sir karl popper. philosophical problems & their roots in science

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The British Society for the Philosophy of Science

The Nature of Philosophical Problems and Their Roots in ScienceAuthor(s): K. R. PopperSource: The British Journal for the Philosophy of Science, Vol. 3, No. 10 (Aug., 1952), pp.124-156

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THE NATURE OF PHILOSOPHICAL PROBLEMS

of things, do not, I hold, constitute a basis for the distinction of

disciplines. Disciplines are distinguished partly for historical

reasonsand reasonsof administrativeconvenience (such as the or-ganisationof teachingand of appointments),partlybecause hetheories

which we constructto solve our problemshave a tendency1 to growinto unified systems. But all this classificationand distinctionis a

comparativelyunimportantand superficialaffair. We are notstudents

of subjectmatterbut studentsof problems. And problems may cut

right acrossthe bordersof any subjectmatteror discipline.Obvious as this fact may appear o some people, it is so important

for our presentdiscussion hat it is worth while to illustrate t by anexample. It hardly needs mentioning that a problem posed to a

geologist such as the assessmentof the chances of finding depositsof

oil or of uranium n a certaindistrictneedsfor its solution the help of

theories and techniquesusually classifiedas mathematical,physical,and chemical. It is, however, less obvious that even a more 'basic'

science such as atomic physics may have to make use of a geological

survey,and of geologicaltheoriesand techniques, f it wishes to solve

a problem arising n one of its most abstract nd fundamentalheories;for example,the problemof testing predictions oncerning he relative

stabilityor instabilityof atomsof an even or odd atomic number.

I amquiteready o admitthatmanyproblems,even if their solution

involves the most diversedisciplines,nevertheless'belong', in some

sense, to one or anotherof the traditionaldisciplines; for example,the two problemsmentioned' belong' clearlyto geology andphysics

respectively. This is due to the fact that each of them arisesout of a

discussionwhich is characteristic f the tradition of the disciplinein

question. It arises out of the discussionof some theory, or out ofempirical tests bearing upon a theory; and theories,as opposed to

subjectmatter,may constitutea discipline (which might be described

asasomewhat oose clusterof theoriesundergoingaprocessofchallenge,

change,andgrowth). But this doesnot alter the view that the classi-

fication into discipliness comparativelyunimportant,and thatwe are

students,not of disciplines,but of problems.But are therephilosophical roblems The present position of

English philosophy,which I shall take as

my pointof

departure,originates,I believe, from the late ProfessorLudwig Wittgenstein's

1 This tendency can be explained by the principle that theoreticalexplanationsare the more satisfactory he better they can be supported by independentvidence.

(This somewhat cryptic remarkcannot, I fear, be amplified in the presentcontext.)

I 125



K. R. POPPER

influentialdoctrine hat there arenone; that all genuine problemsare

scientific problems; that the alleged problems of philosophy are

pseudo-problems; that the allegedpropositionsor theoriesof philo-sophy are pseudo-propositions r pseudo-theories that they arenotfalse (if false,' theirnegationswould be truepropositionsor theories)but strictlymeaninglesscombinationsof words, no more meaningfulthan the incoherentbabblingof a child who has not yet learnedto

speakproperly.2As a consequence, philosophy cannot contain any theories. Its

true nature,accordingto Wittgenstein, is not that of a theory, but

that of an activity. The task of all genuine philosophy is that ofunmasking philosophicalnonsense, and of teaching people to talk

sense.

My plan is to take this doctrine3 of Wittgenstein'sas my starting1 It is of particular mportancein this connection to realise that Wittgenstein's

use of the term 'meaningless' is notthe usual and somewhat vague one accordingto which an absurdlyfalse assertion(suchas '2 +- 3 = 5427' or 'I can play Bach

on the adding machine') may be called 'meaningless'. He called a statement-like

expression 'meaningless' only if it is not a properly constructedstatement at all,

and therefore neither true nor false. Wittgenstein himself gave the example:'Socrates is identical'.

2SinceWittgensteindescribedhis own Tractatussmeaningless (see also the next

footnote), he distinguished,at least by implication, between revealing and unim-

portantnonsense. But this does not affecthis main doctrinewhich I am discussing,the non-existence of philosophicalproblems. (A discussionof other doctrines of

Wittgenstein'scan be found in the Notes to my OpenSociety,esp. notes 26, 46, 5I,and 52 to ch. ii.)

3 It is easyto detect at once one flaw in this doctrine: the doctrine,it may be said,is itself a philosophic theory, claiming to be true,and not to be meaningless. This

criticism,however, is a little too cheap. It might be countered n at least two ways.

(I) One might say that the doctrine is indeed meaninglessquadoctrine,but not qua

activity. (This is the view of Wittgenstein, who said at the end of his Tractatus

Logico-Philosophicushat whoever understood the book must realiseat the end that

it was itself meaningless,and must discardit like a ladder, after having used it to

reachthe desiredheight.) (2) One might say thatthe doctrineis not a philosophicalbut an empiricalone, that it statesthe historicalfact that all 'theories' proposed

by philosophersarein factungrammatical; that they do not, in fact,conform to the

rules inherent in those languagesin which they appearto be formulated,that this

defect turns out to be impossible to remedy ; and that every attempt to expressthem properly has lead to the loss of their philosophic character(and revealed

them, for example, asempiricaltruisms,or as false statements). These two counter

argumentsrescue,I believe, the threatenedconsistencyof the doctrine,which in this

way indeed becomes 'unassailable', asWittgensteinputs it by the kind of criticism

referred o in this note. (See alsothe next note but one.)

126




point (section 2). I shall try (in section 3) to explain it; to defend it,

to some extent ; and to criticise it. And I shall supportall this (in

sections4 to 6) by some examplesfrom the historyof scientificideas.But before proceedingto carryout this plan, I wish to reaffirm

my conviction that a philosophershould philosophise,that is, try to

solve philosophic problems, rather than talk about philosophy. If

Wittgenstein'sdoctrine s true,thennobody can,in thissense,philoso-

phise. If this were my opinion, I would give up philosophy.But it so happens that I am not only deeply interestedin certain

philosophicalproblems(Ido not much carewhetherthey are' rightly '

called 'philosophical problems'), but possessedby the belief that Imay even contribute--if only a little, andonly by hardwork-to their

solution. And my only excuse for talking here about philosophy-insteadof philosophising-is, in the lastresort,my hopethat, n carryingout my programmefor this address,an opportunitywill offer itself

of doing a little philosophising,afterall.

2

Ever since the rise of Hegelianismtherehas existed a dangerousgulf between science and philosophy. Philosopherswere accused-

rightly, I believe-of ' philosophisingwithout knowledge of fact ',and theirphilosophieswere describedas 'mere fancies,even imbecile

fancies'.1 Although Hegelianismwas theleading nfluence n Englandand on the Continent, opposition to it, and contempt of its preten-tiousness,never died out completely. Itsdownfallwas broughtabout

by aphilosopherwho, likeLeibniz,Kant,andJ.S. Mill beforehim, had

a sound

knowledge

of science,especially

mathematics. I am

speakingof BertrandRussell.

Russell s also the author of the classification(closelyrelatedto his

famous theoryof types) which is the basisofWittgenstein's view of

philosophy,the classification f the expressionsof a language nto

(I) Truestatements

(2) False statements

(3) Meaninglessxpressions,mongwhich therearestatement-like

sequencesf words,whichmaybe called'pseudo-statements.

Russelloperated ith thisdistinctionn connection iththesolution

1 The two quotationsare not the words of a scientificcritic,but, ironicallyenough,

Hegel's own characterisation f the philosophy of his friendand forerunnerSchelling.Cf. my Open Society,note 4 (and text) to ch. 12.

127



K. R. POPPER

of the logical paradoxeswhich he discovered. It was essential,forthis solution, to distinguish,more especially, between (2) and (3).We might say, in ordinaryspeech,that a falsestatement, ike ' 3 times4 equals173 ' or ' All catsare cows ', is meaningless. Russell,however,reservedthis characterisationor expressionssuch as ' 3 times 4 arecows' or ' All cats equal 173 ', that is, for expressions which are better

not describedas false statements(as can easily be seen from the factthat theirprimafacienegations,for example, ' Some cats do not equal173 ' are no more satisfactory han the original expressions)but as

pseudo-statements.

Russell used this distinction mainly for the elimination of theparadoxes (which,he indicated,were meaninglesspseudo-statements).

Wittgenstein went further. Led, perhaps,by the feeling that what

philosophers, especially Hegelian philosophers, were saying wassomewhatsimilar o theparadoxesof logic, he used Russell'sdistinctionin order to denounce all philosophyas meaningless.

As aconsequence,herecouldbeno genuinephilosophical roblems.All allegedphilosophicalproblemscould be classifiednto fourclasses:

(i)

thosewhich arepurely logical

ormathematical,

o be answeredbylogical or mathematicalpropositions,and thereforenot philosophical

(2) thosewhich arefactual,to be answeredby some statementof the

empiricalsciences,and thereforeagain not philosophical; (3) those

which are combinations of (i) and (2), and therefore, again, not

philosophical; and (4) meaninglesspseudo-problemssuch as 'Do

all cows equal 173?' or 'Is Socrates dentical?' or ' Does an invisible,

untouchable,and apparentlyaltogetherunknowable Socratesexist ?'

Wittgenstein's dea of eradicatingphilosophy (and theology) with

the help of anadaptionof Russell'stheory of typeswas ingeniousand

original (and more radical even than Comte's positivism which it

resemblesclosely).2 This ideabecame the inspirationof the powerful

modernmchool of language analystswho have inherited his belief

that there are no genuine philosophical problems, and that all a

1Wittgenstein still upheld the doctrine of the non-existence of philosophical

problemsin the form here describedwhen I saw him last (in 1946,when he presidedover a stormy meeting of the Moral ScienceClub in Cambridge,on the occasionof

my readinga

paperon 'Are there

PhilosophicalProblems ?

').SinceI had never

seen any of his unpublishedmanuscriptswhich were privatelycirculatedby some of

his pupils, I had been wondering whether he had modified what I here call his

'doctrine' ; but I found his views on this most fundamentaland influentialpoint of

his teaching unchanged.2 Cf. note52 (2) to ch. ii of my OpenSociety.

128




philosopheran do is to unmaskanddissolve he linguisticpuzzleswhichhavebeenproposedby traditionalphilosophy.

My own view of the matters thatonlyaslongasI havegenuinephilosophicalroblemso solve shallI continue o takeaninterestn

philosophy. I fail to understand he attractionof a philosophywithoutproblems. I know, of course, hatmanypeopletalknon-

sense; andit is conceivable hatit should becomeone's task (anunpleasant ne) to unmasksomebody'snonsense, or it may be

dangerousonsense. ButI believe hatsomepeoplehavesaidthingswhichwerenotverygoodsense, ndcertainlyotverygoodgrammar,

but which areat the sametimehighly interesting ndexciting,andperhapsmore worth listening o thanthe good sense of others. I

may mentionthe differentialndintegralcalculuswhich,especiallyin its earlyforms,was,no doubt,completelyparadoxicalnd non-sensicalby Wittgenstein'sand other) standards; which became,however,reasonablywell foundedas the resultof some hundred

yearsof greatmathematicalfforts butwhose oundationsvenatthis

very moment are still in need,and in the process,of clarification.'We

mightremember,n thiscontext, hat t was the contrast etween

theapparentbsoluteprecision f mathematicsndthevaguenessnd

inprecisionof philosophicalanguagewhich deeply impressed heearlier ollowersof Wittgenstein.Buthad herebeenaWittgensteinto use his weaponsagainst he pioneersof the calculus, nd had hesucceededn the eradication f theirnonsense,where theircontem-

porarycritics (suchas Berkeleywho was, fundamentally,ight)failed, hen he wouldhavestrangledne of themostfascinatingnd

philosophicallymportantdevelopmentsn the historyof thought.

Wittgensteinonce wrote: 'Whereof one cannotspeak, thereofone mustbesilent.' Itwas, f Irememberightly,ErwinSchroedingerwho replied 'But it is onlyherethatspeaking ecomesinteresting.'Thehistoryof the calculus-andperhaps f his own theory2-bearshim out.

No doubt,we shouldall trainourselves o speakas clearly,as

1 am alluding to G. Kreisel's recent construction of a monotone bounded

sequenceof rationalsevery term of which can be actually computed,but which does

notpossess

acomputable

imit-in contradiction o whatappears

o be theprimaacie

interpretationof the classical heoremof Bolzano and Weierstrass,but in agreementwith Brouwer'sdoubts about this theorem. Cf.Journalof SymbolicLogic,1952, 17,

57-2Before Max Born proposed his famous probability interpretation, Schroe-

dinger'swave equationwas, some might contend, meaningless.

129



K. R. POPPER

precisely, as simply, and as directly as we can. But I believe that

thereis not a classic of science, or of mathematics,or indeed a book

worth reading that could not be shown, by a skilful applicationofthe techniqueof languageanalysis,to be full of meaninglesspseudo-

propositionsand what some people might call ' tautologies'.

3

But I have promisedto say somethingin defence of Wittgenstein'sviews. What I wish to say is, first, that thereis much philosophical

writing (especiallyn theHegelianschool)whichmay ustlybecriticisedas meaninglessverbiage; secondly, that this kind of irresponsible

writingwaschecked, oratimeatleast,by the influenceof Wittgensteinand thelanguageanalysts although t is likely that the mostwholesome

influence in this respectwas the example of Russellwho, by the in-

comparablecharmand the clarityof his writings, established he fact

that subtlety of content was compatiblewith lucidity and unpreten-tiousnessof style).

But I amprepared

toadmit

more. Inpartial

defence of

Wittgenstein'sview, I ampreparedo defend the following two theses.

My first thesis is that every philosophy, and especially every

philosophical'school', is liable to degenerate n such a way that its

problemsbecome practicallyndistinguishablerom pseudo-problems,and its cant,accordingly,practicallyndistinguishablerommeaninglessbabble. This, I shalltry to show, is a consequenceof philosophical

inbreeding. The degenerationof philosophicalschools is the con-

sequenceof the mistakenbeliefthatone canphilosophisewithout being

compelled to turn to philosophyby problemswhichariseoutsidephilo-

sophy-in mathematics, or example,or in cosmology, or in politics,or in religion,or in social ife. To put it in otherwords,my first thesis

is this. Genuinephilosophical roblemsare always rooted in urgent

problems utsidephilosophy,andthey die if theserootsdecay. In their

efforts to solve them, philosophersare liable to pursue what looks

like aphilosophicalmethod or like a techniqueor like an unfailingkeyto philosophicalsuccess.' But no such methodsor techniquesexist;

philosophicalmethodsareunimportant,andany method is legitimate1 It is veryinterestinghattheimitatorswerealways nclined o believe hatthe

'master'didhis work with thehelpof a secretmethodor a trick. It is reportedthat nJ. S. Bach'sdayssomemusicianselieved hathe possessed secret ormulafor theconstructionf fugue hemes.

130




if it leads o resultscapableof being rationallydiscussed. What matters

is neither methods nor techniques-nothing but a sensitiveness to

problems,and a consumingpassionfor them; or as the Greekssaid,the gift of wonder.

There are those who feel the urge to solve a problem, those for

whom the problembecomesreal, ike a disorderwhichtheyhaveto getout of theirsystem.' They will make a contributioneven if they use

a method or a technique. But there are others who do not feel

this urge, who have no seriousand pressingproblem but who never-

theless produce exercises in fashionablemethods, and for whom

philosophy is applicationof whatever insight or technique you like)rather han search. They areluringphilosophy nto thebog of pseudo-

problemsandverbalpuzzles; eitherby offeringus pseudo-problemsfor real ones (the dangerwhich Wittgensteinsaw), or by persuadingus to concentrateupon the endlessand pointlesstask of unmaskingwhat they rightly or wrongly takefor pseudo-problems thetrapinto

which Wittgenstein fell).

My second thesis s thatwhat appears o be theprimafaciemethod

ofteachingphilosophy

s liabletoproduce

aphilosophy

which answers

Wittgenstein's description. What I mean by 'primafacie method

of teachingphilosophy , and what would seem to be the only method,is that of giving the beginner (whom we take to be unawareof the

history of mathematical,cosmological, and other ideas of science as

well as of politics)the works of the greatphilosopherso read; say,of

Plato and Aristotle, Descartesand Leibniz, Locke, Berkeley, Hume,

Kant, and Mill. What is the effect of such a course of reading?

A new world of astonishinglysubtle and vast abstractionsopens itself

to the reader,abstractionsof an extremely high and difficult level.Thoughts and argumentsare put before his mind which sometimes

arenot only hardto understand,but whose relevanceremainsobscure

since he cannot find out what they may be relevant to. Yet the

studentknows that these are the greatphilosophers,hatthisis the wayof philosophy. Thus he will make aneffortto adjusthismind to what

he believes (mistakenly,as we shallsee) to be their way of thinking.He will attemptto speaktheirqueer anguage,to match the torturous

spiralsof theirargumentation,andperhapseven tie himselfup in theircurious knots. Some may learn these tricks in a superficialway,

1Iamalludingoa remarkyProfessorilbert yle,whosays npage of his

ConceptfMind 'Primarily amtryingo getsomedisordersutof my own

system.

13I



K. R. POPPER

others may begin to become genuinely fascinatedaddicts. Yet I

feel that we ought to respectthe man who, having made his effort,

comes ultimately to what may be described as Wittgenstein'scon-clusion : 'I have learned the jargon as well as anybody. It is veryclever and captivating. In fact, it is dangerouslycaptivating; for

the simpletruthaboutthe matter s thatit is much ado aboutnothing

--just a lot of nonsense.'

Now I believe such a conclusion to be grossly mistaken; it is,

however, the almost inescapableresult, I contend, of the primaaciemethod of teachingphilosophy here described. (I do not deny, of

course, that some particularlygifted studentsmay find very muchmore in the works of the greatphilosophershan thisstoryindicates-

and without deceiving themselves.) For the chance of finding out

the extra-philosophical roblems (the mathematical, cientific,moral

and political problems) which inspired these great philosophersis

very small indeed. These problems can be discovered, as a rule,

only by studyingthe history of, for example, scientific deas,and es-

pecially the problem-situationn mathematicsand the sciencesof the

periodin

question; and this, in turn,

presupposesa considerableac-

quaintancewith mathematicsandscience. Only an understanding f

the contemporaryproblem-situation n the sciences can enable the

student of the great philosophersto understand hat they tried to

solve urgent and concreteproblems; problemswhich, they found,could not be dismissed. And only afterunderstandinghis fact can a

studentattaina differentpictureof the greatphilosophies-one which

makes full sense of the apparentnonsense.I shalltry to establishmy two theseswith the help of examples;

but before turningto theseexamples,I wish to summarisemy theses,and to balancemy account with Wittgenstein.

My two thesesamount to the contentionthatphilosophy s deeplyrooted in non-philosophicalproblems; that Wittengstein'snegative

judgment is correct,by andlarge,as farasphilosophiesare concerned

which have forgotten their extra-philosophicaloots; and that these

roots are easily forgotten by philosopherswho ' study' philosophy,instead of being forced into philosophy by the pressureof non-

philosophicalproblems.My view of Wittgenstein'sdoctrinemay be summedup as follows.

It is true, by andlarge, thatpurephilosophicalproblemsdo not exist ;for indeed, the purer a philosophicalproblem becomes, the more islost of its originalsense,significance,or meaning,and the more liable

132




is its discussionto degenerate nto empty verbalism. On the other

hand, there exist not only genuine scientificproblems, but genuine

philosophicalproblems. Even if, upon analysis, heseproblemsturnout to-have actualcomponents, heyneednot beclassified sbelongingto science. And even if they shouldbe solubleby, say, purely logicalmeans, they need not be classified as purely logical or tautological.

Analogoussituationsarise n physics. Forexample,the explanationor

prediction of certain spectralterms (with the help of a hypothesis

concerning the structureof atoms) may turn out to be soluble by

purely mathematicalcalculations. But this, again, does not imply

that the problem belonged to pure mathematics rather than tophysics. We are perfectlyjustified in calling a problem 'physical'if it is connected with problems and theories which have been

traditionallydiscussedby physicists (such as the problems of the

constitutionof matter), even if the means used for its solution turn

out to be purely mathematical. As we have seen, the solution of

problemsmaycut throughthe boundaryof manysciences. Similarly,a problem may be rightly called 'philosophical' if we find that,

although originally it may have arisen in connection with, say,atomic theory, it is more closely connected with the problems and

theories which have been discussed by philosophers than with

theoriesnowadaystreatedby physicists. And again, t does not matter

in the leastwhat kind of methods we use in solving suchaproblem.

Cosmology, for example,will alwaysbe of greatphilosophicalnterest,eventhough by someof itsmethods t hasbecomecloselyalliedto what

is perhapsbetter called 'physics '. To say that, since it deals with

factualissues,it must belong to science ratherthan to philosophy,is

not only pedanticbut clearlythe result of an epistemological,andthusof a philosophical, dogma. Similarly, there is no reason why a

problem soluble by logical means should be denied the attribute

'philosophical'. It may well be typicallyphilosophical,or physical,or biological. For example, logical analysisplayed a considerable

part in Einstein's special theory of relativity; and it was,

partly, this fact which made this theory philosophically interesting,and which gave rise to a wide range of philosophical problems

connectedwith it.Wittgenstein's doctrine turns out to be the result of the thesis

that all genuine statements(and thereforeall genuineproblems)can

be classifiedinto one of two exclusive classes: factual statements

(synthetic posteriori), nd logical statements(analytic priori). This

'33



K. R. POPPER

simpledichotomy, lthough xtremely aluableor thepurposes f a

roughsurvey, urnsout to be for manypurposesoo simple.1 But

althought is, as it were,specially esignedo excludeheexistence fphilosophicalroblems,t is veryfarfromachieving venthisaim;for evenif we accept he dichotomy,we can stillclaimthatfactualor logicalor mixedproblemsmayturnout,in certain ircumstances,to be philosophical.

4

I now turntomy

firstexample

: Platoand heCrisis nEarly

Greek

Atomism.

My thesis here is that Plato's centralphilosophicaldoctrine, the

so-called Theory of Forms or Ideas,cannot be properlyunderstood

except in an extra-philosophical ontext 2; more expecially in the

context of the criticalproblem situationin Greek science3(mainly

1 Alreadyin my LogicderForschungVienna, 1935), I pointed out that a theorysuch as Newton's may be interpretedither as factual or as consisting of implicitdefinitions(in thesenseofPoincar6 andEddington),and thattheinterpretationwhich

a physicistadoptsexhibits itself in hisattitudeowardstestswhich go againsthis theoryrather hanin what he says. The dogma of the simple dichotomy has been recently

attacked,on very different ines, by F. H. Heinemann(Proc.oftheXth Intern.Congress

of Philosophy Amsterdam, 1949), Fasc.2, 629, Amsterdam,1949), by W. van 0.

Quine, and by Morton G. White. It may be remarked,again from a different

point of view, that the dichotomy applies, in a precisesense, only to a formalised

language, and therefore is liable to break down for those languages in which

we must speak prior to any formalisation,i.e. in those languages in which all

the traditionalproblems were conceived. Some members of the school of the

language analysts,however, still believe it a sound method to unmaska theory as

' tautological'.

2 Inmy OpenSocietyandItsEnemies, have triedto explainin some detailanother

extra-philosophicalroot of the same doctrine,viz. a political root. I also discussed

there (in note 9 to ch. 6 of the revised4th edition, 1952) the problem with which I

am concerned in the presentsection, but from a somewhat differentangle. The

note referred o and the presentsection partly overlap ; but they arelargely supple-

mentary to each other. Relevant references(esp. to Plato) omitted here will be

found there.3There arehistorianswho deny that the term 'science' can be properlyapplied

toany development

which is olderthan thesixteenthor even the seventeenthcentury.But quite apartfrom the fact that controversiesabout labelsshould be avoided,there

can, I believe, no longer be a doubt nowadays about the astonishing similarity,not to say identity, of the aims, interests,activities,arguments,and methods,of, say,Galileo and Archimedes,or Copernicusand Plato, or Kepler and Aristarchus(the'Copernicusof antiquity '). And any doubt concerning he extremeageof scientific

134




in the theoryof matter)which developedas a consequence f the

discoveryf the irrationalityf the square ootof two. If my thesis

is correct, hen Plato'stheoryhas not so farbeenfully understood.(Whethera 'full' understandingan everbe achieveds, of course,mostquestionable.)But the moreimportant onsequence ouldbethat t canneverbe understoodby philosophersrainedn accordancewiththeprimaaciemethoddescribedn theforegoing ection-unless,of course, heyarespecially ndadhoc nformed f the relevantacts

(whichthey mayhaveto accepton authority).It is well known thatPlato'sTheoryof Forms s historicallys

well as in its contentcloselyconnectedwith the Pythagoreanheorythatallthingsare, n essence, umbers.The details f thisconnection,and the connectionbetween Atomism and Pythagoreanism,re

perhapsnot so well known. I shall thereforeell the whole storyin brief,as I seeit atpresent.

It appearshat the founderof the Pythagoreanrderor sectwas

deeply mpressed y two discoveries.The firstdiscoverywas thataprimaaciepurelyqualitativehenomenonuchas musicalharmonywas, in essence,based

uponthe

purelynumericalatios

I :2; 2:

3 ;3 : 4. The secondwas that he right or' straightangle(obtainableforexamplebyfoldingaleaftwice,sothat hetwofolds orma cross)wasconnectedwith thepurelynumerical atios3 :4 : 5, or 5 : 12 : 13

(thesidesof rectangularriangles).These wo discoveries,t appears,led Pythagoraso thesomewhat antasticgeneralisationhatallthingsare, in essence,numbers,or ratiosof numbers or thatnumberwasthe ratio(logos reason),he rational ssenceof things,or theirrealnature.

Fantastic s thisideawas,it proved n manywaysfruitful. Oneof its mostsuccessfulapplications as to simplegeometricaligures,

observation,and of carefulcomputationsbasedupon observation,has been dispellednowadays by the discovery of new evidence concerning the history of ancient

astronomy. We can now draw not only a parallelbetween Tycho and Hipparchus,but even one between Hansen (1857) and Cidenasthe Chaldean(314 B.c.), whose

computationsof the 'constants for the motion of Sun and Moon' are without ex-

ception comparable n precisionto those of the bestnineteenth-centuryastronomers,

'Cidenas' value for the motion of the Sun from the Node (o -5to great), althoughinferiorto Brown's,is superior o atleast one of the most widely usedmodem values',wrote J. K. Fotheringham n 1928, in his most admirablearticle 'The Indebtednessof Greekto ChaldeanAstronomy' (The Observatory,928,51, No. 653), upon which

my contention concerningthe age of astronomyis based.1From Aristotle'sMetaphysics

135



K. R. POPPER

such as squares,rectangularandisoscelestriangles,and also to certain

simple solids, such as pyramids. The treatmentof some of these

geometricalproblemswas basedupon the so-calledgnomon.This can be explainedas follows. If we indicate a squareby four

dots,

we may interpretthis as the result of adding three dots to the onedot on the upperleft corner. These threedots are the firstgnomon;

we may indicate t thus:

By addinga secondgnomon, onsistingof five moredots,we obtain

One seesat once thateverynumberof thesequenceof theoddnumbers,

I, 3, 5, 7 . . . , each forms a gnomonof a square,and that the sums

I, I + 3, I + 3 + 5, I + 3 +- 5 7, . . . are the squarenumbers,and that, if n is the (numberof dots in the) side of a square, ts area

(total number of dots = n2) will be equal to the sum of the first nodd numbers.

As with the treatmentof squares, o with the treatmentof isosceles

triangles.

Here eachgnomon s a lasthorizontal ine of points, and each element

of thesequence , 2, 3,4, . . . is agnomdn. The' triangular umbers'

are the sums I + 2; I + 2 + 3 ; I + 2 + 3 + 4, etc., that is, the

136




plausible but uncertain and prejudiced opinions (doxa) of falliblemortals.' In his interpretationof the Table of Opposites,Plato was

influencedby Parmenides, he man who stimulated he developmentof Democritus'atomic theory.

Returningnow to the original Pythagoreanview, there is one

thing in it which is of decisiveimportance or our story. Itwill havebeen observed that the Pythagoreanemphasis upon Number wasfruitful from the point of view of the development of scientificideas. This is often but somewhat loosely expressedby saying thatthe Pythagoreans encouraged numerical scientific measurements.

Now the point which we mustrealise s that,for the Pythagoreans,allthiswas countingatherhanmeasuring. t was the countingof numbers,of invisibleessencesor 'Natures ' which were Numbers of little dotsor stigmata. Admittedly, we cannot count theselittle dots directly,since they are invisible. What we actually do is not to countthe Numbers or NaturalUnits, but to measure, .e. to count arbitraryvisible units. But the significanceof measurementswas interpretedas revealing, indirectly, the true Ratiosof the NaturalUnits orof theNaturalNumbers.

Thus Euclid's methods of proving the so-called 'Theorem of

Pythagoras' (Euclid'sI, 47) accordingto which, if a is the side of a

triangle oppositeto its right anglebetween bandc,

(I)a2= b2+ C2,

was completely foreign to the spirit of Pythagoreanmathematics.In spite of the fact that the theorem was known to the Babyloniansand geometrically proved by them, neither Pythagorasnor Plato

appearo have known the

generalgeometricalroof;for the

problemfor which they offered solutions, the arithmeticalne of finding the

integral solutionsfor the sides of rectangular riangles,can be easily1 Plato's distinction (epistemEs. doxa) derives, I think, from Parmenides (truth

vs. seeming). Plato clearly realised that all knowledge of the visible world, the

changing world of appearance,consists of doxa; that it is tainted by uncertaintyeven if it utilisesthe epistemF,he knowledge of the unchanging' Forms' and of puremathematics, o the utmost, and even if it interprets he visible world with the helpof a theory of the invisible world. Cf. Calylus, 439b f., Rep. 476d f. ; and

especiallyTimaeus,9bff.,where the distinction s appliedto thosepartsof Plato'sown

theorywhich we shouldnowadayscall' physics'or 'cosmology ', or,more generally,'natural science '. They belong, Platosays,to the realm of doxa(in spiteof the factthatscience= scientia epistemF cf. my remarkson thisproblemin ThePhilosophicalQuarterly,April 1952, p. 168). For a differentview concerning Plato's relation to

Parmenides,see Sir David Ross, Plato'sTheory fIdeas,Oxford, 1951, p. 164.

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K. R. POPPER

solved, if (i) is known by the formula (mand n arenaturalnumbers,and m > n)

(2) a = m2 - n2 ; b= 2mn ; c = m2- n2.

But formula (2) was unknown to Pythagorasandeven to Plato. This

emerges from the traditionaccordingto which Pythagorasproposedthe formula

(3) a = 2m(m + I) + I; b = 2m(m + I) ; c = 2m + I

which can be readoff thegn-omonf the squarenumbers,but which is

less generalthan (2), since it fails, for example, for 8 : 15 : 17. To

Plato, who is reportedto have improved Pythagoras' ormula (3), isattributed another formula which still falls short of the generalsolution (2).

We now come to the discoveryof the irrationalityf thesquareoot

of two. According to tradition,this discoverywas made within the

Pythagoreanorder,but waskeptsecret. (Thisis suggestedby the old

term for 'irrational',' arrhetos, that is, 'unspeakable',which might

well have meant 'the unspeakable mystery'.) This discoverystruckat the root of

Pythagoreanismfor it meant that sucha

simplegeometrical entity as the diagonal d of the squarewith the side a

could demonstrablynot be characterisedby any ratio of natural

numbers; d: a was no ratio. The traditionhasit that the memberof

the school who gave away the secret was killed for his treachery.However thismaybe, there s little doubt that the realisation f thefact

that irrationalmagnitudes (they were, of course, not recognisedas

numbers)existed,andthat their existencecould be proved, led to the

downfall of the Pythagoreanorder.

The Pythagorean theory, with its dot-diagrams, contains, nodoubt, the suggestion of a very primitive atomism. How far the

atomic theory of Democritus was influencedby Pythagoreanism s

difficultto assess. Its main influencescame, one can say for certain,

from the Eleatic School: from Parmenidesand from Zeno. The

basic problem of this school, and of Democritus, was that of the

rationalunderstandingof change. (I differ here from the interpreta-tions of Cornford and others.) I think that this problem derives

from Ionian ratherthanfrom Pythagoreanthought, and that it hasremainedthe fundamentalproblem of NaturalPhilosophy.

Although Parmenideshimself was not a physicist (as opposed to

his great Ionian predecessors),he may be described,I believe, as

having fathered theoretical hysics. He produced an anti-physical

14o




theory which, however, was the first hypothetical-deductive ystem.And it was the beginning of a long series of suchsystemsof physical

theorieseachof which was an improvementon its predecessor. As arule the improvementwas found necessaryby the realisation hat the

earliersystem was falsifiedby certainfacts of experience. Such an

empiricalrefutationof the consequencesof a deductivesystemleadsto

efforts at its reconstruction,and thus to a new and improved theorywhich, as a rule, clearly bears the mark of its ancestry,of the older

theory as well as of the refuting experience.These experiencesor observationswere, we shall see, very crude

at first,but they became more and more subtleasthe theoriesbecamemore and more capableof accounting or the cruderobservations. In

the case of Parmenides'theory, the clash with observation was so

obvious that it would seem perhaps ancifulto describethe theory as

the first hypothetical-deductive ystem of physics. We may, there-

fore, describe it as the last pre-physical deductive system, whose

falsificationgave rise to the first truly physical theory, the atomistic

theory of Democritus.Parmenides'

theoryis

simple.He finds it

impossibleto under-

stand change or movement rationally, and concludes that there is

really no change-or that change is only apparent. But before we

indulge in feelings of superiority,in the face of such a hopelesslyunrealistictheory,we should first realise hat thereis a seriousproblemhere. If a thing X changes, then clearly it is no longer the same

thing X. On the otherhand,we cannotsay that X changeswithout

implying that X persistsduringthe change; that it is the same thingX, at the beginning and at the end of the change. Thus, it appears

that we arrive at a contradiction,and that the idea of a thing thatchanges,and thereforethe idea of change, is impossible.

All this soundsvery philosophicaland abstract,and so it is. But

it is a fact that the difficultyhere indicatedhas never ceased to make

itself felt in the developmentof physics.1 And a deterministicsystemsuch as that of Einstein's field theory might even be describedas a

four-dimensional ersionof Parmenides'unchanging hree-dimensionaluniverse. For, in a sense, no change occurs in Einstein's four-

dimensionalblock-universe. Everything is therejust as it is, in itsfour-dimensional locus; change becomes a kind of 'apparent'change; it is 'only' the observerwho, as it were, glides along his

1This maybe seen fromEmileMeyerson'sdentityndReality, ne of the most

interesting hilosophicaltudies f the development f physicalheories.

K 141




Democritus' theory of changewas of tremendous mportancefor

the developmentof physicalscience. It was partly acceptedby Plato,

who retainedmuch of atomism but explained change not only byunchangingyet moving atoms, but also by the 'Forms' which were

subjectneitherto changenor to motion. But it was condemnedbyAristotlewho taught n its stead hatallchangewasthe unfoldingof theinherent potentialitiesof essentiallyunchangingsubstances.But al-

thoughAristotle'stheoryof substances s thesubjects f changebecame

dominant, t provedbarren 1 andDemocritus'theorythat all changemust be explained by movement became the tacitly acceptedofficial

programmeof physicsdown to our own day. It is still part of thephilosophy of physics, in spite of the fact that physics itself has

outgrown it (to say nothing of the biological and social sciences).Forwith Newton, in addition o moving mass-points,forcesf changing

intensity (anddirection)enter the scene. True, thesechangescan be

explained as due to, or dependent upon, motion, that is upon the

changingposition of particles,but they are neverthelessnot identical

with the changesn position; owing to the squareaw, the dependenceis not even a linear one. And with Faradayand Maxwell, changingfieldsof forcesbecomeasimportantas materialatomicparticles. That

our modern atoms turnout to be composite s a minormatter; from

Democritus'point of view, not our atoms but ratherour elementary

particleswould be realatoms--exceptthatthesetoo turnout to be liable

to change. Thus we have a most interestingsituation. A philosophyof change, designed to meet the difficultyof understandingchange

rationally, serves sciences for thousandsof years, but is ultimately

supersededby the developmentof scienceitself; and this fact passes

practicallyunnoticed by philosopherswho are busily denying theexistence of philosophicalproblems.

Democritus'theorywas a marvellousachievement. It provideda

theoretical rameworkfor the explanationof most of the empiricallyknown propertiesof matter (discussedalready by the lonians), such

as compressibility,degreesof hardnessand resilience,rarefactionand

condensation, oherence,disintegration, ombustion,andmanyothers.

1The barrennessof the 'essentialist' (cf. note 2 above) theory of substance s

connected with itsanthropomorphism;

for substances(as Locke saw)

taketheir

plausibilityfrom the experienceof a self-identical but changing and unfolding self.

But although we may welcome the fact that Aristotle'ssubstanceshave disappearedfrom physics, here is nothing wrong, as ProfessorHayek says,in thinking anthropo-

morphically about man and there is no reasonwhy they should disappear rom

psychology.

143



K. R. POPPER

But apartfrom being importantas an explanationof the phenomenaof experience, the theory was important in other ways. First, it

established he methodological principle that a deductive theory orexplanationmust' save the phenomena , thatis, mustbe in agreementwith experience. Secondly, it showed that a theory may be specu-lative, and basedupon the fundamental(Parmenidean) rinciplethat

the world as it must be understoodby argumentative hought turns

out to be different rom the world of primaacieexperience,from the

world as seen, heard, smelled, tasted, touched; 1 and that such a

speculativetheory may neverthelessaccept the empiricist criterion'

that it is the visiblethatdecidesthe acceptanceor rejectionof a theoryof the invisible (such as the atoms). This philosophyhasremainedfundamental o the whole developmentof physics,andhascontinuedto conflict with all ' relativistic' and 'positivistic' 4 tendencies.

Furthermore,Democritus'theory led to the first successesof the

method of exhaustion(the forerunnerof the calculusof integration),sinceArchimedeshimselfacknowledgedthat Democrituswas the first

to formulate the theory of the volumes of cones and pyramids.5But

perhapsthe most fascinatingelement in Democritus' theory is

his doctrineof the quantisationof spaceand time. I have in mind

the doctrine,now extensivelydiscussed,6hatthere s a shortestdistance

and a smallesttime interval,thatis to say, distances n spaceandtime

(elementsof length andtime, Democritus'ameres in contradistinction

to his atoms)suchthatno smallerones are measurable.

1 Cf. Democritus, Diels, fragm. 11 (cf. Anaxagoras,Dield fragm. 21 ; see also

fragm. 7).2 Cf. SextusEmpiricus,Adv. mathem.(Bekker)vii. 140, p. 221, 23B.

'Relativistic' in the sense of philosophical relativism, e.g. of Protagoras'homomensuraoctrine. It is, unfortunately, tillnecessaryo emphasise hatEinstein's

theory has nothing in common with philosophicalrelativism.

4 Such as those of Bacon; the theory (but fortunatelynot the practice)of the

early RoyalSociety and in our time, of Mach (who opposedatomic theory) ; and

of the sense-data heorists.

5 Cf. Diels, fragm. 155, which must be interpreted n the light of Archimedes

(ed. Heiberg) II2, p. 428 f. Cf. S. Luria'smost importantarticle'Die Infinitesimal-

methodederantikenAtomisten' (Quellen& StudienurGesch.d. Math.Abt. B. Bd. 2,

Heft 2 (1932), p. 142).

6 Cf. A. March,NaturundErkenntnis,Vienna, 1948, p. 193f.7 Cf. S. Luria,op. cit., esp. pp. 148ff., 172if.

144



K. R. POPPER

these two books do not contain any reference to the problem of

irrationality.1

My beliefthat Democritus did not know

aboutirrationalities s

basedon the fact that there are no tracesof a defence of his theory

against he fatal blow which it received from this discovery. For the

blow was as fatal to Atomism as it was to Pythagoreanism. Both

theorieswere basedon the doctrinethat allmeasurements, ultimately,

countingof naturalunits,so thateverymeasurementmustbe reducible

to purenumbers. The distancebetweenany two atomicpointsmust,

therefore,consist of a certain number of atomic distances; thus all

distancesmustbe commensurable. But this turnsout to be impossibleeven in the simplecase of the distancesbetween the cornersof a square,becauseof the incommensurability f its diagonalwith its side.

It was Plato who realised this fact, and who in the Laws stressed

its importancein the strongestpossibleterms, denouncinghis com-

patriotsfor their failureto realisewhat it meant. It is my contention

that his whole philosophy, and especiallyhis theory of' Forms' or

'Ideas', was influencedby it.Plato was very close to the Pythagoreansas well as to the Eleatic

Schools; andalthoughhe appears o have felt antipathetico Demo-critus, he was himself an atomist of a kind. (Atomist teachingremained as one of the school traditionsof the Academy.2) This is

not surprisingn view of the close relationbetween Pythagoreanand

atomisticideas. But all this was threatenedby the discoveryof the

irrational. I suggest that Plato'smain contributionto sciencesprangfrom his realisationof the problem of the irrational,and from the

modification of Pythagoreanismand atomism which he undertook

in orderto rescuesciencefrom acatastrophic

ituation.

He realisedthat the purely arithmeticaltheory of nature was de-

feated, and that a new mathematicalmethod for description and

explanation of the world was needed. Thus he encouraged the

developmentof an autonomousgeometricalmethod which found its

fulfilmentin the 'Elements' of the PlatonistEuclid.

What are the facts ? I shall try to put them all brieflytogether.

(I) Pythagoreanism nd atomismin Democritus'form were both

fundamentally

basedon arithmetic, hat is on counting.

1 This would be in keeping with the fact mentioned in the note cited from the

Open Society, that the term 'alogos' is only much later known to be used for

'irrational', and that Plato who (Repub. 34d) alludesto Democritus'title, neverthe-

less never uses'alogos' as a synonym for 'arrhetos'.2 See S. Luria,esp. on Plutarch,loc.cit.

146




(2) Platoemphasisedhe catastrophicharacterf the discoveryof theirrationals.

(3) He inscribed ver thedoorof the Academy: 'Nobody un-trainedn geometrymayentermy house'. Butgeometry,ccordingto Plato's mmediatepupilAristotleas well as to Euclid,treatsofincommensurablesr irrationals,n contradistinctiono arithmeticwhichtreats f' the odd andthe even'.

(4) Withina shorttime afterPlato'sdeath,his schoolproduced,in Euclid'sElements, work whose mainpoint was that it freedmathematicsromthe' arithmetical'assumptionf commensurability

or rationality.(5) Plato himselfcontributedo thisdevelopment,ndespecially

to thedevelopmentf solidgeometry.(6) More especially,he gave in the Timaeus specifically eo-

metricalversionof the formerlypurelyarithmeticaltomictheory,that s,aversionwhichconstructedheelementaryarticlesthe amousPlatonicbodies)out of triangleswhich incorporatedhe irrational

squareroots of two and of three. (See below.) In most other

respects, e preserved othPythagoreandeasaswell as someof themost importantdeas of Democritus.' At the sametime, he triedto eliminateDemocritus'oid; for he realised hat motionremains

possible ven n a ' full' world,providedmotion sconceived sof thecharacter f vortices n a liquid. Thushe retainedome of themostfundamentaldeasof Parmenides.2

(7) Platoencouragedhe constructionf geometricalmodels oftheworld,andespeciallymodelsexplainingheplanetarymovements.Euclid's

geometrywasnot intendedas an exercisen

puregeometry(asnow usuallyassumed), ut as a theory f theworld. Eversince3

1 Plato took over, more especially,Democritus'theory of vortices (Diels, fragm.167, 164 ; cf. Anaxagoras,Diels 9 ; and 12, 13) ; see also the next footnote, and his

theory of what we nowadays would call gravitational phenomena (Diels, 164;

Anaxagoras,12, 13, I5, and 2)-a theory which, slightly modified by Aristotle,was

ultimately discardedby Galileo.

2 Plato's reconciliationof atomism and the theory of the plenum 'nature abhors

the void') became of the greatest mportance or the history of physicsdown to our

ownday.

For it influencedDescartesstrongly,

became the basisof thetheory

of

ether andlight, and thusultimately,via HuyghensandMaxwell, of de Broglie'sand of

Schroedinger'swave mechanics.

3The only exception is the partial reappearance f arithmeticalmethods in the

New QuantumTheory, e.g. in the electronshell theory of the periodic systembased

upon Pauli's exclusion principle.

147



K. R. POPPER

Plato and Euclid, but not before, it has been taken for grantedthat

geometry (rather hanarithmetic) s the fundamental nstrumentof all

physical explanationsand descriptions,of the theory of matteraswellas of cosmology.'

These are the historical facts. They go a long way, I believe,to establishmy contention thatwhat I have describedas theprima aciemethodof philosophycannot eadto anunderstanding f the problemswhich inspiredPlato. Nor canit leadto anappreciation f what maybe justly claimed to be his greatestphilosophicalachievement, the

geometrical heory of the world which becamethe basisof the works

of Euclid, Aristarchus,Archimedes, Copernicus, Kepler, Galileo,Descartes,Newton, Maxwell, and Einstein.

But is this achievement properly described as philosophical?

Does it not ratherbelong to physics-a factual science-and to puremathematics-a branch, Wittgenstein's school would contend, of

tautologicallogic ?

I believe that we can at this stage see fairly clearly why Plato's

achievement (although it has no doubt its physical, its logical, its

mixed, and its nonsensical

components)

was a

philosophical

achieve-

ment ; why at leastpartof his philosophyof natureand of physicshas

lastedand,I believe, will last.

What we find in Plato and his predecessors s the conscious

constructionandinvention of a new approach owards the world and

towards the knowledge of the world. This approachtransformsa

fundamentally theological idea, the ideaof explaininghevisibleworld

by a postulatednvisibleworld,2 nto the fundamental nstrument of

theoreticalscience. The idea was explicitly formulatedby Anaxa-

gorasandDemocritus3as theprincipleof investigationsnto the natureof matter or body ; visible matterwas to be explainedby hypotheses

Concerningthe moderntendencytowards what is sometimescalled ' arithmetisa-

tion of geometry' (a tendency which is hardly characteristic f all modern work

on geometry), it should be noted that there is little similaritywith the Pythagorean

approach inceinfinite equencesfnaturalnumbersare its main instrumentrather han

the natural numbersthemselves.

1 Fora similarview of Plato's andEuclid's nfluence,see G. F. Hemens,Proc.ofthe

Xth Intern.Congress f Philosophy Amsterdam I949), Fasc.2, 847.2 Cf. Homer'sexplanationof the visible world beforeTroy with the help of the

invisible world of the Olympus. The idea loses, with Democritus, some of its

theological character(which is still strongin Parmenides,although less so in Anaxa-

goras)but regains t with Plato, only to lose it soon afterwards.3 See the referencesgiven above.

148




about nvisibles, boutan invisibletructurehichs toosmallo beseen.With Plato,this ideas s consciously ccepted ndgeneralised; the

visibleworld of change s ultimatelyo be explained y aninvisibleworldof unchangingForms (orsubstancesressences,r 'natures-as we shallsee,geometricalhapes r figures).

Is this ideaabout he invisible tructuref mattera physical r a

philosophicaldea? If a physicist ctsuponthistheory, hat s to say,if he acceptst, perhaps venwithoutbecomingconscious f it, byacceptinghe traditionalproblems f his subject, spresented y the

problem-situationith which he is confronted and f he, so acting,

producesnewspecificheoryof thestructurefmatter then shouldnotcallhimaphilosopher.But f hereflects pon t,and, orexample,rejects t (like Berkeleyor Mach),preferring phenomenologicalor positivisticphysicsto the theoretical nd somewhattheologicalapproach,henhe maybecalledaphilosopher. Similarly,hosewho

consciouslyearchedor thetheoreticalapproach, ho constructedt,andwhoexplicitlyormulatedt, and hus ransferredhehypothetical-deductivemethod rom thefield of theology o thatof physics,were

philosophers,venthough heywerephysicistsn so far astheyactedupontheirown precepts nd tried to produceactual heoriesof theinvisible tructure f matter.

But I shallnot pursuehequestion s to theproperapplicationfthe label 'philosophy' any further; for this problem,which is

Wittgenstein's roblem, clearlyturnsout to be one of linguisticusage,a pseudo-problemhichby now mustbe rapidlydevelopinginto a boreto my audience. ButI wish to adda few morewordsonPlato'stheoryof Formsor Ideas,or moreprecisely, n point (6) of

thelist of historicalactsgivenabove.Plato'stheoryof the structure f mattercan be found in the

Timaeus.It hasatleastasuperficialimilarity ith the moderntheoryof solidswhichinterpretshem as crystals. His physicalbodiesare

composedof invisibleelementaryparticlesof variousshapes,the

shapesbeing responsibleor the macroscopic ropertiesof visiblematter. The shapesof the elementaryparticles,n theirturn,aredeterminedby theshapes f theplane igureswhich formtheirsides.

Andtheseplane igures,n theirturn,areultimately llcomposed ftwoelementaryriangles, iz. the half-squareor soscelesectangular)trianglewhichincorporateshesquareoot ftwo,and hehalf-equilateralrectangularrianglewhichincorporateshe square-rootf three,bothof themirrationals.

149



K. R. POPPER

These triangles, n their turn, are describedas the copies 1 of un-

changing 'Forms' or 'Ideas', which means that specificallygeo-

metrical Forms' are admitted into the company of the Pythagoreanarithmeticalorm-Numbers.

There is little doubt that the motive of this construction s the

attemptto solve the crisisof atomismby incorporating he irrationals

into the last elementsof which the world is built. Once this has been

done, the difficultyof the existenceof irrationaldistances s overcome.

But why did Plato choose just these two triangles? I have

elsewhere2 expressedthe view, as a conjecture,that Plato believed

that all other irrationalsmight be obtainedby addingto the rationalsmultiples of the squareroots of two and three. I now feel quiteconfident that the crucialpassagein the Timaeus learly implies this

doctrine (which was mistaken,as Euclid later showed). For in the

passagein question, Plato says quite clearly that 'All trianglesare

derived from two, each having a right angle', going on to specifythese two as the half-squareand half-equilateral. But this can onlymean,in the context, that all trianglescanbe composedby combiningthese two, a view which is

equivalent

to the mistakentheory of the

relativecommensurabilityof all irrationalswith sumsof rationalsand

the squareroots of two andthree.3

But Plato did not pretendthat he had a proof of the theory in

question. On the contrary,he saysthathe assumes he two trianglesasprinciples in accordancewith an accountwhich combinesprobable

conjecturewith necessity . And a little later,afterexplainingthat he

takes the half-equilateral riangleas the second of his principles,he

says, 'The reason s too long a story ; but if anybodyshould test this

matter, and prove that it has this property' (I supposethe propertythat all other trianglescan be composedof thesetwo) ' then the prizeis his, with all our good will '.4 The languageis somewhatobscure,and no doubt the reason s that Plato lackeda proof of his conjecture

1 For the processby which the trianglesarestampedout of space(the 'mother ')

by the ideas (the 'father '), cf. my Open Society,note 15 to ch. 3, and the references

there given, as well as note 9 to ch. 6.2 In the lastquotednote

3In the note referredto I also

conjectured

that it was the close approximationof the sum of these two squareroots to

riwhich encouragedPlato in his mistaken

theory. Although I have no new evidence, I believe that this conjectureis much

strengthenedby the view that Plato in fact believedin the mistaken heory described

here.

4 The two quotationsarefrom the Timaeus, 3c/d and 54a/b

150




concerning these two triangles, and felt it should be supplied by

somebody.The obscurity of the passagehad the strangeeffect that Plato's

quite clearly stated choice of trianglesintroduce irrationals nto his

world of Forms seems to have escaped notice, in spite of Plato's

emphasisupon the problem in other places. And this fact, in turn,

may perhapsexplain why Plato'sTheory of Forms could appearto

Aristotleto be fundamentally he same as the Pythagoreantheory of

form-numbers,1 ndwhy Plato'satomismappearedo Aristotlemerely

1 I believe that our considerationmay throw some light on the problemof Plato's

famous' two principles -' The One 'and' The IndeterminateDyad'. The follow-ing interpretationdevelops a suggestionmadeby van der Wielen (De Ideegetallenan

Plato,1941, p. 132f.)

andbrilliantlydefendedagainstvan derWielen's own criticism

by Ross (Plato'sTheory fIdeas,p. 201). We assumethat the 'IndeterminateDyad '

is a straight ine or distance,not to be interpretedas a unit distance,or ashaving yetbeen measured at all. We assume that a point (limit, monas,' One') is placed

successivelyin such positions that it divides the Dyad accordingto the ratio I :n,for any naturalnumber n. Then we can describe the 'generation' of the numbers

as follows. Forn-=

I, the Dyad is divided into two parts whose ratio is I :I.This maybe interpretedas the ' generation of Twoness out of Onenessand the Dyad,

since we have divided the Dyad into two equal parts. Having thus 'generated'the number 2, we can divide the Dyad accordingto the ratio I :2 (and the largersection, as before, accordingto the ratio I :I), thus generatingthreeequal partsand

the number 3 ; generally, the 'generation' of a number n gives rise to a division

of the Dyad in the ratioI : n, and with this, to the ' generation of the number n + I.(And in each stageintervenesthe ' One ', the point which introducesa limit or form

or measure nto the otherwise indeterminate' Dyad, afresh, o create he new number;this remark s intendedto strengthenRoss' caseagainstvan derWielen's.)

Now it should be noted that this procedure,although it 'generates' (in the first

instance, at least) only the series of naturalnumbers, nevertheless contains a geo-

metrical lement-the division of a line, first into two equal parts,and then into two

parts accordingto a certainproportion I : n. Both kinds of division are in need of

geometrical methods, and the second, more especially, needs a method such as

Eudoxus' Theory of Proportions. Now I suggest that Plato began to ask himself

why he shouldnot divide the Dyad also in the proportionofI :V and of I : 1/3.

This, he must have felt, was a departurefrom the method by which the natural

numbers are generated; it is less'arithmetical' still, and it needs more specifically'geometrical' methods. But it would' generate , in the place of naturalnumbers,

linearelementsin the proportionI :V2and I :V-3,

which may be identicalwith the

'atomic lines'

(Metaphysics,92ai9)from which the atomic

trianglesareconstructed.At the sametime, the characterisationf the Dyad as 'indeterminate'would become

highly appropriate,ii view of the Pythagoreanattitude(cf. Philolaos,Diels fragm.2and 3) towards the irrational. (Perhaps he name 'The Greatand the Small' beganto be replaced by 'The IndeterminateDyad' when irrational proportions were

generatedin addition to rationalones.)

ISI



K. R. POPPER

as a comparativelyminor variationof that of Democritus. Aristotle,in spite of his associationof arithmeticwith the odd and even, and of

geometry with the irrational,does not appear to have taken the

problem of the irrationalsseriously. It appears hat he took Plato'sreform programme for geometry for granted; it had been partlycarried out by Eudoxus before Aristotle entered the Academy,andAristotle was only superficiallynterested n mathematics. He neveralludes to the inscriptionon the Academy.

To sum up, it seems probablethat Plato's theory of Formswas,like his theory of matter, a re-statementof the theories of his pre-

decessors, he Pythagoreansand Democritusrespectively, n the lightof his realisationhat the existenceof irrationalsdemanded he emanci-

pation of geometry from arithmetic. By encouragingthis emanci-

pation, Plato contributedto the developmentof Euclid'ssystem, themost important and influential deductive theory ever constructed.

By his adoptionof geometry as the theory of the world, he providedAristarchus,Newton, and Einstein with their intellectual toolbox.The calamityof Greekatomismwas thustransformednto a momen-tous achievement. But Plato'sscientific nterestsare

partly forgotten.The problem-situationn sciencewhich gave rise to his philosophical

problems is little understood. And his greatest achievement, the

geometrical theory of the world, has influencedour world-pictureto such an extent that we unconsciously ake it for granted.

6

One example never suffices. As my second example, out of a

great manyinterestingpossibilities, chooseKant. His CritiquefPureReasons one of the most difficultbooks everwritten. Kantwrote in

undue haste, and about a problem which, I shall try to show, was

insoluble. Nevertheless it was not a pseudo-problem,but an in-

Assuming this view to be correct, we might conjecturethat Plato approached

slowly (beginning n the HippiasMajor,and thuslong beforethe Republic-asopposedto a remarkmade by Ross op. cit.,top of page 56) to the view that the irrationalsre

numbers, ince both the natural numbers and the irrationalsare 'generated' bysimilarand

essentiallygeometricprocesses.But once this view is reached

(andit was

firstreached, t appears, n the Epinomis 9od-e, whether or not this work is Plato's),theneven the irrationaltrianglesof the Timaeus ecome 'numbers' (i.e. characterised

by numerical,if irrational,propositions). But with this, the peculiarcontribution

of Plato,and thedifferencebetweenhis and thePythagorean heory,is liableto become

indiscernible and this may explainwhy it has been lost sight of, even by Aristotle.

152




escapableproblem which arose out of the contemporarysituation of

physicaltheory.His book was written for people who knew some Newtonian

stellar dynamicsand who had at least some idea of its history-of

Copernicus,Tycho, Brahe,Kepler,and Galileo.

It is perhapshard for intellectualsof our own day, spoilt andblas6

as we areby the spectacleof scientificsuccess, o realisewhat Newton's

theory meant, not just for Kant, but for any eighteenth centurythinker. After the unmatcheddaringwith which the Ancients had

tackledthe riddle of the Universe, there had come a period of long

decay,recovery,and thenastaggering uccess. Newton had discoveredthelong soughtsecret. His geometrical heory,basedon andmodelled

afterEuclid,had been at firstreceivedwith greatmisgivings,even byits own originator.' The reason was that the gravitational orce of

attractionwas felt to be ' occult', or at leastsomethingwhich needed

anexplanation. But althoughno plausible xplanationwasfound (andNewton scorned recourseto ad-hochypotheses), all misgivings had

disappeared ong before Kant made his own importantcontribution

to Newtoniantheory, 78 years after

thePrincipia.2No qualifiedjudge 3 of the situationcould doubt any longer that the theory was

true. It has been testedby the most precisemeasurements, nd it had

always been right. It had led to the predictionof minute deviations

from Kepler's aws, and to new discoveries. In a time like ours,when

theoriescome andgo like the buses n Piccadilly,andwhen everyschool-

boy has heard thatNewton haslong been supersededby Einstein, t is

hard to recapturethe sense of conviction which Newton's theory

inspired,or the senseof elation,and of liberation. A unique venthad

happened n thehistoryof thought,onewhich couldnever be repeated:the first and final discoveryof the absolutetruthabout the universe.An age-old dream hadcome true. Mankindhad obtainedknowledge,real, certain, indubitable, and demonstrable knowledge--divinescientiaor episteme, nd not merely doxa,human opinion.

Thus for Kant, Newton's theory was simply true, and the beliefin its truthremainedunshaken or a centuryafterKant'sdeath. Kant

1 See Newton's letter to Bentley, 1693.2

The so-calledKant-LaplaceanHypothesis published by Kant in I755.3Therehadbeensomeverypertinentcriticism(especiallybyLeibnizandBerkeley)

but in view of the successof the theory it was-I believe rightly-felt that the criticshad somehow missed the point of the theory. We must not forget thateven todaythe theory stillstands,with only minor modifications,asanexcellentfirst (or, in viewof Kepler, perhapsas a second) approximation.

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K. R. POPPER

to the end acceptedwhat he and everybody else took for a fact, the

attainmentof scientiaor episteme. At first he accepted it without

question. This state he called his 'dogmatic slumber'. He wasrousedfrom it by Hume.

Hume had taught that there could be no such thing as certain

knowledge of universallaws, or episteme that all we knew was

obtained with the help of observationwhich could be only of par-ticulars,so that our knowledge was uncertain. His argumentswere

convincing (andhe was, of course,right). But herewas the fact, or

what appearedas a fact-Newton's attainmentof episteme.

Hume rousedKant to the realisationof the nearabsurdityof whathe never doubtedto be a fact. Here was a problemwhich could not

be dismissed. How could a man have got hold of suchknowledge?

Knowledge which was general, precise,mathematical,demonstrable,

indubitable,and yet explanatoryof observedfacts?

Thus arose the centralproblem of the Critique 'How is purenatural science possible?'. By 'pure natural

science'--scientia,episteme--Kantmeans, simply, Newton's theory.

Althoughthe

Critiques

badlywritten, and

althoughit abounds

in bad grammar, hisproblemwas not a linguistic puzzle. Here was

knowledge. How could we ever attain it ? The question was

inescapable. But it was also insoluble. For the apparent act of the

attainmentof epistem-was no fact. As we now know, or believe,

Newton's theory is no more than a marvelloushypothesis, an as-

tonishingly good approximation; unique indeed, but not as divine

truth,only asa unique nventionof a humangenius; not episteme, ut

belonging to the realmof doxa. With this,Kant'sproblem, ' How is

pure natural science possible', collapses, and the most disturbingof his perplexitiesdisappear.

Kant's proposed solution of his insoluble problem consisted of

what he proudly calledhis ' CopernicanRevolution' of the problemof knowledge. Knowledge-episteme-was possiblebecausewe arenot

passivereceptorsof sensedata,but theiractivedigestors. By digestingand assimilating hem, we form and organisethem into a Universe.

In this process,we impose upon the materialpresentedto our senses

the mathematicalaws which arepartof our digestiveandorganisingmechanism. Thus our intellect does not discover universal laws

in nature, but it prescribesits own laws and imposes them uponnatures.

This theory is a strange mixture of absurdityand truth. It is

154




as absurd s the mistakenproblemt attemptso solve; for it provestoo much,being designedo provetoo much. Accordingo Kant's

theory, purenatural cience' s not onlypossible althoughhe doesnot realise his,it becomes he necessaryesult f our mentaloutfit.For if the fact of our attainmentf epistemeanbe explained t all

by the fact that our intellectlegislatesor andimposests own laws

uponnature, henthe firstof thesetwo factscannotbe contingentanymore han hesecond.Thus heproblems nolongerhowNewtoncould makehis discoverybut how everybody lse couldhavefailedto make it. How is it thatourdigestivemechanism id not work

muchearlierThis is a patentlyabsurdconsequence f Kant's dea. But to

disnmisst offhand,and to dismisshis problemas a pseudo-problemis not good enough. Forwe canfindanelementof truthn his dea

(anda muchneeded orrection f someHumeanviews),after educinghisproblem o its properdimensions. His question,we now know,or believewe know, shouldhavebeen: 'How aresuccessfulhypo-thesespossible?' And our answer,n the spiritof his Copernican

Revolution,might,I

suggest,be

somethingike

this: Because,as

yousaid,we arenotpassiveeceptorsfsensedata,butactiveorganisms.Becausewe react o ourenvironmentotalwaysmerelynstinctively,but sometimesconsciouslyndfreely. Becausewe can nventmyths,stories, heories; becausewe have a thirstfor explanation, n in-satiablecuriosity,a wish to know. Becausewe not only inventstoriesandtheories, uttrythemout andseewhethertheywork andhow theywork. Because y a great ffort,by tryinghardanderringoften,we maysometimes,f we arelucky,succeedn hittingupona

story,an explanation,which 'saves the phenomena'; perhapsbymakingup a mythabout invisibles',such asatomsor gravitationalforces,whichexplain hevisible. Becauseknowledges anadventureof ideas. These deas, t is true,areproduced y us, andnot by theworldaround s ; theyarenotmerely hetraces f repeatedensationsorstimulior whatnot ; hereyouwereright. Butwe aremoreactiveandfree thanevenyou believed; forsimilarobservationsr similarenvironmentalituationsdo not, as your theoryimplied,produce

similarexplanationsndifferentmen. Nor isthefact hatwe originateourtheories, ndthatwe attempto impose hemupontheworld,an

explanationf theirsuccess, syou believed. Fortheoverwhelmingmajority f ourtheories, f ourfreelynventeddeas,areunsuccessful;theydonotstandupto searchingests,andarediscardedsfalsified y

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K. R. POPPER

experience. Only a very few of themsucceed, or a time, in the

competitive truggle orsurvival.'

7

Few of Kant's uccessorsappear verto haveclearlyunderstoodtheprecise roblem-situationhichgaverise o hiswork. Thereweretwo suchproblemsor him,Newton'sdynamics f theheavens, ndthe absolutetandardsf humanbrotherhoodndjustice o which theFrench evolutionariesappealed,r,asKantputs t, ' thestarry eavens

aboveme,andthe moralaw withinme'. ButKant'sstarry eavensare eldomdentifiedsanallusionoNewton.= FromFichte nward,3

many have copiedKant's method' and the dreadful tyle of his

Critique.But most of these mitators aveforgottenKant'soriginalinterestsndproblems, usily rying ithero tighten,orelsetoexplainaway,theGordian notin whichKant, hroughno faultof hisown,had tied himselfup.

We mustbewareof mistakinghewell-nigh enselessndpointlesssubtleties f the imitatorsorthe

pressingnd

genuine roblemsf the

pioneer. We shouldrememberhathis problem,althoughnot an

empiricalone in the ordinaryense,neverthelessurnedout, unex-

pectedly, o be in somesensefactual Kantcalledsuch facts trans-cendental), since t arose rom anapparentutnon-existentnstanceof a scientia r epistm-e.Andwe should, submit, eriouslyonsiderthe suggestion hat Kant'sanswer, n spiteof its partialabsurdity,contained henucleus f a philosophy f science.

IThe ideasof this 'answer'were elaboratedn

my Logikder

Forschung1935).2 That this identifications correctedmaybe seenfrom the lastten linesof the

venultimate aragraphf the CritiquefPracticaleason.

3Cf. my OpenSociety, ote58to ch. 12

The LondonSchoolof Economics nd PoliticalScience

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sir karl popper. philosophical problems & their roots in science

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