The British Society for the Philosophy of Science

The Nature of Philosophical Problems and Their Roots in Science Author(s): K. R. Popper Source: The British Journal for the Philosophy of Science, Vol. 3, No. 10 (Aug., 1952), pp. 124-156 Published by: Oxford University Press on behalf of The British Society for the Philosophy of Science Stable URL: http://www.jstor.org/stable/685553 . Accessed: 13/09/2013 04:37
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THE NATURE OF PHILOSOPHICAL PROBLEMS AND THEIR ROOTS IN SCIENCE * K. R. POPPER
I

IT was after some hesitation that I decided to take as my point of departurethe present position of English philosophy. For I believe that the function of a scientistor of a philosopheris to solve scientific or philosophicalproblems,ratherthan to talk about what he or other philosophersare doing or might do. Even an unsuccessfulattempt to solve a scientific or philosophicalproblem, if it is an honest and devoted attempt, appearsto me more significantthan any discussion of a question such as ' What is science ?' or 'What is philosophy ? '. And even if we put this latter question,as we should, in the somewhat improved form 'What is the characterof philosophicalproblems ?', I for one should not bother much about it; I should feel that it has little weight if comparedwith even such a minor problem of philosophy, as, say, the question whether every discussionmust always proceed from 'assumptions' or 'suppositions' which themselves are beyond argument.' of When describing'What is the character philosophical problems?' as a somewhat improved form of ' What is philosophy ?', I wished to hint at one of the reasonsfor the futility of the currentcontroversy concerningthe natureof philosophy-the naive belief that there is an entity such as 'philosophy', or perhaps 'philosophical activity', and that it has a certaincharacteror 'nature '. The belief that there is such a thing as physics, or biology, or archaeology,and that these 'studies' or 'disciplines' are distinguishableby the subject matter which they investigate, appearsto me to be a residue from the time when one believed that a theory had to proceedfrom a definitionof its own subjectmatter.2 But subjectmatter,or kinds of things, or classes
* The Chairman's address, delivered at the meeting of 28 April 1952, to the Philosophy of Science Group of the British Society for the History of Science. 1 call this a minor problem because I believe that it can easily be solved, by refuting the (' relativistic') doctrine indicated in the text. 2 This view is part of what I have called 'essentialism'. Cf. for example my N.S., 1944, II, Open Society,ch. II, or 'The Poverty of Historicism I' (Economica No. 42). 124

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THE

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of things, do not, I hold, constitute a basis for the distinction of disciplines. Disciplines are distinguished partly for historical reasons and reasons of administrativeconvenience (such as the organisationof teachingand of appointments),partlybecausethe theories which we constructto solve our problemshave a tendency1 to grow into unified systems. But all this classificationand distinction is a comparativelyunimportantand superficialaffair. We arenot students of subjectmatterbut studentsof problems. And problems may cut right acrossthe bordersof any subject matter or discipline. Obvious as this fact may appearto some people, it is so important for our present discussionthat it is worth while to illustrateit by an example. It hardly needs mentioning that a problem posed to a geologist such as the assessmentof the chances of finding deposits of oil or of uraniumin a certaindistrictneeds for its solution the help of theories and techniques usually classified as 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 geological theoriesand techniques,if it wishes to solve a problem arisingin one of its most abstract fundamental and theories ; for example, the problem of testingpredictionsconcerningthe relative stabilityor instabilityof atoms of an even or odd atomic number. I am quitereadyto admit that many problems,even if theirsolution involves the most diverse disciplines,nevertheless'belong', in some sense, to one or another of the traditionaldisciplines; for example, the two problemsmentioned ' belong' clearly to geology and physics respectively. This is due to the fact that each of them arisesout of a discussionwhich is characteristic the tradition of the discipline in of It arises out of the discussionof some theory, or out of question. empirical tests bearing upon a theory; and theories, as opposed to subject matter, may constitutea discipline (which might be described as a somewhatloose clusterof theoriesundergoinga processofchallenge, change, and growth). But this does not alterthe view that the classification into disciplinesis comparativelyunimportant,and that we are students,not of disciplines,but of problems. But are there philosophical ? problems The present position of English philosophy, which I shall take as my point of departure, originates, I believe, from the late Professor Ludwig Wittgenstein's
1 This tendency can be explained by the principle that theoretical explanations are the more satisfactorythe better they can be supported by independent evidence. (This somewhat cryptic remarkcannot, I fear, be amplified in the presentcontext.) I 125

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K. R. POPPER

influentialdoctrinethat there are none; that all genuine problems are scientific problems; that the alleged problems of philosophy are pseudo-problems; that the alleged propositionsor theories of philoor ; sophy are pseudo-propositions pseudo-theories that they are not false (if false,' their negationswould be true propositionsor theories) but strictlymeaninglesscombinationsof words, no more meaningful than the incoherent babbling of a child who has not yet learned to speak properly.2 As a consequence, philosophy cannot contain any theories. Its true nature, according to Wittgenstein, is not that of a theory, but that of an activity. The task of all genuine philosophy is that of unmasking philosophical nonsense, and of teaching people to talk sense. My plan is to take this doctrine3 of Wittgenstein'sas my starting
1 It is of particularimportance in this connection to realise that Wittgenstein's use of the term 'meaningless' is not the usual and somewhat vague one according to which an absurdly false assertion (such as '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'. 2 Since Wittgenstein describedhis own Tractatus meaningless(see also the next as footnote), he distinguished, at least by implication, between revealing and unimportant nonsense. But this does not affecthis main doctrine which I am discussing, the non-existence of philosophical problems. (A discussion of other doctrines of Wittgenstein's can be found in the Notes to my Open Society,esp. notes 26, 46, 5I, and 52 to ch. ii.) 3 It is easy to 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 counteredin 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 that Logico-Philosophicus whoever understood the book must realise at the end that it was itself meaningless, and must discard it like a ladder, after having used it to reachthe desiredheight.) (2) One might say that the doctrineis not a philosophical but an empirical one, that it states the historical fact that all 'theories' proposed by philosophersare in fact ungrammatical; that they do not, in fact, conform to the rules inherent in those languages in which they appearto be formulated, that this defect turns out to be impossible to remedy ; and that every attempt to express them properly has lead to the loss of their philosophic character (and revealed them, for example, as empiricaltruisms,or as false statements). These two counter argumentsrescue,I believe, the threatenedconsistencyof the doctrine, which in this way indeed becomes 'unassailable', as Wittgenstein puts it by the kind of criticism referredto in this note. (See also the next note but one.) 126

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THE NATURE point (section 2).

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

to some extent ; and to criticise it. And I shall support all this (in sections4 to 6) by some examples from the history of scientific ideas. But before proceeding to carry out 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'sdoctrineis true, then nobody can, in this sense,philosophise. If this were my opinion, I would give up philosophy. But it so happens that I am not only deeply interested in certain philosophical problems (I do not much carewhether they are ' rightly ' called 'philosophical problems'), but possessedby the belief that I may even contribute--if only a little, and only by hardwork-to their solution. And my only excuse for talking here about philosophyinsteadof philosophising-is, in the lastresort,my hope that,in carrying out my programme for this address,an opportunity will offer itself of doing a little philosophising,after all.
2

Ever since the rise of Hegelianism there has existed a dangerous gulf between science and philosophy. Philosopherswere accusedrightly, I believe-of ' philosophising without knowledge of fact ', and their philosophieswere describedas 'mere fancies,even imbecile fancies'.1 Although Hegelianismwas the leadinginfluencein England and on the Continent, opposition to it, and contempt of its pretentiousness,never died out completely. Its downfallwas brought about by a philosopherwho, like Leibniz,Kant,andJ. S. Mill beforehim, had a sound knowledge of science,especiallymathematics. I am speaking of BertrandRussell. Russellis also the author of the classification(closely relatedto his famous theoryof types) which is the basis ofWittgenstein's view of of philosophy, the classification the expressionsof a languageinto
(I) True statements (2) False statements

(3) Meaningless expressions, amongwhich therearestatement-like of words,whichmay be called'pseudo-statements '. sequences Russell with in with operated thisdistinction connection thesolution quotationsarenot the words of a scientificcritic,but, ironicallyenough, own characterisation the philosophy of his friend and forerunnerSchelling. of Hegel's Cf. my Open Society,note 4 (and text) to ch. 12. 127
1 The two

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K. R. POPPER

of the logical paradoxeswhich he discovered. It was essential,for this solution, to distinguish, more especially, between (2) and (3). We might say, in ordinaryspeech, that a false statement,like ' 3 times 4 equals173 ' or ' All catsarecows ', is meaningless. Russell,however, reserved this characterisation expressionssuch as ' 3 times 4 are for not describedas false statements (as can easily be seen from the fact that theirprimafacienegations, for example, ' Some cats do not equal 173 ' are no more satisfactorythan the original expressions)but as pseudo-statements. Russell used this distinction mainly for the elimination of the paradoxes(which, he indicated,were meaninglesspseudo-statements). Wittgenstein went further. Led, perhaps,by the feeling that what philosophers, especially Hegelian philosophers, were saying was somewhatsimilarto the paradoxesof logic, he usedRussell'sdistinction in order to denounce all philosophy as meaningless. As a consequence, therecouldbe no genuinephilosophical problems. 1 All allegedphilosophical into four classes: problemscould be classified to (i) those which are purely logical or mathematical, be answeredby and thereforenot philosophical; logical or mathematical propositions, (2) those which are factual,to be answeredby some statementof the empirical sciences, and therefore again not philosophical; (3) those which are combinations of (i) and (2), and therefore, again, not cows' or ' All cats equal 173 ', that is, for expressions which are better

philosophical; and (4) meaningless pseudo-problems such as 'Do all cows equal 173 ?' or 'Is Socrates identical?' or ' Does an invisible, and apparentlyaltogether unknowable Socratesexist ?' untouchable, Wittgenstein'sidea of eradicatingphilosophy (and theology) with the help of an adaptionof Russell'stheory of types was ingenious and original (and more radical even than Comte's positivism which it resemblesclosely).2 This idea became the inspirationof the powerful school of language analysts who have inherited his belief modernm that there are no genuine philosophical problems, and that all a
Wittgenstein still upheld the doctrine of the non-existence of philosophical problems in the form here describedwhen I saw him last (in 1946, when he presided over a stormy meeting of the Moral Science Club in Cambridge, on the occasion of my reading a paper on 'Are there PhilosophicalProblems ? '). Since I had never which were privately circulatedby some of seen any of his unpublishedmanuscripts 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. note 52 (2) to ch. ii of my Open Society. 128 1

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him out.
1

can philosopher do is to unmaskand dissolvethe linguisticpuzzles which have been proposed traditional by philosophy. is My own view of the matter thatonly as long as I have genuine to in philosophical problems solve shallI continueto takean interest I fail to understand attractionof a philosophy the philosophy. without problems. I know, of course,that manypeople talk nonsense; and it is conceivablethat it should become one's task (an one) to unmasksomebody'snonsense,for it may be unpleasant nonsense. But I believethatsomepeoplehavesaidthings dangerous whichwerenotverygoodsense, certainly verygoodgrammar, and not but which are at the sametime highly interesting exciting,and and more worth listeningto than the good senseof others. I perhaps and which, especially may mentionthe differential integralcalculus in its earlyforms,was, no doubt,completelyparadoxical nonand sensicalby Wittgenstein's which became, (and other) standards; well foundedas the result of some hundred however, reasonably of mathematical efforts butwhosefoundations atthis even ; years great momentare still in need, and in the process,of clarification.' very We mightremember, thiscontext,thatit was the contrast in between the apparent of mathematics the vagueness absolute and and precision the languagewhich deeply impressed inprecisionof philosophical earlier followersof Wittgenstein. But hadtherebeena Wittgenstein to use his weaponsagainstthe pioneersof the calculus, had he and in of where their contemsucceeded the eradication theirnonsense, porary critics (such as Berkeley who was, fundamentally, right) one and failed,then he would have strangled of the most fascinating in the history of thought. philosophically importantdevelopments Wittgensteinonce wrote: 'Whereof one cannot speak, thereof one mustbe silent.' Itwas,if I remember rightly,ErwinSchroedinger ' But it is only herethatspeaking : who replied becomes interesting.' of The historyof the calculus-andperhaps his own theory2-bears

No doubt, we shouldall trainourselvesto speakas clearly,as am alluding to G. Kreisel's recent construction of a monotone bounded sequence of rationalsevery term of which can be actually computed, but which does to not possessa computablelimit-in contradictionto what appears be the prima facie interpretationof the classicaltheorem of Bolzano and Weierstrass,but in agreement with Brouwer's doubts about this theorem. Cf. Journalof Symbolic Logic, 1952, 17, 572 Before Max Born proposed his famous probability interpretation, Schroedinger's wave equation was, some might contend, meaningless. 129

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K. R. POPPER

precisely, as simply, and as directly as we can. But I believe that there is not a classic of science, or of mathematics,or indeed a book worth reading that could not be shown, by a skilful application of the technique of language analysis,to be full of meaninglesspseudopropositionsand what some people might call ' tautologies'.
3

But I have promisedto say somethingin defenceof Wittgenstein's views. What I wish to say is, first, that there is much philosophical in writing (especially the Hegelianschool)which mayjustly be criticised as meaningless verbiage; secondly, that this kind of irresponsible writing was checked,for a time at least,by the influenceof Wittgenstein and the languageanalysts(althoughit is likely thatthe most wholesome influence in this respectwas the example of Russellwho, by the incomparablecharm and the clarity of his writings, establishedthe fact that subtlety of content was compatible with lucidity and unpretentiousnessof style). But I am prepared to admit more. In partial defence of to Wittgenstein'sview, I am prepared defendthe following two theses. first thesis is that every philosophy, and especially every My philosophical 'school', is liable to degeneratein such a way that its from pseudo-problems, problems become practicallyindistinguishable and its cant, accordingly,practically from meaningless indistinguishable babble. This, I shall try to show, is a consequence of philosophical inbreeding. The degeneration of philosophical schools is the conwithout being sequenceof the mistakenbelief that one can philosophise to turn to philosophy by problems whichariseoutside compelled philosophy-in mathematics,for example, or in cosmology, or in politics, or in religion, or in sociallife. To put it in otherwords, my first thesis is this. Genuinephilosophical problemsare always rooted in urgent outside and they die if theserootsdecay. In their problems philosophy, efforts to solve them, philosophersare liable to pursue what looks like a philosophicalmethod or like a techniqueor like an unfailingkey to philosophicalsuccess.' But no such methods or techniquesexist; philosophicalmethods are unimportant,and any method is legitimate
1 It is veryinteresting thatthe imitators werealwaysinclinedto believethatthe 'master' did his work with the help of a secretmethodor a trick. It is reported a thatin J. S. Bach'sdayssome musicians believedthathe possessed secretformula for the construction fuguethemes. of

130

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if it leadsto resultscapableof being rationallydiscussed. What matters is neither methods nor techniques-nothing but a sensitivenessto problems, and a consuming passionfor 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,like a disorderwhich they have to get out of their system.' 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 nevertheless produce exercises in fashionable methods, and for whom philosophy is application whatever insight or technique you like) (of ratherthansearch. They areluring philosophyinto the bog of pseudoproblems and verbal puzzles; either by offering us pseudo-problems for real ones (the danger which Wittgenstein saw), or by persuading us to concentrate upon the endless and pointless task of unmasking what they rightly or wrongly take for pseudo-problems(the trapinto which Wittgenstein fell). My second thesisis that what appearsto be the primafaciemethod of teachingphilosophyis liable to producea philosophywhich 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 greatphilosophersto read ; say, of Plato and Aristotle, Descartes and Leibniz, Locke, Berkeley, Hume, Kant, and Mill. What is the effect of such a course of reading? A new world of astonishinglysubtle and vast abstractions opens itself to the reader, abstractionsof an extremely high and difficult level. Thoughts and arguments are put before his mind which sometimes are not only hard to understand, whose relevanceremainsobscure but since he cannot find out what they may be relevant to. Yet the studentknows that these are the greatphilosophers, that this is the way of philosophy. Thus he will make an effortto adjusthis mind to what he believes (mistakenly,as we shall see) to be their way of thinking. He will attemptto speaktheir queerlanguage, to match the torturous spiralsof their argumentation,and perhapseven tie himself up in their curious knots. Some may learn these tricks in a superficialway, to on Gilbert whosays page of his 9 by Ryle, 1I amalluding a remark Professor I : to out Concept Mind 'Primarily am trying get somedisorders of my own of system.
13I

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K. R. POPPER others may begin to become genuinely fascinated addicts. Yet I feel that we ought to respect the man who, having made his effort, comes ultimately to what may be describedas Wittgenstein's conclusion : 'I have learned the jargon as well as anybody. It is very clever and captivating. In fact, it is dangerously captivating; for the simple truthabout the matteris that it is much ado about nothing --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 prima facie method of teaching philosophy here described. (I do not deny, of course, that some particularlygifted students may find very much more in the works of the greatphilosophers than this story indicatesand without deceiving themselves.) For the chance of finding out the extra-philosophical problems (the mathematical,scientific, moral and political problems) which inspired these great philosophers is very small indeed. These problems can be discovered, as a rule, only by studying the history of, for example, scientificideas, and esin pecially the problem-situation mathematicsand the sciences of the in question ; and this, in turn, presupposesa considerableacperiod of quaintancewith mathematicsand science. Only an understanding the contemporary problem-situationin the sciences can enable the student of the great philosophers to understandthat they tried to solve urgent and concrete problems; problems which, they found, could not be dismissed. And only after understanding fact can a this student attaina differentpictureof the great philosophies-one which makes full sense of the apparentnonsense. I shall try to establishmy two theses with the help of examples; but before turning to these examples,I wish to summarisemy theses, and to balancemy account with Wittgenstein. My two thesesamount to the contentionthat philosophyis deeply rooted in non-philosophicalproblems; that Wittengstein'snegative judgment is correct,by and large, as far as philosophiesare concerned which have forgotten their extra-philosophical roots; and that these roots are easily forgotten by philosopherswho ' study' philosophy, instead of being forced into philosophy by the pressure of nonphilosophicalproblems. My view of Wittgenstein'sdoctrinemay be summedup as follows. It is true, by and large, that pure philosophicalproblemsdo not exist ; for indeed, the purer a philosophicalproblem becomes, the more is lost of its original sense, significance,or meaning, and the more liable
132

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THE NATURE OF PHILOSOPHICAL PROBLEMS is its discussionto degenerateinto empty verbalism. On the other hand, there exist not only genuine scientific problems, but genuine philosophicalproblems. Even if, upon analysis,these problems turn out to-havefactualcomponents,they need not be classified belonging as to science. And even if they should be soluble by, say, purely logical means, they need not be classified as purely logical or tautological. Analogoussituationsarisein physics. For example,the explanationor prediction of certain spectral terms (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 to physics. We are perfectlyjustified in calling a problem 'physical' if it is connected with problems and theories which have been traditionally discussed by physicists (such as the problems of the constitution of matter), even if the means used for its solution turn out to be purely mathematical. As we have seen, the solution of problemsmay cut through the boundaryof many sciences. 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,it does not matter in the least what kind of methods we use in solving such a problem. interest, Cosmology, for example,will alwaysbe of greatphilosophical even though by some of its methodsit hasbecome closely alliedto what is perhaps better called 'physics '. To say that, since it deals with factual issues, it must belong to science ratherthan to philosophy, is not only pedanticbut clearlythe resultof an epistemological,and thus of 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 typically philosophical,or physical, or biological. For example, logical analysis played 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 therefore all genuine problems) can be classified into one of two exclusive classes: factual statements a and a (synthetic posteriori), logical statements (analytic priori). This
'33

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K. R. POPPER

of for valuable the purposes a simpledichotomy, extremely although too turnsout to be for manypurposes simple.1 But roughsurvey, of the to designed exclude existence althoughit is, as it were, specially even this aim; it philosophical problems, is very far from achieving for even if we acceptthe dichotomy, can still claimthat factual we or logicalor mixedproblems turnout, in certain circumstances, may to be philosophical.
4 I now turn to my first example : Platoandthe Crisisin EarlyGreek Atomism. My thesis here is that Plato's central philosophical doctrine, the so-called Theory of Forms or Ideas, cannot be properly understood except in an extra-philosophicalcontext 2; more expecially in the context of the critical problem situation in Greek science3 (mainly
1 Already in my LogicderForschung (Vienna, 1935), I pointed out that a theory such as Newton's may be interpreted either as factual or as consisting of implicit which definitions (in the senseofPoincar6 and Eddington), and that the interpretation a physicistadoptsexhibits itself in his attitude towardstestswhich go againsthis theory ratherthan in what he says. The dogma of the simple dichotomy has been recently attacked,on very differentlines, by F. H. Heinemann (Proc.of theXth 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 precise sense, 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 unmask a theory as ' tautological '. I andIts Enemies, have tried to explain in some detailanother 2 In my OpenSociety extra-philosophicalroot of the same doctrine, viz. a political root. I also discussed there (in note 9 to ch. 6 of the revised 4th edition, 1952) the problem with which I am concerned in the present section, but from a somewhat different angle. The note referredto and the presentsection partly overlap ; but they are largely supplementary to each other. Relevant references (esp. to Plato) omitted here will be found there. 3 There are historianswho deny that the term 'science' can be properly applied to any developmentwhich is older than the sixteenth or even the seventeenthcentury. But quite apartfrom the fact that controversiesabout labels should 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 Copernicus and Plato, or Kepler and Aristarchus(the ' Copernicusof antiquity '). And any doubt concerningthe extreme age of scientific 134

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in the theory of matter)which developedas a consequence the of If my thesis root of discovery the irrationality the square of two. of is correct,then Plato'stheoryhas not so far been fully understood. can is, (Whethera 'full' understanding ever be achieved of course, most questionable.)But the moreimportant would be consequence thatit can neverbe understood philosophers trained accordance in by with theprima methoddescribed the foregoing in section-unless, facie of course,they arespecially adhocinformed the relevant of and facts have to accepton authority). (whichthey may It is well known1 that Plato'sTheoryof Formsis historically as well as in its contentcloselyconnected with the Pythagorean theory thatall thingsare,in essence, of numbers.The details thisconnection, and the connectionbetween Atomism and Pythagoreanism, are not so well known. I shalltherefore the whole story tell perhaps in brief,asI seeit at present. It appears the founderof the Pythagorean that orderor sectwas two discoveries. The first discovery was that deeplyimpressed by a prima purelyqualitative suchas musical facie phenomenon harmony basedupon the purelynumerical ratiosI : 2; 2 : 3 ; was, in essence, ' was 3 : 4. The second thatthe' right' or ' straight angle(obtainable for example foldinga leaftwice,so thatthe two foldsforma cross) by was connected with the purelynumerical ratios3 : 4 : 5, or 5 : 12 : 13 sidesof rectangular Thesetwo discoveries, appears, it (the triangles). led Pythagoras the somewhat to fantastic that generalisation all things was are, in essence,numbers,or ratiosof numbers or thatnumber ; the ratio(logos reason), rational the essenceof things,or theirreal = nature. Fantastic this ideawas, it provedin manyways fruitful. One as of its most successful was applications to simplegeometrical figures, observation,and of carefulcomputationsbased upon observation,has been dispelled nowadays 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 Cidenas the Chaldean (314 B.c.), whose computations of the 'constants for the motion of Sun and Moon' are without exception comparablein precision to those of the best nineteenth-centuryastronomers, 'Cidenas' value for the motion of the Sun from the Node (o"-5to great), although inferiorto Brown's, is superiorto at leastone of the most widely used modem values ', wrote J. K. Fotheringhamin 1928, in his most admirablearticle 'The Indebtedness of Greekto ChaldeanAstronomy' (The Observatory, 1928, 51, No. 653), upon which my contention concerning the age of astronomy is based. 1 From Aristotle's Metaphysics 135

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such as squares,rectangularand isoscelestriangles,and also to certain simple solids, such as pyramids. The treatment of some of these geometricalproblems was based upon the so-calledgnomon. This can be explainedas follows. If we indicatea squareby four dots,

we may interpret this as the result of adding three dots to the one dot on the upper left corner. These three dots are the firstgnomon; we may indicateit thus :

By addinga secondgnomon, consistingof five more dots, we obtain

One seesat once thatevery numberof the sequenceof the odd numbers, 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,its area (total number of dots = n2) will be equal to the sum of the first n odd numbers. As with the treatmentof squares,so with the treatmentof isosceles triangles.

Here each gnomonis a last horizontalline of points, and each element numbers' of the sequenceI, 2, 3, 4, . . . is agnomdn. The' triangular are the sums I + 2; I + 2 + 3 ; I + 2 + 3 + 4, etc., that is, the
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sums of the first n naturalnumbers. By putting two such triangles side by side

we obtainthe parallelogram with the horizontalsiden + I andthe other side n, containing n(n + I) dots. Since it consists of two isosceles triangles,its number is 2(I + + . . . + n), so that we obtain the equation (3) I + 2 + . . . + n = In(n + I) and
(4) d(I +2 + .... +n) . 2

n(n +

).

From this it is easy to obtain the general formula for the sum of an arithmeticalseries. We also obtain 'oblong numbers', thatis the numbersof oblong rectangularfigures, of which the simplestis

with the oblong numbers 2 + 4 + 6 . . . , i.e. the gn'im-inof an oblong is an even number,and the oblong numbersare the sums of the even number. These considerations were extended to solids; for example, by summing the first triangular number, pyramid numbers were obtained. But the main applicationwas to plain figures,or shapes,or ' Forms'. These, it was believed, are characterised the by appropriate sequence of numbers, and thus by the numerical ratios of the consecutive numbers of the sequence. In other words, 'Forms' are or numbers ratiosof numbers. On the other hand, not only shapes
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of things, but also abstract properties,such as harmony, and ' straightare numbers. In this way, the general theory that numbers ness' arethe rationalessencesof all things,is arrivedat with some plausibility. It is very probablethat the developmentof thisview was influenced the similarityof the dot-diagrams with the diagramof a constellation by suchas the Lion, or the Scorpion,or the Virgo. If a Lion is an arrangement of dots, it must have a number. In this way the belief seems to have arisen that the numbers, or 'Forms', are heavenly shapes of things. One of the main elements of this early theory was the so-called distinctionbetween 'Table of Opposites', basedupon the fundamental odd and even numbers. It containssuch things as ONE ODD MALE MANY EVEN FEMALE CHANGE (BECOMING) INDETERMINATE OBLONG CROOKED LEFT DARKNESS BAD

DETERMINATE SQUARE STRAIGHT RIGHT LIGHT GOOD

REST (BEING)

In reading through this strange table one gets some idea of the working of the Pythagoreanmind, and why not only the 'Forms' or shapes of geometrical figures were consideredto be numbers, in ideas,suchasJusticeand, of course,Harmony, essence,but also abstract and Health, Beauty and Knowledge. The table is interesting also becauseit was taken over, with very little alteration,by Plato. Plato's famous theory of 'Forms' or 'Ideas' may indeed be described, somewhat roughly, as the doctrine that the ' Good' side of the Table of Oppositesconstitutesan (invisible)Universe, a Universe of Higher Reality, of the Unchanging and Determinate 'Forms' of all things, and that True and Certain Knowledge (epistwme scientia science) = = can be of this Unchanging and Real Universe only, while the visible world of change and flux in which we live and die, the world of generationand destruction,the world of experience,is only a kind of reflectionor copy of thatRealWorld. It is only a world of appearance, of which no True and Certain Knowledge can be obtained. What are can be obtained in the place of Knowledge (episteme) only the
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plausible but uncertain and prejudiced opinions (doxa) of fallible mortals.' In his interpretationof the Table of Opposites, Plato was influencedby Parmenides,the man who stimulatedthe development of Democritus' atomic theory. Returning now to the original Pythagorean view, there is one thing in it which is of decisive importancefor our story. It will have been observed that the Pythagorean emphasis upon Number was fruitful from the point of view of the development of scientific ideas. This is often but somewhat loosely expressedby saying that the Pythagoreans encouraged numerical scientific measurements. Now the point which we must realiseis that, for the Pythagoreans,all this was counting rather measuring.It was the counting of numbers, than of invisible essencesor 'Natures ' which were Numbers of little dots or stigmata. Admittedly, we cannot count these little dots directly, since they are invisible. What we actually do is not to count the Numbers or Natural Units, but to measure,i.e. to count arbitrary visible units. But the significanceof measurementswas interpreted as revealing, indirectly, the true Ratios of the Natural Units or of the NaturalNumbers. Thus Euclid's methods of proving the so-called 'Theorem of Pythagoras' (Euclid'sI, 47) according to which, if a is the side of a triangleopposite to its right angle between b and c, a2 = b2+ C2, (I) was completely foreign to the spirit of Pythagorean mathematics. In spite of the fact that the theorem was known to the Babylonians and geometrically proved by them, neither Pythagoras nor Plato appearto have known the generalgeometrical proof; for the problem for which they offered solutions, the arithmetical of finding the one integral solutions for the sides of rectangulartriangles,can be easily
1 Plato's distinction (epistemE doxa) derives, I think, from Parmenides (truth vs. 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 uncertainty even if it utilisesthe epistemF, knowledge of the unchanging' Forms' and of pure the mathematics,to the utmost, and even if it interpretsthe visible world with the help of a theory of the invisible world. Cf. Calylus, 439b f., Rep. 476d f. ; and 29b especially Timaeus, ff., where the distinctionis appliedto those partsof Plato'sown theory which we should nowadays call' physics' or 'cosmology ', or, more generally, 'natural science '. They belong, Plato says, to the realm of doxa (in spite of the fact = that science= scientia epistemF cf. my remarkson this problem in ThePhilosophical ; April 1952, p. 168). For a different view concerning Plato's relation to Quarterly, Parmenides,see Sir David Ross, Plato's Theory Ideas,Oxford, 1951, p. 164. of 139

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solved, if (i) is known by the formula (m and n are naturalnumbers, and m > n) a = m2 - n2 ; b = 2mn ; c = m2- n2. (2) But formula (2) was unknown to Pythagorasand even to Plato. This emerges from the traditionaccording to which Pythagorasproposed the formula
(3) a = 2m(m
+ I) + I; b = 2m(m + I) ; c = 2m + I

of which can be read off the gn-omon the squarenumbers,but which is less general than (2), since it fails, for example, for 8 : 15 : 17. To Plato, who is reportedto have improved Pythagoras'formula (3), is attributed another formula which still falls short of the general solution (2). We now come to the discovery of the irrationality thesquare root of two. According to tradition, this discovery was made within the of Pythagoreanorder,but was kept secret. (This is suggestedby the old term for 'irrational', ' arrhetos that is, 'unspeakable', which might ', well have meant 'the unspeakable mystery '.) This discovery struckat the root of Pythagoreanism for it meant that such a simple ; geometrical entity as the diagonal d of the square with the side a could demonstrably not be characterisedby any ratio of natural numbers; d: a was no ratio. The traditionhas it that the member of the school who gave away the secret was killed for his treachery. of However this may be, thereis little doubt that the realisation the fact that irrationalmagnitudes (they were, of course, not recognised as numbers)existed, and that their existencecould be proved, led to the downfall of the Pythagoreanorder. The Pythagorean theory, with its dot-diagrams, contains, no doubt, the suggestion of a very primitive atomism. How far the atomic theory of Democritus was influenced by Pythagoreanismis 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 rational understandingof change. (I differ here from the interpretations of Cornford and others.) I think that this problem derives from Ionian rather than from Pythagorean thought, and that it has remainedthe fundamentalproblem of Natural Philosophy. Although Parmenideshimself was not a physicist (as opposed to his great Ionian predecessors),he may be described, I believe, as physics. He produced an anti-physical having fathered theoretical
14o

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THE NATURE OF PHILOSOPHICAL PROBLEMS system. theory which, however, was the first hypothetical-deductive And it was the beginning of a long series of such systems of physical theories each of which was an improvement on its predecessor. As a rule the improvementwas found necessaryby the realisationthat the earlier system was falsified by certain facts of experience. Such an empiricalrefutationof the consequencesof a deductivesystem leadsto efforts at its reconstruction,and thus to a new and improved theory which, 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 subtle as the theoriesbecame more and more capableof accountingfor the cruderobservations. In the case of Parmenides'theory, the clash with observation was so obvious that it would seem perhapsfanciful to describethe theory as the first hypothetical-deductivesystem of physics. We may, therefore, describe it as the last pre-physical deductive system, whose falsificationgave rise to the first truly physical theory, the atomistic theory of Democritus. Parmenides'theory is simple. He finds it impossible to understand 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 hopelessly unrealistic theory, we shouldfirst realisethat thereis a seriousproblem here. If a thing X changes, then clearly it is no longer the same thing X. On the other hand, we cannot say that X changeswithout implying that X persistsduring the change; that it is the same thing X, 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 that changes, and thereforethe idea of change, is impossible. All this sounds very philosophicaland abstract,and so it is. But it is a fact that the difficultyhere indicatedhas never ceased to make itself felt in the development of physics.1 And a deterministic system such as that of Einstein'sfield theory might even be describedas a four-dimensional versionof Parmenides' unchangingthree-dimensional universe. For, in a sense, no change occurs in Einstein's fourdimensionalblock-universe. Everything is there just as it is, in its four-dimensional locus; change becomes a kind of 'apparent' change; it is 'only' the observerwho, as it were, glides along his and one 1This may be seenfrom EmileMeyerson's Identity Reality, of the most studies the development physical of of theories. interesting philosophical
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world-line and becomes successivelyconscious of the different loci . along this world-line, that is, of his spatio-temporal surrounding . . To ieturn from this new Parmenides the older fatherof theoretito cal physics, we may presenthis deductivetheory roughly as follows. (I) Only what is, is. (2) What is not does not exist. (3) Non-being, that is, the void, does not exist. it is full). Motion is impossible (sincethereis no empty spaceinto which (6) anything could move). The conclusions (5) and (6) were obviously contradictedby facts. Thus Democritus argued from the falsity of the conclusion to that of the premises: (6') There is motion (thus motion is possible). (5') The world has parts; it is not one, but many. (4') Thus the world cannot be full.' (3') The void or (non-being) exist. So far the theory had to be altered. With regardto being, or to the many existing things (asopposedto the void), Democritusadopted Parmenides'theory that they had no parts. They were indivisible (atoms), because they were full, because they had no void inside. The centralpoint of this theory is that it gives a rationalaccount of change. The world consistsof empty space (the void) with atoms in it. The atoms do not change; they are Parmenideanindivisible of block universesin miniature.2 All changeis due to rearrangement is atoms in space. Accordingly, all change movement. Since the only it kind of novelty possibleis novelty of arrangement, is, in principle, in possible to predictallfuture changes the world,provided we manage to predict the motion of mass-points.
1 The inference from the existence of motion to that of a void does not follow, because Parmenides'inference from the fullness of the world to the impossibility of motion does not follow. Plato seems to have been the first to see, if only dimly, that in a full world circularor vortex-like motion is possible, provided that there is a liquid-like medium in the world. (Peas can move with the vortices of pea-soup.) becomes the basisof This idea, first offered somewhat half-heartedlyin the Timaeus, and of the light-ether theory as it was held down to 19o5. Cartesianism 2Democritus' theory admitted also large block-atoms, but the vast majority of his atoms were invisibly small.

(4) The worldis full. (5) The world has no parts; it is one huge block (because

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Democritus' theory of change was of tremendousimportancefor the development of physicalscience. It was partly acceptedby Plato, who retained much of atomism but explained change not only by unchangingyet moving atoms, but also by the 'Forms' which were subjectneither to change nor to motion. But it was condemned by Aristotlewho taughtin its steadthatall changewas the unfoldingof the inherent potentialities of essentially unchanging substances.But alas of though Aristotle'stheory of substances the subjects changebecame dominant,it proved barren; 1 and Democritus'theory that all change must be explained by movement became the tacitly accepted official programme of physics down to our own day. It is still part of the philosophy of physics, in spite of the fact that physics itself has outgrown it (to say nothing of the biological and social sciences). of Forwith Newton, in additionto moving mass-points,forces changing scene. True, these changescan be intensity (and direction) enter the explained as due to, or dependent upon, motion, that is upon the changing position of particles,but they are neverthelessnot identical with the changesin position; owing to the squarelaw, the dependence is not even a linear one. And with Faradayand Maxwell, changing fields of forcesbecome as importantas materialatomic particles. That our modern atoms turn out to be compositeis a minor matter; from Democritus'point of view, not our atoms but ratherour elementary would be realatoms--except thatthesetoo turnout to be liable particles to change. Thus we have a most interestingsituation. A philosophy of change, designed to meet the difficulty of understandingchange rationally, serves sciences for thousands of years, but is ultimately supersededby the development of science itself; and this fact passes practically unnoticed by philosophers who are busily denying the existence of philosophicalproblems. Democritus'theory was a marvellousachievement. It provided a theoreticalframework for the explanationof most of the empirically known propertiesof matter (discussedalready by the lonians), such as compressibility,degrees of hardnessand resilience,rarefactionand condensation, coherence,disintegration, combustion,and many others.
1 The barrennessof the ' essentialist' (cf. note 2 above) theory of substanceis connected with its anthropomorphism; for substances(as Locke saw) take their plausibility from the experience of a self-identicalbut changing and unfolding self. But although we may welcome the fact that Aristotle'ssubstanceshave disappeared from physics,there is nothing wrong, as ProfessorHayek says,in thinking anthropomorphically about man; and there is no reason why they should disappearfrom psychology.

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But apart from being importantas an explanationof the phenomena of experience, the theory was important in other ways. First, it establishedthe methodological principle that a deductive theory or explanationmust ' save the phenomena', that is, must be in agreement with experience. Secondly, it showed that a theory may be speculative, and based upon the fundamental (Parmenidean) principle that world as it must be understoodby argumentativethought turns the out to be differentfrom the world of prima facie experience,from the world as seen, heard, smelled, tasted, touched; 1 and that such a speculativetheory may neverthelessaccept the empiricist'criterion' that it is the visible that decidesthe acceptanceor rejectionof a theory of the invisible2 (such as the atoms). This philosophy has remained fundamentalto the whole development of physics, and has continued 3 to conflict with all ' relativistic' and 'positivistic' 4 tendencies. Furthermore,Democritus' theory led to the first successesof the method of exhaustion(the forerunnerof the calculus of integration), since Archimedeshimself acknowledgedthat Democrituswas the first to formulate the theory of the volumes of cones and pyramids.5 But perhaps the most fascinatingelement in Democritus' theory is his doctrine of the quantisationof space and time. I have in mind that the doctrine,now extensivelydiscussed,6 thereis a shortestdistance and a smallesttime interval, that is to say, distancesin space and time (elementsof length and time, Democritus'ameres7 in contradistinction to his atoms) such that no smallerones are measurable.
Cf. Democritus, Diels, fragm. 11 (cf. Anaxagoras,Dield fragm. 21 ; see also fragm. 7). 2 (Bekker)vii. 140, p. 221, 23B. S Cf. Sextus Empiricus,Adv. mathem. 'Relativistic' in the sense of philosophical relativism, e.g. of Protagoras' to doctrine. It is, unfortunately,still necessary emphasisethat Einstein's homomensura has nothing in common with philosophicalrelativism. theory 4 Such as those of Bacon; the theory (but fortunately not the practice) of the early Royal Society; and in our time, of Mach (who opposed atomic theory) ; and of the sense-datatheorists. 5 Cf. Diels, fragm. 155, which must be interpretedin the light of Archimedes (ed. Heiberg) II2, p. 428 f. Cf. S. Luria'smost importantarticle 'Die Infinitesimalzur & methode der antikenAtomisten ' (Quellen Studien Gesch.d. Math.Abt. B. Bd. 2, Heft 2 (1932), p. 142). 6 Cf. A. March, Natur undErkenntnis, Vienna, 1948, p. 193 f. 7 Cf. S. Luria, op. cit., esp. pp. 148 ff., 172 if.
1

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THE NATURE OF PHILOSOPHICAL PROBLEMS 5 was developed and expounded as a point Democritus' atomism for point reply 1 to the detailedargumentsof his Eleatic predecessors, of Parmenides and of his pupil, Zeno. EspeciallyDemocritus'theory of atomic distancesand time intervals is the direct result of Zeno's arguments,or more precisely, of the rejection of Zeno's conclusions. But nowhere in Zeno is there an allusionto the discoveryof irrationals. We do not know the date of the proof of the irrationalityof the squareroot of two, or the date when the discovery became publicly known. Although there existed a traditionascribingit to Pythagoras 2 someauthors callit the ' theorem (sixthcentury B.C.), andalthough can be little doubt that the discovery was not of Pythagoras', there made, and certainlynot publicly known, before 450 B.C., andprobably not before 420. Whether Democritus knew about it is uncertain. I now feel inclined to believe that he did not; and that the title of kai Democritus' two lost books, Peri alogangrammbn kastan,should be translated On IllogicalLines and Full Bodies (Atoms),3and that
1 This point for point reply is preserved Aristotle'sOn Generation Corruption, in and 316a 14 ff., a very important passagefirst identified as Democritean by I. Hammer by Jensenin 1910 and carefullydiscussed Luriawho says (op.cit. 135) ofthe Parmenides and Zeno : 'Democritus borrows their deductive arguments,but he arrivesat the opposite conclusion.' 2 Cf. G. H. Hardy and H. M. Wright, Introduction the to of Theory Numbers (1938), 39, 42, where a very interestinghistoricalremarkon Theodorus'proof, as reported pp. will in Plato's Theaeteteus, be found. 3 Rather than On Irrational Lines and Atoms, as I translatedin note 9 to ch. 6 of my Open Society(reviseded.). What is probably meant by the title (considering Plato's passagementioned in the next note) might, I think, be best renderedby ' On and Crazy Distances Atoms'. Cf. H. Vogt, Bibl. Math., 190Io, 10, 147, and S. Luria, op. cit. pp. I68 ff., where it is convincingly suggested that (Arist.) De insec. lin. 968b 17 and Plutarch,De comm. notit.,38, 2, p. 1078 f., contain tracesof Democritus' work. According to these sources, Democritus' argument was this. If lines are infinitelydivisible,then they are composed of an infinity of ultimate units and are therefore all relatedlike oo : oo, that is to say, they are all 'non-comparable ' (there is no proportion). Indeed, if lines are consideredas classesof points, the 'number' (potency) of the points of a line is, according to modem views, equal for all lines, whether the lines are finite or infinite. This fact has been describedas ' paradoxical' (for example, by Bolzano) and might well have been described as 'crazy' by Democritus. It may be noted that, according to Brouwer, even the classicaltheory of the measure a continuum leadsto fundamentallythe same results; sincehe asserts of that all classicalcontinua have zero measure,the absence of a ratio is here expressed by o : o. Democritus' result (and his theory of ameres) appearsto be inescapableas i.e. method, on the counting long as geometry is based on the Pythagoreanarithmetical of dots.

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these two books do not contain any reference to the problem of irrationality.1 My belief that Democritus did not know about irrationalitiesis based on the fact that there are no traces of a defence of his theory againstthe 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 all measurement ultimately, is, of naturalunits,so that every measurement must be reducible counting to pure numbers. The distancebetween any two atomic points must, therefore, consist of a certain number of atomic distances; thus all must be commensurable. But this turnsout to be impossible distances even in the simple caseof the distances between the cornersof a square, of because of the incommensurability its diagonal with its side. It was Plato who realisedthis fact, and who in the Laws stressed its importancein the strongest possible terms, denouncing his compatriotsfor their failureto realisewhat it meant. It is my contention that his whole philosophy, and especially his theory of' Forms' or 'Ideas', was influencedby it. Plato was very close to the Pythagoreansas well as to the Eleatic Schools ; and although he appearsto have felt antipathetic Demoto critus, he was himself an atomist of a kind. (Atomist teaching remainedas one of the school traditionsof the Academy.2) This is not surprisingin view of the close relation between Pythagoreanand atomistic ideas. But all this was threatenedby the discovery of the irrational. I suggest that Plato's main contributionto science sprang from his realisationof the problem of the irrational,and from the modification of Pythagoreanismand atomism which he undertook situation. in order to rescuescience from a catastrophic He realisedthat the purely arithmeticaltheory of nature was defeated, and that a new mathematical method for description and explanation of the world was needed. Thus he encouraged the development of an autonomous geometricalmethod which found its fulfilment in the 'Elements' of the PlatonistEuclid. What are the facts? I shall try to put them all briefly together. and atomism in Democritus'form were both (I) Pythagoreanism based on arithmetic,that is on counting. fundamentally 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. 534d) alludesto Democritus' title, nevertheless never uses 'alogos' as a synonym for 'arrhetos'. 2 See S. Luria, esp. on Plutarch, loc. cit.
1 This

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of the character the discovery (2) Plato emphasised catastrophic of the irrationals. over the door of the Academy: 'Nobody un(3) He inscribed trained geometrymay entermy house'. But geometry, in according to Plato'simmediatepupil Aristotleas well as to Euclid,treatsof or in to incommensurables irrationals, contradistinction arithmetic whichtreats the odd andthe even'. of' (4) Within a shorttime afterPlato'sdeath,his schoolproduced, a in Euclid'sElements, work whose main point was that it freed of mathematics the ' arithmetical' from assumption commensurability or rationality. and to (5) Plato himselfcontributed this development, especially to the development solid geometry. of a he geo(6) More especially, gave in the Timaeus specifically atomictheory, metricalversionof the formerlypurelyarithmetical the thatis, a version whichconstructed elementary (the particles famous the which incorporated irrational Platonicbodies) out of triangles squareroots of two and of three. (See below.) In most other ideasas well as some of the he both Pythagorean respects, preserved most important ideas of Democritus.' At the same time, he tried that to eliminate Democritus' void; for he realised motionremains motionis conceived of the as evenin a ' full ' world,provided possible some of the most character vorticesin a liquid. Thushe retained of fundamental ideasof Parmenides.2 of the models of (7) Plato encouraged construction geometrical modelsexplaining planetary the movements. the world,andespecially as in Euclid'sgeometrywas not intended an exercise puregeometry but now usuallyassumed), as a theory the world. Ever since3 of (as
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 greatestimportancefor the history of physicsdown to our own day. For it influenced Descartes strongly, became the basis of the theory of ether and light, and thus ultimately,via Huyghens and Maxwell, of de Broglie'sand of Schroedinger'swave mechanics. of 3The only exception is the partial reappearance arithmeticalmethods in the New Quantum Theory, e.g. in the electron shell theory of the periodic system based upon Pauli's exclusion principle.

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Plato and Euclid, but not before, it has been taken for granted that instrumentof all geometry (ratherthan arithmetic)is the fundamental physicalexplanationsand descriptions,of the theory of matteras well as of cosmology.' These are the historical facts. They go a long way, I believe, to establishmy contentionthat what I have describedas the prima facie method of philosophycannotlead to an understanding the problems of which inspiredPlato. Nor can it lead to an appreciation what may of be justly claimed to be his greatest philosophical achievement, the geometricaltheory of the world which became the 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 pure mathematics-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 nonsensicalcomponents) was a philosophicalachievement ; why at least part of his philosophyof natureand of physicshas lasted and, I believe, will last. What we find in Plato and his predecessorsis the conscious constructionand invention of a new approachtowards the world and towards the knowledge of the world. This approach transformsa the fundamentally theological idea, the ideaof explaining visibleworld a postulated invisibleworld,2into the fundamentalinstrument of by theoretical science. The idea was explicitly formulated by Anaxainto the nature gorasandDemocritus3 as the principleof investigations of matter or body ; visible matterwas to be explainedby hypotheses
Concerning the modern tendency towards what is sometimescalled ' arithmetisation of geometry ' (a tendency which is hardly characteristic all modern work of on geometry), it should be noted that there is little similaritywith the Pythagorean approachsince infinite of sequences naturalnumbersare its main instrumentratherthan the naturalnumbers themselves. 1 For a similarview of Plato'sand Euclid'sinfluence,see G. F. Hemens, Proc.of the Xth Intern.Congress Philosophy(AmsterdamI949), Fasc. 2, 847. of 2 Cf. Homer's explanation of the visible world before Troy with the help of the invisible world of the Olympus. The idea loses, with Democritus, some of its theological character(which is still strong in Parmenides,although less so in Anaxagoras) but regainsit with Plato, only to lose it soon afterwards. 3 See the references given above.

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aboutinvisibles, aboutan invisible structure is toosmall beseen. which to With Plato, this ideasis consciously and generalised; the accepted visibleworld of changeis ultimately be explained an invisible to by worldof unchangingForms (orsubstances essences, 'natures ' ' or or -as we shallsee, geometrical or shapes figures). Is this idea aboutthe invisiblestructure mattera physical a of or idea? If a physicist uponthistheory,thatis to say, acts philosophical if he accepts perhaps even without becomingconscious it, by of it, the of as accepting traditional problems his subject, presented the by with ; problem-situation whichhe is confronted andif he, so acting, a of ; produces new specific theoryof thestructure matter thenI should not callhima philosopher. Butif he reflects uponit, and,forexample, it (like Berkeleyor Mach), preferring phenomenological a rejects or positivistic and somewhattheological physicsto the theoretical then thosewho approach, he maybe calleda philosopher. Similarly, searched the theoretical for who constructed it, consciously approach, andwho explicitly formulated andthustransferred hypotheticalthe it, deductive methodfromthe field of theologyto thatof physics, were even thoughthey werephysicists so faras they acted in philosophers, and upon theirown precepts triedto produceactualtheoriesof the invisible structure matter. of But I shallnot pursue question to the properapplication the as of the label 'philosophy' any further; for this problem, which is Wittgenstein's problem, clearly turns out to be one of linguistic a pseudo-problem which by now mustbe rapidlydeveloping usage, into a boreto my audience. But I wish to adda few morewordson Plato'stheoryof Formsor Ideas,or moreprecisely, point (6) of on the list of historical factsgivenabove. Plato's theory of the structure mattercan be found in the of It hasat leasta superficial with Timaeus. similarity themoderntheory of solidswhich interprets them as crystals. His physicalbodiesare of invisibleelementaryparticlesof variousshapes,the composed for shapesbeing responsible the macroscopic propertiesof visible matter. The shapesof the elementary in particles, their turn, are determined the shapes the planefigures of which form theirsides. by And theseplanefigures,in theirturn,areultimately composed all of twoelementary viz. the half-square isosceles triangles, (or rectangular) whichincorporates square oftwo,andthehalf-equilateral the root triangle the rectangular trianglewhich incorporates square-root three,both of of themirrationals.
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These triangles,in their turn, are describedas the copies 1 of unchanging 'Forms' or 'Ideas', which means that specifically geo' metrical Forms' are admittedinto the company of the Pythagorean arithmetical Form-Numbers. There is little doubt that the motive of this constructionis the attempt to solve the crisisof atomism by incorporatingthe irrationals into the last elements of which the world is built. Once this has been is done, the difficultyof the existenceof irrationaldistances overcome. But why did Plato choose just these two triangles? I have elsewhere 2 expressed the view, as a conjecture, that Plato believed that all other irrationalsmight be obtainedby adding to the rationals multiples of the square roots of two and three. I now feel quite confident that the crucial passagein the Timaeus clearly implies this doctrine (which was mistaken, as Euclid later showed). For in the passage in question, Plato says quite clearly that 'All triangles are derived from two, each having a right angle', going on to specify these two as the half-squareand half-equilateral. But this can only mean, in the context, that all trianglescan be composed by combining these two, a view which is equivalent to the mistakentheory of the with sums of rationalsand relative commensurabilityof all irrationals the squareroots of two and three.3 But Plato did not pretend that he had a proof of the theory in question. On the contrary,he says that he assumesthe two triangles as principles' in accordance with an accountwhich combinesprobable '. And a little later, afterexplainingthat he conjecturewith necessity takes the half-equilateraltriangle as the second of his principles,he says, 'The reasonis too long a story ; but if anybody should test this matter, and prove that it has this property' (I suppose the property that all other trianglescan be composed of these two) ' then the prize is his, with all our good will '.4 The language is somewhat obscure, and no doubt the reasonis that Plato lacked a proof of his conjecture
1 For the processby which the trianglesare stampedout of space (the 'mother ') the ideas (the 'father '), cf. my Open Society,note 15 to ch. 3, and the references by there given, as well as note 9 to ch. 6. 2 In the last quoted note 3 In the note referredto I also conjectured that it was the close approximation of the sum of these two squareroots to which encouragedPlato in his mistaken ri theory. Although I have no new evidence, I believe that this conjecture is much strengthenedby the view that Plato in fact believed in the mistakentheory described here. 53c/d and 54a/b 4 The two quotations are from the Timaeus,

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concerning these two triangles, and felt it should be supplied by somebody. The obscurity of the passage had the strange effect that Plato's quite clearly stated choice of triangles introduce irrationalsinto 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's Theory of Forms could appear to Aristotle to be fundamentallythe same as the Pythagoreantheory of to and form-numbers,1 why Plato'satomismappeared Aristotlemerely
1 I believe that our consideration may throw some light on the problem of Plato's famous' two principles'-' The One 'and' The Indeterminate Dyad'. The followvan ing interpretationdevelops a suggestion made by van der Wielen (De Ideegetallen and brilliantlydefendedagainstvan der Wielen's own criticism Plato, 1941, p. 132 f.) by Ross (Plato's Theory Ideas,p. 201). We assumethat the 'IndeterminateDyad ' of is a straightline or distance,not to be interpretedas a unit distance,or as having yet been measured at all. We assume that a point (limit, monas, ' One') is placed successivelyin such positions that it divides the Dyad according to the ratio I : n, for any naturalnumber n. Then we can describethe ' generation' of the numbers as follows. For I, the Dyad is divided into two parts whose ratio is I :I. n-= This may be 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 according to the ratio I :2 (and the larger section, as before, according to the ratio I :I), thus generating threeequal parts and the number 3 ; generally, the 'generation' of a number n gives rise to a division of the Dyad in the ratio I : n, and with this, to the ' generation' of the number n + I. (And in each stage intervenesthe ' One ', the point which introducesa limit or form or measureinto the otherwise' indeterminate'Dyad, afresh,to createthe new number; this remarkis intended to strengthenRoss' case againstvan der Wielen's.) Now it should be noted that this procedure,although it 'generates' (in the first instance, at least) only the series of natural numbers, nevertheless contains a geometrical element-the division of a line, first into two equal parts,and then into two parts according to a certain proportion 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 should not divide the Dyad also in the proportion of I :V and of I : 1/3. This, he must have felt, was a departure from 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 natural numbers, linear elements in the proportionI : V2 and I : V-3, which may be identicalwith the 'atomic lines ' (Metaphysics, 992ai9) from which the atomic trianglesare constructed. At the same time, the characterisation the Dyad as 'indeterminate 'would become of highly appropriate, view of the Pythagoreanattitude (cf. Philolaos, Diels fragm. 2 iii and 3) towards the irrational. (Perhapsthe name 'The Greatand the Small' began to be replaced by 'The Indeterminate Dyad' when irrational proportions were generated in addition to rationalones.)

ISI

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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 appearsthat he took Plato's reform programme for geometry for granted; it had been partly carried out by Eudoxus before Aristotle entered the Academy, and Aristotle was only superficiallyinterestedin mathematics. He never alludes to the inscriptionon the Academy. To sum up, it seems probable that Plato's theory of Forms was, like his theory of matter, a re-statementof the theories of his predecessors,the Pythagoreansand Democritus respectively,in the light of his realisation that the existenceof irrationals demandedthe emanciof geometry from arithmetic. By encouraging this emancipation pation, Plato contributedto the development of Euclid's system, the most important and influential deductive theory ever constructed. By his adoption of geometry as the theory of the world, he provided Aristarchus,Newton, and Einstein with their intellectual toolbox. The calamityof Greekatomism was thus transformed into a momentous achievement. But Plato'sscientificinterestsare partly forgotten. The problem-situationin science which gave rise to his philosophical problems is little understood. And his greatest achievement, the geometrical theory of the world, has influenced our world-picture to such an extent that we unconsciouslytake it for granted.
6

One example never suffices. As my second example, out of a greatmany interestingpossibilities,I choose Kant. His Critique ofPure Reason one of the most difficultbooks ever written. Kant wrote in is undue haste, and about a problem which, I shall try to show, was insoluble. Nevertheless it was not a pseudo-problem, but an inAssuming this view to be correct, we might conjecture that Plato approached slowly (beginningin the HippiasMajor,and thus long before the Republic-as opposed to a remark made by Ross op. cit., top of page 56) to the view that the irrationals are numbers, since both the natural numbers and the irrationals are 'generated' by similarand essentiallygeometric processes. But once this view is reached(and it was first reached,it appears,in the Epinomis 99od-e, whether or not this work is Plato's), then even the irrationaltrianglesof the Timaeus become 'numbers' (i.e. characterised by numerical, if irrational,propositions). But with this, the peculiar contribution of Plato, andthe differencebetween his andthe Pythagoreantheory, is liableto become indiscernible; and this may explain why it has been lost sight of, even by Aristotle.

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escapableproblem which arose out of the contemporarysituation of physical theory. His book was written for people who knew some Newtonian stellar dynamics and 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 and blas6 as we are by the spectacleof scientificsuccess,to realisewhat Newton's theory meant, not just for Kant, but for any eighteenth century thinker. After the unmatched daring with which the Ancients had tackled the riddle of the Universe, there had come a period of long decay, recovery,and then a staggeringsuccess. Newton had discovered the long sought secret. His geometricaltheory,basedon and modelled after Euclid, had been at first receivedwith great misgivings, even by its own originator.' The reason was that the gravitationalforce of attractionwas felt to be ' occult', or at least something which needed was found (and an explanation. But althoughno plausibleexplanation Newton scorned recourse to ad-hochypotheses), all misgivings had disappeared long before Kant made his own important contribution to Newtonian theory, 78 years after the Principia.2 No qualified judge 3 of the situation could doubt any longer that the theory was true. It has been testedby the most precisemeasurements, it had and been right. It had led to the prediction of minute deviations always from Kepler'slaws, and to new discoveries. In a time like ours,when theoriescome and go like the busesin Piccadilly,andwhen everyschoolboy has heard that Newton has long been superseded Einstein,it is by hard to recapture the sense of conviction which Newton's theory eventhad inspired,or the sense of elation, and of liberation. A unique in the historyof thought, one which could never be repeated: happened the first and final discovery of the absolute truth about the universe. An age-old dreamhad come true. Mankindhad obtainedknowledge, real, certain, indubitable, and demonstrable knowledge--divine scientiaor episteme, not merely doxa,human opinion. and Thus for Kant, Newton's theory was simply true, and the belief in its truth remainedunshakenfor a century after Kant'sdeath. Kant
See Newton's letter to Bentley, 1693. The so-called Kant-Laplacean Hypothesis published by Kant in I755. 3There had been some very pertinentcriticism(especiallyby Leibnizand Berkeley) but in view of the successof the theory it was-I believe rightly-felt that the critics had somehow missed the point of the theory. We must not forget that even today the theory still stands,with only minor modifications,as an excellent first (or, in view of Kepler, perhapsas a second) approximation. 153
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to the end acceptedwhat he and everybody else took for a fact, the attainment of scientiaor episteme. At first he accepted it without question. This state he called his 'dogmatic slumber'. He was roused from it by Hume. Hume had taught that there could be no such thing as certain ; knowledge of universal laws, or episteme that all we knew was obtained with the help of observationwhich could be only of particulars,so that our knowledge was uncertain. His argumentswere convincing (and he was, of course, right). But here was the fact, or what appearedas a fact-Newton's attainmentof episteme. Hume rousedKant to the realisationof the near absurdityof what he never doubted to be a fact. Here was a problem which could not be dismissed. How could a man have got hold of such knowledge ? Knowledge which was general, precise, mathematical,demonstrable, indubitable,and yet explanatoryof observed facts? Thus arose the central problem of the Critique 'How is pure natural science possible? '. By 'pure natural science'--scientia, episteme--Kantmeans, simply, Newton's theory. is Although the Critique badly written, and although it abounds in bad grammar,this problem was not a linguistic puzzle. Here was knowledge. How could we ever attain it ? The question was inescapable. But it was also insoluble. For the apparentfact of the attainment of epistem-was no fact. As we now know, or believe, Newton's theory is no more than a marvellous hypothesis, an astonishingly good approximation; unique indeed, but not as divine but truth, only as a uniqueinvention of a human genius ; not episteme, to the realm of doxa. With this, Kant'sproblem, ' How is belonging pure natural science possible', collapses, and the most disturbing of his perplexitiesdisappear. Kant's proposed solution of his insoluble problem consisted of what he proudly called his ' CopernicanRevolution' of the problem we of knowledge. Knowledge-episteme-was possiblebecause arenot active digestors. By digesting passivereceptorsof sensedata,but their and assimilatingthem, we form and organise them into a Universe. In this process,we impose upon the materialpresentedto our senses the mathematicallaws which are part of our digestive and organising mechanism. Thus our intellect does not discover universal laws in nature, but it prescribes its own laws and imposes them upon natures. This theory is a strange mixture of absurdity and truth. It is
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as absurd the mistaken as it to problem attempts solve; for it proves too much,being designed provetoo much. According Kant's to to natural science'is not only possible although does he ; theory,'pure result not realisethis, it becomesthe necessary of our mentaloutfit. of For if the fact of our attainment episteme be explained all can at for by the fact that our intellectlegislates andimposesits own laws nature,then the first of these two factscannotbe contingent upon is anymorethanthesecond.Thustheproblem no longerhow Newton could makehis discovery how everybodyelse couldhave failed but to makeit. How is it that our digestivemechanism not work did muchearlier ? This is a patentlyabsurdconsequence Kant'sidea. But to of it offhand,and to dismiss problemas a pseudo-problem his disnmiss is not good enough. Forwe canfindan elementof truthin hisidea correction someHumean of (anda muchneeded views),after reducing his problemto its properdimensions. His question, now know, we or believewe know, shouldhavebeen: 'How aresuccessful hypothesespossible?' And our answer,in the spiritof his Copernican Revolution,might, I suggest,be somethinglike this: Because,as we of but data, active you said, arenotpassive receptors sense organisms. Because reactto our environment alwaysmerelyinstinctively, we not but sometimes and we consciously freely. Because caninventmyths, becausewe have a thirstfor explanation, inan stories,theories; satiablecuriosity,a wish to know. Becausewe not only invent storiesand theories, try themout andsee whethertheywork and but how theywork. Because a greateffort,by tryinghardanderring by if in often,we may sometimes, we arelucky,succeed hittingupona an explanation, which 'saves the phenomena'; perhaps story, by makingup a myth about'invisibles', suchas atomsor gravitational is forces,whichexplainthevisible. Because knowledge an adventure of ideas. Theseideas,it is true,areproduced us, andnot by the by worldaround ; theyarenot merelythetraces repeated us of sensations or stimulior whatnot ; hereyou wereright. Butwe aremoreactive and free thaneven you believed; for similar observations similar or environmental situations not, as your theory implied,produce do similar in men. Nor is thefactthatwe originate explanations different our theories, thatwe attempt imposethemuponthe world,an and to of theirsuccess, you believed. Forthe overwhelming as explanation of of ideas,areunsuccessful; majority ourtheories, ourfreelyinvented andarediscarded falsified as they do not stand to searching up tests, by 155

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experience. Only a very few of them succeed,for a time, in the competitive strugglefor survival.'
7

Few of Kant'ssuccessors ever to have clearlyunderstood appear the precise whichgaveriseto hiswork. There were problem-situation two suchproblems him, Newton'sdynamics the heavens, for of and the absolute of standards humanbrotherhood justiceto whichthe and French revolutionaries heavens or, appealed, asKantputsit, ' the starry aboveme, andthe morallaw withinme'. But Kant's heavens starry areseldom as identified anallusion Newton.= From to Fichte onward,3 many have copied Kant's'method' and the dreadful style of his have forgottenKant'soriginal Critique.But most of theseimitators and interests problems, or eitherto tighten, elseto explain busilytrying the Gordian knot in whichKant,throughno faultof his own, away, had tied himselfup. We mustbeware mistaking well-nighsenseless pointless of and the of for subtleties the imitators thepressing genuine of and problems the We shouldremember that his problem,althoughnot an pioneer. turnedout, unexsense,nevertheless empiricalone in the ordinary to be in some sensefactual(Kantcalledsuchfacts'transpectedly, but instance cendental sinceit arosefrom an apparent non-existent '), consider of a scientia epistm-e. Andwe should,I submit,seriously or the suggestionthat Kant'sanswer,in spite of its partialabsurdity, of of contained nucleus a philosophy science. the may be seenfrom the last ten lines of the of the Critique Practical Reason. venultimate paragraph of note 58 to ch. 12 3 Cf. my OpenSociety, and The LondonSchoolof Economics PoliticalScience LondonW C I HoughtonStreet,Aldwych,
2 That this identification corrected is

der in (1935). I The ideasof this 'answer' were elaborated my Logik Forschung

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