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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
Published by: Oxford University Press on behalf of The British Society for the Philosophy ofScienceStable URL: http://www.jstor.org/stable/685347
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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.
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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.
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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".'
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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
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PROOFS AND REFUTATIONS (I)
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.
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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].
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PROOFS AND REFUTATIONS (I)
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.
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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.
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PROOFS AND REFUTATIONS (I)
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.
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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
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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'.
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PROOFS AND REFUTATIONS (I)
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.
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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
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(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).
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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).
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PROOFS AND REFUTATIONS (I)
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.
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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
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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.
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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.
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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.
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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.
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PROOFS AND 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 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).
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