YOU will ask me why I wish to speak about rationalism and empiricism to teachers of philosophy. Each of us has explained often enough to his students what rationalism is and what empiricism is - so why take up the old question anew?

I should like to answer that it is this very fact of our common profession which induces me to discuss a subject as old as the history of philosophy. Everyone who has taught the history of philosophy, whether he did it with more or less enthusiasm, is familiar with the feeling of dissatisfaction with which he often went home from his classes. This story of human endeavour to orient oneself in the world of knowledge; of the power of abstract thought and of intellectual constructions too remote from concrete evidence to be acceptable; of brilliant and of dry writers; of remarks of profound wisdom and of obscure formulations that no one has been able to translate into comprehensible language; of so much talent and genius and so few common results - this story of ever renewed attempts, but never of lasting success - what is it good for? Why should we teach it, if there is no outcome, no recognized truth?

An interpretation has been suggested according to which the diversity of system does not express an ultimate diversity of opinion. All philosophical systems, it has been said, mean the same thing; they are but different languages revealing the same wisdom merely in different versions. I do not think that this relativistic interpretation of its history renders great service to philosophy. It does not promote philosophical research to attenuate differences by reading into rival philosophical books opinions that can be adjusted to each other. The contradictions of the systems are too obvious, and we shall gain more from the study of historical systems by trying to explain their errors than by trying to explain their errors away. The philosopher of the twentieth century should have enough intellectual distance from the constructions of his predecessors to be capable of an objective critique, and he should have the courage to say what is wrong with philosophy - since it is evident that philosophy has been unable to develop a common doctrine that could be taught to students with the general consent of all those who teach philosophy.

Those among us who have taught one of the sciences will know what it means to teach on a common ground. The sciences have developed a general body of knowledge, carried by universal recognition, and he who teaches a science does so with the proud feeling of introducing his students into a realm of well-established truth. Why must the philosopher renounce the teaching of established truth? Why must he qualify all his teachings by the clause 'according to the view of philosopher X' and restrict his objectivity to the statement of what was the view of philosopher X? Incidentally, not even that statement can be made with universal consent, since the interpretation of philosophic systems is but another field of disagreement. Imagine a scientist who were to teach electronics in the form of a report on views of different physicists, never telling his students what are the laws governing electrons. The idea appears ridiculous. Though the physicist does mention the history of his field of study, the views of individual physicists appear as contributions to a common result established with a superpersonal validity and universally accepted. Why must the philosopher forgo a generally accepted philosophy?

It will help to control the diversity of opinion if the history of philosophy is reconsidered with the eyes of the critic. If philosophers have produced a great many contradicting systems, all except one must be wrong; and it is even probable that all are wrong. The history of philosophy should therefore include a history of the errors of philosophers; in uncovering the sources of error historical research will contribute to truth.

It is with this programme in mind that I should like to rediscuss the issues of rationalism and empiricism. I think that the study of this historical controversy reveals one of the fundamental sources of philosophical error, and that it is worth while to take up this study at a meeting of a philosophical association.

Both empiricism and rationalism found their first formulations as philosophical systems in Greek philosophy. While empiricism seems to be the somewhat earlier form, it was rationalism that made Greek philosophy famous, since the classical period, symbolized by the names of Socrates, Plato, Aristotle, must be regarded as representing a philosophic rationalism. This notation will appear justified if I say that by rationalism I understand a philosophy which makes reason prior to sense observation, or which regards reason as a source of knowledge independent of and superior to empirical observation.

There seems to be no doubt that the source of the rationalist turn in Greek philosophy was the success of the deductive method in mathematics, particularly in geometry. Geometry as a deductive science is a discovery of the Greeks. What had been, in the hands of the Egyptians, a system of practical rules employed by surveyors and architects, became a deductive science through the acuity of the Greek mind. It is difficult for us to imagine the transition which in those days must have taken place in the Greek conception of knowledge. The possibility of a knowledge derived from reason alone was demonstrated for the first time. What made this knowledge superior to empirical knowledge was its precision and dependability.

On the other hand, this knowledge was obviously not empty: it controlled physical reality, since it enabled the mathematician to foretell geometrical relations which could be verified by actual measurement. The practical usefulness of geometry is for the philosopher a proof of its physically descriptive nature. Thus was laid the foundation of a conception of mathematics for which, at a later stage, Kant coined the name of the synthetic a priori. Mathematics was discovered as presenting the key which unlocks the door to the secrets of physical reality. Compared with the power of abstract thought, the contribution of observation to knowledge appeared an insignificant addition. Philosophic rationalism grew from the discovery that reason can be used to control empirical observation, that what the senses report can be ordered into a logical system in terms of which future observation is predictable.

We should keep this historical situation in mind when we wish to understand Plato's theory of ideas. Seen with the eyes of the modern logician, the theory appears absurd, incredibly involved, and inappropriate to solve the problem of knowledge. But for Plato it appeared to be a solution of the problem of mathematical knowledge. The discovery of mathematical laws was conceived as analogous to empirical observation: through the eyes of the mind we see the properties of ideal objects in the same way as we see those of empirical objects with our empirical eyes. Since the laws of the ideal objects control the behaviour of empirical objects, the ideal objects must possess a higher form of reality, which is but incompletely mirrored in the properties of the observables.

The logician will argue that this is picture language instead of explanation. It is; but what can be expected of a philosophy which, for the first time, is confronted by the riddle of the coincidence of reason and reality? Later philosophies have tried to answer the question by speaking of a pre-established harmony or constructing a transcendental logic according to which mind contributes necessary presuppositions to experience. Such philosophies may represent better statements of the problem, but I do not think that they are solutions either. Kant's philosophy, in particular, though advanced with the claim of offering a solution by embracing rationalism and empiricism in the form of a higher synthesis, appears to the critical eye as a rationalism of the purest form without any injection of empiricism. No rationalist has ever denied the fact that sense observation supplies useful contributions to knowledge, and it does not make Kant an empiricist that he admits such contributions. What distinguishes the rationalist from the empiricist is the doctrine that there are some fundamental truths controlling physical reality which reason, and reason alone, can find out. This is the essential content of Kant's philosophy. It is Kant's great merit that he has stated the issue more clearly than any of his predecessors in his theory of the synthetic a priori. That he solved the problem will be admitted by nobody who has studied the development of the mathematical sciences in the one and a half centuries that have passed since Kant's death.

I said that the problems of rationalism derive from the study of the nature of mathematical knowledge. I have to add that they are immensely multiplied by an extension of the concept of knowledge springing from the same source. Once more we have to go back to Plato to see the origin of this extension. All knowledge, for Plato, is ultimately mathematical knowledge. A knowledge of the physical world that has not yet assumed the mathematical form is not regarded by Plato as genuine knowledge; it is opinion, helpful for practical purposes, but has to wait for its transformation into mathematics before it can claim to deserve the epithet of truth. It is this extension which has made rationalism so dangerous for the empirical sciences. From it springs the depreciation of empirical observation; the neglect of the use of the senses for the finding of truth; the attitude of a philosophy which found its caricature in certain periods of medieval philosophy, when learned men attempted to answer questions of physics by the study of the books of Aristotle rather than by an observation of physical phenomena. The derogatory connotation of the word 'empirical', surviving in such phrases as 'empirical school of medicine', bears witness to the rationalist distortion of the nature of knowledge. After two thousand years of dominating philosophic systems, this rationalist extension once more found its classical formulation in Kant's philosophy of the synthetic a priori, with its extension of the synthetic a priori character to the fundamentals of physics. The 'metaphysical foundations of science' reflect the dominance of a rationalist conception of knowledge at a time when mathematical physics, in the theory of Newton, seemed to have discovered the ultimate law of nature in a mathematical formula.

There is a second extension of the concept of mathematical knowledge, which has done as much mischief in the history of philosophy as the extension of the concept of mathematical knowledge to physics. I refer to the elaboration of a philosophy of ethics by analogy with mathematics. Plato's theory of ideas was meant to serve this very purpose. The Socratic doctrine that virtue is knowledge received its substantiation by a philosophy according to which there exist ideas of good actions as well as ideas of physical and mathematical objects. What is good is perceived as well as what is true, and perceived, like truth, in an act of vision of ideas and not through sense perception. From Socrates and Plato derives the ethico-cognifive parallelism, which has ever since remained an indispensable constituent of rationalism. The desire to establish ethics on an unquestionable basis was believed to have its most powerful support in a conception of knowledge which made reason a lawgiver. The giver of the moral law was identified with the giver of the cognitive law, and the moral law in me seemed to call for the same kind of recognition as the starry heavens above me. The construction of an ethics by the geometrical method became the programme of rationalism.

These developments would have a limited significance if they merely concerned the conceptions of a philosophical school. They would restrict the predominance of this school to times past, so that room would be left, at later times, for other schools that kept clear of such errors. What has made rationalism so dangerous is the fact that its fundamental errors did not remain restricted to its own school; that its principal opponent, empiricism, was infected by the same kind of error; and that empiricist philosophy was to end in failure because it did not overcome the very mistake which made rationalism incompatible with science - the mistake of identifying knowledge with mathematical knowledge.

It is true that in the empiricist philosophies this mistake is not overtly made. The empiricist has always attacked rationalism by the argument that the rationalist neglects the contribution of sense observation to knowledge. But in developing his own philosophy, the empiricist unconsciously accepted the fundamental thesis of rationalism, according to which genuine knowledge has to be as reliable as mathematical knowledge, and thus was pushed into the hopeless position of proving that empirical knowledge was as good as mathematical knowledge. Historically speaking, this rationalist commitment of empiricism found its expression in the fact that empiricists were always on the defence, that they had to prove that their philosophy was as good as rationalism for the establishment of absolute truth. A product of such a hopeless defence is what we call naive empiricism, or materialism. On the other hand, those empiricists who were too honest to deceive themselves had to end up as sceptics. The empiricist sceptic is the philosopher who has freed himself from the intoxication of rationalism but still feels committed to its challenge; who regards empiricism as a failure because it cannot attain the very aim which was set up by rationalism, the aim of an absolutely reliable knowledge. Through the tacit presupposition of his criticism, the sceptic becomes the victim of the very error which has led the rationalist to his logical cobwebs, the error of regarding mathematical knowledge as the gauge by which all other knowledge has to be measured. That is true of the ancient sceptics, who believed to have proved that there is no truth because they had proved that there is no mathematical truth for empirical knowledge. And that is true of the most prominent among the empiricist sceptics, of David Hume, whose philosophy is both the first consistent construction of empiricism and its breakdown.

The classical period of modern empiricism shows all the symptoms of an infection with rationalist fundamentals. Francis Bacon, the prophet of induction, makes it clear through the title of his major work that he intends by inductive methods to parallel the deductive logic of Aristotle. His empiricism is carried by the naive confidence that induction can establish a knowledge as good as that of the mathematician or logician. John Locke makes the hopeless attempt of combining the necessary and nonempty truth of the rationalist with an empirical origin of concepts, and preaches an ethics according to which the fundamentals of political democracy are derivable from reason. With the greatest personality of the trio, with David Hume, British empiricism reaches its climax. In Hume, empiricism finds its final formulation: all a priori knowledge is analytic, all synthetic knowledge derives from sense perception; in other words, sense perception is the only criterion of nonanalytic truth. But the very formulation of the principle of empiricism spells the failure of empirical method. Knowledge concerning the future is unattainable, because observation merely informs us about the past. Predictive knowledge can only be constructed by inferential methods; but since the implication leading from past to future observation is synthetic, deductive inference cannot be applied to the derivation of statements about the future, and the only instrument of prediction is the inductive inference. And since the inductive inference can neither be validated by deductive logic nor by reference to experience, we have no reason to accept it: the inductive inference appears unjustifiable. Empirical methods cannot establish knowledge.

I have seen many a textbook on the history of philosophy which belittles Hume's sceptical result. I have read remarks voicing the opinion that Hume's criticism has been overestimated, that few philosophers are troubled by the impossibility of justifying induction, or that induction represents a minor difficulty for scientific method which need not concern the philosopher. A philosopher who writes such things has not understood the history of philosophy. He has not realized that the rationalist attempt to account for scientific knowledge is incompatible with modern science, and that, if Hume's criticism cannot be overcome, empiricism is likewise unable to supply a theory of science. Hume's criticism is the most serious objection that has ever been raised against the possibility of knowledge. If Hume's objection cannot be answered, the history of philosophy is the history of error. The search for truth would end with the admission of ignorance; all I know is that I know nothing - the aphorism of Socrates, less characteristic for him than for the school of the sceptics, has found its final elaboration in the Enquiry concerning Human Understanding.

There are other philosophers, who hope to escape Humean scepsis by a resort to belief. We have many beliefs, they argue, which we hold without criticism; why should we not accept the belief in induction? We have had so much success with it, why should we abandon it? A friendly Humean smile should suffice to do away with this argument, but you find it in modern books on induction. Belief has become very fashionable. Some call it animal faith; others prefer the name of rational belief; and the trick of the defence is to argue that without belief we could not live and that even the empiricist believes something. Belief is a comfortable beauty rest - why not sleep on it and have our peace?

I do not think that anyone who takes logic seriously can defend the escape into belief. That everyday life is full of beliefs is a truism; but it is the philosopher's task to examine such habits in the light of logic. If he arrives at the result that the most fundamental belief is not justifiable, it does not follow that we had better stop logical inquiry but that we should stop claiming to have a philosophy. To speak of the metaphysics of empiricism is a poor defence for all those who, in the face of modern science, look for an excuse to continue rationalist speculation. The argument of the empiricist is that we have no means to prove the truth of a statement predicting future observations. It is ridiculous to call this argument a metaphysics based on belief. The quandaries of empiricism do not entitle the metaphysician to preach a new philosophy of faith. If the empiricist fails, philosophy has failed; there is no return to the beliefs of pre-Humean eras.

The solution of Hume's problem must be sought within the frame of empiricism; otherwise it would be no solution. In order to find it we must formulate more precisely what Hume's argument proves. It proves that there is no knowledge of the future if knowledge is to mean absolutely reliable knowledge. The argument is conclusive only if knowledge is what the rationalist understands by it: what Hume has proved is that the rationalist aim of knowledge is unattainable by empiricist methods. It is the unconscious adherence to the rationalist programme of knowledge, to the interpretation of knowledge in terms of mathematical knowledge, which leads to the Humean scepsis. The empiricist who turns sceptic has not sufficiently stripped himself of the creed of rationalism.

The way to a consistent empiricism is open only to those who are ready to interpret empirical knowledge in its own right, to abandon the prejudice that mathematics is the prototype of all knowledge. It was a long way to this liberation, which would never have been found without the help of mathematical science itself. The story of this liberation is, in fact, a story of the mathematics and mathematical physics of the nineteenth and twentieth centuries. I should like to outline this story in a short review of its prominent chapters, before I turn to the answer we can give today to Hume's criticism.

The disintegration of the synthetic a priori began with developments within mathematics. Twenty years after Kant's death, Bolyai and Lobachefski discovered a non-Euclidean geometry. The significance of these discoveries was immediately recognized by Gauss, who had independently come upon similar geometrical results: Gauss saw that if the mathematician knows more than one geometrical system, the question which system applies to physical reality is an empirical question.

The logical development which pushed mathematics from its throne is amazingly simple. As long as there was only one geometry, the mathematician seemed to have the key unlocking physical space, and reason appeared to be the lawgiver of physical reality. If there is a plurality of possible geometries, the mathematician is unable to tell which of them fits physical space, and the selection of the one geometry that describes the physical universe is left to the physicist. It is true, the process of this selection is somewhat complicated because it includes a decision for certain definitions, called co-ordinative definitions; but this fact does not make the determination of the geometry of physical space less empirical. As in the case of all physical hypotheses, mathematics merely presents us with a set of possibilities, among which observation singles out the one that corresponds to reality. The criterion of synthetic truth is not reason, but observation - the empiricist principle includes the application of mathematics to physical reality. The development introduced with the discovery of the non-Euclidean geometries reaches its final stage in Russell's analysis of arithmetic: mathematical truth is analytic, mathematics is not descriptive of physical reality.

Analytic truth offers no problem to the philosopher. That reason is able to establish analytic statements is possible because analytic statements do not anticipate future experience, do not restrict the realm of what is observable. The tautologies of a language are merely the mirror image of the rules in terms of which the language is set up; they say in the object language what is expressed through the structure of propositions and what is thus a consequence of metalinguistic rules. The tertium non datur, for instance, is merely the object-language equivalent of a rule laying down a two-valued character of the language. The analytic a priori does not call for rationalism and is compatible with the most critical empiricism.

In reducing mathematical truth to analytic relations, modern analysis of mathematics destroys the basis on which rationalism is erected - that is the simple fact which no philosopher can ignore. To regard mathematics as the ideal which the physical sciences should try to approximate means misunderstanding the nature of mathematics. A physical science made to the model of mathematics would be empty, could not inform us about the physical world. All synthetic knowledge derives from observation: the empiricist is finally victorious because the mathematician himself has renounced the claim of knowing synthetic truth. The philosopher who in the twentieth century still attempts to derive knowledge from reason has lost his most potent support, the support by the mathematician, and resembles the man who still looks for the perpetual motion machine - it cannot be done, that is the answer which modern science gives to two thousand years of struggle for a rationalist interpretation of knowledge.

The destructive effect of modern analysis of mathematics extends into the field of ethics. The plan to derive ethical principles, like mathematical principles, from reason appeared plausible as long as mathematics was believed to be rational synthetic truth. But if all knowledge springing from reason is analytic and reason cannot select among different systems, each of which is consistent while it contradicts every other one, the ethico-cognitive parallelism renders a bad service to ethics. What would the author of an ethics constructed by the geometrical method have answered, had he been told that the geometrical method would justify, just as well, a non-Spinozist ethics? But I will not venture prophecies about the reaction of a Spinoza to the mathematics of the twentieth century and rather say what he should have answered: he should have admitted that if moral principles have a cognitive character, they are empty, since they would be knowledge derived from reason alone.

The answer modern logic gives to the problem of ethics resembles the answer given to the problem of geometry: only the implications between axioms and theorems are capable of logical proof, but the axioms themselves are not demonstrable by reason. The fundamental ethical principles can be accounted for by the philosopher as little as the axioms of physical space. But while the axioms of physical space at least have cognitive character and are capable of empirical verification by the physicist, the fundamental principles of ethics are non-cognitive and require a different treatment, of which I shall speak presently.

I have first to go into a closer analysis of empirical knowledge. The reduction of mathematics to analytic relations is a negative result. It has to be supplemented by a conception of synthetic knowledge which at the same time satisfies the empiricist criterion of truth and overcomes Hume's criticism.

It is well known that the answer to this question leads into the theory of probability. Empiricist philosophers, from the time of Hume's own studies in probability to our days, have therefore repeatedly tried to construct theories of probability that can account for empirical knowledge. One should believe that all these theories of probability were constructed along empiricist principles. A survey of the present scene of the discussion on probability shows that this is not the case. The analysis of probability is confused by remnants of a rationalist interpretation of knowledge; the germs of rationalism have so deeply penetrated into philosophical thought as even to infect empiricist-minded thinkers of our time. Rationalism revives in the attempts of logicians at constructing a theory of probability from pure reason, a theory of inductive logic for which degrees of probability are derivable from the logical structure of propositions, like theorems of deductive logic. These theories are sometimes constructed by the help of the so-called principle of indifference, sometimes by methods determining a so-called degree of confirmation. The common feature of all these theories is that their proponents believe they have an analytic rule which can determine, on the basis of observational material, with what degree of probability future observations of a certain kind will occur.

The rationalistic root of these theories is obvious. If logic is unable to foretell the future, logic should at least be able to tell the probabilities of the various possible forms of the future - in this attenuated form the rationalist desire of a physical world controlled by reason has slipped into the mathematical philosophy of our time. In the new form it is as active and as dangerous as in the old form of promising certainty; promising probabilities is perhaps an even more dangerous form of rationalism because it looks so modest and so modern.

The criticism of all these theories is easily given. Like all mathematical systems, the calculus of probability is analytic; all it can do is to derive probabilities from other probabilities, and the other probabilities must be given. In order to be applicable to physical reality, the calculus must be supplemented by a rule telling how to find the first probabilities. This rule cannot be mathematical because it cannot be analytic. If it were analytic, it would not tell anything about the future and thus could not be used as a guide for action. Conversely, since the probability statement is meant to be an advice how to act, it must say something about the future; therefore it cannot be derived by means of deductive logic from observational material referring to the past. This criticism is simple; that there are modern logicians who will not accept it proves that the urge for rationalism is not always controlled by logic.

The empiricist theory of probability is based on the frequency interpretation. The probability statement predicts a frequency of events; it says something about the future and is therefore verifiable in terms of the predicted events. The synthetic rule furnishing probabilities before the total sequence of events has occurred is the rule of induction by enumeration. Although frequency refers to a class, the probability statement is applicable to a single case, since the life of every person includes many single cases; following the rules of probability leads to the greatest possible number of successes.

The most important tool for this consideration is the concept of posit. A probability statement does not permit us to assert the sentence about the predicted event as true; the sentence can only be asserted in the sense of a posit. Positing a statement means dealing with it as true, although we do not know its truth. To justify a posit we need not prove that it is true; all we have to prove is that it is advantageous in some sense to deal with the statement as true.

The concept of posit is the key to the understanding of predictive knowledge. It can be proved that all forms of induction, including the so-called inference by confirmation, are reducible to induction by enumeration, and the enormous amount of mathematics employed in modern physics can be shown to supply merely a concatenation of individual inferences in terms of such simplest inductions. The significance of analytic thought for empirical science is thus explained: the function of deductive operations is to tie together simple inductions into a network, which as a whole represents but another posit, a posit that can be proved superior to each of the individual posits from which it is constructed. Predictive knowledge is therefore accounted for when it is possible to justify induction by enumeration. This justification, which Hume considered impossible, can be given when the conclusion of the inductive inference is regarded, not as asserted with the claim of truth, but as asserted in the sense of a posit. It can be shown that if it is possible at all to make predictions, the inductive inference is an instrument to find them; and the inference is justified because its applicability represents a necessary condition of success.

The rationalist conception of inductive logic breaks down because it is faced by the impossible task of validating a synthetic principle through reason. The empiricist conception of inductive logic is essentially different. The principle of induction by enumeration, which constitutes its only synthetic principle, is not regarded as self-evident, or as a postulate which logic could validate. What logic can prove is that the use of the principle is advisable if a certain aim is envisaged, the aim of predicting the future. This proof, the justification of induction, is constructed in terms of analytic considerations. The empiricist is allowed to use a synthetic principle, because he does not assert that the principle is true or must lead to true conclusions or to correct probabilities or to any kind of success; all he asserts is that employing the principle is the best he can do. This renunciation of any truth claim enables him to incorporate a synthetic principle in an analytic logic and to satisfy the condition that what he asserts on the basis of his logic is analytic truth only. He can do so because the conclusion of the inductive inference is not asserted by him, but only posited; what he asserts is that positing the conclusion is a means to his end. The empiricist principle that reason cannot make other than analytic contributions to knowledge, that there is no synthetic self-evidence, is thus fully carried through.

The crisis of empiricism, expressed in David Hume's scepticism, was the product of a misinterpretation of knowledge and vanishes for a correct interpretation - such is the outcome of a philosophy grown from the soil of modern science. The rationalist has not only presented the world with a series of untenable systems of speculative philosophy - he had also poisoned the empiricist interpretation of knowledge by inducing the empiricist to strive for unattainable aims. The conception of knowledge as a system of statements that are demonstrable as true had to be overcome by the evolution of science before a solution of the problem of predictive knowledge could be found. The search for certainty had to die down within the most precise of all sciences of nature, within mathematical physics, before the philosopher could account for scientific method.

I have given you a short account of the historical controversy between rationalism and empiricism with the intention of studying the roots of philosophical error. I came to the result that the errors of traditional philosophy spring from a misinterpretation of knowledge, from regarding mathematical knowledge as the prototype of all knowledge. You will ask me: if that is the fundamental mistake, why was it committed? In other words, can we push the explanation one step further back and name, not only the ground of the error, but also the ground of the ground of the error? This question, in fact, can be answered. There have been two causes at work which are responsible for the philosophical misinterpretation of knowledge.

The first is given by the historical evolution of science. It took several thousand years until science assumed a form which made possible a consistent empiricism. The philosopher cannot be blamed for identifying mathematical with physical geometry as long as the mathematician of his time holds the same view. If the non-Euclidean geometry had been discovered at Euclid's time, philosophy might have taken a very different course. Incidentally, looking in retrospect at the evolution of mathematics, one must admit that such a development might have been possible. The fundamentals of non-Euclidean geometry can be constructed with rather simple mathematical means, and one could very well imagine a Bolyai among the disciples of Euclid. The Hellenistic and later the Roman era were sophisticated enough to lend an intellectual background to such enterprise; we know that the heliocentric system was conceived as early as 280 B.C. Since the history of geometry records no such anticipation of non-Euclidean systems, we can do nothing but point to the delay in the evolution of mathematics and make it responsible for the delayed evolution of philosophy. On the other hand, the development of physics leading to the abandonment of causality and to the primacy of probability could not possibly have taken place before the growth of modern experimental science. In its ultimate transition to a conception of knowledge as a system of posits, philosophy was therefore dependent on developments outside the control of the philosopher.

But the dependence on the evolution of science is not the only ground that can be adduced for the rationalist misinterpretation of knowledge. Science has many aspects, its mathematical aspect being only one of them; and the existence of empirical philosophies at all times shows that the empirical side of science has always been noticed. If the philosopher was inclined to overemphasize the mathematical nature of science, if he regarded mathematics as the ideal to which all other forms of science should aspire, such attitude must be explained in terms of the psychology of the philosopher. And here, in fact, we discover the deepest root of the misconception of knowledge that led to the errors of philosophy.

Looking through the long list of philosophical systems, one cannot get away from the impression that the men who made these systems were not primarily interested in an unprejudiced interpretation of science. For the philosopher, analysis of science has at all times been a means to an end; he needed science as a foothold from which to reach for other goals. He undertook the study of science with a preconceived aim, reading into science his own objectives: He wanted certainty and saw only those aspects of science which lent themselves to an interpretation in terms of absolutely certain knowledge; he wanted moral directives and saw only those aspects of science which lent themselves to an interpretation of knowledge in terms of an ethico-cognitive parallelism. Even the empiricist, who did not accept the rationalist solution, could not free himself from the rationalist objective and, if his logic was unbiased, turned sceptic. The rationalist misinterpretation of knowledge sprang from extra-logical motives, from the intention of establishing certainty for human knowledge and moral directives for human behaviour. The philosopher erred because his analysis was not impartial: science was for him what he wanted it to be.

What, then, can we do to build up a better philosophy? The study of error should help us to find the truth. Since philosophy is dependent on science, we should make this dependence the conscious condition of our work: we should know that the nature of knowledge can be studied only through analysis of science. The idea of a philosophical theory of knowledge that derives the general outlines of knowledge from the structure of the mind, or from an insight into the nature of being, should forever be abandoned. There is no ontology, no separate realm of philosophical knowledge that precedes science. Theory of knowledge is analysis of science. Philosophy does not contribute any content to knowledge; it merely studies the form of knowledge as exhibited in the work of the scientist and examines all claims to validity. In so doing, the philosopher will know that all he can strive for is a philosophy of the knowledge of his time.

Theory of knowledge is about one half of the subject of traditional philosophy; the other half is concerned with value judgments, in particular, with ethics. I said that the axioms of ethics, like those of geometry, are not subject to the critique of the philosopher; only the implications between various moral rules are controlled by logic, while the fundamental rules are of a noncognitiye nature. They are imperatives and thus volitional decisions. The philosopher, who had to renounce setting up the principles of physics, will be ready also to renounce setting up the fundamental ethical imperatives. His task can only consist in a logical analysis of moral behaviour, comparable to his analysis of cognitive behaviour. He will point out the significance of implications for moral behaviour, of the relations connecting the different volitional aims, making one aim subordinate to another; and he will emphasize the necessity of studying psychology and sociology to everyone who wants to combine volitional aims into an ordered system. Without the results of these sciences, implications between volitional aims cannot be set up, since these implications depend on the synthetic knowledge comprised by psychology and sociology. As in the theory of knowledge, the work of the philosopher in the field of ethics will essentially consist in establishing order, in making explicit the logical controls of a field of behaviour so controversial and so indispensable.

A philosophy of the kind described, which is analysis rather than independent construction, may appear unsatisfactory to all those who indulge in admiration of the philosophy of the systems. Philosophical systems have a persuasive power which it is difficult to resist, a power comparable to the magic force of great works of literature; and renouncing philosophical speculation may appear like renouncing the beauty and greatness which has endowed philosophy with its leading position in civilization. I will not argue about the aesthetic value of the systems; but I should like to draw attention to the fact that this positive side is inseparably connected with the negative side of traditional philosophy: with the absence of general agreement of philosophic theories. Philosophic systems are individualistic creations; like certain works of art, they do not call for the consent of all but for the admiration of the few to whom they mean something. The sober study of truth is deprived of the glamour of artistic creation; but it carries the advantage of paving the path to universal agreement, of setting up results that eventually will be exempt from controversy and attack. It is the path of science on which the philosophy of logical analysis is marching. Though less attractive to the romantic mind, the adoption of scientific method will appear the inescapable consequence of an unprejudiced study of the history of philosophy; it is the only successful path open to the philosophy of the twentieth century.

Those who have tried to go along this path have had an experience which I would like to regard as the equivalent of aesthetic values: the experience of the power of clarification that is the concomitant of analysis. To know what one means, and to know what knowledge means, is a goal worthy of the philosopher; and I think that we may all be happy if our work contributes to clarity of meanings. The scientist discovers synthetic truth; let it be our job to tell what this truth means and where it is to be incorporated in the totality of human civilization. The philosopher has always regarded it as his task to be an educator; let us be educators clarifying meanings, by putting cognition and volition in their right places, by making explicit what is understood, by making distinct what is active in the dim background of consciousness. Clarity of thought is one of the most refined intellectual enjoyments - let us make it the supreme goal of the philosopher, not only to say the truth, but also to say it with the clarity that springs from logical analysis.