Conjectures and Refutations

I

It was after some hesitation that I decided to take as my point of departure the present position of English philosophy. For I believe that the function of a scientist or of a philosopher is to solve scientific or philosophical problems, rather than to talk about what he or other philosophers are doing or might do. Any unsuccessful attempt to solve a scientific or philosophical problem, if it is an honest and devoted attempt, appears to me more significant than a discussion of such a question as ‘What is science?’ or ‘What is philosophy?’ And even if we put this latter question, as we should, in the slightly better form, ‘What is the character of philosophical problems?’, I for one should not bother much about it; I should feel that it had little weight, even compared with such a minor problem of philosophy as the question whether every discussion or every criticism must always proceed from ‘assumptions’ or ‘suppositions’ which themselves are beyond argument.1

When describing ‘What is the character of philosophical problems?’ as a slightly better form of ‘What is philosophy?’ I wished to hint at one of the reasons for the futility of the current controversy concerning the nature of philosophy: the naïve belief that there is an entity such as ‘philosophy’, or perhaps ‘philosophical activity’, and that it has a certain character or essence or ‘nature’. The belief that there is such a thing as physics, or biology, or archaeology, and that these ‘studies’ or ‘disciplines’ are distinguishable by the subject matter which they investigate, appears to me to be a residue from the time when one believed that a theory had to proceed from a definition of its own subject matter.² But subject matter, or kinds of things, do not, I hold, constitute a basis for distinguishing disciplines. Disciplines are distinguished partly for historical reasons and reasons of administrative convenience (such as the organization of teaching and of appointments), and partly because the theories which we construct to solve our problems have a tendency³ to grow into unified systems. But all this classification and distinction is a comparatively unimportant and superficial affair. We are not students of some subject matter but students of problems. And problems may cut right across the borders of any subject matter or discipline.

Obvious as this fact may appear to some people, it is so important for our present discussion that it is worth while illustrating it by an example. I need hardly mention that a geologist’s problem such as assessing the chances of finding deposits of oil or uranium in a certain district has to be solved with 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 theories and techniques, to solve a problem in one of its most abstract and fundamental theories; for example the problem of testing predictions about the relative stability or instability of atoms of an even or odd atomic number.

I am quite ready to admit that many problems, even if their solution involves the most diverse disciplines, nevertheless ‘belong’ in some sense to one or another of the traditional disciplines; the two problems just mentioned clearly ‘belong’ to geology and physics respectively. This is because each of them arises out of a discussion characteristic of the tradition of the discipline in question. Each arises out of the discussion of some theory, or out of empirical tests bearing upon a theory; and theories, as opposed to subject matter, may constitute a discipline (which might be described as a somewhat loose cluster of theories undergoing challenge, change, and growth). But this does not affect my point that the classification into disciplines is comparatively unimportant, and that we are students not of disciplines but of problems.

But are there philosophical problems? The present position of English philosophy—my point of departure—originates, I believe, in the late Professor Ludwig Wittgenstein’s doctrine that there are none; that all genuine problems are scientific problems; that the alleged problems of philosophy are pseudo-problems; that the alleged propositions or theories of philosophy are pseudo-propositions or pseudo-theories; that they are not false (if they were false, their negations would be true propositions or theories) but strictly meaningless combinations of words,4 no more meaningful than the incoherent babbling of a child who has not yet learned to speak properly.⁵

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 to unmask philosophical nonsense, and to teach people to talk sense.

My plan is to take this doctrine 6 of Wittgenstein’s as my starting point. I shall try to explain it (in section ii); to defend it, to some extent; and to criticize it (in section iii). And I shall illustrate all this (in sections iv to xi) 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 philosopher should philosophize: he should try to solve philosophical problems, rather than talk about philosophy. If Wittgenstein’s doctrine is true, then nobody can philosophize, in my sense. Were this 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 care whether they are ‘rightly’ called ‘philosophical problems’), but inspired by the hope that I may contribute—if only a little, and only by hard work—to their solution. My only excuse for talking here about philosophy—instead of philosophizing— is my hope that in carrying out my programme for this address an opportunity may turn up to do a little philosophizing after all.

II

Ever since the rise of Hegelianism there has existed a dangerous gulf between science and philosophy. Philosophers were accused—rightly, I believe—of ‘philosophizing without knowledge of fact’, and their philosophies were described as ‘mere fancies, even imbecile fancies’.7 Although Hegelianism was the leading influence in England and on the Continent, opposition to it, and contempt of its pretentiousness, never died out completely. Its downfall was brought about by a philosopher who like Leibniz, Berkeley, and Kant before him had a sound knowledge of science, and especially of mathematics. I am speaking of Bertrand Russell.

Russell is also the author of the classification, closely related to his famous theory of types, which is the basis of Wittgenstein’s view of philosophy: the classification (criticized below on p. 309) of the expressions of a language into

(1) True statements
(2) False statements
(3) Meaningless expressions, among which there are statement-like sequences of words, the so-called ‘pseudo-statements’.

Russell used this distinction to solve the problem of the logical paradoxes which he discovered. For his solution it was essential to distinguish more especially between (2) and (3). We might say, in ordinary speech, that a false statement like, ‘3 times 4 equals 173,’ or, ‘All cats are cows’, is meaningless. Russell, however, reserved the term ‘meaningless’ for expressions such as, ‘3 times 4 are cows,’ or ‘All cats equal 173’, that is for expressions of a sort which it is better not to describe as false statements. They are better not described as false because the negation of a meaningful but false statement will always be true. But the prima facie negation of the pseudo-statement, ‘All cats equal 173’, is, ‘Some cats do not equal 173’, and this is just as unsatisfactory a pseudo-statement as the original statement. Negations of pseudo-statements are again pseudo-statements, just as negations of proper statements (true or false) are proper statements (false or true, respectively).

This distinction allowed Russell to eliminate the paradoxes (which, he said, were meaningless pseudo-statements). Wittgenstein went further. Led perhaps by the feeling that what philosophers, especially Hegelian philosophers, were saying was somewhat similar to the paradoxes of logic, he used Russell’s distinction in order to denounce all philosophy as strictly meaningless.

As a result there could be no genuine philosophical problems. All alleged philosophical problems could be classified under four heads:8 (1) those which are purely logical or mathematical, to be answered by logical or mathematical propositions, and therefore not philosophical; (2) those which are factual, to be answered by some statement belonging to empirical science, and therefore again not philosophical; (3) those which are combinations of (1) and (2), and therefore again not philosophical; and (4) meaningless pseudo-problems such as, ‘Do all cats equal 173?’ or, ‘Is Socrates identical?’ or, ‘Does an invisible, untouchable, and apparently altogether unknowable Socrates exist?’.

Wittgenstein’s idea of eradicating philosophy (and theology) with the help of an adaptation of Russell’s theory of types was ingenious and original (and more radical even than Comte’s positivism, which it resembles closely).⁹ This idea became the inspiration of a powerful modern school of language analysts who have inherited his belief that there are no genuine philosophical problems, and that all a philosopher can do is to unmask and dissolve the linguistic puzzles which have been proposed by traditional philosophy.

My own view of the matter is that only as long as I have genuine philosophical problems to solve shall I continue to take an interest in philosophy. I fail to understand the attraction of a philosophy without problems. I know, of course, that many people talk nonsense; and it is conceivable that it should become one’s task (an unpleasant one) to unmask somebody’s nonsense, for it may be dangerous nonsense. But I believe that some people have said things which were not very good sense, and certainly not very good grammar, but which were all the same highly interesting and exciting, and perhaps more worth listening to than the good sense of others. I may mention the differential and integral calculus which, especially in its early forms, was no doubt completely paradoxical and nonsensical by Wittgenstein’s (and other) standards; which became, however, reasonably well founded as the result of some hundred years of great mathematical efforts; but whose foundations even at this very moment are still in need, and in the process, of clarification.10 We might remember in this context that it was the contrast between the apparent absolute precision of mathematics and the vagueness and imprecision of philosophical language which deeply impressed the earlier followers of Wittgenstein. But had there been a Wittgenstein to use his weapons against the pioneers of the calculus, and had he succeeded in eliminating their nonsense where their contemporary critics (such as Berkeley, who was fundamentally right) failed, he would have strangled one of the most fascinating and philosophically important developments in the history of thought. Wittgenstein once wrote: ‘Whereof one cannot speak, thereof one must be silent.’ It was, if I remember rightly, Erwin Schrödinger who replied: ‘But it is only here that speaking becomes worth while.’^10a The history of the calculus—and perhaps of Schrödinger’s own theory 11— bears him out.

No doubt we should all train ourselves to speak as clearly, as precisely, as simply, and as directly, as we can. Yet 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 contain many meaningless pseudo-propositions and what some people might call ‘tautologies’.

Moreover, I believe that even Wittgenstein’s original adaptation of Russell’s theory rests upon a logical mistake. From the point of view of modern logic there no longer appears to be any justification for speaking of pseudo-statements or type mistakes or category-mistakes within ordinary, naturally grown languages (as opposed to artificial calculi) so long as the conventional rules of custom and grammar are observed. One may even say that the positivist who tells us with the air of the initiated that we are using meaningless words, or that we are talking nonsense, literally does not know what he is talking about—he simply repeats what he has heard from others who also did not know. But this raises a technical question which I cannot deal with here. (It is dealt with, however, in chapters 11 to 14, below.)

IV

I now turn to my first example: Plato and the Crisis in Early Greek 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-philosophical context;15 more especially in the context of the critical problem situation in Greek science¹⁶ (mainly in the theory of matter) which developed as a result of the discovery of the irrationality of the square root of two. If my thesis is correct, Plato’s theory has not so far been fully understood. (Whether a ‘full’ understanding can ever be achieved is, of course, most questionable.) But a more important consequence would be that it can never be understood by philosophers trained in accordance with the prima facie method described in the foregoing section—unless, of course, they are specially and ad hoc informed of the relevant facts. (These they may have to accept on authority—which means abandoning the prima facie method of teaching philosophy described above.)

It seems likely 17 that Plato’s Theory of Forms is both in origin and in content closely connected with the Pythagorean theory that all things are in essence numbers. The details of this connection and the connection between Atomism and Pythagoreanism are perhaps not so well known. I will therefore briefly tell the story here, as I see it at present.

It appears that the founder of the Pythagorean order or sect was deeply impressed by two discoveries. The first was that a prima facie purely qualitative phenomenon such as musical harmony was, in essence, based upon the purely numerical ratios 1 : 2; 2 : 3; 3 : 4. The second was that the ‘right’ or ‘straight’ angle (obtainable for example by folding a leaf twice so that the two folds form a cross) was connected with the purely numerical ratios 3 : 4 : 5, or 5 : 12 : 13 (the sides of rectangular triangles). These two discoveries, it appears, led Pythagoras to the somewhat fantastic generalization that all things are, in essence, numbers or ratios of numbers; or that number is the ratio (logos = reason), the rational essence, of things, or their real nature.

Fantastic as this idea was, it proved fruitful in many ways. One of its most successful applications was to simple geometrical figures such as squares, rectangular and isosceles triangles, and also to certain simple solids such as pyramids. The treatment of some of these geometrical problems was based upon the so-called gnōmōn.

This can be explained as follows. If we indicate a square by 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 first gnōmōn; we may indicate it thus:

By adding a second gnōmōn, consisting of five more dots, we obtain

One sees at once that every number of the sequence of odd numbers, 1, 3, 5, 7 …, forms the gnōmōn of a square, and that the sums 1, 1 + 3, 1 + 3 + 5, 1 + 3 + 5 + 7, … are the square numbers, and that if n is the (number of 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 treatment of squares, so with the treatment of equilateral triangles. The following figure may be regarded as representing a growing triangle—growing downwards through the addition of ever new horizontal lines of dots.

Here each gnōmōn is a last horizontal line of dots and each element of the sequence 1, 2, 3, 4, … is a gnōmōn. The ‘triangular numbers’ are the sums 1 + 2; 1 + 2 + 3; 1 + 2 + 3 + 4, etc., that is, the sums of the first n natural numbers. By putting two such triangles side by side

we obtain the parallelogram with the horizontal side n + 1 and the other side n, containing n(n + 1) dots. Since it consists of two isosceles triangles its number is 2(1 + 2 + … + n), so that we obtain the equation

and thus

From this it is easy to obtain the general formula for the sum of an arithmetical series.

We also obtain ‘oblong numbers’, that is the numbers of oblong rectangular figures of which the simplest is

with the oblong numbers 2 + 4 + 6 …; the gnōmōn of an oblong is an even number, and the oblong numbers are the sums of the even numbers.

These considerations were extended to solids; for example, by summing the first triangular numbers, pyramid numbers were obtained. But the main application was to plane figures, or shapes, or ‘Forms’. These were believed to be characterized by the appropriate sequence of numbers, and thus by the numerical ratios of the consecutive numbers of the sequence. In other words, ‘Forms’ are numbers or ratios of numbers. On the other hand, not only shapes of things, but also abstract properties, such as harmony and ‘straightness’ are numbers. In this way the general theory that numbers are the rational essences of all things is arrived at.

It seems probable that the development of this view was influenced by the similarity of the dot-diagrams with the diagram of a constellation such as the Lion, or the Scorpion, or the Virgo. If a Lion is an arrangement of dots it must have a number. In this way Pythagoreanism seems to be connected with the belief that the numbers, or ‘Forms’, are heavenly shapes of things.

VI

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 Pythagoreanism is difficult to assess. Its main influences came, so much seems certain, from the Eleatic School: from Parmenides and from Zeno. The basic problem of this school, and of Democritus, was that of the rational understanding of change. (I differ here from the interpretations of Cornford and others.) I think that this problem derives from Heraclitus, and thus from Ionian rather than from Pythagorean thought,19 and that it still remains the fundamental problem of Natural Philosophy.

Although Parmenides was perhaps not a physicist (unlike his great Ionian predecessors), he may be described, I believe, as having fathered theoretical physics. He produced an anti-physical²⁰ (rather than a-physical, as Aristotle said) theory which, however, was the first hypothetico-deductive system. 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 improvement was found necessary because it was realized that the earlier system was falsified by certain facts of experience. Such an empirical refutation of the consequences of a deductive system leads to efforts at its reconstruction, and thus to a new and improved theory which as a rule clearly bears the marks of its ancestry, of the older theory as well as of the refuting experience.

These experiences or observations were, we shall see, very crude at first, but they became more and more subtle as the theories became more and more capable of accounting for the cruder observations. In the case of Parmenides’ theory the clash with observation was so obvious that it would seem perhaps fanciful to describe the theory as the first hypothetico-deductive system of physics. We may, therefore, describe it as the last pre-physical deductive system, whose refutation or falsification gave rise to the first physical theory of matter, 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 should first realize that there is a serious problem 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 changes without implying that X persists during 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 therefore the idea of change, is impossible.

All this sounds very philosophical and abstract, and so it is. But it is a fact that the difficulty here indicated has never ceased to make itself felt in the development of physics.21 And a deterministic system such as the field theory of Einstein might even be described as a four-dimensional version of Parmenides’ unchanging three-dimensional universe. For in a sense no change occurs in Einstein’s four-dimensional block-universe. Everything is there just as it is, in its four-dimensional locus; change becomes a kind of ‘apparent’ change; it is ‘only’ the observer who as it were glides along his world-line and becomes successively conscious of the different loci along this world-line; that is, of his spatiotemporal surroundings …

To return from this new Parmenides to the older father of theoretical physics, we may paraphrase his deductive theory roughly as follows.

(1) Only what is, is.
(2) What is not does not exist.
(3) Non-being, that is, the void, does not exist.

(4) The world is full.

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

(6) Motion is impossible (since there is no empty space within which anything could move).

The conclusions (5) and (6) were obviously contradicted by 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.22
(3’) The void (or non-being) exists.

So far the theory had to be altered. With regard to being, or to the many existing things (as opposed to the void), Democritus adopted Parmenides’ theory that they had no parts. They were indivisible (atoms), because they were full, because they had no void inside.

The main point about this theory is that it gives a rational account of change. The world consists of empty space (the void) with atoms in it. The atoms do not change; they are Parmenidean indivisible block universes in miniature.²³ All change is due to rearrangement of atoms in space. Accordingly all change is movement. Since the only kind of novelty which can arise on this view is novelty of arrangement,²⁴ it will be possible, in principle, to predict all future changes in the world, provided we manage to predict the motion of all atoms (or, in modern parlance, all mass-points).

Democritus’ theory of change was of tremendous importance for the development of physical science. It was partly accepted by Plato, who retained much of atomism, explaining change, however, not only by unchanging yet moving atoms but also by other ‘Forms’ which were subject neither to change nor to motion. But it was condemned by Aristotle who taught in its stead 25 that all change was the unfolding of the inherent potentialities of essentially unchanging substances. Aristotle’s theory of substances as the subjects of change became dominant; but it proved barren;²⁶ and Democritus’ metaphysical theory that all change must be explained by movement became the tacitly accepted programme of work in 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). For with Newton, in addition to moving mass-points, forces of changing intensity (and direction) enter the scene. True, the changes of the Newtonian forces can be explained as due to, or dependent upon, motion; that is, upon the changing position of particles. But they are nevertheless not identical with changes in the position of particles; owing to the inverse square law the dependence is not even a linear one. And with Faraday and Maxwell, changing fields of forces become as important as material atomic particles. That our modern atoms turn out to be composite is a minor matter; from Democritus’ point of view not our atoms but rather our elementary particles would be the real atoms—except that these too turn out to be liable to change. Thus we have a most interesting situation. A philosophy of change, designed to meet the difficulty of understanding change rationally, serves science for thousands of years, but is ultimately superseded by the development of science itself; and this fact passes practically unnoticed by philosophers who are busily denying the existence of philosophical problems.

Democritus’ theory was a marvellous achievement. It provided a theoretical framework for the explanation of most of the empirically known properties of matter (discussed already by the Ionians), such as compressibility, degrees of hardness and resilience, rarefaction and condensation, coherence, disintegration, combustion, and many others. But the theory was important not only as an explanation of the phenomena of experience. First, it established the methodological principle that a deductive theory or explanation must ‘save the phenomena’;27 that is, be in agreement with experience. Secondly, it showed that a theory may be speculative, and based upon the fundamental (Parmenidean) principle that the world as it must be understood by argumentative thought turns out to be different from the world of prima facie experience, from the world as seen, heard, smelled, tasted, touched;²⁸ and that such a speculative theory may nevertheless accept the empiricist ‘criterion’ that it is the visible that decides the acceptance or rejection of a theory of the invisible²⁹ (such as the atoms). This philosophy has remained fundamental to the whole development of physics, and has continued to conflict with all ‘relativistic’ and ‘positivistic’³⁰ philosophical tendencies.

Furthermore, Democritus’ theory led to the first successes of the method of exhaustion (the forerunner of the calculus of integration), since Archimedes himself acknowledged that Democritus was the first to formulate the theory of the volumes of cones and pyramids.31 But perhaps the most fascinating element in Democritus’ theory is his doctrine of the quantization of space and time. I have in mind the doctrine, now extensively discussed,³² that there is a shortest distance and a smallest time interval; that is to say, that there are distances in space and time (elements of length and time, Democritus’ amerēs³³ in contradistinction to his atoms) such that no smaller ones are possible.

VIII

It was Plato who realized this fact, and who in the Laws stressed its importance in the strongest possible terms, denouncing his compatriots for their failure to gauge its implications. I believe that his whole philosophy, and especially his theory of ‘Forms’ or ‘Ideas’, was influenced by it.

Plato was very close to the Pythagorean as well as to the Eleatic School; and although he appears to have felt antipathetic to Democritus he was himself a kind of atomist. (Atomist teaching remained as one of the school traditions of the Academy.41) This is not surprising in view of the close relation between Pythagorean and atomistic ideas. But all this was threatened by the discovery of the irrationals. I suggest that Plato’s main contribution to science sprang from his realization of the problem of the irrational, and from the modification of Pythagoreanism and atomism which he undertook in order to rescue science from a catastrophic situation.

He realized that the purely arithmetical theory of nature was defeated, and that a new mathematical method for the description and explanation of the world was needed. Thus he encouraged the development of an autonomous geometrical method. It found its fulfilment in the ‘Elements’ of the Platonist Euclid.

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

(1) Pythagoreanism and atomism in Democritus’ form were both fundamentally based on arithmetic; that is to say on counting.

(2) Plato emphasized the catastrophic character of the discovery of the irrationals.

(3) He inscribed over the gates of the Academy: ‘Nobody Untrained in Geometry May Enter My House’. But geometry, according to Plato’s immediate pupil Aristotle⁴² as well as Euclid, typically treats of incommensurables or irrationals, in contradistinction to arithmetic which treats of ‘the odd and the even’ (i.e. of integers and their relations).

(4) Within a short time after Plato’s death his school produced, in Euclid’s Elements, a work one of whose main points was that it freed mathematics from the ‘arithmetical’ assumption of commensurability or rationality.

(5) Plato himself contributed to this development, and especially to the development of solid geometry.

(6) More especially, he gave in the Timaeus a specifically geometrical version of the formerly purely arithmetical atomic theory; a version which constructed the elementary particles (the famous Platonic bodies) out of triangles which incorporated the irrational square roots of two and of three. (See below.) In nearly all other respects he preserved Pythagorean ideas as well as some of the most important ideas of Democritus.43 At the same time he tried to eliminate Democritus’ void; for he realized⁴⁴ that motion remains possible even in a ‘full’ world, provided motion is conceived as of the character of vortices in a liquid. Thus he retained some of the most fundamental ideas of Parmenides.⁴⁵

(7) Plato encouraged the construction of geometrical models of the world, and especially models explaining the planetary movements. And I believe that Euclid’s geometry was not intended as an exercise in pure geometry (as is now usually assumed), but as an organon of a theory of the world. According to this view the ‘Elements’ is not a ‘textbook of geometry’ but an attempt to solve systematically the main problems of Plato’s cosmology. This was done with such success that the problems, having been solved, disappeared and were almost forgotten; though a trace remains in Proclus who writes, ‘Some have thought that the subject matter of the various books [of Euclid] pertains to the cosmos, and that they are intended to help us in our contemplation of, and theorizing about, the universe’ (op. cit., note 38 above, Prologus, II, p. 71, 2–5). Yet even Proclus does not mention in this context the main problem—that of the irrationals (of course he does mention it elsewhere); though he points out, rightly, that the ‘Elements’ culminate with the construction of the ‘cosmic’ or ‘Platonic’ regular polyhedra. Ever since 46 Plato and Euclid, but not before, geometry (rather than arithmetic) appears as the fundamental instrument of all physical explanations and descriptions, in the theory of matter as well as in cosmology.⁴⁷

IX

These are the historical facts. They go a long way, I believe, towards establishing my main thesis: that what I have called the prima facie method of teaching philosophy cannot lead to an understanding of the problems that inspired Plato. Nor can it lead to an appreciation of what may be justly claimed to be his greatest philosophical achievement, the geometrical theory of the world. The great physicists of the Renaissance—Copernicus, Galileo, Kepler, Gilbert—who turned from Aristotle to Plato intended by this move to replace the Aristotelian qualitative substances or potentialities by a geometrical method of cosmology. Indeed, that is what the Renaissance (in science) largely meant: a renaissance of the geometrical method, which was the basis of the works of Euclid, Aristarchus, Archimedes, Copernicus, Kepler, Galileo, Descartes, and became the basis of the works of Newton, Maxwell, and Einstein.

But is this achievement properly described as philosophical? Does it not rather belong to physics—a factual science; and to pure mathematics—a branch, as Wittgenstein’s school would contend, of tautological logic?

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 achievement; why at least part of his philosophy of nature and of physics has lasted and, I believe, will last.

What we find in Plato and his predecessors is the conscious construction and invention of a new approach towards the world and towards knowledge of the world. This approach transforms an originally theological idea, the idea of explaining the visible world by a postulated invisible world,48 into the fundamental instrument of theoretical science. The idea was explicitly formulated by Anaxagoras and Democritus⁴⁹ as the principle of investigation into the nature of matter or body; visible matter was to be explained by hypotheses about invisibles, about an invisible structure which is too small to be seen. With Plato this idea is consciously accepted and generalized; the visible world of change is ultimately to be explained by an invisible world of unchanging ‘Forms’ (or substances, or essences, or ‘natures’; that is, as I shall try to show in more detail, geometrical shapes or figures).

Is this idea about the invisible structure of matter a physical or a philosophical idea? If a physicist merely acts upon this theory, if he accepts it, perhaps unconsciously, by accepting the traditional problems of his subject as furnished by the problem-situation with which he is confronted; and if he, so acting, produces a new specific theory of the structure of matter, then I should not call him a philosopher. But if he reflects upon it, and, for example, rejects it (like Berkeley or Mach), preferring a phenomenological or positivistic physics to the theoretical and somewhat theological approach, then he may be called a philosopher. Similarly, those who consciously sought the theoretical approach, who constructed it, and who explicitly formulated it, and thus transferred the hypothetical and deductive method from theology to physics, were philosophers, even though they were physicists in so far as they acted upon their own precepts and tried to produce actual theories of the invisible structure of matter.

But I shall not pursue the question of the proper application of the label ‘philosophy’ any further; for this problem, which is Wittgenstein’s problem, clearly turns out to be one of linguistic usage; it is indeed a pseudo-problem, and one which by now must be rapidly degenerating into a bore to my audience. Yet I wish to add a few words on Plato’s theory of Forms or Ideas, or, to be more precise, on point (6) in the list of historical facts given above.

Plato’s theory of the structure of matter can be found in the Timaeus. It has at least a superficial similarity to the modern theory of solids which interprets them as crystals. His physical bodies are composed of invisible elementary particles of various shapes, the shapes being responsible for the macroscopic properties of visible matter. The shapes of the elementary particles are determined in their turn by the shapes of the plane figures which form their sides. And these plane figures, in their turn, are ultimately all composed of two elementary triangles: the half-square (or isosceles rectangular) triangle which incorporates the square root of two, and the half-equilateral rectangular triangle which incorporates the square root of three, both of them irrationals.

These triangles, in their turn, are described as the copies 50 of unchanging ‘Forms’ or ‘Ideas’, which means that specifically geometrical ‘Forms’ are admitted into the heaven of the Pythagorean arithmetical Form-Numbers.

There is little doubt that the motive of this construction is the attempt to solve the crisis of atomism by incorporating irrationals into the last elements of which the world is built. Once this has been done the difficulty arising from the existence of irrational distances is overcome.

But why did Plato choose just these two triangles? I have elsewhere 51 expressed the view, as a conjecture, that Plato believed that all other irrationals might be obtained by adding to the rationals multiples of the square roots of two and three.⁵² I now feel more confident that the crucial passage in the Timaeus does imply this doctrine (which was mistaken, as Euclid later showed). For in the passage in question Plato says quite clearly, ‘All triangles originate from two, each having a right angle’, going on to specify these two as the half-square and half-equilateral. But in the context this can only mean that all triangles originate somehow from these two. This seems to be a hint at the mistaken theory of the relative commensurability of all irrationals with sums of rationals and the square roots of two and three.⁵³

But Plato does not pretend that he has a proof of the theory in question. On the contrary, he says that he assumes the two triangles as principles, ‘in accordance with an account which combines likely conjecture with necessity’. And a little later, after explaining that he takes the half-equilateral triangle as the second of his principles, he says, ‘The reason is too long a story; but if anybody should probe into this matter, and prove that it has this property’ (I suppose the property that all other triangles can be composed of these two) ‘then the prize is his, with all our good will’.⁵⁴ The language is somewhat obscure, and the likely reason is that Plato was conscious that he lacked a proof of his (mistaken) conjecture concerning these two triangles, and felt it should be supplied by somebody.

The obscurity of the passage had, it appears, the strange effect that Plato’s quite clearly stated choice of triangles which introduce irrationals into his world of Forms escaped the notice of most of his readers and commentators in spite of Plato’s emphasis upon the problem of irrationality in other places. And this in turn may perhaps explain why Plato’s Theory of Forms could appear to Aristotle to be fundamentally the same as the Pythagorean theory of form-numbers,55 and why Plato’s atomism appeared to Aristotle merely as a comparatively minor variation on that of Democritus.56 Aristotle, in spite of taking for granted both the association of arithmetic with the odd and even, and of geometry with the irrational, does not appear to have taken the problem of the irrationals seriously. Proceeding as he did from an interpretation of the Timaeus which identified Plato’s Space with matter, he seems to have taken 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 superficially interested in mathematics. He never alludes to the inscription over the Academy gates.

To sum up,^56a it seems probable that Plato’s theory of Forms and also his theory of matter were both restatements of the theories of his predecessors, the Pythagoreans and Democritus respectively, in the light of his realization that the irrationals demanded that geometry should come before arithmetic. By encouraging this emancipation of geometry Plato contributed to 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 calamity of Greek atomism was thus transformed into a momentous achievement. But Plato’s scientific interests are partly forgotten. The problem-situation in 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 unreflectingly take it for granted.

X

One example never suffices. As my second example, out of a great many interesting possibilities, I choose Kant. His Critique of Pure Reason is one of the most difficult books ever written. Kant wrote in great haste,57 and about a problem which, I shall try to show, was not only insoluble but also misconceived. Nevertheless it was not a pseudo-problem, but an inescapable problem which arose out of the contemporary situation in science.

His book was written for people who knew something about Newton’s stellar dynamics and who had at least some idea of his forerunners—of Copernicus, Tycho Brahe, Kepler and Galileo.

It is perhaps hard for intellectuals of our own day, spoilt and blasé as we are by the spectacle of scientific success, to realize what 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 long periods of decay and recovery, and then a staggering success. Newton had discovered the long sought secret. His geometrical theory, based on and modelled after Euclid, had been received at first with great misgivings, even by its own originator.⁵⁸ The reason was that the gravitational force of attraction was felt to be ‘occult’, or at least something which needed an explanation. But although no plausible explanation was found (and Newton scorned recourse to ad hoc hypotheses) all misgivings had disappeared long before Kant made his own important contribution to Newtonian theory, 68 years after the Principia.⁵⁹ No qualified judge⁶⁰ of the situation could doubt any longer that Newton’s theory was true. It had been tested by the most precise measurements, and it had always been right. It had led to the prediction of minute deviations from Kepler’s laws, and to new discoveries. In a time like ours, when theories come and go like the buses in Piccadilly, and when every schoolboy has heard that Newton has long been superseded by Einstein, it is hard to recapture the sense of conviction which Newton’s theory inspired, or the sense of elation, and of liberation. A unique event had happened in the history of thought, one which could never be repeated: the first and final discovery of the absolute truth about the universe. An age-old dream had come true. Mankind had obtained knowledge, real, certain, indubitable, and demonstrable knowledge—divine scientia or epistēmē, and not merely doxa, human opinion. This sense of conviction became—through Voltaire—the origin of the Enlightenment.60a

Thus for Kant Newton’s theory was simply true, and the belief in its truth remained unshaken for a century after Kant’s death. Kant to the end accepted what he and everybody else took for a fact, the attainment of scientia or epistēmē. 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 epistēmē; that all we knew was obtained with the help of observation which could be only of singular (or particular) instances, so that all theoretical knowledge was uncertain. His arguments were convincing (and he was, of course, right). Yet there was a fact, or what appeared as a fact—Newton’s attainment of epistēmē.

Hume roused Kant to the realization of the near absurdity of 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, and indubitable, like Euclidean geometry, and yet capable of giving a causal explanation of observed facts?

Thus arose the central problem of the Critique: ‘How is pure natural science possible?’ By ‘pure natural science’—scientia, epistēmē—Kant simply meant Newton’s theory. (This he does not say, unfortunately; and I do not see how a student reading the first Critique, 1781 and 1787, could possibly find out. But that Kant has Newton’s theory in mind is clear from the Metaphysical Foundations of Natural Science, 1786, where he gives an a priori deduction of Newton’s theory, see especially the eight theorems of the Second Main Part, with its Additions, especially Addition 2, Note 1, paragraph 2. Kant relates Newton’s theory, in the fifth paragraph of the final ‘General Note on Phenomenology’, to the ‘starry heavens’. It is also clear from the ‘Conclusion’ of the Critique of Practical Reason, 1788, where the appeal to the ‘starry heavens’ is explained, at the end of the second paragraph, by a reference to the a priori character of the new astronomy.61)

Although the Critique is badly written, and although bad grammar abounds in it, Kant’s central problem was not a linguistic puzzle. Here was knowledge. How could Newton ever attain it? The question was inescapable.⁶²

But it was also insoluble. For the apparent fact of the attainment of epistēmē was no fact. As we now know, or believe we know, Newton’s theory is no more than a marvellous conjecture, an astonishingly good approximation; unique indeed, but not as divine truth, only as a unique invention of a human genius: not epistēmē, but belonging to the realm of doxa. With this Kant’s problem, ‘How is pure natural science possible’, collapses, and the most disturbing of his perplexities disappears.

Kant’s proposed solution of his insoluble problem consisted of what he proudly called his ‘Copernican Revolution’ of the problem of knowledge. Knowledge—epistēmē—was possible because we are not passive receptors of sense data, but their active digestors. By digesting and assimilating them we form and organize them into a Cosmos, the Universe of Nature. In this process we impose upon the material presented to our senses the mathematical laws which are part of our digestive and organizing mechanism. Thus our intellect does not discover universal laws in nature, but it prescribes its own laws and imposes them upon nature.

This theory is a strange mixture of absurdity and truth. It is as absurd as the mistaken problem it attempts to solve; for it proves too much, being designed to prove too much. According to Kant’s theory, ‘pure natural science’ is not only possible; although he does not always realize this, it becomes, contrary to his intention, the necessary result of our mental outfit. For if the fact of our attainment of epistēmē can be explained at all by the fact that our intellect legislates for nature, and imposes its own laws upon it, then the first of these two facts cannot be contingent any more than the second.63 Thus the problem is no longer how Newton could make his discovery but how everybody else could have failed to make it. How is it that our digestive mechanism did not work much earlier?

This is a patently absurd consequence of Kant’s idea. But to dismiss it offhand, and to dismiss his problem as a pseudo-problem, is not good enough. For we can find an element of truth in his idea (and a much needed correction of some Humean views) after reducing his problem to its proper dimensions. His question, we now know, or believe we know, should have been: ‘How are successful conjectures possible?’ And our answer, in the spirit of his Copernican Revolution, might, I suggest, be something like this: Because, as you said, we are not passive receptors of sense data, but active organisms. Because we react to our environment not always merely instinctively, but sometimes consciously and freely. Because we can invent myths, stories, theories; because we have a thirst for explanation, an insatiable curiosity, a wish to know. Because we not only invent stories and theories, but try them out and see whether they work and how they work. Because by a great effort, by trying hard and making many mistakes, we may sometimes, if we are lucky, succeed in hitting upon a story, an explanation, which ‘saves the phenomena’; perhaps by making up a myth about ‘invisibles’, such as atoms or gravitational forces, which explain the visible. Because knowledge is an adventure of ideas. These ideas, it is true, are produced by us, and not by the world around us; they are not merely the traces of repeated sensations or stimuli or what not; here you were right. But we are more active and free than even you believed; for similar observations or similar environmental situations do not, as your theory implied, produce similar explanations in different men. Nor is the fact that we create our theories, and that we attempt to impose them upon the world, an explanation of their success,64 as you believed. For the overwhelming majority of our theories, of our freely invented ideas, are unsuccessful; they do not stand up to searching tests, and are discarded as falsified by experience. Only a very few of them succeed, for a time, in the competitive struggle for survival.⁶⁵

1 I call this a minor problem because I believe that it can easily be solved, by refuting the (‘relativistic’) doctrine which gives rise to the question. (Thus the answer to the question is negative. See the Addendum to vol. ii of my Open Society, added to the fourth edition of 1962.)

² This view is part of what I have called ‘essentialism’. Cf. for example my Open Society, chs. 2 and 11, or The Poverty of Historicism, section 10.

³ This tendency can be explained by the principle that theoretical explanations are the more satisfactory the better they can be supported by independent evidence. For in order to be supported by mutually independent pieces of evidence, a theory must be sweeping.

⁴ ‘All animals are equal but some are more equal than others’ is an excellent example of an expression which would be ‘meaningless’ in the technical sense of Russell and Wittgenstein, though clearly far from meaningless (in the sense of pointless) in the context of Orwell’s Animal Farm. It is interesting that later Orwell considered the possibility of introducing a language, and enforcing its use, in which ‘All men are equal’ would become meaningless in Wittgenstein’s technical sense.

⁵ Since Wittgenstein described his own Tractatus as meaningless (see also the next footnote) he distinguished, at least by implication, between revealing or important and worthless or unimportant nonsense. But this does not affect his 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, especially notes 26, 46, 51, and 52 to ch. 11.)

⁶ 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 perhaps a little cheap. It might be countered in at least two ways. (1) One might say that the doctrine is indeed meaningless qua doctrine, but not qua activity. (This is the view of Wittgenstein, who said at the end of his Tractatus Logico-Philosophicus that whoever understood the book must realize at the end that it was itself meaningless, and must discard it like a ladder, after having used it to reach the desired height.) (2) One might say that the doctrine is not a philosophical but an empirical one; that is states the historical fact that all apparent ‘theories’ proposed by philosophers are in fact ungrammatical; that these do not, in fact, conform to the rules inherent in those languages in which they appear to be formulated; that it turns out to be impossible to remedy this defect; and that every attempt to express them properly has led to the loss of their philosophic character (and revealed them as, for example, empirical truisms, or as false statements). These two counter arguments do, I believe, rescue the threatened consistency of the doctrine, which in this way indeed becomes ‘unassailable’—to use Wittgenstein’s term—by the kind of criticism referred to in this note. (See also the next note but one.)

⁷ The two quotations are not the words of a scientific critic but, ironically enough, Hegel’s own characterization of the Natural Philosophy of his forerunner and one-time friend Schelling. Cf. my Open Society, note 4 (and text) to ch. 12.

⁸ Wittgenstein still upheld the doctrine of the non-existence of philosophical problems in the form here described when I saw him last (in 1946, when he presided over a stormy meeting of the Moral Sciences Club in Cambridge, on the occasion of my reading a paper on ‘Are there Philosophical Problems?’). Since I had never seen any of his unpublished manuscripts which were privately circulated by some of his pupils I had been wondering whether he had modified what I here call his ‘doctrine’; but on this, the most fundamental and influential part of his teaching, I found his views unchanged.

¹⁰ I am alluding to G. Kreisel’s recent construction (Journal of Symbolic Logic, 17, 1952, 57) of a monotone bounded sequence of rationals every term of which can be actually computed, but which does not possess a computable limit—in contradiction to what appears to be the prima facie interpretation of the classical theorem of Bolzano and Weierstrass, but in agreement, it seems, with Brouwer’s doubts about this theorem.

^10a After this paper was first published Schrödinger told me that he could not remember saying this, and that he did not believe that he ever said it; but he liked the remark. (Added 1964: I have found since that its real author was my old friend Franz Urbach.)

¹¹ Before Max Born proposed his famous probability interpretation, Schrödinger’s wave equation was, some might contend, meaningless. (This is not, however, my opinion.)

¹² It is very interesting that the imitators were always inclined to believe that the ‘master’ did his work with the help of a secret method or a trick. It is reported that in J. S. Bach’s days some musicians believed that he possessed a secret formula for the construction of fugue themes.
It is also interesting to note that all the philosophies which have become fashionable (so far as I am aware) have offered their disciples a kind of method for producing philosophical results. This is true of Hegelian essentialism which teaches its adherents to produce essays on the essence or nature or idea of everything—the soul, the universe, or the University; it is true of Husserl’s phenomenology; of existentialism; and also of language analysis.

¹³ I am alluding to a remark by Professor Gilbert Ryle, who says on page 9 of his Concept of Mind: ‘Primarily I am trying to get some disorders out of my own system.’

¹⁴ Already in my L.Sc.D. of 1934 I had pointed out that a theory such as Newton’s may be interpreted either as factual or as consisting of implicit definitions (in the sense of Poincaré and Eddington), and that the interpretation which a physicist adopts exhibits itself in his attitude towards tests which go against his theory rather than in what he says. I also pointed out that there are non-analytical theories which are not testable (and therefore not a posteriori) but which had a great influence on science. (Examples are the early atomic theory, or the early theory of action by contact.) I called such untestable theories ‘meta-physical’, and asserted that they were not meaningless. The dogma of the simple dichotomy has been recently attacked, on very different lines, by F. H. Heinemann (Proc. of the Xth Intern. Congress of Philosophy, Fasc. 2, 629, Amsterdam, 1949), by W. V. 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 formalized language, and therefore is liable to break down for those languages in which we must speak prior to any formalization, i.e. in those languages in which all the traditional problems were conceived.

¹⁵ In my Open Society and its Enemies I have tried to explain in some detail another extra-philosophical root of the same doctrine—its political root. I also discussed there (in note 9 to ch. 6 of the revised 4th edn., 1962) the problem with which I am concerned in the present section, but from a somewhat different angle. The note referred to and the present section overlap a little; but they largely supplement each other. Relevant references (especially to Plato) omitted here will be found there.

¹⁶ There are historians who deny that the term ‘science’ can be properly applied to any development which is older than the sixteenth or even the seventeenth century. But quite apart from the fact that controversies about labels should be avoided, there can, I believe, no longer be any 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 ‘Copernicus of antiquity’). And any doubt concerning the extreme age of scientific observation and of careful computations based upon observation has been dispelled by the discovery of new evidence concerning the history of ancient astronomy. We can now draw a parallel not only between Tycho and Hipparchus, but even 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 comparable in precision to those of the best nineteenth-century astronomers. ‘Cidenas’ value for the motion of the Sun from the Node (0².5 too great), although inferior to Brown’s, is superior to at least one of the most widely used modern values’, wrote J. K. Fotheringham in 1928, in his admirable article ‘The Indebtedness of Greek to Chaldean Astronomy’ (The Observatory, 1928, 51, No. 653), upon which my contention concerning the age of metrical astronomy is based.

¹⁸ Plato’s distinction (epistēmē vs. doxa) is derived through Parmenides from Xenophanes (truth vs. conjecture or seeming). Plato clearly realized that all knowledge of the visible world, the changing world of appearance, consists of doxa; that it is tainted by uncertainty even if it utilizes the epistēmē, the knowledge of the unchanging ‘Forms’ and of pure mathematics, to the utmost; and even if it interprets the visible world with the help of a theory of the invisible world. Cf. Cratylus, 439b ff., Republic, 476d ff.; and especially Timaeus, 29b ff., where the distinction is applied to those parts of Plato’s own 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 = epistēmē; cf. my remarks on this problem in ch. 20 below). For a different view concerning Plato’s relation to Parmenides see Sir David Ross, Plato’s Theory of Ideas, Oxford, 1951, p. 164.

¹⁹ Karl Reinhardt in his Parmenides (1916; second edition 1959, p. 220) says very forcefully: ‘The history of philosophy is a history of its problems. If you want to explain Heraclitus, tell us first what his problem was.’ I fully agree; and I believe, as against Reinhardt, that Heraclitus’ problem was the problem of change—or more precisely, of the self-identity (and non-identity) of the changing thing during change. (See also my Open Society, ch. 2.) If we accept Reinhardt’s evidence about the close link between Heraclitus and Parmenides, then this view of Heraclitus’ problem makes of Parmenides’ system an attempt to solve the problem of the paradoxes of change by making change unreal. As against this, Cornford and his disciples follow Burnet’s doctrine that Parmenides was a (dissident) Pythagorean. This may well be true, but the evidence in its favour does not show that he did not also have an Ionian teacher. (See also ch. 5, below.)

²⁰ Cp. Plato, Theaetetus, 181a, and Sextus Empiricus, Adv. Mathem. (Bekker), X. 46, p. 485, 25.

²¹ This may be seen from Emile Meyerson’s Identity and Reality, one of the most interesting philosophical studies of the development of physical theories. Hegel (following Heraclitus, or Aristotle’s account of him) took the fact of change (which he considered self-contradictory) to prove the existence of contradictions in the world, and therefore to disprove the ‘law of contradiction’; i.e. the principle that our theories must avoid contradictions at all cost. Hegel and his followers (especially Engels, Lenin, and other Marxists) began to see ‘contradictions’ everywhere in the world, and denounced all philosophies upholding the law of contradiction as ‘metaphysical’, a term which they used to imply that these philosophies ignore the fact that the world changes. See ch. 15, below.

²² The inference from the existence of motion to that of a void is not valid because Parmenides’ inference from the fullness of the world to the impossibility of motion is not valid. Plato seems to have been the first to see, if only dimly, that in a full world circular or vortex-like motion is possible, provided that there is a liquid-like medium in the world. (Tea leaves can move with the vortex of tea in the cup.) This idea, first offered somewhat half-heartedly in the Timaeus (where space is ‘filled’, 52c) becomes the basis of Cartesianism and of the theory of the ‘luminiferous ether’ as it was held down to 1905. (See also note 44, below.)

²³ Democritus’ theory also admitted large block-atoms, but the vast majority of his atoms were invisibly small.

²⁵ Inspired by Plato’s Timaeus, 55, where the potentialities of the elements are explained by the geometrical properties (and thus the substantial forms) of the corresponding solids.

²⁶ The barrenness of the ‘essentialist’ (cf. note 2 above) theory of substance is connected with its anthropomorphism; for substances (as Locke saw) take their plausibility from the experience of a self-identical but changing and unfolding ego. But although we may welcome the fact that Aristotle’s substance has disappeared from physics, there is nothing wrong, as Professor Hayek says, in thinking anthropomorphically about man. So there is no philosophical or a priori reason why Aristotle’s substance—the psyche—ought to disappear from psychology.

²⁸ Cf. Democritus, Diels, fragm. 11 (cf. Anaxagoras, Diels, fragm. 21; see also fragm. 7).

³⁰ ‘Relativistic’ in the sense of philosophical relativism, e.g. of Protagoras’ homo mensura doctrine. It is, unfortunately, still necessary to emphasize that Einstein’s theory has nothing in common with this philosophical relativism.
‘Positivistic’ as were the tendencies of Bacon; of 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-data theorists.

³¹ Cf. Diels, fragm. 155, which must be interpreted in the light of Archimedes (ed. Heiberg) II², p. 428 f. Cf. S. Luria’s most important article ‘Die Infinitesimalmethode der antiken Atomisten’ (Quellen & Studien zur Gesch. d. Math., B., 2, Heft 2, 1932, p. 142).

³³ Cf. S. Luria, op. cit., especially pp. 148 ff., 172 ff. Miss A. T. Nicols in ‘Indivisible Lines’ (Class. Quarterly, xxx, 1936, 120 f.) argues that ‘two passages, one from Plutarch, the other from Simplicius’ show why Democritus ‘could not believe in indivisible lines’; she does not however discuss Luria’s opposing views of 1932, which I find much more convincing, especially if we remember that Democritus tried to answer Zeno (see next note). But whatever Democritus’ views on indivisible or atomic distances, Plato appears to have thought that Democritus’ atomism needed revision in the light of the discovery of the irrationals. Heath however (Greek Mathematics, 1, 1921, p. 181, referring to Simplicius and Aristotle) also believes that Democritus did not teach the existence of indivisible lines.

³⁴ This point for point reply is preserved in Aristotle’s On Generation and Corruption, 316a, 14 ff., a very important passage first identified as Democritean by I. Hammer Jensen in 1910 and carefully discussed by Luria who says (op. cit., 135) of Parmenides and Zeno: ‘Democritus borrows their deductive arguments, but he arrives at the opposite conclusion.’

³⁵ Cf. G. H. Hardy and E. M. Wright, Introduction to the Theory of Numbers, 1938, pp. 39, 42, where a very interesting historical remark on Theodorus’ proof, as reported in Plato’s Theaetetus, will be found. See now also the article by A. Wasserstein, ‘Theaetetus and the History of the Theory of Numbers’, Classical Quarterly, 8, N.S., 1958, pp. 165–79, the best discussion of the subject known to me.

³⁶ Rather than On Irrational Lines and Atoms, as I translated it in note 9 to ch. 6 of my Open Society (second edn.). What is probably meant by the title (considering Plato’s passage mentioned in the next note) might, I think, be best rendered by ‘On Crazy Lines and Atoms’. Cf. H. Vogt, Bibl. Math., 1910, 10, 147 (against whom Heath argues, op. cit., 156 f., but not I think quite successfully) and S. Luria, op. cit., pp. 168 ff., where it is convincingly suggested that (Arist.) De insec. lin., 968b 17 and Plutarch, De comm. notit., 38, 2, p. 1078 f., contain traces of Democritus’ work. According to these sources, Democritus’ argument was this. If lines are infinitely divisible then they are composed of an infinity of ultimate units and are therefore all related like ∞ : ∞, that is to say, they are all ‘noncomparable’ (there is no proportion). Indeed, if lines are considered as classes of points, the cardinal ‘number’ (potency) of the points of a line is, according to modern views, equal for all lines, whether the lines are finite or infinite. This fact has been described as ‘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 classical theory of the Lebesgue measure of a continuum leads to fundamentally the same results; for Brouwer asserts that all classical continua have zero measure, so that the absence of a ratio is here expressed by o : o. Democritus’ result (and his theory of amerēs) appears to be inescapable as long as geometry is based on the Pythagorean arithmetical method, i.e. on the counting of dots.

³⁷ This would be in keeping with the fact, mentioned in the note cited from the Open Society, that the term ‘alogos’ was, it seems, only much later used for ‘irrational’, and that Plato who alludes (Republic 534d) to Democritus’ title, uses ‘alogos’ there in the sense of ‘crazy’; he never uses it as a synonym for ‘arrhētos’ as far as I know.

³⁸ Procli Diadochi in primum Euclidis Elementorum librum commentarii, ed. G. Friedlein, Leipzig, 1873, p. 487, 7–21.

⁴⁰ The story is told of one Hippasus, a somewhat shadowy figure; he is said to have died at sea (cf. Diels⁶, 4). See also A. Wasserstein’s article mentioned in note 35, above.

⁴³ Plato took over, more especially, Democritus’ theory of vortices (Diels, fragm. 167, 164; cf. Anaxagoras, Diels, 9, and 12, 13; see also the next two footnotes) and his theory of what we nowadays would call gravitational phenomena (Diels, Democritus 164; Anaxagoras, 12, 13, 15, and 2)—a theory which, slightly modified by Aristotle, was ultimately discarded by Galileo.

⁴⁴ The clearest passage is Timaeus, 80c, where it is said that neither in the case of (rubbed) amber nor of the ‘Heraclean stone’ (magnet) is there any real attraction; ‘there is no void and these things push themselves around, one upon another’. On the other hand Plato was not too clear on this point, since his elementary particles (other than the cube and the pyramid) cannot be packed without leaving some (empty?) space between them, as Aristotle observed in De Caelo, 306b5. See also note 22 above (and Timaeus 52e).

⁴⁵ Plato’s reconciliation of atomism and the theory of the plenum (‘nature abhors the void’) became of the greatest importance for the history of physics down to our own day. For it strongly influenced Descartes, became the basis of the theory of ether and light, and thus ultimately, via Huyghens and Maxwell, of de Broglie’s and of Schrödinger’s wave mechanics. See my report in Atti d. Congr. Intern. di Filosofia (1958), 2, 1960, pp. 367 ff.

⁴⁶ An exception is the reappearance of arithmetical methods in Quantum Theory, e.g. in the electron shell theory of the periodic system based upon Pauli’s exclusion principle; an inversion of Plato’s tendency to geometrize arithmetic (see below).
Concerning the modern tendency towards what is sometimes called ‘arithmetization of geometry’ (a tendency which is by no means characteristic of all modern work on geometry), it should be noted that it shows little similarity to the Pythagorean approach since sets, or infinite sequences, of natural numbers are its main instruments, rather than the natural numbers themselves.
Only those who confine themselves to ‘constructive’ or ‘finitist’ or ‘intuitionist’ methods of number theory—as opposed to set-theoretic methods—might claim that their attempts to reduce geometry to number theory resemble Pythagorean or pre-Platonic ideas of arithmetization. A great step in this direction has been achieved quite recently, it seems, by the German mathematician E. de Wette.

⁴⁷ For a similar view of Plato’s and Euclid’s influence, see G. F. Hemens, Proc. of the Xth Intern. Congress of Philosophy (Amsterdam, 1949), Fasc. 2, 847.

⁴⁸ Cf. Homer’s explanation of the visible world around 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 regains it with Plato, only to lose it again soon afterwards.

⁵⁰ For the process by which the triangles are stamped out of space (the ‘mother’) by the ideas (the ‘fathers’), cf. my Open Society, note 15 to ch. 3, and the reference there given, as well as note 9 to ch. 6. In admitting irrational triangles into his heaven of divine Forms Plato admits something ‘indeterminable’ in the sense of the Pythagoreans, i.e. something belonging to the ‘Bad’ side of the Table of Opposites. That ‘Bad’ things may have to be admitted seems to be first stated in Plato’s Parmenides, 130b–e; the admission is put into the mouth of Parmenides himself.

⁵² This would mean that all geometrical distances (magnitudes) are commensurable with one of three ‘measures’ (or a sum of two or all of them) related as 1 : √2: √3. It seems likely that Aristotle even believed that all geometrical magnitudes are commensurable with one of two measures, viz. 1 and √2. For he writes (Metaphysics, 1053a17): ‘The diagonal and the side of a square and all (geometrical) magnitudes are measured by two (measures).’ (Cp. Ross’ note on this passage.)

⁵³ In note 9 to ch. 6 of my Open Society, mentioned above, I also conjectured that the approximation of √2 + √3 to π encouraged Plato in his mistaken theory.

⁵⁵ 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 following interpretation develops a suggestion made by van der Wielen (De Ideegetallen van Plato, 1941, p. 132 f.) and brilliantly defended against van der Wielen’s own criticism by Ross (Plato’s Theory of Ideas, p. 201). We assume that the ‘Indeterminate Dyad’ is a straight line or distance, not to be interpreted as a unit distance, or as having yet been measured at all. We assume that a point (limit, monas, ‘One’) is placed successively in such positions that it divides the Dyad according to the ratio 1: n, for any natural number n. Then we can describe the ‘generation’ of the numbers as follows. For n = 1, the Dyad is divided into two parts whose ratio is 1 : 1. This may be interpreted as the ‘generation’ of Twoness out of Oneness (1 : 1 = 1) and 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 1 : 2 (and the larger of the ensuing sections, as before, according to the ratio 1 : 1), thus generating three equal parts and the number 3; generally, the ‘generation’ of a number n gives rise to a division of the Dyad in the ratio 1 : n, and with this, to the ‘generation’ of the number n + 1. (And in each stage the ‘One’ intervenes afresh as the point which introduces a limit or form or measure into the otherwise ‘indeterminate’ Dyad to create the new number; this remark may strengthen Ross’ case against van der Wielen’s. Cp. also Toeplitz’s, Stenzel’s, and Becker’s papers in Quellen & Studien z. Gesch. d. Math., 1, 1931. None of them, however, hint at a geometrization of arithmetic—in spite of the figures on pp. 476f.)
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 1 : 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 1 : √2 and of 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 proportion 1 : √2 and 1: √3, which may be identical with the ‘atomic lines’ (Metaphysics, 992a19) from which the atomic triangles are constructed. At the same time the characterization of the Dyad as ‘indeterminate’ would become highly appropriate, in view of the Pythagorean attitude (cf. Philolaos, Diels, fragm. 2 and 3) towards the irrational. (Perhaps the name ‘The Great and the Small’ began to be replaced by ‘The Indeterminate Dyad’ when irrational proportions were generated in addition to rational ones.)
Assuming this view to be correct, we might conjecture that Plato slowly approached (beginning in the Hippias Major, and thus long before the Republic—as opposed to a remark made by Ross, op. cit., top of p. 56) the view that the irrationals are numbers (a) since they are comparable with numbers (Met., 1021a 4 f.) and (b) since both the natural numbers and the irrationals are ‘generated’ by similar and essentially geometric processes. Yet once this view is reached (and it was first reached, it appears, in the Epinomis, 990d-e, whether or not this work is, as I am inclined to believe, Plato’s), then even the irrational triangles of the Timaeus become ‘numbers’ (i.e. characterized by numerical, if irrational, proportions). But at this point the peculiar contribution of Plato, and the difference between his and the Pythagorean theory, may become indiscernible; which may explain why it has been lost sight of even by Aristotle (who suspected both ‘geometrization’ and ‘arithmetization’).

⁵⁶ That this was Aristotle’s view has been pointed out by Luria; see above, Note 31.

⁵⁹ The so-called Kant-Laplacean Hypothesis, published by Kant in 1755. (It fell dead from the press.)

⁶⁰ There had been some very pertinent criticism (especially by Leibniz and Berkeley) but in view of the success of 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 approximation (or, in view of Kepler, perhaps as a second approximation).

⁶¹ Kant says there that Newton gave us so clear ‘an insight into the structure of the universe that it will remain unchanged in all time; and though there is hope that our insight will ever grow through continued observation, there never need be fear of a setback’. (Added 1989.)

⁶³ A crucial requirement which any adequate theory of knowledge must satisfy is that it must not explain too much. Any non-historical theory explaining why a certain discovery had to be made must fail because it could not possibly explain why it was not made somewhat earlier.

⁶⁴ Applying note 63, no theory can explain why our search for explanatory theories is successful. Successful explanation must retain, on any valid theory, the probability zero, assuming that we measure this probability, approximately, by the ratio of the ‘successful’ explanatory hypotheses to all hypotheses which might be designed by man.

⁶⁵ The ideas of this ‘answer’ were elaborated in my L.Sc.D. (1934, 1959, and later editions).

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