It is clear that such a philosophy ought to end in unqualified Agnosticism. The Hegelians, to be sure, made merry over the Unknowable of Mr. Spencer, but their own Absolute is really just the Unknowable in its 'Sunday best'. Nothing that we can say about anything which is not the Absolute is really true, because there really is nothing but the Absolute to speak about, and nothing that we can say about the Absolute is quite true either, because we can never succeed in saying itself of it. Mr. Bradley, far the most eminent of the philosophers of the Absolute, has made persistent and brilliant attempts to show that, in spite of this, we do know enough to be sure that our own mind is more like the Absolute than a cray-fish, and a cray-fish more like it than a crystal. But when all is said, though I owe more to Mr. Bradley than I can ever acknowledge adequately, I cannot help feeling that there are two men in Mr. Bradley, a great constructive thinker and a subtle destructive critic, and that the destructive Hyde is perpetually pulling to pieces all that the constructive Jekyll has built up. Of course it is obvious that the truth of mathematics, if mathematics are true, is a fatal stone of stumbling for this type of philosophy. Mathematics never attempts to say anything about the 'Absolute'—the only 'Absolute' of which it knows is only a 'degraded conic'—yet it claims that its statements, if once they have been correctly expressed, are not 'partial' but complete.
Over against the Hegelianizing philosophers, we had, of course, the men of science. No one could wish to speak of the scientific men of the days of Huxley without deep respect for their success in adding to our positive knowledge of facts. But it may perhaps be said at this distance of time that it was not precisely the greatest among them who were most prominent as mystagogues of Science with the big S, and it may certainly be said that when the mystagogues, the Cliffords, Huxleys, and the rest, undertook to improvise a theory of first principles, their achievement was little better than infantile. They took it on trust from Hume that the whole of knowledge is built up of sensations, actual or 'revived', and quite missed Kant's point that their empiricism left the formal constituent in knowledge, the type of order by which data are organized into an intelligible pattern, wholly out of account. Even when they deigned to read Kant, they read him without any inkling of the character of the 'critical' problem. Hence they taught dogmatically as true a theory of scientific method which Hume himself had elaborately proved impossible. It was just because Hume had seen so clearly that no universal scientific truths can be derived from premisses which merely record particular facts that he professed himself a follower of the 'academic' or 'sceptical' philosophy. He recognized the impossibility of constructing scientific knowledge out of its material constituent alone, but did not see where the formal constituent could come from, and so resigned himself to regarding the actual successes of science as a kind of standing miracle.
The men of the 'seventies were, after all, in many cases more anxious to damage theology than to build up Philosophy. They read Hume without any delicate sense for his urbane ironies, and believed in good faith that he and John Stuart Mill between them had shown that by a mysterious process called 'induction' it is possible to prove rigorously universal conclusions in science without universal premisses. A scientific law, according to them, is only a convenient short-hand notation in which to register the 'routine of our perceptions'. Thus we have known of a great many men who have died, and have never known of any man who lived to much over a hundred without dying. The universal proposition 'all men are mortal' is a short expression for this information, and it is nothing more. It ought to have been obvious that, if this is a true account of science, all scientific 'generalizations' are infinitely improbable. The number of men of whom we know that they have died is insignificant by comparison with the multitude of those who have lived, are living, or will live, and we have no guarantee that this insignificant number is a fair average sample. So again, unless there are true universal propositions which are not 'short-hand' for any plurality of observed facts whatever, we cannot with any confidence, however faint, infer that a 'regular sequence' or 'routine' which has been observed from the dawn of recorded time up to, say, midnight, August 4, 1919, will continue to be observed on August 5, 1919. How, except by relying on the truth of some principle which does not depend itself on the validity of 'generalization', can we tell that it is even slightly probable that the nature of things will not change suddenly at the moment of midnight between August 4 and August 5, 1919? What is called 'inductive' science certainly has 'pulled off' remarkable successes in the past, but we can have no confidence that these successes will be repeated unless there are much better reasons for believing in its methods and initial assumptions than any which the scientific man who is an amateur 'empiricist' in his philosophy can offer us. We may note, in particular, that this empiricism, which has been expounded most carefully by Pearson and Mach, coincides with Hegelian Absolutism in leading to the denial of the truth of mathematics. It would be a superfluous task to argue at length that, e.g., De Moivre's theorem or Taylor's theorem is not a short-hand formula for recording the 'routine of our perceptions'.
The general state of things at the time of which I am speaking was thus that relations were decidedly strained between a body of philosophers and a body of scientific men who ought at least to have met on the common ground of a complete Agnosticism. The philosophers were, in general, shy of Science, mainly, no doubt, because they were modest men who knew their own limitations, but they had a way of being condescending to Science, which naturally annoyed the scientific men. These latter professed a theory of the structure of knowledge which the philosophers could easily show to be grotesque, but the retort was always ready to hand that at any rate Science seemed somehow to be getting somewhere while Philosophy appeared to lead nowhere in particular.
The conditions for mutual understanding have now greatly improved, thanks mainly to the labour of mathematicians with philosophical minds on the principles of their own science. If we admit that mathematics is true—and it seems quite impossible to avoid the admission—we can now see that neither the traditional Kant-Hegel doctrine nor the traditional sensationalistic empiricism can be sound. Not to speak of inquiries which have been actually created within our own life-time, it may fairly be said that the whole of pure mathematics has been shown, or is on the verge of being shown, to form a body of conclusions rigidly deduced from a few unproved postulates which are of a purely logical character. Descartes has proved to be right in his view that the exceptional certainty men have always ascribed to mathematical knowledge is not due to the supposed restriction of the science to relation of number and magnitude—there is a good deal of pure mathematics which deals with neither—but to the simplicity of its undefined notions and the high plausibility of its unproved postulates. Bit by bit the bad logic has been purged out of the Calculus and the Theory of Functions and these branches of study have been made into patterns of accurate reasoning on exactly stated premisses. It has appeared in the process that the alleged contradictions in mathematics upon which the followers of Kant and Hegel laid stress do not really exist at all, and only seemed to exist because mathematicians in the past expressed their meaning so awkwardly. Further, it has been established that the most fundamental idea of all in mathematics is not that of number or magnitude but that of order in a series and that the whole doctrine of series is only a branch of the logic of Relations. From the logical doctrine of serial order we seem to be able to deduce the whole arithmetic of integers, and from this it is easy to deduce further the arithmetic of fractions and the arithmetic or algebra of the 'real' and 'complex' numbers. As the logical principles of serial order enable us to deal with infinite as well as with finite series, it further follows that the Calculus and the Theory of Functions can now be built up without a single contradiction or breach of logic. The puzzles about the infinitely great and infinitely small, which used to throw a cloud of mystery over the 'higher' branches of Mathematics, have been finally dissipated by the discovery that the 'infinite' is readily definable in purely ordinal terms and that the 'infinitesimal' does not really enter into the misnamed 'Infinitesimal Calculus' at all. Arithmetic and the theory of serial order have been shown to be the sufficient basis of the whole science which, as Plato long ago remarked, is 'very inappropriately called geometry'. A résumé of the work which has been thus done may be found in the stately volumes of the Principia Mathematica of Whitehead and Russell, or—to a large extent—in the Formulario Matematico of Professor Peano. Of other works dealing with the subject, the finest from the strictly philosophical point of view is probably that of Professor G. Frege on The Fundamental Laws of Arithmetic. The general result of the whole development is that we are now at last definitely freed from the haunting fear that there is some hidden contradiction in the principles of the exact sciences which would vitiate all our knowledge of universal truths. This removes the chief, if not the only ground for the view that all the truths of Science are only 'partial'. At the same time, the proof that pure mathematics is a strictly logical development and that all its conclusions are of the hypothetical form, 'if a b c ..., then x' definitely disproves the popular Kantian doctrine that sense-data are a necessary constituent of scientific knowledge. And with this dogma falls the main ground for the denial that knowledge about the soul and God is attainable. The recovery of a sounder philosophical method has, as Mr. Russell himself says, disposed of what was yesterday the accepted view that the function of Philosophy is to narrow down the range of possible interpretations of facts until only one is left. Philosophy rather opens doors than shuts them. It multiplies the number of logically possible sets of premisses from which consequences agreeing with empirical facts may be inferred. Mr. Russell's unreasoned anti-theism seems to me to make him curiously blind to an obvious application of this principle. On the other side, the revived attention to the logical methods of the sciences is killing the crude sensationalism of the days which saw the first publication of Mach's Science of Mechanics and Pearson's Grammar of Science. The claims of 'induction' to be a method of establishing truths may be fairly said to have been completely exposed. It is clearer now than it was when Kant made the observation that each of the 'sciences' contains just so much science as it contains mathematics, and that the Critical Philosophy was fully justified in insisting that all science implies universal à priori postulates, though it went wrong in thinking that these postulates are laws of the working of the human mind or are 'put into' things by the human mind. How far Science has moved away from crude sensationalistic empiricism may be estimated by a comparison of the successive editions of the Grammar of Science. It must always have been apparent to an attentive reader that the chapters of that fascinating book which deal directly with the leading principles of Physics and Biology are of very different quality from the earlier chapters which expound, with many self-contradictions and much wrath against metaphysicians and theologians whom the writer seems never to have tried to understand, the fantastic 'metaphysics of the telephone-exchange'. But the difference of quality is more marked in the second edition than in the first, and in the (alas!) unfinished third edition than in the second. So far, then, as the problem of the unification of the sciences is concerned, the old prejudices which divided the rationalist philosopher from the sensationalist scientific man seem to have been, in the main, dissipated. We can see now that what used to be called Philosophy and what used to be called Science are both parts of one task, that they have a common method and presuppose a common body of principles.
So far it may be said with truth that Philosophy is becoming more faithful than Kant was himself to the leading ideas of 'Criticism', and again that it is reverting once more, as it reverted in the days of Galileo, to the positions of Plato. I do not mean that the whole programme has been completely executed and that there is nothing for a successor of Frege or Russell to do. It is instructive to observe that at the very end of the great work on arithmetic to which I have referred Frege found himself compelled by difficulties which had been overlooked until Russell called attention to them to add an appendix confessing that there was a single important flaw in his elaborate logical construction of the principles of arithmetic. He had shown that if there are certain things called 'integers', defined as he had defined them, the whole of arithmetic follows. But he had not shown that there is any object answering to his definition of an integer, and the logical researches of Russell had thrown some doubt on the point. This proved that some restatement of the initial assumptions of the theory was needed. Since the date of Frege's appendix (1903), Mr. Russell and others have done something towards the necessary rectification, and the resulting 'Theory of Types' is pretty certainly one of the most important contributions ever made to logical doctrine, but it may still be reasonably doubted whether the 'Theory of Types', as expounded by Whitehead and Russell in their Principia Mathematica, is the last word required. At any rate, it seems clear that it is a great step on the right road to the solution of a most difficult problem.
There still remains the greatest problem of all, the harmonization of Science and Life. I cannot believe that this problem is an illegitimate one, or that we must sit down content to accept the severance of 'fact' and 'value' as final for our thought. Even the unification of the sciences itself remains imperfect so long as we treat it as merely something which 'happens to be the case' that there are many things and many kinds of things in the universe and also a number of relations in which they 'happen' to stand. It is significant that in his later writings Mr. Russell has been driven to abandon the concept of personal identity, which is so fundamental for practical life, and to assert that each of us is not one man but an infinite series of men of whom each only exists for a mathematical instant. I am sure that such a theory requires the abandonment of the whole notion of value as an illusion, and even more sure that it is ruinous to any practical rule of living, and I cannot believe in the 'philosophy' of any man who is satisfied to base his practice on what he regards as detected illusion. Hence I find myself strongly in sympathy with my eminent Italian colleague Professor Varisco, who has devoted his two chief works (I Massimi Problemi and Conosci Te Stesso) to an exceedingly subtle attempt to show that 'what ought to be', in Platonic phrase 'the Good', is in the end the single principle from which all things derive their existence as well as their value. Mr. Russell's philosophy saves us half Plato, and that is much, but I am convinced that it is whole and entire Plato whom a profounder philosophy would preserve for us. I believe personally that such a philosophy will be led, as Plato was in the end led, to a theistic interpretation of life, that it is in the living God Who is over all, blessed for ever, that it will find the common source of fact and value. And again I believe that it will be led to its result very largely by what is, after all, perhaps the profoundest thought of Kant, the conviction that the most illuminating fact of all is the fact of the absolute and unconditional obligatoriness of the law of right. It is precisely here that fact and value most obviously meet. For when we ask ourselves what in fact we are, we shall assuredly find no true answer to this question about what is if we forget that we are first and foremost beings who ought to follow a certain way of life, and to follow it for no other reason than that it is good. But I cannot, of course, offer reasons here for this conviction, though I am sure that adequate reasons can be given. Here I must be content to state this ultimate conviction as a 'theological superstition', or, as I should prefer to put it with a little more certainty, as a matter of faith. The alternative is to treat the world as a stupid, and possibly malicious, bad joke.
Note.—It may be thought that something should have been said about the revolt against authority and tradition which has styled itself variously 'Pragmatism' and 'Humanism', and also about the recent vogue of Bergsonianism. I may in part excuse my silence by the plea that both movements are, in my judgement, already spent forces. If I must say more than this, I would only remark about Pragmatism that I could speak of it with more confidence if its representatives themselves were more agreed as to its precise principles. At present I can discern little agreement among them about anything except that they all show a great impatience with the business of thinking things quietly and steadily out, and that none of them seems to appreciate the importance of the 'critical' problem. 'Pragmatism' thus seems to me less a definite way of thinking than a collective name for a series of 'guesses at truth'. Some of the guesses may be very lucky ones, but I, at least, can hardly take the claims of unmethodic guessing to be a philosophy very seriously. To 'give and receive argument' appears to me to be of the very essence of Philosophy. As for M. Bergson, I yield to no one in admiration for his brilliancy as a stylist and the happiness of many of his illustrations. But I have always found it difficult to grasp his central idea—if he really has one—because his whole doctrine has always seemed to me to be based upon a couple of elementary blunders which will be found in the opening chapter of his Données Immédiates de la Conscience. We are there called on to reject the intellect in Philosophy on the grounds (1) that, being originally developed in the services of practical needs, it can at best tell us how to find our way about among the bodies around us, and is thus debarred from knowing more than the outsides of things; (2) that its typical achievement is therefore geometry, and geometry, because it can measure only straight lines, necessarily misconceives the true character of 'real duration'. Now, as to the first point, I should have thought it obvious that the establishment of a modus vivendi with one's fellows has always been as much of a practical need as the avoidance of stones and pit-falls, and the alleged conclusion about the defects of the intellect does not therefore seem to me to follow from M. Bergson's premisses, even if we had any reason, as I do not see that we have, to accept the premisses. And as to the second point, I would ask whether M. Bergson possesses a clock or a watch, and if he has, how he supposes time is measured on them? He seems to me to have forgotten the elementary fact that angles can be measured as well as straight lines. (I might add that he makes the further curious assumption that all geometry is metrical.) It may be that something would be left of the Bergsonian philosophy if one eliminated the consequences of these initial blunders, but I do not know what the remainder would be. At any rate, the anti-intellectualism which M. Bergson and his disciple, Professor Carr, seem to regard as fundamental will have to go, unless different and better grounds can be found for it. I must leave it to others to judge of the adequacy of this apology.
Varisco, The Great Problem (Macmillan).