundations,thereby restoringthe image of authoritative,infallible,irrefutablemathematics,'the only Science thatit haspleasedGodhithertoto bestowon mankind'(Hobbes[I65i], of this p. I5). Mostscepticsresignedthemselvesto theimpregnability of A is now overdue. stronghold dogmatistepistemology.2 challenge The coreof thiscase-studywill challengemathematical formalism, but will not challengedirectlythe ultimatepositionsof mathematical dogmatism. Its modestaim is to elaboratethe point that informal, mathematics does not grow througha monotonous quasi-empirical, increaseof thenumberof indubitably established theoremsbutthrough the incessantimprovementof guessesby speculationandcriticism,by the logic of proofsandrefutations. Sincehowevermetamathematics is a paradigmof informal,quasi-empirical mathematics just now in the will also rapidgrowth, essay,by implication, challengemodem mathematicaldogmatism. The studentof recenthistory of metamathematics will recognisethepatternsdescribed herein hisown field. Both H. Poincare and G. Polya propose to apply E. Haeckel's 'fundamental ' biogenetic law about ontogeny recapitulatingphylogeny to mental development, in particular to mathematical mental development. (Poincare [I908], p. I35, and Polya [1962b].) To quote Poincare: 'Zoologists maintain that the embryonic development of an animal recapitulatesin brief the whole history of its ancestors throughoutgeologic time. It seems it is the same in the development ofminds. ... For this reason, the history of science should be our first guide' (C. B. Halsted's authorisedtranslation,p. 437). 2 For a discussionof the role of mathematicsin the dogmatist-scepticcontroversy, cf. my [I962]. 6

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The dialogue form should reflect the dialectic of the story; it is meant to contain a sort of rationallyreconstructed or ' distilled'history. The realhistorywill chimein in thefootnotes,mostof whichareto be taken, as an organicpart of the essay. therefore, I A Problemand a Conjecture The dialoguetakesplace in an imaginaryclassroom. The classgets interestedin a PROBLEM: is there a relationbetween the number of vertices V, the number of edges E and the number of faces F of polyhedra-particularly of regularpolyhedra-analogous to the trivial relationbetween the numberof verticesand edges ofpolygons,namely, that there are as many edges as vertices: V=E? This latter relation enables us to classifypolygonsaccording to the number of edges (or vertices): triangles, quadrangles, pentagons, etc. An analogous relation would help to classifypolyhedra. Aftermuch trial and errorthey notice that for all regularpolyhedra V-

E+

F=

1 First noticed

2.1

Somebody guesses that this may apply for any

by Euler [I750]. His original problem was the classificationof polyhedra, the difficultyof which was pointed out in the editorialsummary: ' While in plane geometry polygons (figuraerectilineae) could be classifiedvery easily accordto number of which is always equal to the number the their of course sides, ing of their angles, in stereometry the classificationof polyhedra (corporahedrisplanis inclusa) represents a much more difficult problem, since the number of faces alone is insufficient for this purpose.' The key to Euler's result was just the invention of the concepts of vertexand edge: it was he who first pointed out that besides the number of faces the number of points and lines on the surface of the polyhedron determinesits (topological) character. It is interesting that on the one hand he was eager to stressthe novelty of his conceptualframework, and that he had to invent the term ' acies' (edge) instead of the old ' latus' (side), since latuswas a polygonal concept while he wanted a polyhedral one, on the other hand he still retained the term ' angulussolidus' (solid angle) for his point-like vertices. It has been recentlygenerallyacceptedthat the priority of the resultgoes to Descartes. The ground for this claim is a manuscriptof Descartes [ca. I639] copied by Leibniz in Paris from the original in 1675-6, and rediscovered and published by Foucher de Careil in i860. The priority should not be granted to Descartes without a minor qualification. It is true that Descartesstates that the number of plane angles equals 2S + 2a - 4 where by b he meansthe number of faces and by a the number of solid angles. It is also true that he statesthat there are twice as many plane angles as edges (latera). The trivial conjunction of these two statementsof course yields the Euler formula. But Descartesdid not see the point of doing so, since he still thought in terms of angles(planeand solid) and faces, and did not make a consciousrevolutionary change to the concepts of o-dimensionalvertices, i-dimensional edges and 2-dimensional faces as a necessaryand sufficientbasisfor the full topological characterisationof polyhedra. 7

I. LAKATOS

polyhedron whatsoever. Others try to falsify this conjecture, try to test it in many differentways-it holds good. The results the conjecture,and suggestthat it could be proved. It is corroborate and conjecture-that we enter at this point-after the stagesproblem the classroom.1The teacherisjust goingto offera proof 2. A Proof TEACHER: In our last lesson we arrived at a conjecture concerning

polyhedra,namely, that for all polyhedra V-- E+ F= 2, where V is the number of vertices,E the number of edges and F the number of faces. We tested it by variousmethods. But we haven't yet proved it. Has anybody found a proof? PUPIL SIGMA: 'I for one have to admit that I have not yet been

able to devise a strict proof of this theorem. ...

As however the

truth of it has been establishedin so many cases,there can be no doubt that it holds good for any solid. Thus the proposition seems to be satisfactorilydemonstrated.'2 But if you have a proof, please do present it.

In fact I have one. It consistsof the following thoughti: Let us imagine the polyhedronto be hollow, with Step experiment. a surfacemade of thin rubber. If we cut out one of the faces, we can stretchthe remainingsurfaceflat on the blackboard,without tearingit. The faces and edges will be deformed,the edges may become curved, but V, E and F will not alter,so that if and only if V -E + F = 2 for TEACHER:

the original polyhedron, then V-

E+

F=

I for this flat network-

remember that we have removed one face. (Fig. I shows the flat network for the caseof a cube.) Step 2: Now we triangulateour map -it does indeed look like a geographicalmap. We draw (possibly curvilinear)diagonals in those (possibly curvilinear)polygons which 1Eulertestedthe conjecturequitethoroughlyfor consequences.He checkedit for prisms,pyramidsand so on. He could have addedthat the propositionthat of the conjecture. Another thereareonly five regularbodiesis alsoa consequence is the hithertocorroborated propositionthatfourcoloursare suspectedconsequence sufficientto coloura map. and testingin the caseof V-- E + F= 2 is discussedin The phase of conjecturing

P6lya([I954],Vol. I, the firstfive sectionsof the thirdchapter,pp. 35-4I). Polya stoppedhere,anddoesnot dealwiththephaseofproving-thoughof coursehe points out the needfor a heuristicof' problemsto prove' ([1945],p. I44). Ourdiscussion startswherePolyastops. 2 Euler ([I750],

p. 119 and p. 124). But later [I751] he proposed a proof.

8

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are not already(possiblycurvilinear)triangles. By drawing each diagonalwe increasebothE andF by one,so thatthe totalV-- E + F will not be altered (Fig. 2).

Step 3: From the triangulatednetwork

v1

\

we now removethe trianglesone by one. To removea trianglewe eitherremovean edge-upon whichone faceandone edge disappear (Fig.3a),or we removetwo edgesanda vertex-upon whichone face, two edgesandonevertexdisappear (Fig.3b). Thusif V- E+ F= I

FIG. I

' FIG. 2

FIG.3a

FIG.3b

beforea triangleis removed,it remainsso afterthetriangleis removed. At the end of this procedurewe get a single triangle. For this V- E-+ F = I holds true. Thus we have provedour conjecture.l PUPIL DELTA: You shouldnow callit a theorem.Thereis nothing aboutit any more.2 conjectural PUPILALPHA: I wonder. I see that this experiment can be per-

formedfor a cubeor for a tetrahedron, but how am I to know thatit can be performedfor any polyhedron? For instance,are you sure, canbestretched Sir,thatanypolyhedron, afterhavingafaceremoved, flat on theblackboard? I am dubiousaboutyourfirststep. 1 This proof-idea stems from Cauchy [I8I]. Delta's view that this proof has establishedthe 'theorem' beyond doubt was sharedby many mathematiciansin the nineteenth century, e.g. Crelle [I826-27], II, pp. 668-671, Matthiessen[1863], p. 449, Jonquieres[89goa]and [I89ob]. To quote a characteristicpassage: ' After Cauchy's proof, it became absolutely indubitablethat the elegant relation V+ F = E+ 2 applies to all sorts of polyhedra,just as Euler statedin 1752. In I8II all indecision should have disappeared.' Jonquieres[I89oa], 2

pp. III-II2.

9

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PUPILBETA: Are you sure that in triangulating the map one will

always get a new face for any new edge? I am dubious about your second step. PUPIL GAMMA: Are you sure that there are only two alternatives-

the disappearance of one edgeor else of two edgesand a vertex-when one the drops trianglesone by one? Are you even sure that one is left with a single triangleat the end of this process? I am dubious about your -third step.1 TEACHER:Of course I am not sure.

ALPHA:But then we are worse off than before! Insteadof one conjecturewe now have at least three! And this you call a ' proof'! I admit that the traditional name 'proof' for this TEACHER: thought-experiment may rightly be considered a bit misleading. I do not think that it establishesthe truth of the conjecture. DELTA: What does it do then? What do you think a mathematical proof proves? TEACHER: This is a subtle questionwhich we shall try to answer later. Till then I propose to retain the time-honouredtechnicalterm 'proof' for a thought-experiment-or' quasi-experiment'-whichsuggests a decomposition intosubconjectures or lemmas,thus of the originalconjecture embeddingit in a possibly quite distant body of knowledge. Our 'proof', for instance, has embedded the original conjecture-about

crystals, or, say, solids-in

the theory of rubber sheets. Descartes or

Euler, the fathers of the original conjecture, certainly did not even dream of this.2 1 The class is a rather advanced one. To Cauchy, Poinsot, and to many other excellent mathematiciansof the nineteenth century these questionsdid not occur. 2 Thought-experiment (deiknymi)was the most ancient pattern of mathematical proof. It prevailedin pre-EuclideanGreek mathematics(cf. A. Szab6 [1958]). That conjectures (or theorems) precede proofs in the heuristic order was a commonplace for ancient mathematicians. This followed from the heuristic precedence of 'analysis' over 'synthesis'. (For an excellent discussion see Robinson [1936].) According to Proclos, ... it is . . . necessaryto know beforehandwhat is sought' (Heath [1925], I, p. I29). 'They said that a theorem is that which is proposed with a view to the demonstration of the very thing proposed'-says Pappus(ibid. I, p. IO). The Greeks did not think much of propositionswhich they happened to hit upon in the deductive direction without having previously guessed them. They called them porisms,corollaries, incidental results springing from the proof of a theorem or the solution of a problem, results not directly sought but appearing,as it were, by chance, without any additionallabour, and constituting, as Proclus says, a sort of windfall (ermaion)or bonus (kerdos)(ibid. I, p. 278). We read in the editorialsummaryto Euler [1753]that arithmeticaltheorems'were discovered I0

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whichareLocalbutnotGlobal 3. Criticismof theProofby Counterexamples TEACHER: This decomposition of the conjecture suggested by the proof opens new vistas for testing. The decomposition deploys the conjectureon a wider front, so that our criticismhas more targets. We now have at least three opportunitiesfor counterexamplesinstead of one! GAMMA:I alreadyexpressedmy dislike of your third lemma (viz. that in removing trianglesfrom the network which resultedfrom the stretching and subsequent triangulation, we have only two possibilities: either we remove an edge or we remove two edges and a vertex). I suspectthat other patternsmay emerge when removing a triangle. TEACHER: Suspicionis not criticism. criticism? GAMMA: Then is a counterexample TEACHER: Certainly. Conjecturesignore dislikeand suspicion,but they cannot ignore counterexamples. THETA(aside): Conjecturesare obviously very differentfrom those who representthem. GAMMA: I propose a trivial counterexample. Take the triangular network which resultsfrom performingthe first two operationson a cube (Fig. 2). Now if I remove a trianglefrom the insideof this network, as one might take a piece out of a jigsaw puzzle, I remove one trianglewithout removing a single edge or vertex. So the thirdlemma long beforetheir truthhas been confirmedby rigid demonstrations'.Both the ' instead EditorandEulerusefor thisprocessof discoverythe modernterm' induction of the ancient' analysis'(ibid.). The heuristicprecedenceof the resultover the folklore. argument,of the theoremover the proof,hasdeeprootsin mathematical is saidto havewritten Letus quotesomevariationson a familiartheme: Chrysippus to Cleanthes: 'Justsendme the theorems,then1 shallfindthe proofs'(cf. Diogenes Laertius [ca. 200], VII. I79).

Gauss is said to have complained:

'I have had my

results for a long time; but I do not yet know how I am to arrive at them' (cf. Arber [I954], p. 47), and Riemann: 'If only I had the theorems! Then I should find the proofs easily enough.' (Cf. Holder [I924], p. 487.) Polya stresses: 'You have to guess a mathematicaltheorem before you prove it' ([1954], Vol. I, p. vi). The term 'quasi-experiment'is from the above-mentioned editorial summary to Euler [I753]. According to the Editor: 'As we must refer the numbersto the pure can intellect alone, we can hardly understandhow observationsand quasi-experiments be of use in investigating the nature of the numbers. Yet, in fact, as I shall show here with very good reasons,the propertiesof the numbers known today have been mostly discoveredby observation . . ' (Polya'stranslation; he mistakenlyattributes the quotation to Euler in his [I954], I, p. 3). II

I. LAKATOS

is false-and not only in the caseof the cube, but for all polyhedra in the flatnetworkof which all the triangles exceptthe tetrahedron, are boundarytriangles. Your proof thus provesthe Eulertheorem for the tetrahedron.But we alreadyknewthat V- E+ F= 2 for so why proveit? the tetrahedron, TEACHER:You are right. But notice that the cube which is a

to to the thirdlemmais not also a counterexample counterexample the mainconjecture,sincefor the cube V- E+ F= 2. You have shownthe povertyof the argument-theproof-but not the falsityof our conjecture. ALPHA: Will you scrapyourproofthen? TEACHER:No.

Criticism is not necessarilydestruction. I shall

improvemy proofso thatit will standup to the criticism. GAMMA: How?

TEACHER:Before showing how, let me introduce the following

an examplewhich terminology. I shallcall a 'local counterexample' refutesa lemma (withoutnecessarilyrefutingthe main conjecture), an examplewhichrefutesthe andI shallcalla 'globalcounterexample' is local but not main conjectureitself Thus your counterexample a is criticismof the global. A local,but not global,counterexample proof,but not of the conjecture. GAMMA:So, the conjecturemay be true, but your proof does not

proveit.

TEACHER:But I can easilyelaborate,improvetheproof,by replacing

the false lemma by a slightlymodifiedone, which your counterof any examplewill not refute. I no longercontendthat theremoval but merelythatat each mentioned, followsoneof thetwopatterns triangle theremoval trianglefollows of anyboundary operation stageof theremoving all one of thesepatterns.Coming back to my thought-experiment, thatI haveto do is to inserta singlewordin my thirdstep,to wit, that ' fromthetriangulated networkwe now removethe boundary triangles You will one '. oneby agreethatit onlyneededa triflingobservation to put the proofright.1 GAMMA:I do not think your observationwas so trifling; in fact it

was quiteingenious. To makethis clearI shallshow thatit is false. Takethe flatnetworkof the cubeagainandremoveeight of the ten 1 Lhuilier, when correctingin a similar way a proof of Euler, says that he made only a 'trifling observation' ([I8I2-I3], p. I79). Euler himself, however, gave the proof up, since he noticed the trouble but could not make that' triflingobservation'. I2

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trianglesin the order given in Fig. 4. At the removal of the eighth triangle, which is certainlyby then a boundarytriangle, we removed two edges and no vertex-this changesV- E+ F by I. And we are left with the two disconnectedtriangles9 and Io.

\ 7 ?,/ /66

\

FIG.4 TEACHER: Well, I might save face by saying that I meant by a

boundary triangle a triangle whose removal does not disconnect the network. But intellectual honesty prevents me from making surreptitiouschangesin my position by sentencesstartingwith 'I meant . .' so I admit that now I must replacethe second version of the triangle-removingoperationwith a thirdversion: that we remove the trianglesone by one in such a way that V- E + F does not alter. KAPPA: I generously agree that the lemma correspondingto this operationis true: namely, that if we remove the trianglesone by one in such a way that V- E+ F does not alter, then V- E+ F does not alter. TEACHER: No.

The lemma is that the trianglesin our network can

thatin removingthemin therightorderV- E+- F will not beso numbered altertill we reachthe lasttriangle. KAPPA: But how should one construct this right order, if it exists

at all?1 Your original thought-experiment gave the instructions: remove the triangles in any order. Your modified thought-experiment gave the instruction: remove boundary trianglesin any order. Now you say we should follow a definite order, but you do not say which and whether that order exists at all. Thus the thought-experiment breaksdown. You improved the proof-analysis,i.e. the list of lemmas; but the thought-experimentwhich you called 'the proof' has disappeared. RHO: Only the third step has disappeared. 1 Cauchy thought that the instructionto find at each stage a trianglewhich can be removed either by removing two edges and a vertex or one edge can be trivially carriedout for any polyhedron ([i8II], p. 79). This is of course connected with his inability to imagine a polyhedron that is not homeomorphic with the sphere. I3

I. LAKATOS KAPPA:

Moreover, did you improvethe lemma? Your first two

simpleversionsat leastlookedtriviallytruebeforethey wererefuted; your lengthy,patchedup versiondoesnot even look plausible. Can you reallybelievethatit will escaperefutation? TEACHER:'Plausible' or even 'trivially true' propositions are

maturedin usuallysoonrefuted:sophisticated, conjectures, implausible hit criticism,might on the truth. OMEGA: And what happens if even your 'sophisticated con-

jectures' are falsifiedand if this time you cannotreplacethem by unfalsifiedones? Or, if you do notsucceedin improvingthe argumentfurtherby localpatching? You havesucceededin gettingovera local counterexample which was not globalby replacingthe refuted lemma. Whatif you do not succeednext time? TEACHER:Good question-it will be put on the agenda for to-

morrow.

4. Criticism of theConjecture by GlobalCounterexamples ALPHA:I have a counterexamplewhich will falsifyyour firstlemma

to the mainconjecture,i.e. -but this will also be a counterexample thiswill be a globalcounterexample as well.

TEACHER:Indeed! Interesting. Let us see. ALPHA: Imagine a solid bounded by a pair of nested cubes-a pair

of cubes,one of whichis inside,but doesnot touchthe other(Fig.5).

*

I

i,,,~

J

FIG. 5

This hollow cube falsifiesyour firstlemma,becauseon removinga on to facefromthe innercube,the polyhedronwill not be stretchable a plane. Nor will it helpto removea facefromtheoutercubeinstead. Besides,for eachcube V- E+ F= 2, so that for the hollow cube V-E+F=4. 1.1 Now TEACHER:Good show. Let us call it Counterexample

what?

1This Counterexample 1 was first noticed by Lhuilier ([I812-I3], p. 194).

But

Gergonne,the Editor, added(p. I86) that he himself noticed this long beforeLhuilier's I4

PROOFS

AND

REFUTATIONS

(I)

of theconjecture.Themethodof surrender (a) Rejection GAMMA:Sir, your composure baffles me. A single counterexample refutesa conjectureas effectivelyas ten. The conjectureand its proof have completely misfired. Hands up! You have to surrender. Scrapthe false conjecture,forget about it and try a radically new approach. has received a severe I agree with you that the conjecture TEACHER: criticism by Alpha's counterexample. But it is untrue that the proof has 'completely misfired'. If, for the time being, you agree to my earlierproposal to use the word 'proof' for a 'thought-experiment which leads to decomposition of the original conjecture into subconjectures', insteadof using it in the sense of a ' guaranteeof certain truth', you need not draw this conclusion. My proof certainly proved Euler's conjecturein the first sense, but not necessarilyin the second. You are interestedonly in proofs which 'prove' what they have set out to prove. I am interestedin proofs even if they do not accomplishtheir intendedtask. Columbus did not reachIndiabut he discoveredsomething quite interesting. ALPHA: So accordingto your philosophy-while a local counterexample (if it is not global at the same time) is a criticismof the proof, but not of the conjecture-a global counterexampleis a criticismof the conjecture,but not necessarilyof the proof. You agree to surrender as regards the conjecture, but you defend the proof. But if the conjectureis false, what on earth does the proof prove? GAMMA: Your analogy with Columbus breaksdown. Accepting a global counterexamplemust mean total surrender. (b) Rejectionof the counterexample.The methodof monster-barring DELTA: But why accept the counterexample? We proved our conjecture-now it is a theorem. I admit that it clashes with this so-called 'counterexample'. One of them has to give way. But why should the theorem give way, when it has been proved? It is the 'criticism' that should retreat. It is fake criticism. This pair of paper. Not so Cauchy,who publishedhis proofjust a year before. And this counterexamplewas to be rediscoveredtwenty years later by Hessel ([I832], p. I6).

Both LhuilierandHesselwere led to theirdiscoveryby mineralogical collectionsin whichthey noticedsomedoublecrystals,wherethe innercrystalis not translucent, the stimulusof the crystalcollectionof his but the outeris. Lhuilieracknowledges friend ProfessorPictet ([I8I2-I3], p. i88). Hesselrefersto lead sulphidecubes enclosed in translucentcalcium fluoride crystals([i832], p. i6).

I5

I. LAKATOS

nested cubes is not a polyhedron at all. It is a monster,a pathological case, not a counterexample. GAMMA:Why not? A polyhedronis a solidwhosesurfaceconsistsof polygonalfaces. And my counterexample is a solid bounded by polygonal faces. TEACHER:Let us call this definition Def. 1.1 DELTA: Your definition is incorrect. A polyhedron must be a surface:it has faces, edges, vertices, it can be deformed, stretched out on a blackboard, and has nothing to do with the concept of' solid'. A polyhedronis a surfaceconsistingof a system of polygons. TEACHER: Call this Def. 2.2

DELTA: So really you showed us two polyhedra-two

surfaces, one

completely inside the other. A woman with a child in her womb is not a counterexample to the thesis that human beings have one head. ALPHA: So! My counterexample has bred a new concept of polyhedron. Or do you dare to assert that by polyhedron you always meant a surface? TEACHER:For the moment let us accept Delta's Def. 2. Can you refute our conjecture now if by polyhedron we mean a surface? ALPHA: Certainly. Take two tetrahedra which have an edge in common (Fig. 6a). Or, take two tetrahedra which have a vertex in common (Fig. 6b). Both these twins are connected, both constitute one single surface. And, you may check that for both V-- E + F = 3 TEACHER:Counterexamples2a and 2b.3 1 Definition1 occurs first in the

' eighteenth century; e.g.: One gives the name polyhedralsolid,or simply polyhedron,to any solid bounded by planes or plane faces' (Legendre[I794], p. I60). A similar definition is given by Euler ([I75o]). Euclid, while defining cube, octahedron,pyramid, prism, does not define the general term polyhedron, but occasionallyuses it (e.g. Book XII, Second Problem, Prop. I7). 2 We find Definition2 implicitly in one of Jonquieres'papersread to the French Academy against those who meant to refute Euler's theorem. These papers are a thesaurusof monsterbarringtechniques. He thunders against Lhuilier'smonstrous pair of nested cubes: ' Such a system is not really a polyhedron, but a pair of distinct polyhedra, each independent of the other. ... A polyhedron, at least from the classicalpoint of view, deservesthe name only if, before all else, a point can move continuouslyover its entiresurface; here this is not the case . . . This first exception ofLhuilier can thereforebe discarded' ([89gob],p. I70). This definition-as opposed to Definition I-goes down very well with analyticaltopologists who are not interested at all in the theory of polyhedraas such but as a handmaidenfor the theory of surfaces. 3 2a and 2b were missed by Lhuilierand first discoveredonly by Counterexamples Hessel ([I832], p. 13).

i6

PROOFS

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(I)

DELTA:I admire your pervertedimagination, but of course I did not mean that anysystemof polygons is a polyhedron. By polyhedron I meant a systemof polygonsarrangedin sucha way that (1) exactlytwo polygonsmeetat everyedgeand(2) it is possibletogetfrom the insideof any polygonto the insideof any otherpolygonby a routewhichnevercrossesany edgeat a vertex. Your firsttwins will be excludedby the first criterion in my definition, your second twins by the second criterion.

FIG.6b

FIG.6a

TEACHER:Def: 3.1 ALPHA: I admire your perverted ingenuity in inventing one definition after another as barricades against the falsification of your

pet ideas. Why don't you just define a polyhedron as a system of polygons for which the equation V- E + F= 2 holds, and this Perfect Definition .... KAPPA: Def P.2 ALPHA: . . . would settle the dispute for ever? There would be no need to investigate the subject any further. DELTA: But there isn't a theorem in the world which couldn't be falsified by monsters. 1 Definition3 first turns up to keep out twintetrahedrain M6bius ([i865], p. 32). We find his cumbersome definition reproduced in some modem textbooks in the usual authoritarian' take it or leave it' way; the story of its monsterbarringbackground-that would at least explain it-is not told (e.g. Hilbert-Cohn Vossen [I956], p. 290). 2 DefinitionP according to which Euleriannesswould be a definitionalcharacteristic of polyhedra was in fact suggested by R. Baltzer: 'Ordinary polyhedra are occasionally (following Hessel) called Eulerian polyhedra. It would be more appropriateto find a specialname for non-genuine (uneigentliche) polyhedra' ([I860], Vol. II, p. 207). The referenceto Hessel is unfair: Hessel used the term 'Eulerian' simply as an abbreviationfor polyhedra for which Euler's relation holds in contra-

distinction to the non-Eulerian ones ([1832], p. I9). quotation in footnote pp. I8-I9.

B

17

For Def. P see also the Schlifli

I. LAKATOS TEACHER:I am sorry to interrupt you.

As we have seen, refuta-

tion by counterexamplesdepends on the meaning of the terms in question. If a counterexampleis to be an objectivecriticism,we have to agree on the meaning of our terms. We may achieve such an agreementby defining the term where communicationbroke down. I, for one, didn't define ' polyhedron'. I assumedfamiliaritywith the concept, i.e. the ability to distinguisha thing which is a polyhedron from a thing which is not a polyhedron-what some logicians call knowing the extension of the concept of polyhedron. It turned out that the extension of the concept wasn't at all obvious: definitionsare emerge. I frequentlyproposedand arguedabout when counterexamples suggest that we now considerthe rival definitionstogether, and leave until later the discussion of the differences in the results which will follow from choosing differentdefinitions. Can anybody offer something which even the most restrictivedefinitionwould allow as a real counterexample? KAPPA: Including Def: P? TEACHER: Excluding Def P.

GAMMA:I can. Look at this Counterexample 3: a star-polyhedron

-I

shall call it urchin(Fig. 7). This consists of

(Fig. 8).

12

star-pentagons

It has 12 vertices, 30 edges, and I2 pentagonal faces-you

E

B

?

A

D

Kepler (Fig. 7) shaded each face in a differentway to show which trianglesbelong to the same pentagonalface.

FIGS.7 and 8.

may check it if you like by counting. Thus the Descartes-Eulerthesis

is not true at all, since for this polyhedron V - E +- F-

- 6.1

1 The 'urchin' was first discussedby Kepler in his cosmological theory ([I619], Lib. II, XIX and XXVI, on p. 52 and p. 60 and Lib. V, Cap. I, p. I82, Cap. III, p. 187 and Cap. IX, XLVII).

The name' urchin' is Kepler's (' cui nomenEchinofeci ').

Fig. 7 is copied from his book (p. 52) which containsalso another picture on p. 182.

I8

PROOFS

AND

REFUTATIONS

(I)

DELTA: Why do you think that your 'urchin ' is a polyhedron?

GAMMA: Do you not see? Thisis a polyhedron,whosefacesare the twelve star-pentagons.It satisfiesyour last definition: it is 'a system of polygons arrangedin such a way that (I) exactly two polygonsmeet at everyedge, and (2) it is possibleto get from every polygonto everyotherpolygonwithoutevercrossinga vertexof the pclyhedron' DELTA: But then you do not even know what a polygonis! A is certainlynot a polygon! A polygonis a systemof star-pentagon in edgesarrangedsucha waythat(1) exactlytwoedgesmeetat everyvertex, and(2) theedgeshavenopointsin commonexceptthe vertices. TEACHER: Let us call this Def:4. GAMMA: I don't see why you includethe secondclause. The

rightdefinitionof the polygonshouldcontainthe firstclauseonly. TEACHER:Def. 4'. GAMMA:The second clausehas nothing to

do with the essenceof is already a polygon. Look: if I lift an edgea little,the star-pentagon a polygonevenin yoursense. You imaginea polygonto be drawnin chalk on the blackboard,but you shouldimagineit as a wooden structure:thenit is clearthatwhatyou thinkto be a pointin common is not reallyone point, but two differentpointslying one abovethe other. You aremisledby your embeddingthe polygonin a planeyou shouldlet its limbsstretchout in space!1 rediscovered Poinsotindependently it, andit washewhopointedoutthattheEuler

formuladid not applyto it ([1809],p. 48). The now standardterm'small stellated in general, polyhedron'is Cayley's([I859],p. 125). Schlafliadmittedstar-polyhedra but nevertheless as a monster. According rejectedoursmallstellateddodecahedron to him 'this is not a genuinepolyhedron,for it does not satisfythe condition V-

E + F=

2'([I852],

? 34).

1The disputewhetherpolygonshouldbe definedso as to includestar-polygons ornot(Def.4 or Def.4')is a veryold one. Theargumentputforwardin ourdialogue -that star-polygonscan be embeddedas ordinarypolygonsin a spaceof higher dimensions-isa modemtopologicalargument,butonecanputforwardmanyothers. ThusPoinsotdefendinghis star-polyhedra arguedfor the admissionof star-polygons takenfromanalytical with arguments (between geometry:'... all thesedistinctions " ordinary" and " star"-polygons)aremoreapparent thanreal,andtheycompletely in the analyticaltreatment,in which the variousspeciesof polygonsare disappear an equation quiteinseparable.To the edge of a regularpolygontherecorresponds with realroots,whichsimultaneously yieldsthe edgesof all the regularpolygonsof the sameorder. Thusit is not possibleto obtainthe edgesof a regularinscribed heptagon,withoutat thesametimefindingedgesof heptagonsof thesecondandthird species. Conversely,giventhe edge of a regularheptagon,one may determinethe I9

I. LAKATOS

Would you mind telling me what is the area of a star-pentagon? Or would you say that some polygons have no area? GAMMA:Was it not you yourself who said that a polyhedron has nothing to do with the idea of solidity? Why now suggest that the idea of polygon should be linked with the idea of area? We agreed that a polyhedron is a closed surface with edges and vertices-then why not agree that a polygon is simply a closed curve with vertices? But if you stick to your idea I am willing to define the area of a starDELTA:

polygon.1 TEACHER: Let us leave this disputefor a moment, and proceed as before. Consider the last two definitions together-Def. 4 and Def. 4. Can anyone give a counterexampleto our conjecturethat will comply with bothdefinitionsof polygons? ALPHA: Here is one. Consider a picture-frame like this (Fig. 9). This is a polyhedron accordingto any of the definitionshitherto proposed. Nonethelessyou will find, on counting the vertices,edges and

faces, that V-

E+

F = o.

radius of a circle in which it can be inscribed, but in so doing, one will find three differentcircles correspondingto the three species of heptagon which may be constructed on the given edge; similarly for other polygons. Thus we arejustified in giving the name " polygon " to these new starredfigures' ([1809], p. 26). Schrbder uses the Hankelian argument: 'The extension to rational fractions of the power concept originally associatedonly with the integershas been very fruitfulin Algebra; this suggeststhat we try to do the same thing in geometry whenever the opportunity presentsitself . . .' ([1862], p. 56). Then he shows that we may find a geometrical interpretationfor the concept of p/q-sided polygons in the star-polygons. 1 Gamma'sclaim that he can define the areafor star-polygonsis not a bluff. Some of those who defended the wider concept of polygon solved the problem by putting forward a wider concept of the areaof polygon. There is an especiallyobvious way to do this in the case of regularstar-polygons. We may take the area of a polygon as the sum of the areasof the isoscelestriangleswhich join the centre of the inscribed or circumscribedcircle to the sides. In this case, of course, some ' portions' of the star-polygonwill count more than once. In the case of irregularpolygons where we have not got any one distinguishedpoint, we may still take any point as origin and treatnegatively oriented trianglesas having negative areas(Meister[1769-70], p. I79). It turns out-and this can certainly be expected from an 'area '-that the area thus definedwill not depend on the choice of the origin (M6bius [1827], p. 218). Of course there is liable to be a disputewith those who think that one is not justified in calling the number yielded by this calculationan' area'; though the defendersof the Meister-Mobiusdefinition calledit ' theright definition ' which ' alone is scientifically justified' (R. Haussner'snotes [1906], pp. 114-115). Essentialismhas been a permanent feature of definitionalquarrels. 20

PROOFS

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REFUTATIONS

(I)

TEACHER: Counterexample4.1

So that's the end of our conjecture. It really is a pity, since it held good for so many cases. But it seems that we have just wasted our time. Can't you ALPHA: Delta, I am flabbergasted. You say nothing? define this new counterexample out of existence? I thought there was no hypothesis in the world which you could not save from falsification with a suitable linguistic trick. Are you giving up now? Do you agree at last that there exist non-Eulerian polyhedra? Incredible! BETA:

/l7

FIG. 9

DELTA: You should really find a more appropriate name for your non-Eulerian pests and not mislead us all by calling them ' polyhedra '. But I am gradually losing interest in your monsters. I turn in disgust from your lamentable 'polyhedra', for which Euler's beautiful theorem doesn't hold.2 I look for order and harmony in mathematics, but you only propagate anarchy and chaos.3 Our attitudes are irreconcilable. 1 We find Counterexample 4 too in Lhuilier'sclassical[1812-I3], on p. 185-Gergonne

againaddedthat he knew it. But Grunertdid not know it fourteenyearslater

([I827]) and Poinsot forty-five years later ([1858],p. 67).

2 Thisis paraphrased froma letterof Hermite'swrittento Stieltjes:'I turnaside with a shudderof horrorfrom this lamentableplagueof functionswhich haveno derivatives' ([I893]). 3' Researches dealing with . . . functions violating laws which one hoped were

universal,were regardedalmost as the propagationof anarchyand chaos where past generations had sought order and harmony' (Saks [I933], Preface). Saks refers here to the fierce battles of monsterbarrers(like Hermite!) and of refutationiststhat characterised in the last decades of the nineteenth century (and indeed in the

beginningof the twentieth)the developmentof modernrealfunctiontheory,'the

branchof mathematicswhich dealswith counterexamples' (Munroe [I953], Preface). The similarlyfierce battle that raged later between the opponents and protagonistsof moder mathematicallogic and set-theory was a direct continuation of this. See also footnote 2 on p. 24 and i on p. 25. 21

I. LAKATOS

ALPHA: You are a real old-fashioned Tory! You blame the wickedness of anarchistsfor the spoiling of your 'order' and 'harmony', and you 'solve' the difficultiesby verbal recommendations. TEACHER: Let us hear the latest rescue-definition.

ALPHA:You mean the latest linguistic trick, the latest contraction of the concept of 'polyhedron'! Delta dissolves real problems, insteadof solving them.

FIG. IO

FIG. Iib

FIG. IIa

DELTA: I do not contractconcepts.

It is you who expand them.

is not a genuinepolyhedronat all. Forinstance,thispicture-frame ALPHA:

Why?

DELTA:Take an arbitrarypoint inthe' tunnel'-the spacebounded

by the frame. Lay a planethroughthis point. You will find that any such plane has always two differentcross-sectionswith the 22

PROOFS

AND

REFUTATIONS

(I)

picture-frame, making two distinct, completely disconnected polygons! (Fig. Io). ALPHA: So what? DELTA: In the case of a genuinepolyhedron,throughany arbitrary with the point in spacetherewill be at leastone plane whosecross-section In of one case convex the will consist of single polygon. polyhedron wherever we with this all will comply requirement, polyhedra planes take the point. In the case of ordinaryconcave polyhedrasome planes will have more intersections,but there will always be some that have only one. (Figs. IIa and Iib.) In the case of this picture-frame,if we take the point in the tunnel, all the planes will have two crosssections. How then can you call this a polyhedron? TEACHER: This looks like another definition, this time an implicit one.

Call it Def. 5.1

A series of counterexamples,a matching series of definitions, definitions that are alleged to contain nothing new, but to be merely new revelationsof the richnessof that one old concept, which seems to have as many ' hidden' clausesas there are counterexamples. ForallpolyhedraV- E+ F= 2 seemsunshakable,an old and' eternal' truth. It is strangeto think that once upon a time it was a wonderful guess, full of challengeand excitement. Now, becauseof your weird shifts of meaning, it has turned into a poor convention, a despicable piece of dogma. (He leavesthe classroom.) DELTA: I cannot understandhow an able man like Alpha can waste his talent on mere heckling. He seemsengrossedin the productionof monstrosities. But monstrositiesnever foster growth, either in the world of natureor in the world of thought. Evolution alwaysfollows an harmoniousand orderly pattern. GAMMA: Geneticistscan easily refute that. Have you not heard that mutations producing monstrositiesplay a considerablerole in macroevolution? They call such monstrous mutants 'hopeful ALPHA:

1 Definition 5 was

monsterbarrer E. deJonquieres putforwardby the indefatigable to get Lhuilier'spolyhedronwith a tunnel(picture-frame) out of the way: ' Neither is thispolyhedralcomplexa truepolyhedronin the ordinarysenseof theword,for if one takesany planethroughan arbitrary pointinsideone of the tunnelswhichpass will be composedof two distinct rightthroughthe solid,the resultingcross-section polygonscompletelyunconnectedwith each other; this can occurin an ordinary polyhedronfor certain positionsof the intersecting plane,namelyin the caseof some concavepolyhedra,but not for all of them' ([89gob],pp. I70-I71). One wonders hasnoticedthathisDef.5 excludesalsosomeconcavespheroid whetherdeJonquieres polyhedra. 23

I. LAKATOS

monsters'. It seems to me that Alpha's counterexamples,though monsters, are 'hopeful monsters '.1

DELTA:Anyway, Alpha has given up the struggle. No more monstersnow. GAMMA:I have a new one. It complies with all the restrictionsin Defs. I, 2, 3, 4, and 5, but V-

E+

F=

I.

This Counterexample5

is a simple cylinder. It has 3 faces (the top, the bottom and the jacket), 2 edges (two circles) and no vertices. It is a polyhedron accordingto your definition: (I) exactly two polygons at every edge and (2) it is possible to get from the inside of any polygon to the insideof any other polygon by a route which never crossesany edge at a vertex. And you have to acceptthe facesas genuinepolygons, as they comply with your requirements: (I) exactly two edges meet at every vertex and (2) the edgeshave no points in common except the vertices. DELTA:Alpha stretched concepts, but you tear them! Your 'edges' are not edges! An edgehas two vertices! TEACHER: Def. 6?

GAMMA:But why deny the statusof' edge' to edges with one or possibly zero vertices? You used to contractconcepts, but now you mutilate them so that scarcelyanything remains! DELTA:But don't you see the futility of theseso-calledrefutations? 'Hitherto, when a new polyhedron was invented, it was for some practicalend; today they are invented expressly to put at fault the reasoningsof our fathers,and one never will get from them anything more than that. Our subject is turned into a teratologicalmuseum where decent ordinary polyhedra may be happy if they can retain a very small comer.' 1 'We must not forget that what appearsto-day as a monster will be to-morrow the origin of a line of specialadaptations.... I further emphasizedthe importance of rare but extremely consequentialmutations affecting rates of decisive embryonic processeswhich might give rise to what one might term hopeful monsters, monsters which would starta new evolutionary line if fitting into some empty environmental niche' (Goldschmidt[I933], pp. 544 and 547). My attentionwas drawnto this paper by Karl Popper. 2 Paraphrased from Poincare ([1908], pp. 131-132).

The original full text is this:

'Logic sometimes makes monsters. Since half a century we have seen arisea crowd of bizarre functions which seem to try to resemble as little as possible the honest functions which serve some purpose. No longer continuity, or perhapscontinuity, but no derivatives,etc. Nay more, from the logical point of view, it is these strange functions which are the most general, those one meets without seeking no longer appearexcept as particularcases. There remainsfor them only a very small corner. 24

PROOFS

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REFUTATIONS

(I)

GAMMA: I think that if we want to learn about anything really

deep,we haveto studyit not in its 'normal', regular,usualform,but in itscriticalstate,in fever,in passion. Ifyou wantto knowthenormal healthybody, study it when it is abnormal,when it is ill. If you want to know functions,study their singularities.If you want to know ordinarypolyhedra,study their lunaticfringe. This is how one cancarrymathematical analysisinto the veryheartof the subject.' But evenif you werebasicallyright,don'tyou see the futilityof your ad hocmethod? If you wantto drawa borderlinebetweencounterexamplesandmonsters,you cannotdo it in fits andstarts. TEACHER: I thinkwe shouldrefuseto acceptDelta'sstrategyfor dealingwith globalcounterexamples, althoughwe shouldcongratulate him on his skilfulexecutionof it. We couldaptlylabelhis method themethod ofmonsterbarring. Usingthismethodone caneliminateany to the counterexample originalconjectureby a sometimesdeftbut alwaysadhocredefinitionof the polyhedron,of its definng terms,or of the defining terms of its defining terms. We should somehow treatcounterexamples with more respect,and not stubbornly exorcisethem by dubbingthem monsters. Delta'smain mistakeis of mathematical perhapshisdogmatistbiasin the interpretation proof: he thinksthata proofnecessarily proveswhat it hasset out to prove. of proof will allow for a false conjectureto be My interpretation i.e. to be 'proved', decomposedinto subconjectures.If the conI is to be jecture false, certainlyexpectat leastone of thesubconjectures false. But the decompositionmight still be interesting! I am not to a 'proved' conjecture; perturbedat finding a counterexample I am evenwillingto set out to 'prove' a falseconjecture! I don't follow you. KAPPA: He just follows the New Testament: 'Prove all things; hold fast that which is good' (I Thessalonians5: 2I). THETA:

(to be continued) 'Heretofore when a new function was invented, it was for some practicalend; to-day they are invented expressly to put at fault the reasoningsof our fathers, and

one neverwill get fromthemanythingmorethanthat. 'If logic werethe sole guideof the teacher,it would be necessaryto beginwith the most generalfunctions,thatis to say with the most bizarre. It is the beginner that would have to be set grapplingwith this teratologicalmuseum. . .' (G. B. Halsted'sauthorisedtranslation, pp. 435-436). Poincarediscussesthe problemwith respectto the situationin the theoryof realfunctions-butthatdoesnot makeany difference. 1Paraphrased from Denjoy ([I919], p. 2I). 25