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The British Society for the Philosophy of Science

Proofs and Refutations (I)Author(s): I. LakatosSource: The British Journal for the Philosophy of Science, Vol. 14, No. 53 (May, 1963), pp. 1-25

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T h e B r i t i s h J o u r n a l f o r t h e

Philosophy o f S c i e n c e

VOLUME XIV May, I963 No. 53

PROOFS AND REFUTATIONS (I)*t

I. LAKATOS

For GeorgePdlya's75th 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 monster-

barring.(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 ora-

tion. Proof-generated theorem versus naive conjecture.

* Received 3.x.60

t This paperwas 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

mimeographedand widely circulated. An improved version has been included inthe author'sCambridgePh.D. thesis preparedunder ProfessorR. B. Braithwaite's

supervision(Essaysin theLogicof MathematicalDiscovery,1961). The author also

received much help, encouragementand valuable criticism from Dr T. J. Smiley.The thesiswould not have been written but for the generous help of the Rockefeller

Foundation.

When preparingthis latestversionat the London School of Economicsthe author

tried to take note especiallyof the criticismsand suggestionsof Dr J. Agassi, Dr I.

Hacking, ProfessorsW. C. Kneale and R. Montague, A. Musgrave, ProfessorM.

Polanyi andJ.W. N. Watkins. The treatmentof

the exception-barringmethod

wasimproved underthe stimulus of the criticalremarksof ProfessorsG. Polya and B. L.

van der Waerden. The distinction between the methods of monster-barringand

monster-adjustmentwas suggestedby B. MacLennan.

The papershouldbe seen againstthe backgroundof Polya'srevival of mathema-

tical heuristic,and of Popper'scriticalphilosophy.

A I



I. LAKATOS

? 5. Criticismof the Proofby Counterexampleswhich are Globalbut notLocal.

(a)The role of refutationsin

proof-elaborationand theorem-

formation. The allianceof proofandrefutations.

(b)Reductivestructures.

(c) Fallibilism.

?6. Concept-formation.

Introduction

IT

frequentlyhappens

in the

history

of thought that when a powerfulnew methodemerges the studyof thoseproblemswhich can be dealt

with by the new method advancesrapidlyand attractsthe limelight,while the rest tendsto be ignoredor even forgotten,its studydespised.

This situationseemsto have arisenin our centuryin the Philosophyof Mathematicsas a result of the dynamic development of meta-

mathematics.

The subjectmatterof metamathematicsis an abstractionof mathe-

maticsin which mathematicaltheoriesarereplacedby formalsystems,

proofs by certainsequencesof well-formed formulae,definitionsby

'abbreviatory devices' which are 'theoretically dispensable' but

'typographically convenient '. This abstraction was devised byHilbert to providea powerful techniquefor approachingsome of the

problemsof the methodologyof mathematics. At the sametime there

are problems which fall outside the range of metamathematical

abstractions. Among these are all problems relating to informal

(inhaltliche)mathematicsandto its growth, and all problemsrelatingto

the situationallogic of mathematicalproblem-solving.I shallreferto the schoolof mathematicalphilosophywhichtendsto

identify mathematicswith its metamathematicalabstraction(and the

philosophyof mathematicswith metamathematics)as the ' formalist'

school. One of the cleareststatementsof the formalistposition is to

be found in Carnap[I937].2 Camapdemandsthat (a) 'philosophy is

to be replaced by the logic of science . . . ', (b) ' the logic of science is

nothing otherthanthe logical syntaxof the languageof science . . .

1 Church[1956]I, pp. 76-77. Also cf. Peano[1894],p. 49 andWhitehead-Russell

[I9I0-I3], I, p. 12. Thisis anintegralpartof the Euclideanprogrammeas formulated

by Pascal[I657-58]: cf. Lakatos[I962], p. I58.2 For full detailsof this andsimilarreferencessee the list of works at the end of Part

IIIof this article.

2



PROOFS AND REFUTATIONS (I)

(c)' metamathematicsis thesyntaxof mathematicallanguage'(pp.xiiiand9). Or: philosophyof mathematicsis to be replacedby meta-

mathematics.Formalismdisconnectsthehistoryof mathematicsfromthephilo-sophyof mathematics,since,accordingto the formalistconceptof

mathematics,thereis nohistoryofmathematicsproper. Anyformalistwouldbasicallyagreewith Russell's'romantically'putbutseriouslymeantremark,accordingto whichBoole'sLawsofThought(1854)was'the firstbookeverwrittenon mathematics'.1Formalismdeniesthestatusof mathematicsto mostof whathasbeencommonlyunderstoodto be

mathematics,and can

saynothingaboutits

growth.None of

the 'creative' periodsand hardlyany of the 'critical' periodsofmathematicaltheorieswould be admittedinto the formalistheaven,wheremathematicaltheoriesdwelllike theseraphim,purgedof all the

impuritiesof earthlyuncertainty.Formalists,though,usuallyleave

opena smallbackdoorforfallenangels:if it turnsoutthatforsome'mixturesof mathematicsandsomethingelse' we can find formal

systems' whichincludethemin a certainsense', thentheytoo maybeadmitted

(Curry[1951],pp.

56-57).On thosetermsNewton hadto

wait four centuriesuntilPeano,Russell,andQuinehelpedhim intoheaven by formalisingthe Calculus. Dirac is more fortunate:Schwartzsaved his soul duringhis lifetime. Perhapswe shouldmentionherethe paradoxicalplightof the metamathematician:byformalist,or even by deductivist,standards,he is not an honestmathematician.Dieudonnetalksabout'theabsolutenecessityimposedon anymathematicianwhocaresfor intellectualintegrity'[myitalics]to

present his reasonings in axiomatic form ([I939], p. 225).Under the present dominance of formalism, one is tempted to

paraphraseKant: the historyof mathematics,lackingthe guidanceof

philosophy, has become blind,while the philosophy of mathematics,

turningits back on the most intriguing phenomenain the history of

mathematics,hasbecome empty.' Formalism' is abulwarkof logicalpositivistphilosophy. Accord-

ing to logical positivism, a statement is meaningful only if it is

'tautological' or empirical. Since informal mathematics is neither

1B. Russell[190o]. TheessaywasrepublishedasChapterV of Russell's[I9I8],underthetitle'MathematicsandtheMetaphysicians'. IntheI953 PenguinEditionthequotationcanbe foundonp. 74. Intheprefaceof [9I8] Russellsaysof theessay:'Its toneispartlyexplainedby thefactthattheeditorbeggedme to makethe article

" as romantic as possible".'

3



I. LAKATOS

'tautological'norempirical,it mustbe meaningless,sheernonsense.1The dogmasof logicalpositivismhavebeendetrimentalto thehistoryand

philosophyofmathematics.

Thepurposeof theseessaysis to approachsomeproblemsof the

methodologyofmathematics.I usetheword'methodology' in a senseakin to Polya'sand Bernays''heuristic'2 and Popper's'logic of

discovery' or ' situationallogic '.3 The recent expropriationof the

term 'methodology of mathematics'to serve as a synonymfor'metamathematics'hasundoubtedlya formalisttouch. It indicatesthat in formalistphilosophyof mathematicsthereis no properplacefor

methodologyqualogicof

discovery.4 Accordingto

formalists,1

According to Turquette, G6delian sentences are meaningless ([1950], p. 129).

Turquetteargues againstCopi 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 theydiscussthe statusof informalmathematicsin a particularcase. Neither do they notice

that theorems of informal mathematicsare surely guesses, which one can hardly

classifydogmatist-wiseas ' apriori' and ' a posteriori' guesses.2 P6lya [I945], especially p. I02, and also [I954], [1962a]; Bemays [I947], esp.

p. I873 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]andTarski[193ob].In thefirst

paperTarskiusesthe term' deductivesciences' explicitlyasashorthandfor ' formalised

deductive sciences'. He says: ' Formaliseddeductive disciplinesform the field of

researchof metamathematicsroughly in the same sense in which spatialentities

form the field of researchin geometry.' This sensible formulation is given an

intriguingimperialisttwist in the second

paper:' The deductive

disciplinesconstitute

the subject-matterof the methodology of the deductive sciencesin much the same

sense in which spatialentities constitutethe subject-matterof geometry and animals

that of zoology. Naturally not all deductive disciplinesare presentedin a form

suitable for objectsof scientificinvestigation. Those, for example, are not suitable

which do not reston a definitelogicalbasis,have no preciserules of inference,andthe

theorems of which are formulatedin the usually ambiguous and inexact terms of

colloquiallanguage-in a word those which are not formalised. Metamathematical

investigationsareconfinedin consequenceto the discussionof formaliseddeductive

disciplines.' The innovation is that while the first formulation stated that the

subject-matterof metamathematicsis the formaliseddeductivedisciplines,the secondformulationstatesthatthesubject-matterof metamathematicsisconfinedto formalised

deductivedisciplinesonly becausenon-formaliseddeductive sciencesare not suitable

objectsfor scientificinvestigationat all. This impliesthatthe pre-historyof a forma-

lised disciplinecannot be the subject-matterof a scientificinvestigation-unlike the

pre-historyof a zoological species,which canbe the subject-matterof a very scientific

4




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 programmedTuring 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 betweenthe

rationalism of a machine

and the irrationalism of blind guessing does not hold for live mathe-

matics: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 recognisedand still less, stimulated, by the formalist philosophy.

The history of mathematics and the logic of mathematical discovery,

theoryof evolution. Nobodywill doubtthatsomeproblemsabouta mathematical

theorycanonly be approachedafterit hasbeenformalised,justas someproblemsabouthumanbeings(sayconcerningtheiranatomy)canonly be approachedaftertheirdeath. Butfewwill inferfromthisthathumanbeingsare' suitableforscientific

investigation'onlywhentheyare'presentedin " dead" form', andthatbiologicalinvestigationsareconfinedin consequenceto thediscussionof deadhumanbeings-although,I shouldnot be surprisedif someenthusiasticpupilof Vesaliusin those

gloriousdaysofearlyanatomy,whenthepowerfulnewmethodofdissectionemerged,hadidentifiedbiologywith theanalysisof deadbodies.

In the prefaceof his [1941]Tarskienlargeson hisnegativeattitudetowardsthe

possibilityof

anysortof

methodologyotherthanformal

systems:' A coursein the

methodologyof empiricalsciences. . . mustbelargelyconfinedto evaluationsandcriticismsof tentativegropingsandunsuccessfulefforts.' Thereasonisthatempiricalsciencesareunscientific:for Tarskidefinesa scientifictheory' as a systemof assertedstatementsarrangedaccordingto certainrules'(Ibid.).

1 Oneof themostdangerousvagariesof formalistphilosophyis thehabitof (I)

statingsomething-rightly-aboutformalsystems;(2) thensayingthatthisappliesto ' mathematics'-this is againrightif we accepttheidentificationof mathematicsandformalsystems;(3) subsequently,withasurreptitiousshiftinmeaning,usingtheterm ' mathematics' in the ordinarysense. So Quine says([I951],p. 87), that ' this

reflectsthe characteristicmathematicalsituation:the mathematicianhits uponhisproof by unregimentedinsight and good fortune, but afterwardsother mathema-

ticianscan check his proof'. But often the checking of an ordinaryproof is a verydelicateenterprise,and to hit on a ' mistake' requiresas much insightand luck as to

hit on a proof: the discoveryof' mistakes' in informal proofs may sometimes take

decades-if not centuries.

5



I. LAKATOS

i.e. the phylogenesisand the ontogenesisof mathematicalthought,lcannotbe developedwithout the criticismandultimaterejectionof

formalism.But formalistphilosophyof mathematicshasvery deeproots. Itisthe latestlink in the long chainof dogmatistphilosophiesof mathe-matics. Formorethan two thousandyearstherehas been an argu-ment betweendogmatistsandsceptics.The dogmatistshold that-bythepowerof our humanintellectand/orsenses-we can attaintruthandknow that we haveattainedit. The scepticson the otherhandeitherhold thatwe cannotattainthetruthatall (unlesswith thehelpof

mysticalexperience),or thatwe cannotknowif we can attainit or

thatwe haveattainedit. Inthisgreatdebate,in whichargumentsaretime andagainbroughtup-to-date,mathematicshas been the proudfortressof dogmatism. Wheneverthe mathematicaldogmatismoftheday got intoa 'crisis', anew versiononceagainprovidedgenuinerigour and ultimatefoundations,thereby restoringthe image of

authoritative,infallible,irrefutablemathematics,'the only SciencethatithaspleasedGodhithertotobestowonmankind'(Hobbes[I65i],

p. I5).Most

scepticsresignedthemselvesto the

impregnabilityof this

strongholdof dogmatistepistemology.2A challengeis now overdue.Thecoreof thiscase-studywill challengemathematicalformalism,

but will not challengedirectlytheultimatepositionsof mathematical

dogmatism. Itsmodestaimis to elaboratethe pointthatinformal,

quasi-empirical,mathematicsdoes not grow througha monotonousincreaseof thenumberof indubitablyestablishedtheoremsbutthroughtheincessantimprovementof guessesby speculationandcriticism,bythelogicof proofsandrefutations. Sincehowevermetamathematicsis a paradigmof informal,quasi-empiricalmathematicsjust now in

rapidgrowth,the essay,by implication,will alsochallengemodemmathematicaldogmatism. The studentof recenthistoryof meta-mathematicswillrecognisethepatternsdescribedhereinhisownfield.

Both H. Poincare and G. Polya propose to apply E. Haeckel's 'fundamental

biogeneticlaw ' about ontogeny recapitulatingphylogeny to mentaldevelopment,in

particularto 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 ancestorsthroughout geologic time. It seems it is the samein the developmentofminds. ...

For this reason, the history of science should be our first guide' (C. B. Halsted's

authorisedtranslation,p. 437).2 Fora discussionof the role of mathematicsin the dogmatist-scepticcontroversy,

cf. my [I962].

6




The dialogue form should reflect the dialecticof the story; it is

meant to contain a sort of rationallyreconstructedor ' distilled'history.

The realhistorywill chimein in thefootnotes,mostof whichareto betaken,therefore,as an organicpartof theessay.

I A Problemanda ConjectureThe dialoguetakesplacein animaginaryclassroom. The classgets

interestedin a PROBLEM: is there a relationbetween the numberof

verticesV, the number of edgesE and the numberof facesF of poly-

hedra-particularly of regularpolyhedra-analogous to the trivial

relationbetween the numberof verticesandedgesofpolygons,namely,that there are as many edges as vertices: V=E? This latterrelation

enables us to classifypolygonsaccordingto the number of edges (or

vertices): triangles, quadrangles, pentagons, etc. An analogousrelation would help to classifypolyhedra.

Aftermuch trial anderrortheynotice thatfor all regularpolyhedraV- E+ F= 2.1 Somebody guesses that this may apply for any

1Firstnoticed by Euler [I750]. His original problem was the classificationof

polyhedra,the

difficultyof which was

pointedout in the editorial

summary:' While

in plane geometry polygons(figuraerectilineae)could be classifiedvery easily accord-

ing to the number of their sides, which of course is always equal to the number

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

besidesthe number of faces the number of points and lines on the surface of the

polyhedron determinesits (topological) character. It is interestingthat on the one

hand he was eagerto stressthe novelty of his conceptualframework,and thathe had

to invent the term ' acies' (edge) instead of the old ' latus' (side), since latuswas apolygonal 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

beenrecentlygenerallyacceptedthatthe priorityof 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 rediscoveredand published by Foucher de

Careil in i860. The priority should not be granted to Descartes without a minor

qualification. It is true that Descartesstatesthat the number of plane anglesequals2S + 2a - 4 where by bhe means the numberof faces andby a the numberof solid

angles. It is also truethathe statesthat thereare twice as many planeanglesas edges

(latera). The trivial conjunctionof these two statements of course yields the Eulerformula. But Descartes did not see the point of doing so, since he still thought in

termsof angles(planeandsolid)andfaces,and did not make a consciousrevolutionarychange to the conceptsof o-dimensionalvertices, i-dimensional edges and 2-dimen-

sional facesas a necessaryand sufficientbasisfor the full topologicalcharacterisationof

polyhedra.

7



I. LAKATOS

polyhedron whatsoever. Others try to falsify this conjecture,try to test it in manydifferentways-it holds good. The results

corroboratetheconjecture,and suggestthat it could be proved. It isat this point-after thestagesproblemand conjecture-thatwe enterthe classroom.1The teacherisjustgoingto offeraproof

2. A Proof

TEACHER: In our last lesson we arrived at a conjecture concerning

polyhedra,namely,that for all polyhedraV-- E+ F= 2, where V is

the numberof vertices,E the number of

edges

and F the number of

faces. We testedit by variousmethods. But we haven'tyet provedit. Hasanybody 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 hasbeen establishedin so many cases,there canbe no doubt

that it holds good for any solid. Thus the propositionseems to be

satisfactorilydemonstrated.'2 But if you have a proof, please do

present it.

TEACHER: In factI haveone. It consistsof the following thought-

experiment. Stepi: Let usimaginethepolyhedronto be hollow, with

a surfacemadeof thin rubber. If we cut out one of the faces,we can

stretchthe remainingsurfaceflaton the blackboard,without tearingit.

The faces and edgeswill be deformed,the edges may become curved,but V,E and F will not alter,so thatif andonly if V-E + F = 2 for

the original polyhedron, then V- E+ F= I for this flat network-

remember that we have removed one face. (Fig. I shows the flat

networkfor the caseof a cube.) Step2: Now we triangulateour map-it does indeed look like a geographicalmap. We draw (possibly

curvilinear)diagonalsin those (possiblycurvilinear)polygons which

1Eulertestedtheconjecturequitethoroughlyforconsequences.He checkeditfor prisms,pyramidsand so on. He could have addedthatthe propositionthatthereareonlyfiveregularbodiesis also a consequenceof theconjecture.Another

suspectedconsequenceis thehithertocorroboratedpropositionthatfour coloursaresufficientto coloura map.

The phaseof conjecturingand testingin the caseof V-- E + F= 2 is discussedinP6lya([I954],Vol. I, thefirstfive sectionsof thethirdchapter,pp. 35-4I). Polyastoppedhere,anddoesnotdealwiththephaseofproving-thoughof coursehepointsout theneedforaheuristicof' problemsto prove

'([1945],p. I44). Ourdiscussion

startswherePolyastops.2 Euler([I750], p. 119 andp. 124). But later [I751]he proposeda proof.

8




are not already(possiblycurvilinear)triangles. By drawingeach

diagonalwe increasebothE andF by one,sothatthe totalV-- E+ F

will not be altered(Fig. 2). Step3: From the triangulatednetworkwe now removethetrianglesoneby one. To removea triangleweeitherremoveanedge-upon whichone face andone edgedisappear(Fig.3a),or we removetwo edgesandavertex-upon whichoneface,twoedgesandonevertexdisappear(Fig.3b). Thusif V- E+ F= I

v1 \ '

FIG. I FIG. 2

FIG.3a FIG.3b

beforeatriangleisremoved,it remainssoafterthetriangleisremoved.At the end of this procedurewe get a single triangle. For thisV- E-+ F = I holdstrue. Thuswe haveprovedour conjecture.l

PUPILDELTA: You shouldnow callit a theorem.Thereisnothing

conjecturalaboutit anymore.2PUPILALPHA: I wonder. I see that this experimentcan be per-

formedfora cubeor for a tetrahedron,buthow amI to knowthatitcan be performedfor any polyhedron?Forinstance,areyou sure,Sir,thatanypolyhedron,afterhavingafaceremoved,canbestretchedflatontheblackboard?I am dubiousaboutyourfirststep.

1This proof-ideastemsfrom Cauchy [I8I].2 Delta's view that this proof has establishedthe 'theorem' beyond doubt was

sharedby many mathematiciansin the nineteenthcentury, e.g. Crelle [I826-27], II,pp. 668-671, Matthiessen[1863], p. 449, Jonquieres[89goa]and [I89ob]. To quote acharacteristicpassage: ' After Cauchy'sproof, it became absolutelyindubitablethatthe elegant relation V+ F = E+ 2 appliesto all sorts of polyhedra,just as Eulerstatedin 1752. In I8II all indecision should have disappeared.' Jonquieres[I89oa],pp. III-II2.

9



I. LAKATOS

PUPILBETA: Are you sure that in triangulatingthe mapone will

always get a newfacefor any new edge? I am dubious about yoursecond

step.PUPIL GAMMA: Are you sure that there are only two alternatives-

thedisappearanceof oneedgeor elseof two edgesanda vertex-when one

dropsthetrianglesoneby one? Are you even sure that one is leftwitha singletriangleat the end of thisprocess? I am dubious about your

-third step.1TEACHER:Of course I am not sure.

ALPHA:But then we are worse off than before! Insteadof one

conjecturewe now have at leastthree! And this

youcall a '

proof'!TEACHER:I admit that the traditional name 'proof' for this

thought-experimentmay rightly be considereda bit misleading. I

do not think that it establishesthe truthof the conjecture.DELTA: What does it do then? What do you thinka mathematical

proof proves?TEACHER: This is a subtlequestionwhich we shall try to answer

later. Till then I proposeto retain the time-honouredtechnicalterm

'proof' for athought-experiment-or'quasi-experiment'-whichsuggestsa decompositionof theoriginalconjectureintosubconjecturesor lemmas,thus

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,certainlydid not even

dreamof this.2

1The classis a ratheradvanced one. To Cauchy, Poinsot, and to many other

excellent mathematiciansof the nineteenthcentury

thesequestions

did not occur.

2 Thought-experiment(deiknymi)was the most ancientpattern of mathematical

proof. It prevailedin pre-EuclideanGreekmathematics(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 pre-cedence of 'analysis' over 'synthesis'. (For an excellent discussionsee Robinson

[1936].) Accordingto Proclos, ... it is . . . necessaryto know beforehandwhatis sought' (Heath [1925], I, p. I29). 'They said that a theorem is that which is

proposed with a view to the demonstrationof the very thing proposed '-says

Pappus(ibid. I, p. IO). The Greeks did not think much of propositionswhich they

happenedto hit upon in the deductive direction without having previously guessedthem. They called them porisms,corollaries,incidental resultsspringingfrom 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

Proclussays,a sort of windfall (ermaion)or bonus(kerdos)(ibid. I, p. 278). We read

in the editorialsummaryto Euler[1753]that arithmeticaltheorems 'were discovered

I0



I. LAKATOS

is false-and not only in the case of the cube,but for all polyhedraexceptthe tetrahedron,in the flat networkof whichall the triangles

areboundarytriangles. Yourproofthusprovesthe Eulertheoremfor the tetrahedron.But we alreadyknewthatV- E+ F= 2 forthetetrahedron,so why proveit?

TEACHER:You are right. But notice that the cube which is a

counterexampleto the thirdlemmais not also a counterexampleto

the mainconjecture,sincefor the cubeV- E+ F= 2. You have

shownthepovertyof theargument-theproof-but not thefalsityof

ourconjecture.

ALPHA: Will you scrapyourproofthen?TEACHER:No. Criticismis not necessarilydestruction. I shall

improvemy proofso thatit will standupto the criticism.GAMMA: How?

TEACHER:Before showing how, let me introducethe following

terminology. I shallcalla 'localcounterexample'an examplewhich

refutesa lemma(withoutnecessarilyrefutingthe mainconjecture),andI shallcalla 'globalcounterexample'anexamplewhichrefutesthe

mainconjecture

itself Thusyour counterexample

is local but not

global. A local,but not global,counterexampleis a criticismof the

proof,butnot of theconjecture.GAMMA:So, the conjecturemay be true, but your proof doesnot

proveit.TEACHER:But I caneasilyelaborate,improvetheproof,by replacing

the falselemmaby a slightlymodifiedone, which your counter-

examplewill not refute. I no longercontendthattheremovalof any

trianglefollowsone

ofthetwo

patternsmentioned,but

merelythatateach

stageoftheremovingoperationtheremovalof anyboundarytrianglefollowsoneof thesepatterns.Comingbackto my thought-experiment,all

thatI haveto doisto insertasinglewordinmythirdstep,to wit, that' fromthetriangulatednetworkwe now removetheboundarytrianglesoneby one'. You willagreethatit onlyneededatriflingobservation

to puttheproofright.1GAMMA:I do not thinkyour observationwas so trifling; in factit

wasquiteingenious. To makethisclearI shallshowthatit is false.

Takethe flatnetworkof the cubeagainandremoveeightof theten

1Lhuilier,when correctingin a similarway a proof of Euler, says that he made

only a 'trifling observation' ([I8I2-I3], p. I79). Euler himself, however, gave the

proof up, sincehe noticed the troublebut could not makethat' triflingobservation'.

I2




trianglesin the order given in Fig. 4. At the removal of the eighth

triangle,which is certainlyby then a boundarytriangle,we removed

two edgesandno vertex-this changesV- E+ F by I. And we areleft with the two disconnectedtriangles9 and Io.

\/67 ?,/

6 \

FIG.4

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

boundarytrianglea trianglewhose removal does not disconnectthe

network. But intellectualhonesty prevents me from making sur-

reptitiouschangesin my position by sentencesstartingwith 'I meant

. .' so I admit that now I must replacethe second version of the

triangle-removingoperationwith a thirdversion: thatwe remove the

trianglesone

byone in sucha

waythat V- E

+F does not alter.

KAPPA: I generouslyagree 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 networkcan

be sonumberedthatinremovingthemintherightorderV- E+- F will not

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 trianglesin any order. Your modified thought-experi-ment gave the instruction: remove boundary trianglesin any order.

Now you say we should follow a definiteorder,but you do not saywhich and whetherthat orderexistsat all. Thus the thought-experi-ment breaksdown. You improvedthe proof-analysis,i.e. the list of

lemmas; but the thought-experimentwhich you called 'the proof'hasdisappeared.

RHO: Only the thirdstephasdisappeared.

1Cauchythought that the instructionto find at eachstagea trianglewhich canbe

removed either by removing two edges and a vertex or one edge can be triviallycarriedout for any polyhedron([i8II], p. 79). This is of courseconnected with his

inabilityto imagine a polyhedronthat is not homeomorphicwith the sphere.

I3



I. LAKATOS

KAPPA: Moreover, did you improvethe lemma? Your first two

simpleversionsatleastlookedtriviallytruebeforetheywererefuted;

yourlengthy,patchedupversiondoesnot evenlook plausible. Canyou reallybelievethatit will escaperefutation?

TEACHER:'Plausible' or even 'trivially true' propositionsare

usuallysoonrefuted:sophisticated,implausibleconjectures,maturedin

criticism,mighthit on the truth.OMEGA: And what happens if even your 'sophisticated con-

jectures' are falsifiedand if this time you cannotreplacethem byunfalsifiedones? Or, if you do notsucceedin improvingthe argu-

mentfurtherbylocalpatching? You havesucceededingettingoveralocalcounterexamplewhich wasnot globalby replacingthe refutedlemma. Whatif you do not succeednexttime?

TEACHER:Good question-it will be put on the agenda for to-

morrow.

4. CriticismoftheConjecturebyGlobalCounterexamplesALPHA:Ihaveacounterexamplewhich will falsifyyourfirstlemma

-but thiswill also be acounterexample

to the mainconjecture,

i.e.this will be a globalcounterexampleas well.

TEACHER:Indeed! Interesting. Let us see.

ALPHA:Imaginea solid boundedby a pairof nestedcubes-a pairof cubes,oneof whichis inside,but doesnot touchthe other(Fig.5).

i,,,~ J

* I

FIG.5

Thishollow cubefalsifiesyour firstlemma,becauseon removingafacefrom theinnercube,thepolyhedronwillnot bestretchableon toaplane. Norwill ithelpto removeafacefromthe outercubeinstead.

Besides,for eachcubeV- E+ F= 2, so thatfor the hollow cube

V-E+F=4.TEACHER:Good show. Let us call it Counterexample1.1 Now

what?

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

Gergonne,theEditor,added(p. I86) thathe himselfnoticed thislong beforeLhuilier's

I4




(a) Rejectionof theconjecture.Themethodofsurrender

GAMMA:Sir, your composure baffles me. A single counter-

examplerefutesa conjectureas effectivelyas ten. The conjectureandits proof have completely misfired. Hands up! You have to sur-

render. Scrapthe falseconjecture,forget about it and try a radicallynew approach.

TEACHER:I agreewith you that the conjecturehas received a severe

criticismby Alpha'scounterexample. But it is untrue that the proofhas 'completely misfired'. If, for the time being, you agree to myearlierproposalto use the word 'proof' for a 'thought-experiment

which leads to decomposition of the original conjectureinto sub-conjectures', insteadof using it in the senseof a ' guaranteeof certain

truth', you need not draw this conclusion. My proof certainly

proved Euler'sconjecturein the first sense,but not necessarilyin the

second. You are interestedonly in proofs which 'prove' what theyhave set out to prove. I am interestedin proofs even if they do not

accomplishtheirintendedtask. Columbus did not reachIndiabut he

discoveredsomethingquite interesting.

ALPHA: So accordingto your philosophy-while a local counter-example(if it is not global at the sametime) is a criticismof the proof,but not of the conjecture-a globalcounterexampleis a criticismof the

conjecture,but not necessarilyof the proof. You agreeto surrender

as regards the conjecture,but you defend the proof. But if the

conjectureis false,what on earth does the proof prove?GAMMA: Your analogywith Columbus breaksdown. Accepting

a global counterexamplemust mean total surrender.

(b) Rejectionof the counterexample.The methodof monster-barringDELTA: But why accept the counterexample? We proved our

conjecture-now it is a theorem. I admit that it clasheswith 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 pairof

paper. Not so Cauchy,who publishedhis proof just a year before. And this

counterexamplewas to be rediscoveredtwenty yearslater by Hessel ([I832], p. I6).BothLhuilierandHesselwere led to theirdiscoveryby mineralogicalcollectionsinwhichtheynoticedsomedoublecrystals,wheretheinnercrystalis not translucent,butthe outeris. Lhuilieracknowledgesthestimulusof thecrystalcollectionof hisfriend ProfessorPictet([I8I2-I3], p. i88). Hessel refersto lead sulphidecubesenclosedin translucentcalcium fluoridecrystals([i832], p. i6).

I5



I. LAKATOS

nestedcubesis not a polyhedronat all. It is a monster,a pathologicalcase,not a counterexample.

GAMMA:Why not? A polyhedronis a solidwhosesurfaceconsistsofpolygonalfaces. And my counterexampleis 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. 2.2DELTA: So really you showed us two polyhedra-two surfaces, one

completelyinsidethe 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 yourefute our

conjecturenow if

by polyhedronwe 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

1Definition1 occursfirst in the eighteenthcentury; e.g.:

' One gives the name

polyhedralsolid,or simply polyhedron,to any solid bounded by planesor planefaces'

(Legendre[I794], p. I60). A similardefinitionis 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, SecondProblem,Prop. I7).2 We find Definition2 implicitly in one of Jonquieres'papersread to the French

Academy againstthose who meant to refute Euler'stheorem. These papersare a

thesaurusof monsterbarringtechniques. He thundersagainstLhuilier'smonstrous

pairof nestedcubes: ' Sucha systemis not reallya polyhedron,but a pairof 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; herethisis not the case . . . This firstexception

ofLhuilier can thereforebe discarded'([89gob],p. I70). This definition-as opposedto Definition I-goes down very well with analyticaltopologistswho arenot inter-

estedat all in the theory of polyhedraas suchbut as a handmaidenfor the theory of

surfaces.3 Counterexamples2a and 2b were missedby Lhuilierand firstdiscoveredonly by

Hessel ([I832], p. 13).

i6




DELTA:I admireyour pervertedimagination,but of courseI did

not mean thatanysystemof polygonsis apolyhedron. By polyhedron

I meant a systemofpolygonsarrangedin sucha way that(1) exactlytwopolygonsmeetateveryedgeand(2) it ispossibletogetfromtheinsideofany

polygonto the insideofanyotherpolygonbya routewhichnevercrossesany

edgeat a vertex. Your firsttwinswill be excludedby the first criterion

in my definition,your secondtwins by the secondcriterion.

FIG.6a FIG.6b

TEACHER:Def: 3.1

ALPHA: I admire your perverted ingenuity in inventing onedefinition 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.

1Definition3 first turnsup to keep out twintetrahedrain M6bius ([i865], p. 32).

We find his cumbersome definition reproducedin some modem textbooks in the

usualauthoritarian' take it or leave it' way; the story of its monsterbarringback-

ground-that would at leastexplainit-is not told (e.g. Hilbert-CohnVossen [I956],

p. 290).2 DefinitionP accordingto which Euleriannesswould be a definitional character-

istic of polyhedra was in fact suggested by R. Baltzer: 'Ordinary polyhedra are

occasionally (following Hessel) called Eulerian polyhedra. It would be moreappropriateto find a specialname for non-genuine (uneigentliche)polyhedra' ([I860],Vol. II, p. 207). The referenceto Hessel is unfair: Hesselused the term 'Eulerian'

simply as an abbreviationfor polyhedrafor which Euler'srelationholds in contra-

distinction to the non-Eulerian ones ([1832], p. I9). For Def. P see also the Schlifli

quotation in footnote pp. I8-I9.

B 17




DELTA:Why do you think thatyour 'urchin ' is apolyhedron?GAMMA:Do younot see? Thisis a polyhedron,whose facesare

the twelve star-pentagons.It satisfiesyour lastdefinition:it is 'asystemof polygons arrangedin such a way that (I) exactlytwo

polygonsmeet at everyedge,and(2)it is possibleto get fromeverypolygonto everyotherpolygonwithoutevercrossinga vertexof the

pclyhedron'DELTA:But thenyou do not even knowwhata polygonis! A

star-pentagonis certainlynot a polygon! A polygonis a systemofedgesarrangedinsucha waythat(1)exactlytwoedgesmeetateveryvertex,and

(2)the

edgeshaveno

pointsin common

exceptthe vertices.

TEACHER: Let us call this Def:4.GAMMA: I don't seewhy you includethe second clause. The

rightdefinitionof thepolygonshouldcontainthe firstclauseonly.TEACHER:Def. 4'.GAMMA:The second clausehasnothing to do with theessenceof

apolygon. Look: if I liftanedgealittle,thestar-pentagonisalreadyapolygoneveninyoursense. Youimagineapolygonto bedrawninchalkon the blackboard,but

youshould

imagineit as a wooden

structure:thenit is clearthatwhatyouthinkto beapointin commonis not reallyone point,but two differentpointslying one abovetheother. Youaremisledby yourembeddingthepolygonin a plane-you shouldlet its limbsstretchoutin space!1

Poinsotindependentlyrediscoveredit,andit washewhopointedoutthattheEulerformuladidnot applyto it ([1809],p. 48). Thenow standardterm'smallstellated

polyhedron'is Cayley's([I859],p. 125). Schlafliadmittedstar-polyhedraingeneral,butnevertheless

rejectedoursmallstellateddodecahedronasa monster.

Accordingto him 'this is not a genuinepolyhedron,for it does not satisfythe conditionV- E + F= 2'([I852], ? 34).

1Thedisputewhetherpolygonshouldbe definedso as to includestar-polygonsor not(Def.4orDef.4')is averyoldone. Theargumentputforwardin ourdialogue-that star-polygonscanbe embeddedas ordinarypolygonsin a spaceof higherdimensions-isamodemtopologicalargument,butonecanputforwardmanyothers.ThusPoinsotdefendinghisstar-polyhedraarguedforthe admissionof star-polygonswithargumentstakenfromanalyticalgeometry:'... allthesedistinctions(between" ordinary" and"star"-polygons)aremoreapparentthanreal,andtheycompletely

disappearin the analyticaltreatment,in whichthevariousspeciesof polygonsarequiteinseparable.To theedgeof a regularpolygontherecorrespondsanequationwithrealroots,whichsimultaneouslyyieldstheedgesof alltheregularpolygonsofthe sameorder. Thusit is not possibleto obtainthe edgesof a regularinscribed

heptagon,withoutat thesametimefindingedgesofheptagonsof thesecondandthird

species. Conversely,giventheedgeof a regularheptagon,onemaydeterminethe

I9



I. LAKATOS

DELTA: 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 yourselfwho said that a polyhedronhas

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 agreedthat a polyhedron is a closed surfacewith edges and vertices-then

why not agreethat a polygon is simply a closed curve with vertices?

But if you stick to your ideaI am willing to definethe area of a star-

polygon.1TEACHER: Let us leave this

disputefor a

moment,and

proceedas

before. Consider the last two definitionstogether-Def. 4 and Def.4. Can anyone give a counterexampleto our conjecturethat will

comply with bothdefinitionsof polygons?ALPHA: Here is one. Considera picture-framelike this (Fig. 9).

This is a polyhedronaccordingto any of the definitionshithertopro-posed. Nonethelessyou will find, on countingthe vertices,edgesand

faces, that V- E+ F = o.

radius of a circle in which it can be inscribed,but in so doing, one will find threedifferent circlescorrespondingto the three speciesof heptagon which may be con-

structedon the given edge; similarlyfor other polygons. Thus we arejustified in

giving the name " polygon " to thesenew starredfigures' ([1809], p. 26). Schrbder

uses the Hankelian argument: 'The extension to rational fractions of the power

concept originallyassociatedonly with the integershasbeen very fruitfulin Algebra;this suggeststhat we try to do the samething in geometry wheneverthe 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.1Gamma'sclaimthathe candefinethe areafor

star-polygonsis not abluff. Some

of those who defendedthe wider concept of polygon solved the problem by puttingforward a wider conceptof the area of polygon. Thereis an especiallyobvious wayto do this in the case of regularstar-polygons. We may take the area of a polygonas the sum of the areasof the isoscelestriangleswhichjoin the centreof the inscribed

or circumscribedcircle to the sides. In this case, of course,some ' portions' of the

star-polygonwill count more thanonce. In the case of irregularpolygons where we

have not got any one distinguishedpoint, we may still take anypoint as origin and

treatnegativelyorientedtrianglesashavingnegativeareas(Meister[1769-70], p. I79).It turnsout-and this can certainlybe expected from an 'area '-that the areathus

definedwill not depend on the choice of the origin (M6bius [1827], p. 218). Ofcoursethereis liable to be a disputewith those who think that one is not justifiedin

callingthe numberyielded by this calculationan' area'; though the defendersof the

Meister-Mobiusdefinitioncalled it ' therightdefinition' which ' aloneis scientifically

justified' (R. Haussner'snotes [1906], pp. 114-115). Essentialismhas been a per-manentfeature of definitionalquarrels.

20




TEACHER: Counterexample4.1

BETA: 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 wastedour time.

ALPHA: Delta, I am flabbergasted. You say nothing? Can't youdefine 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!

/l7

FIG.9

DELTA: You should really find a more appropriate name for yournon-Eulerian pests and not mislead us all by calling them ' polyhedra '.

But I am gradually losing interest in your monsters. I turn in disgustfrom 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 ir-

reconcilable.

1We findCounterexample4too in Lhuilier'sclassical[1812-I3], on p. 185-Gergonne

againaddedthat he knewit. But Grunertdid not know it fourteenyearslater

([I827]) and Poinsot forty-five yearslater([1858],p. 67).2Thisis paraphrasedfromaletterof Hermite'swrittento Stieltjes:'I turnaside

with a shudderof horrorfromthislamentableplagueof functionswhich havenoderivatives' ([I893]).

3' Researches dealing with . . . functions violating laws which one hoped were

universal,were regardedalmost as the propagationof anarchyand chaoswhere past

generations had sought order and harmony' (Saks [I933], Preface). Saks refershere to the fierce battlesof monsterbarrers(like Hermite!) and of refutationiststhat

characterised in the last decades of the nineteenth century (and indeed in thebeginningof the twentieth)the developmentof modernrealfunctiontheory,'thebranch of mathematicswhich dealswith counterexamples' (Munroe[I953], Preface).The similarlyfiercebattlethat ragedlaterbetween the opponentsand protagonistsof

moder mathematicallogic and set-theory was a direct continuation of this. Seealso footnote 2 on p. 24 and i on p. 25.

21




picture-frame,making two distinct, completely disconnected

polygons! (Fig. Io).

ALPHA: So what?DELTA: In the caseof a genuinepolyhedron,throughany arbitrary

point in spacetherewill be at least oneplanewhosecross-sectionwith the

polyhedronwill consistof one singlepolygon. In the case of convex

polyhedraall planeswill comply with this requirement,wherever we

take the point. In the caseof ordinaryconcavepolyhedrasome planeswill have more intersections,but therewill alwaysbe some that have

only one. (Figs. IIa and Iib.) In the case of thispicture-frame,if

we take thepoint

in thetunnel,

all theplanes

will have two cross-

sections. How then can you call this a polyhedron?TEACHER: This looks like anotherdefinition,this time an implicit

one. Call it Def. 5.1ALPHA: A seriesof counterexamples,a matchingseriesof defini-

tions, definitions that are alleged to contain nothing new, but to be

merely new revelationsof the richnessof that one old concept, which

seemsto have asmany ' hidden' clausesas therearecounterexamples.For

allpolyhedraV- E+ F= 2 seems

unshakable,an old and' eternal'

truth. It is strangeto thinkthat 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 leavestheclassroom.)DELTA: I cannotunderstandhow anable manlike Alphacanwaste

his talenton mereheckling. He seemsengrossedin the productionof

monstrosities. But monstrositiesnever foster growth, either in the

world of natureor in the world ofthought.

Evolutionalways

follows

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

1Definition5 wasputforwardbytheindefatigablemonsterbarrerE. deJonquieresto getLhuilier'spolyhedronwithatunnel(picture-frame)outof theway: ' Neitheris thispolyhedralcomplexa truepolyhedronin theordinarysenseof theword,forifone takesanyplanethroughanarbitrarypointinsideoneof the tunnelswhichpassright throughthesolid,theresultingcross-sectionwill be composedof two distinct

polygonscompletelyunconnectedwith eachother; this canoccurin an ordinarypolyhedronforcertainpositionsof theintersectingplane,namelyin thecaseof someconcavepolyhedra,butnot for all of them' ([89gob],pp. I70-I71). OnewonderswhetherdeJonquiereshasnoticedthathisDef.5excludesalsosomeconcavespheroidpolyhedra.

23



I. LAKATOS

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

DELTA:Anyway, Alpha has given up the struggle. No moremonstersnow.

GAMMA:I have a new one. It complieswith all the restrictionsin

Defs. I, 2, 3, 4, and 5, but V- E+ F= I. This Counterexample5is 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) exactlytwo polygons at every edgeand (2) it is possible to get from the inside of any polygon to the

insideof anyotherpolygon by a routewhich never crossesany edge ata vertex. Andyou haveto acceptthe facesasgenuinepolygons,asthey

comply with your requirements:(I) exactly two edgesmeet at everyvertexand(2)the edgeshaveno pointsin common exceptthe vertices.

DELTA:Alpha stretched concepts, but you tear them! Your

'edges' are not edges! An edgehas twovertices!

TEACHER: Def. 6?

GAMMA:But why deny the statusof' edge' to edgeswith one or

possiblyzero vertices? You used to contract

concepts,but now

youmutilate them so that scarcelyanythingremains!

DELTA:But don't you seethe futilityof these so-calledrefutations?

'Hitherto, when a new polyhedron was invented, it was for some

practicalend; today they are invented expresslyto put at fault the

reasoningsof our fathers,and one never will get from them anythingmore than that. Our subjectis turned into a teratologicalmuseum

where decent ordinarypolyhedramay be happy if they can retaina

verysmallcomer.'

1 'We must not forget that what appearsto-day as a monsterwill be to-morrow

the origin of a line of special adaptations.... I furtheremphasizedthe importanceof rarebut extremely consequentialmutationsaffectingrates of decisive embryonic

processeswhich might give rise to what one might term hopeful monsters,monsters

which would starta new evolutionaryline if fitting into some empty environmental

niche' (Goldschmidt[I933], pp. 544 and 547). My attentionwas drawnto thispaper

by KarlPopper.2 Paraphrased from Poincare ([1908], pp. 131-132). The original full text is this:

'Logic sometimes makes monsters. Sincehalf a centurywe have seen arise a crowdof bizarrefunctions which seem to try to resemble as little as possible the honest

functionswhich serve some purpose. No longer continuity, or perhapscontinuity,but no derivatives,etc. Nay more, from the logical point of view, it is thesestrangefunctions which are the most general, those one meets without seeking no longer

appearexcept as particularcases. There remainsfor them only a very smallcorner.

24




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

deep,we haveto studyit not in its 'normal', regular,usualform,but

in its criticalstate,infever,inpassion. Ifyouwanttoknowthenormalhealthybody, studyit when it is abnormal,when it is ill. If youwant to know functions,studytheirsingularities.If you want toknow ordinarypolyhedra,studytheirlunaticfringe. This is howone cancarrymathematicalanalysisinto theveryheartof thesubject.'But evenif you werebasicallyright,don'tyouseethefutilityof yourad hocmethod? If you want to drawa borderlinebetweencounter-

examplesandmonsters,you cannotdo it in fitsandstarts.TEACHER: I think

we shouldrefuseto acceptDelta'sstrategyfordealingwithglobalcounterexamples,althoughwe shouldcongratulatehim on hisskilfulexecutionof it. We couldaptlylabelhismethodthemethodofmonsterbarring.Usingthismethodone caneliminateany

counterexampleto the originalconjectureby a sometimesdeft but

alwaysadhocredefinitionof thepolyhedron,of its definng terms,orof the definingterms of its defining terms. We should some-how treatcounterexampleswith more respect,and not stubbornlyexorcisethem

by dubbingthem monsters. Delta'smainmistakeis

perhapshisdogmatistbiasin theinterpretationof mathematicalproof:he thinksthata proofnecessarilyproveswhat it hassetout to prove.My interpretationof proof will allow for a falseconjectureto be

'proved', i.e. to be decomposedinto subconjectures.If the con-

jectureisfalse,I certainlyexpectatleastoneof thesubconjecturesto befalse. But the decompositionmight stillbe interesting! I am not

perturbedat findinga counterexampleto a 'proved' conjecture;I am even

willingto set out to

'prove'a false

conjecture!THETA: I don't follow you.KAPPA: He just follows the New Testament: 'Prove all things;

hold fast that which is good' (I Thessalonians5: 2I).

(tobecontinued)'Heretofore when a new function was invented, it was for some practicalend;

to-day they are invented expresslyto put at fault the reasoningsof our fathers,and

oneneverwill get fromthemanythingmorethanthat.'If logicwerethesoleguideof theteacher,it would benecessaryto beginwith

themostgeneralfunctions,thatis to saywiththemost bizarre. It is thebeginnerthatwouldhaveto be set grapplingwith thisteratologicalmuseum. . .' (G. B.Halsted'sauthorisedtranslation,pp.435-436). Poincarediscussestheproblemwith

respectto the situationin the theoryof realfunctions-butthatdoesnot makeanydifference.

1Paraphrasedfrom Denjoy ([I919], p. 2I).

25

lakatos - proofs and refutations (i)

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