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KANT 169





Water is always like water,
but it has a quite different
taste when drawn at the
fountain head from what it has
when drunk out of a pitcher.


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From the painting by Mignard, from the Castle Howard Collection, now in the National Gallery.



THERE are days and days, and I confess that it is with some hesitation and distrust that I address myself to-day to the task of continuing our observations in common. For now I have to travel with you through regions which it will not be so easy to make clear as it was so long as we had the eye of a Goethe and a Leonardo to lighten us on our way. The comparison with philosophers who were at the same time artists revealed to us much that was of fundamental importance, and gave rise to observations which could not but result in a deep insight into the personality of Kant, in the narrower meaning of the word, but now we must face about, we must once more fix the lenses of our eyes upon a nearer focus; we must bring into comparison philosophers who in their turn will lead us far, but on another road; men, the atmosphere of whose lives does not consist in Beauty and Art, but in research and thought. To-day we will busy ourselves with Descartes the critically empirical, mathematical thinker, and in the next lecture with Bruno the logical schoolman and enthusiastic thinker.

    You must not misunderstand me. There is no such thing as an absolute artist, no such thing as an absolute mathematician, and above all no such thing as an absolute philosopher. This sort of classification into professions will never succeed even with half-important men. Goethe and Leonardo were both of them, as we have seen, great investigators of nature, and thinkers:

Bruno and Descartes on their side possess in a pre-eminent degree the artistic gift of putting into shape: Bruno, in his manner of thinking and speaking, is as much a poet as Plato was; and Descartes, the masterful thinker, is so penetrated with the value of perception and the empirical investigation of nature, that he is the bitter enemy of genuine professional philosophy. We, however, are dealing to-day solely with that which I should like to call the characteristic intellectual attitude. In Goethe and in Leonardo it is distinctly directed outwards: the primacy of the Eye is dominant in both, and indeed of the eye both as a receptive and reproductive machinery of the senses. It is true that we found the result to be very different in the two men; for behind two equally powerful eyes two brains gifted in varying directions take up impressions, and work them up each in its own way. In Leonardo the gift of sight is more precise and, in the widest sense of the word, more correct in its perspective; this he owes to the power, which we recognised in the previous lecture, of referring all that he saw to the inner scheme of perception; before Goethe's eyes, on the other hand, the outlines are uncertain, his power of schematising is insufficient, and he mixes up his thought with everything: but it is exactly this which bestows on him the gift of illuminating the very depths of Nature, depths where without the lamp of creative thought, dark night reigns. Leonardo sees the relationship of things to one another, Goethe sees their relationship to the human intellect; in Leonardo's understanding the masculine element prevails, in Goethe's we find unmistakable feminine or receptive constituents; hence Leonardo's thought is keen, mechanical, scientific, and easily grasped, whereas Goethe's is deeper, more iridescent, baffling conception, because it is pregnant with presentiments too wild to be tamed into words. We shall go further into this in a future lecture; for the moment

we must be contented with recognising the fact that this precise intellectual habit, the method of looking outwards, is the common property of both Leonardo and Goethe. At the same time this habit distinguishes both from Kant, even though a closer examination has revealed to us so many points of contact in the manner of Seeing between him, the artist in thought, and those two artist-sages. But now, for the sake of comparison, we will summon into court two men with essentially different qualifications, — men whose innate intellectual habit points inwards. I say “inwards“ because these thinkers in the first place consult their own thought, and only later on turn to Nature: they do not trust the impression which comes from without, not, that is to say, until they have, as far as may be in any way possible, tested and dissected the whole details of the inner diagnosis: this method of procedure is the exact opposite to that followed by Goethe and Leonardo. This habit I call the method of looking inwards. René Descartes and Giordano Bruno will, as I think, answer our purpose: neither of the two is so nearly akin to Kant as to prevent dark shadows being thrown upon the picture from them upon him, and on the other hand, in respect of talent and feeling, these two great philosophers are just as fundamentally different from one another as Leonardo and Goethe. They have in common only — but this “only“ means very much — the habit of the specific thinker. Bruno, the Goethe of our second pair of philosophers, exclaims, Gli beni de la mente non altronde che dall' istessa mente rostra riportiamo 1 (it is from the mind itself and from no other source that we acquire the riches of the mind), — and Descartes, the strict empiric, the Leonardo, says deliberately, Il n'est aucune question plus importante à résoudre que celle de savoir ce que c'est que la connaissance humaine, et jusqu'où elle s'étend, ... Rien ne me semble plus absurde que de discuter audacieusement sur les mystères de la nature sans

avoir une seule fois cherché si l'esprit humain peut atteindre jusque là. 2
    These few words will have sufficed to show you with what manner of man we have to deal here; at the same time the patent relationship to Kant's objects and methods and convictions is at once striking. The investigation of the essence and of the limits of human knowledge describes exactly a great part — the critical part — of Kant's Life-work, and that the peculiar riches of the mind must be acquired from within and not from without, puts into a few words what Kant looked upon as his positive, practical, and edifying achievement. But even the points of difference will teach us much. The life-stories of the seigneur du Perron (Descartes) and of the man of Nola (Bruno) show conclusively that these two men as regards their intellectual talents are far removed from Kant. In the first lecture we saw how deeply rooted in Kant's method of perception and in his adoption of ideas was that peculiar feature which made him so painfully avoid even the shortest journey; Bruno and Descartes, on the contrary, move restlessly from place to place, and from country to country, as the spirit moves them. Bruno, with his apostle's nature, needs new contacts, new excitements, new disputations; he is bound to strike sparks out of life, to kindle flames in hearts; wherever he goes he arouses glowing love and irreconcilable hatred. Descartes, the reserved man of the world, travels in order to be alone, enjoys in cities “the solitude of the remotest deserts,“ steals away from a place as soon as his presence is noticed, and at the same time, by a systematic observation of the differently constituted men and nations, religions and customs, seeks to free himself from the prejudices which are rooted in us all. Je ne fis autre chose que rouler ça et là dans le monde, tâchant d'y être spectateur plutôt qu'acteur en toutes les comédies qui s'y jouent. 3 Such a funda-

mentally different ordering of life points to far-reaching differences in the essence of the intellect: we may premise without going further that Bruno and Descartes “saw“ otherwise than Kant did. This will be especially clear in the case of Bruno, who, in spite of the purely philosophical tendency of his intellect, is in many respects the veriest antipodes of Kant, and as such can render us valuable service, whereas in Descartes the close kinship leads us to penetrate the inmost secrets of Kant's method of perception, while allowing us to leave on one side the many points of difference between the two as having no value for the object which we have in view.

    Among the very great thinkers of the world's history perhaps none has been so scurvily treated as Descartes; he,— I mean the true Descartes, — is as good as unknown; the shadowy being that under this name is represented to our imagination, is a mere ghost-like caricature. Here was a man who with desperate energy fought to purge himself and us of all philosophical phrases; whose burning endeavour it was to tear philosophy out of the toils of a logic as arrogant as it was impotent, and to open its eyes to the one and only productive authority of pure perception; a man who in open and indignant opposition to the schools cried out, “the whole sum of human science consists in seeing distinctly“; — and of this man the vast majority of cultured people know neither the personality nor the life nor the achievements, with the exception of just one single saying which has been thrashed out until it has become a mere phrase — cogito, ergo sum, — a mere jingle of syllables, unless we knew how it originated in Descartes, and whither it led him. Just think how it would be if some future history should have nothing more to report of Bismarck than that his was the saying, “We Germans fear God, and nothing else in the world,“ as if this very disputable

phrase represented the sum-total of the achievements of his richly active life! Where is the difference, if we only take count of one ambiguous and much misunderstood saying of the pioneer in mathematics, the physicist, the anatomist, the kosmologist, the philosopher, of the man who perhaps more than any other has so enriched our treasure of constructive imagination that to this day philosophy and science are refreshed by the stimulants of his genius? But as though it were not enough that a philosophy resting upon the broadest foundation of an all-embracing, manifest consideration of nature, should have been to such an extent turned topsy-turvy by degradation into mere logical and psychological nut-cracking — beyond all this we are even robbed of the man's personality. Descartes was an aristocrat by birth, — by the bent of his intellect an extreme individualist. He does not only hold himself aloof from his fellow-men, choosing an abode in foreign parts, and leaving a town as soon as he becomes known and gets entangled in social relations, — but even intellectually he surrounds himself with a high wall lest the doctrines of the contemporary philosophical guilds should find their way in, and even for the time being digs a deep moat to keep the wisdom of the ancients at a respectful distance. To treat with scorn the nullities of the professional philosophers — les bagatelles d'école — is for him the distinguishing mark of a “princely character,“ and of himself he confesses, “not the understanding of the arguments of others, but personal investigation on my own account is what constitutes for me the greatest happiness of study.“ It is in a quite different sense from Schopenhauer that Descartes is a great Eremite; for in him there is none of the bitterness or vanity of solitude, it is a proud and peaceful self-contentment. It was only after long years that the incessant pressure of so respected a friend as Pater Mersenne determined him to publish, and it would have

remained at that fragmentary beginning, had not the request of an exalted friend, the Countess Palatine Elizabeth, stood in the light of a royal command to so perfect a man of the world. Je ne recherche point les bonnes grâces de la populace, he writes with quiet disdain in a private letter: but with him populace has a wide meaning; for when Mersenne communicates to him the criticisms of the most learned men in Paris, he answers, “I have long known that there are asses in the world, but I set so little store by their judgment, that it would vex me to be obliged to spend upon it even a minute of my leisure and my peace.“ No more is needed to show that an investigator who so resolutely follows his own road, and avoids all contact with the officially recognised masters of scholastic thought, will not easily develop a system of philosophy fitted to be formulated into a strict scholastic shape. The picture of the world that Descartes unrolls before us, is no grafted scion such as we are used to see in philosophy, but a tree grown from the seed. Plato hangs upon Socrates, and also upon Pythagoras, Anaxagoras, Heraclitus, and others: Aristotle springs from Plato; Bruno from Plotinus, Lucretius, Cusa; Locke, Berkeley, Spinoza, Leibniz from our Descartes; Kant, too, springs from Descartes, and from Leibniz, Locke, Rousseau, Hume; and so it is with all of them; Descartes alone stands by himself. And although he is convinced of the truth of his perceptions, hoping that their victory will result in a new birth of the sciences, still he keeps such jealous watch over his independence, he is so deeply concerned to be left even after his death inviolate in his proud isolation, that he starts by declaring that his method is for himself alone, not for others; mon dessein n'est pas d'enseigner la méthode que chacun doit suivre pour bien conduire sa raison, mais seulement de faire voir en quelle sorte j'ai tâché de conduire la mienne; — and so over and over again he does not shirk the paradox

that his philosophy is void of all originality, which he only admits openly in order that the good people may not fall into the idea of making his name the centre of a school. The idea was to him a scarecrow that there should come men who would imagine that they could in a day compass that of which he had realised the insight after twenty years of study and education, and that upon it they should build up a Philosophy fit to make one's hair stand on end, should delude themselves into the notion that this Philosophy was the result of his “Principles,“ and assure the world that he, Descartes, was its founder. 4 It is touching to hear how he implores posterity, — “never believe that the things of which people are assuring you sum up my teaching, and originate in me: ascribe to me only that which you gather from my own mouth“ — and his real wish, that is to say his wish in opposition to the founding of a school, he tells us clearly enough in the same passage, is ouvrir quelques fenêtres, not to build up a system, but to “tear open the windows and let in the light“ for all those who have eyes to see. You can now distinguish broadly, what occupied this great intellect, and what must needs be his aim when he at last allowed himself to be talked over into appearing in public. Himself a free personality, who at the expense of great labour had torn from his eyes all the bandages which education, parentage, the wisdom of the schools, the doctrines of the Church, had bound round them — his aim is to educate free personalities, and with that object not to teach them, — in the sense that is to say of the schools, — but to lure them on, and to do for them as he had done for himself, namely, to open their eyes, and make them teach themselves by means of perception. By “philosophy“ he understands literally the opening of the eyes, oculos aperire. 5 And since this is the fundamental principle of Descartes' personality and teaching, so he

cares nothing for the fixed establishment of great, universal, irrefutable principles, but gives himself a free hand in the intimate description of his often quaint ideas which only fit in with his own personality. Only look at his portraits! look at his innocently amazed outlook over the world, and his slyly ironical smile at the wisdom of mankind! Why! the man is anti-scholastic to his finger-tips. Even the famous cogito, ergo sum (“I think, therefore I am“) is no logical conclusion, at any rate for him, but the verbal expression, clothed accordingly in the rags of logic, for a definite perception: and when the professional schoolmen want to split hairs with him on the subject he winds up the argument by saying, “I do not argue the question of my being by a syllogism, but I perceive it.“ 6
    This was the man whose fate it was to become — beyond the grave — the sacrifice of the populace in a way no other thinker did. Hardly was the breath out of his body when the European world of learning became divided into two camps, the Cartesians and the anti-Cartesians. The proud Eye, so wise, so lovable in spite of all its distrust, was closed; and now it was to be anatomically dissected and lectured upon. The teaching of Descartes, “perfected“ — as usual — by all manner of insignificant and contradictory minds, was transformed into a system of scholastic definitions and rigid dogmas. Descartes had said, “as for the search after definitions, we can leave that to Messieurs les Professeurs“; in very many cases definitions only serve to make dark what is clear; the professor with his subtle distinctions clouds the natural light of the understanding, and ends by making an obscure problem out of what every peasant knows. Descartes had been indefatigable in confining logic within the narrow bounds of its justified effectiveness, since, as he says, l'art syllogistique ne sert en rien à la découverte de la vérité; whereas the art of logic is a

chief instrument of the schoolmen for talking of things about which they themselves know nothing. 7 A few years after his death there arose a complete logical system, the “Logique de Port Royal,“ which pretended to be founded on his teaching. A very short time elapsed and this so-called Cartesianism was in the very centre of the conflict over the Eucharist: Calvinists and Jansenists, the deniers and the champions of the Real Presence of the Body and Blood of Christ in the bread and wine, both appealed to Descartes: in his grave he was marked as the founder of the philosophia eucharistica; his loftily plain writings, conspicuous for their frankness, were forced to serve, like the arcana disciplinae of the ancient mysteries, as evidence for and against the most abstract cobwebs of the brain, and between whiles the Physicists dragged out the over-hurried hypothesis of a genius on the Gyrations of the Kosmos, fighting for and against it, as if the Personality and nature-teaching of Descartes must stand or fall by it; while Freethinkers and Pietists both took possession of the so-called automatism of beasts, out of which they drew opposite conclusions. For more than a century the world was filled with the roaring of the Cartesians and the bellowing of the anti-Cartesians; of Descartes, the lonely investigator and thinker, there was no longer any talk. And when at last, in no small measure out of seed which he had sown, a new science and a new philosophy had gradually grown up and waxed strong, universal contempt washed away the barren Cartesianism and the equally barren anti-Cartesianism. The great personality of Descartes had long since faded away. Only the ill-starred cogito, ergo sum was bandied about like sea-wrack on the all-devouring ocean of the world's history.
    True, Descartes receives honourable mention in the philosophical histories. Schopenhauer's dictum, “the Father of the Modern Philosophy,“ has been universally

repeated; but it is always in the sense of what is called in stage language un père noble, an honoured but not much noticed person of distinction in the background. I can unhesitatingly recommend to you the first volume of Kuno Fischer's comprehensive work upon the modern philosophy: he gives at any rate a fairly exact biographical account of the man: but even here Descartes is so dealt with that he falls behind the other philosophers; and although there is much material given for a representation of his personality, this very representation, the portrait of such a wholly individual intellect, the plastic bringing into evidence of his special significance, is a failure. In most of the other handbooks you will only find one chapter about him, entitled “Descartes and his school,“ or simply “Cartesianism.“ He who said, “the great intellects talk nonsense as soon as it is their disciples who speak for them, for it is perhaps outside all experience that any pupil should have equalled his master,“ that very man hardly exists any longer save in the title for a School! Nay, more: when all is said and done, few of our professional philosophers are so equipped as to be capable of understanding the true Descartes; for Descartes, as you will already have observed, is far more of a contemplator of nature than a philosopher in the scholastic and still authoritative meaning of the word: indeed we might frankly call him an anti-philosopher. For him philosophy, — this is his own literal definition — is a tree, “the golden tree of life“; its metaphysical roots strike into the dark earth, and as Descartes humorously remarks, it is not upon roots that fruit usually grows; the mighty stem is the science of physics, under which he comprehends the universal laws of all motion, and this stem branches off into the many empirical ramifications of knowledge, at the points of which flowers at last bloom, and the blessing of fruit ripens. 8 You need only look at Descartes' chief systematic work, the

Principles of Philosophy. In Cousin's edition the first part, which contains all the psychological and metaphysical discussions, needs only 57 pages; the three remaining parts, — Physics, Kosmology, and Geognosis, upwards of 400 pages, — while Descartes apologises for not yet being able to publish his Zoology, Botany, and Anthropology. He indeed was the first to put the problem of perception in the foreground, a fact wittily put by Fontenelle in the remark that, avant M. Descartes, on raisonnait plus commodément; les siècles passés sont bien heureux de ne pas avoir eu cet homme là; 9 and so he was the first man to awaken true metaphysical reflection; yet he himself spends but little time over it. It was the distinct perception of his own inner being that served him as the first step towards distinctness in the perception of visible Nature. In the same way he made use of metaphysics as an active help to physics. Anybody who is not competent to follow him in the domain of natural science and mathematics will find it difficult to do him justice. He studies the functions of his brain as a part of the world which directly concerns him, and is therefore of fundamental importance, certainly not in the sense of a professed philosopher in the ordinary modern meaning of the word, whose calling and business it is to think over all matters in the abstract. He has no faith in the professional philosophy: he characterises it as une grande erreur, and says, il est plus facile d'apprendre toutes les sciences à la fois que d'en détacher une seule. A man of this stamp is far removed from our philosophical professors, not only further than their own dearly beloved Spinoza, who never once leaves the domain of the abstract, but further even than a Francis Bacon, who, it is true, constructs a Novum organum for the dissemination of the knowledge of nature, without having ever himself been busied with mathematical and natural-scientific work, and whose first principle it is to

abandon all philosophy in favour of a so-called empiricism; 10 further too than a Locke, or a Berkeley, or a Hume, or a Leibniz, for the chief element of the philosophy of all these men consists in ratiocinatio, that is to say, the pondering in Reason, and progress through pure conclusions of Reason. Here, on the contrary, we see a man whose chief work, unfortunately never finished and only known by fragments, was to carry the title of Le Monde, ou Traité de la Lumière! So it was the whole great world, the Kosmos as we should call it to-day, and in it first and foremost the medium by which it becomes known to us, namely Light, — that it was his aim “to observe, to investigate, to grasp,“ and only the man who keeps this aim before his eyes can hope to gain a correct appreciation of the personality of Descartes, and of the gifts which it bestowed. If we lay a one-sided stress upon the intellectual and theoretical reflections of this man, together with his metaphysical discussions on mind and matter, and his attempts to set forth irrefutably the existence of God and the immortality of the soul, — then we shall not only obtain a crooked picture of him, but we shall at the same time not even be in a position rightly to grasp his peculiar method of looking upon these purely speculative questions. The man who does not study Descartes' physics and does not penetrate their essence, sees his metaphysics in a false perspective; that accounts for the inadequateness of all the representations of Descartes in philosophical books.
    But the same ill luck pursues him elsewhere; for he hardly fares better at the hands of the mathematicians, mechanicians, physicists, and anatomists than he does at those of the philosophers. Inasmuch as we are living under the domination of the extremest specialisation, every single branch of science only enquires after concrete services rendered within its own especial kingdom, and it is upon these that it reports, whereas Descartes' peculiar

domain is the buffer-state. As between metaphysics and physics, so in all cases Descartes is happiest on the frontier. There where union and separation take place, where the coy facts are forced in the interests of combination with other series of facts to become supple and accommodating, — there where everything arises which we call “explaining“ and “understanding“ — there it is that Descartes at last feels himself at home. For that reason, and for that reason only, he devotes himself passionately to the study of mathematics, the great mediator between perception and thought, between things that are visible and thoughts that are invisible. But even mathematics, to the furtherance of which he rendered undying services, are to him “only the husk, not the essence“; to work at pure mathematics for mathematics' sake he looks upon as aimless waste of time, and he hurries so that it is difficult to keep up with him through the technicalities of form and place, in order that he may come at once to Physics and mechanics; but here again it is not the detail of the phenomena which interests him, but the Essence of Light, the Causes of Gravitation, the relationship between the mechanical laws of Matter and the Facts of Life, and so forth. It is true that if he dissects a brain he will give an exact anatomical description of it, 11 but what grips him is the hope of discovering a visible connection between the morphological figure and the function of memory. This last example shows you with special clearness how in this peculiar man theoretical thought and the desire for concrete perception went hand in hand. It followed that Descartes, in the individual sciences, achieved less than might have been expected from a man of his genius. His theorising was detrimental to the freedom of his observation, while at the same time the freedom of his theorising was narrowed by the painstaking detail-work of his observations. Hence it is that even his undeniable

services in the domain of the exact sciences, — his informing thoughts as well as the discovery of facts, — reached their goal for the most part in other hands, not in his own, and therefore are assigned to other names. For example, there is documentary proof, though no notice is taken of it, that he taught the gravity of air and made experiments upon it, when Pascal was a boy and Galilei still maintained the horror vacui as an unassailable dogma, — as also that the famous experiment of the Puy de Dôme was only undertaken under pressure from the unbelieving Pascal; 12 that Descartes should have discovered the circulation of the blood independently of Harvey, and the laws of falling bodies independently of Galilei, are matters of which the specialists take no heed; but for the knowledge of his personality they are of the deepest interest; that he was the first to expound the mathematical laws of the refraction of light, was proved by Humboldt as far back as 1847, but I find no mention of the fact in any later work; in medical books you will find cursory mention of Descartes amongst the leading names under the words “Eye“ and “Brain“ — as you see mere fragments, mere insignificance, or — Nothing. That the perceptible idea of the inertia of matter lies at the bottom of our whole mechanical science, is a matter of common knowledge, but few know that we are indebted to Descartes for it, and there is not one who prefers to base his judgment of the nature of such a mind upon an intellectual feat like this and others, rather than upon the cogito, ergo sum. 13 Just as little is it remembered that it was Descartes who paved the way for a revolution in Physics similar to that of Copernicus in astronomy, when he nourished the inspired conviction — which to his contemporaries was incomprehensible and seemed sheer madness; — Light is motion; and that moreover not the trajectory motion of a body violently flung, as Newton taught, but the motion of an imponderable

matter, the aether, by which our optic nerve is made to oscillate. Under the passive domination of the clumsy Newtonian ideas this thought was forgotten, and when, in order to justify the facts, it had to be taken up again, men preferred to attach themselves to Christian Huyghens — a son and grandson of two most intimate friends of Descartes, — who had grown up under the eyes of the great man, and who had further developed his inspired thoughts as to Aether and Light into the ultimate mathematically and fully developed theory of undulation. And so the constructive thoughts of Descartes are not only the basis of our atomistic physics, but also of our molecular physics. And in spite of all it is but little that we learn about him in the books on natural science, and here too his form remains clouded and distorted before our eyes.
    I hope that I shall incur no displeasure for having shown you so circumstantially how far and why Descartes has seldom been honoured in accordance with his merits, and why his personality is perhaps never rightly judged. I had to introduce this negative method of dealing with the question, because I had it at heart to upset what you might possibly know about him, or rather that is to say, think you know, in order to make way for more correct views. In the meantime I hope that you will yet have learnt something, and feel yourselves nearer to the true Descartes than you did a while ago. And I set great importance upon your knowing exactly what were the views of this remarkable man's brain: for in my lectures this brain constitutes the turning-point of our observations of Kant's personality, just as he himself, in more than one respect, constitutes the turning-point of human thought in general. I purposely use the word Brain, not System, not Metaphysics, not Discoveries: the system of Descartes, that is to say, his Kosmology as it is developed in the Principia and elsewhere, is distasteful, that is to

say, distasteful if we examine it with painstaking accuracy like a dogmatic structure, without paying attention to the author's warning to read his systematic works as fast as possible, comme une fable or ainsi qu'un roman; 14 his metaphysics, in spite of the fact that they are the point from which all later thought proceeds, are at once jejune and extravagant, without ideas and at the same time hyperphantastic; he never, with the single exception of the explanation of the rainbow, 15 followed up and worked out his discoveries to the end in a satisfactory manner: at one moment he allows himself to be choked by empirical detail, in the next he soars into hypotheses which in the plethora of artificially interlaced distinctions of detail are but ill calculated to further the strict beeline of investigation. We will not dispute with him about that, but far rather learn to recognise with Vauvenargues the fact that Descartes has often seen right and guessed right, even where he was in too great a hurry to press forward in the combination of hypothetic causes; ordinary intellects have nothing to fear from such mistakes, les esprits subalternes n'ont point d'erreur en leur privé nom, parce qu'ils sont incapables d'inventer, même en se trompant. 16 Descartes himself, in his wisdom, knew full well how that matter stood, and often gave expression to this appreciation in the words: “it is enough if I clear the road, you must do the rest“ 17 — and therefore I say once more of him his work is of less importance than the Man himself, or, as I said before, the Brain. We men are a right foolish folk: here is the one philosopher of all others, in whom first and foremost personality in the very special character of its intellect, and only in the second place systematic doctrine, forms the driving power and the lasting interest, and yet it is in this very man that we have allowed personality to escape us! Still, in the after life of history certain men enjoy an inexpressible immortality: this Descartes possesses

almost more than any other man; for the thoughts which that brain thought, and even more than the thoughts, the way and manner in which that brain grasped the chief problems of existence, — what therefore we must call the Manner of Seeing, the manner of directing the Eyes outwards and inwards, — all this has so penetrated, impregnated, and informed our philosophy and our natural science, that all of us, no matter to what school we belong, are compelled to weave the warp and woof of our thoughts in the loom of Descartes. Rightly did Huxley, one of the few philosophically trained investigators of Nature of the nineteenth century, remark: “In all thoughts which are characteristically modern, whether in the domain of philosophy or in that of Natural Science, we find, if not always the form, still the spirit of the great Frenchman“; an acknowledgment for which one of the best authorities upon Descartes, Count Foucher de Careil, coined the epigram, On se croit nouveau, on est Cartésien.
    It was first and foremost the whole attitude of the intellect, namely the unconditional enquiring, which made epoch. Descartes' intellectual attitude is sceptical, — but in the old meaning of the word. For the verb skeptomai originally meant to see, to contemplate, to investigate, later to ponder, to reflect upon. In the word sceptic in old days the stress was laid upon investigation and careful contemplation (Gellius called the sceptics quaesitores et consideratores). The instinctive wisdom of the language-forming powers united the perception by the senses with the necessity of exact careful investigation, but not with the meaning of doubt which disintegrates everything, which arose in the decadence of Greek thought, and impressed a new meaning upon the word skepsis. The barrenness of philosophical scepticism is by its narrowed sense confined to logical functions: it neither reaches outwards to

empirical Nature, nor does it reach inwards to confident self-consciousness; the outer Nature as well as the inner essence should have taught the sceptics that that which is a matter of fact does not necessarily hold its own before the logical Forum. The ancient scepticism arose out of shallow thinking, and led to frivolity, whereas the scepticism of Descartes, on the contrary, means an awakening of mankind out of the sleep of dogma to free, enquiring use of the eyes. Descartes did not doubt for doubting's sake, but, on the contrary, in order to help forward the discovery of a possible knowledge. Non que j'imitasse les sceptiques, ... au contraire tout mon dessein ne tendait qu'à m'assurer, et à rejeter la terre mouvante et le sable pour trouver le roc ou l'argile. The old sceptics, however superior they might think themselves, remained snared in superstition up to their necks; while Descartes was in all earnest endeavouring d'entreprendre d'ôter une bonne fois toutes les opinions que j'avais reçues jusques alors en ma créance. Now if Descartes' doubts had contented themselves with leading us back to that perception which he used to clothe in the words cogito, ergo sum, or dubito, ergo sum, or sum, cogito, sum cogitans, and the rest, that of itself would have been something: Kant calls him on that account “a benefactor of the human Reason“: but, in fact, this result of critical reflection simply means the solstice of the Cartesian method of thought: it constitutes the point where motion reverses its direction to cross over from the negative to the positive. The cogito, ergo sum is a perception on the boundary-line, just as with Kant, das ding an sich (“the thing in itself“) is a conception on the boundary line, and it is only fools who find a pleasure in running their heads against boundary stones of this sort. Descartes was no such fool. On this furthest boundary line, upon the “rock“ of his search, he raised a church to the God without whom he could not live; to prove the existence

of God is always a thorny undertaking, for He stands beyond the boundary of Descartes: yet this God not very religiously felt by Descartes, who had been educated in a Jesuit school, is less pressed upon us as something proven than made plausible as a necessary assumption, and has the one advantage that he is a God of truth. Descartes needs Him only in the interests of truth, in order that what is should be true, and for no other purpose. 18 And now the bold investigator addresses himself to constructive intellectual work! He turns his back upon that boundary stone, — in his church he only kneels now and again for short worship: on the other hand he enriches the world with thoughts which are so full of life and freshness by reason of their visibility, that they have defied all the storms of time, and he bestows upon it a wealth of perceptions, which shelter such an inexhaustible symbolical store of truth, that, while reminding us of the oldest traditions of our race, they point to times that are yet to come.
    Pray do not believe that I am using the language of hyperbole: my words are to be taken literally. As examples I will cite a thought introduced by him into philosophy, and an idea introduced into natural science. Descartes' analytical reference of the united subjective and objective experience of man to the two conceptions extension and thought is an idea so simply perceptible that it never can cease working productively: to this day all philosophers fasten on to it. They may use different wool and weave different patterns, still they are weaving at Descartes' loom — as I said before — all of them. On the other hand, a conception like that of the imponderable matter filling the whole universe, the aether, is so rich in symbolical, thoughtful, creative power, that it is only now that, in the light of new discoveries, we are at last beginning to recognise its great fruitfulness. 19

    In his work on the immortality of man Herder remarks: “It is incredible how few special forms in the realm of thought and human activities appear when we put history to the test. There are far fewer Regents who govern the world of the sciences ... than Monarchs who rule over countries.“ There you have, expressed in a short formula, the merit of Descartes. He is one of those incredibly few who produce special forms in the realm of thought — and here, since an exposition of the philosophy of Descartes would lead us too far, we must give up the enumeration of the special forms which he introduced: but what we must keep our keenest sight upon is the way in which this man, receptively and creatively, looked out upon the world, the way in which he came upon “the special forms in the realm of thought.“ Let us now apply ourselves to this task.
    I just now praised the great perceptibility in Descartes' thoughts; at the same time I cited as an example his theory of the aether, an imaginary thing, which when we consider it more nearly defies all perceptibility. An exact analysis will convince us that, as a matter of fact, there are two ways of showing this expression of intellectual satisfaction which in ordinary life we describe as perceptible clearness; we are partly dealing with what is seen, partly with what is thought. The creative power of the informing faculty of sight, directed upon the surrounding universe, was in Descartes of such rare might, that a matter-of-fact contemporary, the great mathematician Christian Huyghens, on receiving the news of his death, exclaimed:

Nature! prends le deuil, viens plaindre la première
Le grand Descartes, et montre ton désespoir;
Quand il perdit le jour, tu perdis la lumière,
Ce n'est qu'à ce flambeau que nous l'avons pu voir. 20

As verses these are not worth much: but coming from the pen of a Huyghens, they have more significance

inasmuch as this investigator belonged to the exactest of the exact. And as you hear, he maintains that the sunlit world was dark and unseen until Descartes lighted a torch over it, the torch of thought. We men see nature all blurred, until clear comprehensions have reduced the chaos of perceptions to order. Our eye sees but dimly, until the thinking brain has fixed it sharply, like an optician's glass, upon the objects in view. In another stanza of the same poem Huyghens makes use of a trope which by the direct opposite completes what he has just said; for he says of Descartes that he

Faisait voir aux esprits ce qui se cache aux yeux.

This implies that Descartes gave visibility to those things which our physical eyes indeed do not see, but which our understanding is compelled to think. And so as in the one case he bestowed thought upon things, so in the other he conferred upon thoughts the representations of the senses: in other words he gave them substance. In the one case it was the turning into thought that which had been indistinctly seen, in the other the turning into something visually perceived an idea which had been indistinctly thought.
    We will at once illustrate these two sayings of Huyghens by examples. Descartes comes to the help of perception when he e.g. explains all the movements of bodies in heaven and on earth by the setting up of certain fundamental conceptions such as inertia, mass, and others; even these simplest phenomena we never knew how to observe aright and see aright before the discovery of such ruling conceptions. To such as these belongs his theory that the Sum of Motion in the universe is once for all immutable, a favourite assertion of Descartes which, for the first time, brings into the chaotic oscillation backwards and forwards and circuitously in the Kosmos, a thought reducing it to order, — a thought which, merely

amplified by an additional sentence, is the foundation of the modern doctrine of the maintenance of energy, which is at the bottom of our whole science of physics. 21 That will suffice for one of Huyghens' affirmations: now for the other. Descartes comes to the assistance of thought through perceptibility, when for example he starts the theory of the above-named aether. This thought-picture leads us on to look upon Light as the movement of an endlessly refined, imponderable, imperceptible matter, which fills the whole world, a movement which the optic nerve betrays to us, without showing it, since, of course, aether is not a thing perceptible and therefore real, but a symbol for something which is presupposed in thought, and undefinable. 22 Another example would be Descartes' doctrine that it is not the Eye but the Brain that sees; all impressions of the senses are in the last instance invisible motions of imperceptible infinitesimal particles inside the Brain. 23 Here, in the case of the hypothetical aether, and in the hypothetical molecular motions of the substance of the brain, the visibility which has been acquired in what are matters of mere thought serves to a consequential observation and concatenation of phenomena; true exact science of nature and of mankind first became possible by means of this and similar symbols.
    Here you have obviously two different intellectual gifts with which our philosopher is accredited, gifts which do not necessarily belong to one another, and both of which, if we see them as purely and absolutely developed as they are here, at once fascinate us as something not easy of comprehension. Descartes knew how to give intelligible form to that which he saw, and at the same time possessed the power of transforming that which was only thought into something visible: that is the fact to which Huyghens calls our attention. And here in very deed he goes straight to the core of the matter,

and for that reason his remark must serve us as a clue to the further analysis of this unique intellect.
    In order swiftly and surely to plumb the depths, I should wish to take the judgment of Huyghens which I have already traced back to its simplest meaning and reduce it to a still more striking, concise, and purposely paradoxical formula. For it is not formula but phrases which are a hindrance to vivid insight, whereas a true formula serves as a skeleton round which the organs of the living figure by degrees arrange themselves. My formula runs thus: — Descartes' distinguishing gift was to make the visible invisible, and the invisible visible.
    If you look around you in the world of your own contemplative consciousness, you will soon observe that the degree of perceptibility of the ideas which fill it is exceedingly various, and the same holds good of the possibility of conceiving them. And you will soon be aware that there exists here a very complicated interchange of displacements, a mutual give and take. We possess thoughts with hardly a shadow of a perception, and we possess perceptions which are attended only by just such a minimum of thought as is necessary for us to be conscious of those perceptions. Our daily life is made up in that way. Without venturing further I will only call your attention to one thing, and that is that a thought that is accompanied by a blurred, hardly realisable perception, therefore an “invisible“ thought, can achieve but little, and that on the contrary pure perception soon grows into something monstrous, intractable, inflexible, unless thought takes the pains to seize upon it and convert it into something unseen. We are in no way embarrassed to find concrete examples, we need only think of our two first lectures: It was by a thought and in the interests of a thought that Goethe brought together the whole incalculable mass of animal and vegetable forms into his idea of metamorphosis: and so he breathed

the artist's soul into what was a mere brutal observation, furthering the investigation of Nature for all time; Helmholtz, the physicist, rightly taught us that the powers with which mathematical science deals cannot be “objects of the perception of the senses,“ but only “objects of the comprehending understanding“; yet Helmholtz, in his work on optics, has none the less to take refuge in plain diagrams, first the wet thread, then the ray, which like the sailor swarming up a rope, “produces itself along the particles of aether,“ and so he goes on from diagram to diagram because this thought of the “comprehending understanding“ could not be realised and appreciated without a perceptible representation. This is the way in which we human beings, half unconsciously, are for ever changing the visible into the invisible — in order to see it better, — and the invisible into the visible, — in order to think it better. Kant, from his metaphysical eminence, has summed up what I am here only concerned to show in a concrete and visible shape into the following pithy sentence: “thoughts without contents are empty; perceptions without comprehensions are blind. Hence it is just as necessary to make our comprehensions perceptible to the senses, as it is to make our perception intelligible, that is to say, to bring it into subjection to comprehensions.“ Kant is here speaking of the common, unconsciously proceeding, necessary functions of all human reason from the moment that it enters into activity in the new-born babe: allow this reason to ripen to such an extent that it desires to build up for itself a science and a philosophy, and you will find this reason standing as conscious intelligence exactly where at its first awakening it stood unconscious. Then it begins to take matters easily; it seems so natural not to follow Kant's warning, but to be busy with empty thoughts and blind perceptions, that three-fourths of all philosophy from the earliest times to the present day

has never busied itself with other things. The writings of St. Thomas Aquinas, for instance, are an inexhaustible arsenal of ideas, which are incapable of exciting the smallest thought — mere “blind perceptions“; and if you skip from the thirteenth to the nineteenth century, you will find that the most popular of all the more modern systems, that of Schopenhauer, takes as its foundation-stone a thought which is, according to Kant, utterly empty, the one which it calls Will and which, according to its definition, is the opposite of an idea and consequently contains nothing capable of being in any way perceptibly understood. All such thought-structures are extravagance, not knowledge: Kant once formulated this very simply. “By mere perception without comprehension the object is certainly given, but not thought; by comprehension without corresponding perception it is thought, but none is given: in neither case, therefore, does any recognition take place.“ How, on the other hand, perception and thought, the visible and invisible, go hand in hand towards the building up of systems of philosophy which explain nature, you may best see from the histories of our natural sciences, the development of which was conditioned by this mutual penetration. Let us here pause for reflection.
    Think of how, at the beginning of the seventeenth century, Copernicus and Kepler are unravelling in its main features the course of the planets round the sun; from the leaning tower of Pisa Galilei makes minute observations of the fall of bodies, — instead of merely reasoning logically upon it as all his predecessors had done, — and pursues his studies upon inclined planes; Descartes and others with keen intellect and patience follow up the mysterious course of the Light-ray, its curves, its refraction, its reflection; Gilbert publishes his observations on magnetism ... from all sides there comes in a stream of additional matter, — that is to say, material of observation, and in

every single sphere the empirical investigators are at work trying to the best of their ability, as Kant demands, to make their perceptions intelligible, that is to subject them to comprehension. Yet here we discover something over which we need not for the moment break our heads, but which we will simply accept as experience; namely that thought cannot directly fasten upon the perception of the senses, but must first with that intent create its own mental perception, — that which we call Symbol when we are wishing rather to bring to the front the perceptible side, Hypothesis when we are dealing with the mental side. Thought must create unity: this is its special function: pure perception only gives a kaleidoscope of special cases. Therefore perceptible thought cannot proceed without Symbol; it cannot, without further help, grasp, comprehend, and absorb the material of perception: without Symbol it remains empty. I can have no thoughts about the courses of the constellations, about the fall of bodies, or about the essence of Light, unless I also possess, besides the empirical material, and for its amplification, a symbolical representation of what takes place in that connection, — in other words something intermediate between perception and thought. And here my intellect makes a further claim. Not only must phenomena, within the individual series of phenomena be joined together by means of symbols, but all the separate series of phenomena with which I have become acquainted by means of empirical perception, must in addition be capable of being understood as one single comprehensive unity. For as Kant will teach you later on, that which we call Nature is “the unity of the multitude of phenomena,“ as it is set forth as a matter of subjective necessity by our thoughts. It is impossible for me to realise a number of natures. The grouping of the planets round the sun, the grouping of the steel filings round the pole of a magnet on my desk must be

taken as energies inside one symmetrical Whole. And here the great Descartes steps in as a creative power: he produces a new “special form in the sphere of thought,“ he changes into visibility that invisible something which our understanding insists upon though it cannot perceive it, — he fills thought with contents: this he is able to do in that he sets up the perceptible hypothesis of a medium filling space, of a matter absolutely refined, invisible, imponderable, fluidly moving — the aether, a symbol, the child of his phantasy. 24 At once all the phenomena mentioned enter the domain of demonstrability and so become accessible to the constructive labours of thought: the aether carries and urges the stars in their courses, the aether as a driving mass becomes the foundation of the phenomena of gravitation, one set of movements of the aether gives birth to what we call the warming of bodies, another set to light, others to electricity and magnetism, and so forth. I refer you to my former lecture and am confident that this one example will show you with extraordinary clearness what is meant by “making visible the invisible.“ At the same time you will learn how indispensable perception is to thought, even to the possibility of thought. Descartes had indeed by his hypothesis poured out such a wealth of visibility over the secrets of Nature, while he

    Faisait voir aux esprits ce qui se cache aux yeux‚

that the eyes of men were dazzled by it. In those days neither the collected empirical material was sufficient, nor was thought adequately trained and refined to be fit for so grandly simple a symbol for all the physical phenomena of movement of the Kosmos. Besides this Descartes in the closer elaboration of the matter had fallen into an error for which he was reproved by Goethe; “he attacks the insoluble problems with a certain hurry, and for the most part enters the subject from the side of

the most complicated phenomena.“ 25 There is much that is artificial and arbitrary in the use which he makes of the conception of the aether. The startling simplicity of the general conception is marred by all sorts of hazardous amplifications in detail. But it is just here, as is the case with every important man, that we learn how far greatness and limitation are set side by side, conditional and conditioned. And so it soon came about that Newton with his keen intellect, at once exact and barren of all imagination, once more seized upon the scholastic fictions of forces working at a distance, and took the old conception of Light as a special Matter: Newton's ideas are in the same relation to Descartes' ideas, as those of a child to those of a man; and yet they corresponded exactly to the requirements of empirical investigation in those days. At the present time, when new matter has been accumulated by the work of centuries, we are gradually going back to Descartes and his symbolical method of thought: in the case of the understanding of Light this took place about a hundred years ago with the introduction of the undulation theory mentioned in the last lecture; in the case of the electric magnetic phenomena about half a century ago; physical experiments to explain gravitation as conditioned by the movement of aether, exactly as Descartes postulated, are the order of the day, 26 and the great Hertz, so early torn from the world, was possessed in death by the dream of reducing “the putative working of the distant forces to conditions of motion in a medium filling space.“ 27 Lord Kelvin — and following him many modern physicists, go still further and contend that the various atoms which chemistry admits are only different gyratory motions of the one and only aether: that there must therefore be no such thing as Matter, but Aether only: in this most exact method of investigation the “Thing“ fades away, the Symbol alone remains. In a symbol so solidly

perspicuous is contained the principle of robust vital power.
    So much for the explanation of the transformation of the invisible into the visible. “Perceptions without conceptions are blind,“ says Kant. Even as I could not budge an inch in the realm of thought unless I possessed a “reasoned“ perception, so I must remain helplessly stuck in the quagmire of perception, unless I should have thoughts to drag me, as horses drag the cart, out of my difficulties. So be it. But how am I to obtain conceptions for my perceptions? Here again an intermediary something is necessary. Perception cannot directly become conception; the intermediary image is the Scheme. We men are incapable of taking into our inner consciousness anything seen or in any way perceived by the senses, unless we have previously in our thoughts reduced it to a Scheme. This is an aptitude which differs greatly in different individuals; yet if a man were altogether unable to generalise, that is to reduce the many perceptions to few schemes, it would certainly be impossible for him to think; for, as Kant hits the point by saying, his perceptions would be blind; he would see, but not recognise. In the last lecture we saw how the great painters schematise: a purely perceptible scheme is still sufficient for their object; only a minimum of conception enters into it. In a somewhat different fashion, but in obedience to precisely the same universal law of human reason, science goes to work. Whereas the painter wishes to see yet more clearly that which is already seen, and calls to his aid conceptions for that sole purpose, the investigator of nature wishes to conceive more clearly that which is seen, and to transform it into something known. When in this process of perceptible reasoning it is that which is perceptible which is preponderant, we speak of a Scheme; when, on the other hand, it is the element of thought which preponderates, we

speak of a Theory. Theory and scheme belong to one another as Hypothesis and Symbol. Now we know exactly with what we have to deal; in order to obtain a concrete example, we must return once more to the seventeenth century.
    This time we must work within narrower limits; we will only take into consideration the works upon the visible movements of perceptible Bodies: for we shall busy ourselves not with hypotheses but with seen facts. Let us then confine our thoughts to the way in which some men in those days busied themselves with the observation of the movements of the heavenly bodies, and how others, — the immortal Galilei in the forefront, instituted eager experiments on the movements of bodies on our earth, that is to say, on the fall, the impetus, the rolling off upon inclined planes, upon the trajectory of projectiles, upon the communication of motion from one body to another, and many other similar matters. The physical acceptations of the ancients proved themselves to be utterly false: new, accurately observed facts accumulated. How to order them? How to “make the perceptions intelligible“ ? How make what took place on earth consistent with what took place in Heaven? the fall of the apple from the tree with the circuit of the moon round the earth? Exactly as man had before, by submitting to thought the perceptible idea of the aether, come to the assistance of thought, so he had to act now in order to make his perceptions visible and capable of being surveyed: he had to remove the cataract from his eye, and that could only be by means of comprehensions, by referring all the single conditions of motion to a scheme which should be in accordance with rule, artificially thought out, and capable of being grasped logically; not given to him by the empirical observation of Nature, but set up autocratically between the eye and Nature by the King in


his Castle of whom I spoke in the first lecture. Here again it was Descartes who laid down the principles of our modern theory of motion, and at the same time of our whole science of mechanics.
    All movements of visible bodies may, as a matter of common knowledge, be referred to three fundamental laws, which we usually call after Newton, because he was the first to crystallise them in words, and has developed them in all their sequence. 28 But the third of these, which is not to be found in Descartes, is by universal consent recognised as a formal amplification of the first, 29 and even so very disputable. 30 We have to deal therefore with two, not three, fundamental laws, and these two laws were not thought out by Newton but by Descartes; Newton took them over almost literally from Descartes, though the latter had not worked them up to such perfect refinement. 31 All that the so-called “first Law“ of Newton contains — that Rest and Motion are not opposites, but only conditions of a body, — that every body left to itself remains perpetually in its own condition whether of Rest or of Motion, — that the body which is set in motion, unless there be some hindrance, will continue to move in a straight line with unaltered speed for all time, — all this stands word for word in Descartes. And I must call your attention to this, that no single one of the thoughts uttered in this law is the result of observation, or even capable of proof by experiment. 32 The second law of Newton too, which treats of the mensuration and direction of the Motion which is communicated by one body to another, is contained without a single omission in Descartes. It is he then, and no other, who perfected this creative work of thought. But here again, as in the case of the aether, Descartes overshot the mark, and like Dürer in his doctrine of proportion, introduced superfluous, and even in the end false, matter, so that the sure tact of a Newton


was sadly needed to purify the core from the slag. But the only thing that is of interest to us here, is the fact that Descartes, by the introduction of a few schematically theoretical conceptions, contrived to unravel and so make available for mental elaboration that which winds itself round our senses from childhood, — that in connection with which the whole united antiquity never achieved clear ideas, — that which the great calculators and experimenters of the fifteenth and sixteenth centuries failed to set free from the entanglements of the whole material of perception; I mean the Phenomena of visible motion. Here again as you see is a “new form in the realm of thought.“ And here as in the former case the value of such a creation for science and philosophy is immeasurable. For just as the symbolical hypothesis of aether paved the roads for thought upon which it was now possible to arrive at a rational appreciation of the phenomena of light, of electricity, etc., by means of a visible representation, so in this case the setting up of a schematic theory of Motion based upon metaphysical conceptions allows us to range the over-rich mass of facts seen into a few schemes of thought, where they can be guarded inclosed in formulae. For there is the turning-point: since the Visible is as fully as possible, — in some lucky cases altogether, — transferred into the realm of the Unseen, of that which is as yet only thought, it possesses a handiness, a pliability, a movability, which otherwise are foreign to its own perceptions, — purely as such — and are dull, inert, awkward: they are, just as Kant taught us, blind, and grope about in the dark; but as soon as the human understanding has arranged them into comprehensible Schemes then it does with them as seems good to it, dissects a Whole into Parts, unites Parts at will, in short behaves as it chooses: it is Lord in its Castle.
    We have now, as I believe, made an important advance in the understanding of the universal relations between

thinking and seeing, — which collaborate in so peculiar and twofold a combination for the building up of a system of philosophy, — as well as in respect of the recognition of Descartes' special aptitude for acting as intermediary between them. Our formula that Descartes' distinguishing gift was to make the visible invisible and the invisible visible, is no longer a formula, but an Insight. But I cannot let the matter rest there. Kant's thinking is a pinnacle of the human intellect; no man can reach him who shirks the trouble of climbing. It is therefore indispensably necessary that you should yourselves now enter upon the region which lies between perceptive seeing (or the sensitive faculty) and the understanding, which binds together comprehensions: otherwise you will only be possessed of partial, not complete, distinctness.
    Let me, however, in a parenthesis introduce a short remark upon the subject of Symbol, Scheme, Hypothesis, and Theory. It is not a question of mere terminological clearness, but of a visible representation, which will also be useful to you philosophically.
    The Symbol, in fullest acceptation of the word, is the perceptive demonstration of that which is thought: the Scheme, in its widest sense, is the rendering into thought of that which has been perceived: the Symbol furnishes thought with a thinkable perception; the Scheme furnishes perception with a visible thought. Within the symbol, however, it is possible to distinguish between a more purely perceptible and a more mental conception of the demonstration: the result of the first is the true Symbol, that of the second is the Hypothesis. In the same way the Scheme splits up into true Scheme and Theory. From this I draw the following explanatory diagram.
    The advantage of this diagram is that it accurately describes the mutual relationships of these different


conceptions — that is to say, if I may so express myself, their mutual position in the Space of Thought. You see at a glance that if, on the one hand, Symbol and Hypothesis are related, on the other the relationship is between Scheme and Theory, while Hypothesis and Theory, Symbol and Scheme in the same way lie close to one another. A very slight mental impulse suffices to turn a Symbol into a Hypothesis, and a Theory into a Scheme; it is a sort of swinging of the pendulum that our intellect

Hypothesis - Symbol - Theory - Scheme

is carrying on the livelong day without paying attention to it. But even the boundary between Symbol and Scheme, as between Hypothesis and Theory, is not insuperable: a small change in the standpoint suffices to give a colour of Scheme to Symbol, and a colour of Symbol to Scheme, and in the sciences Hypotheses have a way of quite quietly, according to seniority, slipping into Theories. On the other hand, as regards the two pairs which stand crosswise to one another, Symbol and Theory, Hypothesis and Scheme, it is a matter of impossi-


bility for them to be changed into one another. But what cannot occur directly may sometimes be effected indirectly, and so it often happens in the Natural Sciences that a Hypothesis by degrees acquires the value of a Symbol, becomes schematised, and at last stands in all the dignity of a Theory. In the course of time that which is really only thought, and as such in a slight degree hypothesised, has managed to assume the character of perceptibility to such a degree, that it is conceived as practical perception, and is then converted into thought, so that it takes the shape of a Scheme, and in the end of a full-grown Theory. With the aether, for example, it is always the case, until often some new discovery suddenly reminds us that this idea only possesses a symbolically hypothetical value; that is the way in which we men befool ourselves without any suspicion that we are doing so. The inverted process from Theory over Scheme, and Symbol over Hypothesis, which hardly occurs in science, is, on that account, common in everyday life. That which is seen is converted into thought by Science, but the layman comprehends scientific schematic thought as true perception: indeed, we have heard a Helmholtz talking of particles of aether “along which“ a Ray moves!
    This, however, is only a side issue. You must draw from it the one distinction between thinking and perceiving which is perpetually being forced to and fro in our brains. Perhaps in addition to that the small artificial Scheme may render us good service.
    And now let us go back to Descartes.
    From the two examples that we have taken, aether and the laws of motion, you will perhaps already have begun to suspect that thought and perception are not merely transiently, but really and permanently divided from one another. A complete fusion between them never takes place. There is never so much as an attempt at


such a fusion. The world, as we perceive it by our senses, does not satisfy thought, and never has satisfied it: for the world is incapable of thought, only our brain is that: and so thought creates for itself a Kosmos of its own, a special perception “converted into thought,“ and discovers at one time the Atoms, at another the Aether which the modern science of physics designates simply as “unperceivable matter and invisible motion.“ 33 And yet thought does perceive the unperceivable because it wills to do so; and thought sees the invisible because in no other way could it build a bridge by which to attain perception, or make a road by which to reach the dreams and works of Reason. We may grant that this aether, this atom, is something perceptible, indeed it is seen with all the special intensity of a dream-picture, and it is only thanks to this vision that thought can climb aloft. In spite of this the aether, like the Atom, is sicklied o'er with the pale cast of thought, and — again like a dream-picture, as we advance they retire and ever elude our grasp: they are indeed not perceptions of the senses, but perception that is thought: a symbol is not a thing: the man who seeks to investigate aether and atom by perception, is tilting against something that does not exist. The analogy holds good with our perception. The schemes upon which we base our experiences in the matter of the movement of bodies have for their aim the transferring of these perceptions into the domain of the comprehensible: here it is, and nowhere else, that thought like a mighty tree must carry and nourish the monstrous rootless liane of empiricism that is “conscious of no bounds.“ In this case our aim is to convert what we have seen into a quantity, that is to say into something so far only thought; colour becomes a quantity of oscillation, and a man born blind can talk as much wisdom about it as a Titian.
    But should you not yet be convinced that it is the


intimate laws of the human intellect, the fundamental facts of metaphysics, that are the informing power that is at work here, — should you imagine that without calling to your help metaphysical discussions you can arrive at clear notions about Time and Space, and about Motion in space and time, I will instead of laying before you arguments for which you are not yet prepared, address one request to you: I would ask you to refer to the scholion on the eighth definition in Newton's mathematical principles of natural science. It is the man of distinctly anti-metaphysical principles who is talking to you, and that indeed in a work of imperishable importance. In the beginning of the passage in question he declares with disconcerting guilelessness — “Time, Space, Place, and Motion, as matters of common knowledge, I do not explain.“ 34 If the question were merely one of dealing with the simple perception of these things, then an explanation of time and space would be as little necessary for the greatest intellect as for the most narrow-minded cow-herd. It seems to me that this postulate was altogether insensate: that which is self-evident cannot gain in value by explanation: on the contrary, it is out of the life that the word comes. Descartes' warning is: il faut mettre au nombre des principales erreurs qui peuvent être commises dans les sciences l'opinion de ceux qui veulent définir ce qu'on ne peut que concevoir. But there is no question of time and space, as they are known to all, — Newton himself will presently teach you that this would not lead us one step further in Science, — but with that intent it is our business to transfer that which is seen into that which is thought, and vice versa, and so we arrive at inextricable confusion until a critique of human Reason has illuminated us. Read a little further in Newton's scholion. You will find there things about “absolute space“ (spatium absolutum) which are not less edifying than the properties of the absolutum quid of


the schoolmen. This “absolute space is without relation to any outward object“ (sine relatione ad externum quodvis); but there would be little to be made of a thing which stands in no relation to anything; therefore, in addition to this absolute space, relative spaces are assumed (in quantity), and these relative spaces are movable in absolute space of which they constitute the parts! I do not think that the human intellect has ever attempted to imagine anything so monstrous as this quantity of spaces, which move about in confusion. It is true that these movements are only a passing idea such as might appeal to the intellect of our aforesaid cow-herd, for immediately afterwards Newton gives utterance to this deep reflection: “if the parts of space are turned out of their place they are, so to speak, removed from themselves“; but even that will not do, and so we receive the amplifying assertion about these relative spaces — “the spaces are their own places“ (spatia sunt sui ipsorum loca). And when you are stuck fast in this utterly senseless empirical jumble, you are taught that this space (of which you were told on the previous page that it is such a matter of common knowledge that it needs no explanation), — is beyond your ken, and that “you are not able to separate its parts by means of your senses“; and therefore, and here comes the gem of the whole, since you are dealing with something not perceptible to the senses, something impossible of distinction, therefore, quoniam, you must assume perceptible mensurations (mensuras sensibiles). So with perception you are to reach the invisible, and to measure something the parts of which you are not able to distinguish! The cause of this confusion which could only be cleared up by the highest critical circumspection and the finest analysis, lies in this, that mankind is not possessed of a clear appreciation of its own intellect: we interchange the Scheme which is only capable of being thought with the


true perception of the senses. There in the case of aether (just think of the theory of undulation and its powerlessness in respect of colour) that which pertains to thought intruded into perception with disastrous results; or perhaps it would be more correct to say, — since the aether is, as you will remember, a thought converted into perception — the human intellect proved incapable of producing out of its own powers a symbol which should equal Nature: here, in the fundamental conceptions of dynamics as developed by Newton, the same intellect proves incapable of freely discovering thoughts in all portions, that is to say, of converting into thoughts its perceptions by the senses. In order to bring our perceptions under a few fundamental conceptions we invented the law of inertia: but the thoughts of absolute space, endless time, the uniformity of a body, which according to definition should be alone, and so removed from all comparison, — all this is not known to us by perception. From empirical perception we borrow that minimum of perceptions of the senses without which our theoretical thoughts would be empty, that minimum without which the scheme could not be fashioned: but true perception never exactly tallies with this theoretical schematisation. And so we come to a standstill as soon as we in all too great simplicity attempt to satisfy the human intellect without a metaphysical critique, although in practice all goes well enough, and a Newton erects a building worthy of everlasting admiration when once we grant him a certain series of premisses as unthinkable as they are imperceptible. 35
    You see from these considerations how important it is accurately to investigate the critical domain between perception and thought, and also how many difficulties throw us into confusion by piling themselves up against our understanding. Happily there is one function of our intellect, one, only one, mathematics, which allows us to


clear up this matter to perfect distinctness. One general explanation, and then I propose to start upon a discussion of Descartes' relationship to mathematics: in this way we shall by degrees reach daylight, and we shall have no difficulty in seeing how all this may be applied to the study of Kant.
    I propose here to insert a diagram which will serve as a pause, and give my words a really comprehensible meaning. If we express the range of the human intellect by a quadrangle, — a circle would be better, and a globe of course still better — we can in general terms affirm that one half belongs to the senses, that is to say, to perception, to that which is perceived, the other to the understanding, thought, the formation of comprehensions; those are the “two quite heterogeneous portions“ of which Kant spoke a while ago. A more minute consideration, however, such as that which the history of our natural sciences has forced upon us, will soon convince us that pure perception and pure thought are not directly in contact, but that there is an intermediate domain which serves to help the crossing over of the one to the other. There are certainly no fixed boundaries; we are not dealing with a machine the wheels of which simply lay hold upon one another, but with a living structure in which every single organ in combination with all the other organs forms a unity at once real and ideal. Whereas in a watch the parts come first, and it is only in the end that the watch as a whole comes into existence by the combination of the parts, — in a living body the Being itself is the first, and that which we are pleased to distinguish as parts or organs, is formed by degrees and has never more than a conditional importance in regard to the Being, since the division of the functions does not take place, as in the watch, according to an immutable stencilled pattern, but one organ can even take up the duties of another. Still a Scheme will serve our present


purpose, and a Scheme is only clear when it is schematic, that is to say, absolutely quadrangular and rectilinear. So we will draw our quadrangle and assign one half to the Senses (the Sinnlichkeit of Kant) and with them to Perception, — the other half to the Understanding (as Kant calls it) with its conceptive Thought. But, towards the middle, pure conceptive thought crosses over to perceptible thought, and in the same way, towards the middle, pure perception of the senses crosses over into thoughtfulness. This boundary land I will denote by hatchings.
    You have already seen how the understanding strove to annex into its own domain the visually seen perceptions in regard to Motion, and how with this intent it drew them over, not without violence, by the help of Schemes to its own special boundary land of perceptible thought; and before that you had seen how the senses had succeeded in awakening to a glorious life scientific thoughts which had up to then remained unfruitful, and when well considered generally unthinkable, by the means of the discovery of a sensible and perfectly perceptible Symbol, the aether.
    The slightest reflection will surely suffice to show you what a travelling backwards and forwards goes on within the human intellect. If, for instance, in our laws of Motion stress should be laid only upon the theoretical and arithmetical, which was the case with Newton the juggler in figures, then these laws end by losing all perceptibility, they leave our middle line for the boundary of the hatched part, they become altogether thoughts: but with Descartes in these very same laws of motion it was the conception of the senses which prevailed, and more recently with Hertz in the same way the geometrically perceptible: by those means the thought shifts towards the middle line, that is to say, towards the Symbol, and Theory becomes relatively more schematic

than theoretic. The same thing takes place with our thoughtful perceptions. They may belong so entirely to the senses, that is to say, they may stand so entirely on the edge of this hatched region, so far therefore from the half assigned to the understanding, that comprehension is not in a position to grasp them. Goethe's metamorphosis is an example of that. Descartes' aether, on the contrary, belongs in an important degree more to the realm of thought, in spite of being still quite concrete. The symbol of the aether can be drawn into itself from


the conceptive portion of our being with such violence that, as you have seen, in the end every concrete conception fades away, and aether subtilises itself into a motion as yet only imagined, dispensing with every perceptible, material foundation (see page 130). In this case then not only is the middle line crossed, and the Symbol turned into Scheme, but this Scheme itself is as yet little more than Thought. I commend to your understanding the Physics of Lord Armstrong and the “Primitive animal“ (Urtier) of Goethe as the two most remote and most opposite ends of our “buffer state.“ In the one case a conception (the movement of the No-Thing in


empty space) which wipes out all conceptibility down to the uttermost remnant, so that it is impossible to think of it any more; in the other case a thought (the original creator of all individuals, itself without any individuality) has so completely materialised itself that there remains not even that minimum of conceptibility without which no form can be clearly recognised.
    From this schematisation and this warning against the misuse of the Scheme, let us now turn to Mathematics.
    The characteristic of the science of Mathematics is that it takes possession of the “buffer state,“ the hatched part of my diagram, and exactly fills it. Here is a case where no scheme can be too uncompromising. Both the two forms of Mathematics (on the one side the perceptible form of the science, — Geometry or the doctrine of Forms, — on the other, the comprehensible form, — Arithmetic or the doctrine of numbers) reach inwardly with exact precision towards the middle line, that is to say, towards the boundary line between the two domains of the understanding and Perception by the Senses. But inasmuch as mathematical science reaches outwards only exactly so far as the boundaries of this intermediate region, and does not cross it, so there arises between its two parts a reciprocal independence, an exact Parallelism which is nowhere else to be found between perception and thought. That which is thought mathematically contains nothing which might not also be perceived, and that which is perceived mathematically embraces no forms which might not also be grasped by thought. Here that unconscious shifting to and fro, of which we spoke just now, does not take place: every mathematical conception, every mathematical representation of ideas, has its appointed and immovable place. The two mathematical fields of intellectual operation are not identical, — the diagram shows how entirely autonomous they are, — and yet they are a matched pair, the one
being the counterpart of the other. On the other hand, the sharp definition of the middle line conditions such an uncompromising antithesis of the two mathematical functions as nowhere else occurs between perception and the representation of conceptions. Here there is no such gradual crossing over as we found between other Schemes and Symbols. Geometry is pure Symbolism; the science of numbers is Schematism devoid of all perception, it is the prototype of what Kant called “thoughts without contents.“ 36 The conversion of the one into the other can only be effected suddenly, and is, as I shall show presently in detail, the result of a purely internal and arbitrary deliberation. Even where the two parts of the middle line are very close to one another — I shall give an example immediately, — there are no means of changing form into numbers gradually; on the contrary, the concordance between thought and perception must be seen directly. If mathematics were not a purely human thought and perception, if we had to derive them from experience, as for instance we do our perceptions of the movements of bodies, then indeed we should be in a bad case; for Nature, as outer experience, gives us no handle whereby we may bring form and numbers into connection. By good luck, however, our empirical shallow pates are at fault, and in geometry we possess our archetypical Symbolism, and in algebra our archetypical Schematism, and therefore, — pray note this therefore — since mathematics are a form of thought and perception dwelling in us, and since they exactly fill that frontier domain of our intellect, therefore it is here, and here only, that we are in a position to convert Symbol into Scheme and Scheme into Symbol in their absolute entirety. 37
    I shall make this conversion clear to you by an example. When a boy receives his first instruction in calculation by letters (Algebra) the poor wretch is in the first place


compelled to learn by heart a series of equations, of which he can make neither head nor tail, not because there is no food for thought in them, but because on the contrary they are matters exclusively of thought, since they deal with pure and therefore empty (“void of contents“) conceptions, absolutely without any perception. The first of these equations runs thus:

(a+b)2=a2+2ab +b2.

That is to say, a and b added together and then multiplied by themselves equal a multiplied by itself, added to twice the product of a multiplied by b, added to b multiplied by itself. Is not that a terror to listen to? But if we take heart, and jump out of schematism into the symbolism of our intellect, we immediately see the truth of the proposition, without wasting a single thought on the matter. Let me show the thing in a diagram, only begging that you will not exercise thought upon it, but just simply open your eyes.

    We take a line a and add to it in a straight line the line b.

lines a+b

And now upon this line we build an equilateral and right-angled quadrangle.

Quadrange (a+b)2


What you see here is (a+b)2. That this square is equal to the square on a increased by the square on b, increased by twice the right angle which consists of the length a and the breadth b, you will see directly from the following construction which I build into our quadrangle.

Quadrangle aa+2ab+bb

In this way Algebra has been converted into Geometry, the scheme of numbers into a Form-Symbol. And you need only invert this simple example, that is to say, think of the square and the construction introduced into it as the starting-point, in order to understand that it must of necessity be possible to convert every geometrical construction, every play of constructive phantasy, into a purely comprehensible, entirely perceptible, in other words algebraical, expression of figures.

    In the case which we have just been talking about mathematical perception and mathematical thought were in close proximity to the dividing middle line: there was therefore no difficulty in grasping the comprehension as material, the perception as abstract: generally, however, they are far removed from that line, and it was Descartes who first taught us how we must set about in order to succeed in revealing the Scheme as Symbol, the Symbol as Scheme, a discovery by means of which he became the founder of the so-called higher mathematics. And


here it is that we must now follow him if we wish once for all to ascertain the relationship between thought and perception, which is indispensable for any understanding of Kant. 38
    The whole course of our considerations up to the present will easily convince you what a special attraction mathematics must have exercised on a man like Descartes, on a man whose distinguishing gift it was to discover Symbols and Schemes, in other words, to make the visible invisible, and the invisible visible. Yet, if we wish to understand Descartes' personal method of perception, it is important that we should be accurately instructed as to his position in regard to mathematics, and that is just where our school-books lead us astray. In order, therefore, to be able to speak of Descartes' mathematical achievements, my first business must be to dispel the common, and almost without exception ruling, misunderstanding about Descartes' conception of mathematics, and about the place which they occupy in his whole thought. This is the only way in which we can extricate ourselves out of the jungle of meaningless phrases into the free Pamir of clear insight.
    In our scientific knowledge of Nature mathematics play the part of the mechanism which electric engineer call a commutator or current reverser. As soon as we succeed in arriving at phenomena, — even should it be in so arbitrary and contradictory a way as was the case with Newton in his doctrine of gravitation, — the game is won; we go on turning the current, i.e. the perceived into the conceptible, and the conceptible into perception, exactly as in the (a+b)2. The one helps the other forward, and so we are ever rising higher and higher, and that without ever falling into error, for the simple reason that we are only working within our own intellect, and so make images and thoughts take their proper places in regard to one another. That was what Descartes, after


Plato, was the first to see; he it is who endowed us with the thought of analytical geometry, with which we shall immediately busy ourselves more closely: yet he did not remain caught in the meshes of purely mathematical ideas, but his masterful intellect stretched out far above the science of mathematics. If it is absurd to follow Schopenhauer in representing Descartes as undervaluing mathematics, so it is hardly less full of misunderstanding and misleading to exaggerate the significance of mathematics in his thought and for his philosophy. The image of the aether and the thought of the law of Inertia are sufficient proof that his development of this mathematical juggling only served him as a preliminary exercise, and so he holds that it is to be understood by others, as his Règles pour la direction de l'esprit clearly set out. The doctrine of numbers and forms does not contain truths; rather is it in one respect quite empty, the emptiest thing that one can imagine: for in it neither is perception nourished by comprehensions from outside, nor do its conceptive gymnastics allow of enrichment by special thoughts; mathematics are simply a system of formal principles of perception and the concatenation of conceptions. 39 Descartes is continually laughing at the professional calculateurs and géomètres, and says that their business is de s'occuper de bagatelles. Open any work on philosophical history, you will find everywhere that Descartes declared that mathematics are the “origin and source of all truths.“ Nothing has done so much to turn good brains amongst us from Descartes as this reputed saying. For what is one to think of so silly an assertion — at best a sort of mythical Pythagorean symbol of Nature, something which was in truth further from this man than from all others? And yet no man doubts the authenticity of the position, otherwise it would not be quoted with the usual inverted commas in one learned German work after another, and the whole


thing is just a matter of mistranslation. The passage in question occurs in the XIth volume, p. 219, of Cousin's edition. Descartes has just set out the first principles of his Method, which he reduces to two principles only: first and foremost, and as indispensable, the clear perception of the object (l'intuition); next, as second, the consistent and unbroken deduction of the propositions (la déduction). Here the perception of the senses and understanding appear in their first and most elementary relationship. 40 Still their reciprocally conditioning interplay cannot but lead us much further. So Descartes points to Mathematics as an example, and as the only safe schooling for the application of this quite universally adopted Method, — mathematics which he holds to be incomparable and indispensable as an exercise of the alliance between the most manifest perception and the strictest logic — and then comes the sentence which has given rise to the misunderstanding to which I have alluded: je suis convaincu qu'elle est supérieure à tout autre moyen humain de connaître, parce qu'elle est l'origine et la source de toutes les vérités. The pronoun elle refers to the Method, the great universal Method, the Method of the reversion of the current, — not to Mathematics! The Method of the reciprocal interpenetration between perception and thought is the source of all true knowledge — this Method! In no way mathematics by themselves and of themselves, of which Descartes on the following page assures us that there is nothing more empty. Rien de plus vide. Even as a matter of grammar the thing is out of court. Elle could not refer to mathematics which are almost always spoken of in the plural, and in this very passage are without exception given as les mathématiques and elles. How little Descartes was inclined to look upon mathematics as the “source of all truths“ is sufficiently manifest from the fact that he reckons les nombres et les figures among those ideas qui ne peuvent pas être estimées


un pur néant, quoique peut-être elles n'aient aucune existence hors de ma pensée, and that in another place he says of them, elles ne peuvent pas étre considérées comme des substances, mais seulement comme des termes sous lesquels la substance est contenue. 41 But that is the way in which we treat our great men; instead of adopting an infinitely subtle, vivid, pregnant knowledge, we accredit the genius with any manner of patent absurdity at which every commonplace man runs a tilt with solemn self-satisfaction. 42 Just as little truth is there in the affirmation that Descartes taught that philosophy was destined to become a “universal system of mathematics,“ an affirmation which we in the same way meet everywhere. He, on the contrary, called attention to the fact, as Plato had already done, that in a series of Sciences, — he mentions optics, astronomy, mechanics, acoustics, everything must at last come to a question of mensuration and figures, and this remark leads him to the affirmation that all these sciences in combination with geometry and arithmetic form une science mathématique en général, or une science mathématique universelle. But this description holds good only in contradistinction to the other sciences, and so far from saying that the universal science of mathematics is all-embracing, Descartes asserts expressly, “I have busied myself so much with it that I think that I may henceforth devote myself to higher sciences, without having to fear being over-hasty.“ Descartes would have agreed with Kant, “Philosophy makes use of mathematics only as an instrument.“ For the rest he himself clenches the question into a convenient and correct formula when he says, “In my method the science of mathematics is the husk and not the core.“
    It was indispensable to replace a conception that is meaningless and false into the bargain by a true appreciation of Descartes' conception. So much for that. There is only one more thing which ought to be brought out in


this connection, and that is the strong insistence which he lays upon perception as the source and fountain of all truths, for that is the true conception of Descartes' teaching. It would be quite imaginable that a philosopher might have set up this “mathematical method,“ and yet have taken the abstract side as his starting-point. Descartes did not do that. On the contrary, just as in mathematics he takes his stand upon geometry, so he consistently insists that perception (l'intuition as he calls it) is the one and only indispensable foundation of all knowledge. What he prizes above all in mathematics is that “they exercise the phantasy in the right conception of forms and motions, and so accustom us to represent phenomena to ourselves correctly.“ 43 It is not the least of the achievements of the pioneer that he introduced the principle of perception into philosophy in the stead of the method of tyrannical and sterile logic which up to his time was alone dominant. It you read the writings of Descartes, you will at once be struck by the frequency with which such expressions as voir clairement, concevoir fort clairement et fort distinctement, imaginer clairement, la conception évidente d'un esprit sain, etc., occur: the foundation-stone upon which the whole of this philosophy rests, is simply clear perception, and so it is that the first power of man which must be methodically developed, is la perspicacité en envisageant distinctement chaque chose, which means, “the piercing glance which shows itself herein that we should see everything clearly.“ Yes! but “perceptions without conceptions are blind“; it is conceptions that first make them intelligible. Thus it is that in Descartes the algebra of déduction follows upon the geometry of intuition, and that the sagacité à observer rigoureusement l'enchaînement des choses follows upon perspicacité. It is characteristic of geometry that by itself it does not carry us very far. It is true that a carefully planned geometrical


construction contains all the connections which may later be drawn from it, still the eye is clumsy and confused, and the more we succeed in converting that which is seen into that which is thought — in this case connecting forms into symbols of figures, — the richer will be the results. This experience drawn from the practice of mathematics was applied by Descartes to all other spheres of thought, exacting that we should first see clearly, and then dissect with flawless logical keenness. Without a brilliantly powerful perception of the material empirical world, no true knowledge, — nothing but cobwebs of the brain! Without an “algebraically“ dexterous analysis of that which has been seen clearly and lightly, no true science, no philosophy! It is always the same principle: the interplay between understanding and the senses, between conception and perception, between Scheme and Symbol. And of all importance is the doctrine that perception always takes the lead, while logical dissection exclusively comes into play in the second place. Pure intuitions of reason and pure logical arguments have no value for Descartes; they are objectless. In contradistinction to the schoolmen not only of his own time, but also of the nineteenth century, Descartes declares roundly, “logical forms and syllogisms are of absolutely no use for the discovery of truth,“ — “Dialectics are rather a hindrance than a help.“ They can only play a part secondarily, — only in the analytical investigation of that which has been discovered by direct and experimental perception. 44
    That is what Descartes understands by his “mathematical method.“ Fundamentally his attitude towards mathematics is precisely the same as that of Plato, who had already suspected and preached the intermediary position of mathematics, and on that account ascribes to the exercise of mathematical methods an incomparable significance for the development of the power of knowledge, but nevertheless laughs at the professional mathe-


maticians when he says, “they make themselves ridiculous with their fussing, as if with their complicated calculations and barbarous terminology they were achieving some mighty thing, whereas the whole significance of mathematics lies in the fact that they serve as a medium of philosophical thought and as a road leading to knowledge.“ 45 Descartes was conscious of this historical connection. According to him the thinkers of antiquity would have found it impossible to recommend mathematics as a philosophical instrument, if by them they had only understood calculation; he was more inclined to believe qu'ils reconnaissaient une certaine science mathématique différente de celle de notre âge, and it was this other science of mathematics which he once more took up.
    I think we have now quite intelligibly shown how there is no inconsistency in Descartes when he at one and the same time declares that there is nothing “more empty“ than mathematics, and in spite of that holds that the philosopher is bound to spend much time over their study. And since you now know that when he busied himself with mathematics it was not on account of any formal whim, not on account of any Pythagorean cobwebs of the brain, but on the one hand in the interest of the precedence of perception over thought in every investigation of nature and mankind, and on the other hand, in the interest of the conscious handling of that method by which perception and thought reciprocally help one another. Since you also are in possession of the comforting assurance that it is no barren philosophy, but scientific and living perception of the world that is at work here, so I hope that you will have the courage to climb one last rocky peak with me where the sharp pure air of the glaciers will be wafted around us. If Descartes has by others been misunderstood, and has remained unrecognised, there is one act of justice rendered to him


by every cyclopaedia. He is the first inventor of analytical geometry, with which he revolutionised our whole doctrine of geometry and numbers, and gave the impetus to the discovery of the so-called higher mathematics, upon which again our modern sciences of Physics, Mechanics, and Astronomy are based. It is now necessary that you should see Descartes, — who made his discovery not as a professed mathematician, but as an amateur after a few months of self-taught studìes, — at work in this direction; the detestation in which we hold all verbosity, should steel you not to rest before you have grasped in its solid significance the question which lies at the bottom of our observations of to-day. I admit that we shall here have to tread the special path of mathematics, and that is distasteful to the man who is no mathematician; yet I hope we shall succeed in applying ourselves to the subject in such a way that even those who are absolutely ignorant of mathematics will be able to see exactly what we are driving at. And with this we shall in the first place gain the advantage of obtaining a quite exact idea of Descartes' individual method of Seeing: in the second place we shall gain the knowledge, not merely theoretical but absolutely concrete resulting from practical perception, that every transition from thought to perception and vice versa, — even where (as in mathematics only) it takes place with absolute precision — has in itself something artificial and arbitrary, from which it results that perception which is thought always remains more or less a Scheme, and thought which is perceived always remains more or less a Symbol; last not least, we shall be driven on a purely perceptible and therefore entirely safe road, to the very central point of that Kantian perception to which it is otherwise so difficult to gain access, and which is so dark and difficult to illuminate. That point is the conception of the Transcendental. In this way Kant's method of Seeing the world will no longer


be so foreign to us, and we shall have gained in addition an advantageous standpoint for a later study of his philosophy. For these reasons I urge you to follow me for a while in the pure domain of mathematics. 46

* * * * * *

    In order that you may make your way with some pleasure into the subject of analytical geometry, which touches the innermost essence of mathematics, I must at the outset tell you what was the aim of this discovery of Descartes. It is necessary that you should know this, otherwise you would see nothing but a sort of ipse dixit in the proceeding, and that might mean astonishment, but it could not mean understanding.
    With the help of perceptible mathematics, namely geometry, simple problems may be solved, but not complex problems; human imagination soon gives up the task: a very complicated system of lines and points and bodies, which assert themselves in various ways, is something which we cannot put with perfect clearness before our eyes; we are not in a position to compare differently formed bodies directly with one another; we are not able to see, to recognise with our eyes, the outcome of it all. But in a quite different measure we are able to deal with the mathematics of conception, that is to say with numbers or the symbols of numbers; for in this case the master law-giver is not perception, but Logic, and that implies the opportune succession of a linked chain of insight into facts, instead of a Present only to be deciphered by a direct and simultaneous combination. If we deal with numbers logically we need not trouble ourselves about the perceptible meaning of each single operation of calculation; the correctness of the result is the important matter. That is why men very early came to reduce lines and rectangular figures to numbers, as, for instance, expounding the relationship


of the square on one side of a rectangular triangle to the squares on the two other sides, not perceptibly by drawing figures, but arithmetically and algebraically. But how arrive at a universally valid expression in numbers for complex figures, such, for instance, as curves? That was the question upon which many men busied themselves, and no one found the solution.
    Here it was that Descartes came to the front as a creative genius. He perceived that to reduce a curved line to a symbolical expression in numbers, the first necessity must be to bring the particular curve (circle, ellipse, volute, etc.), into relation with straight lines. The next task to be solved was the discovery of these straight lines. Once solve that difficulty and discover the relations between the curved line and the straight line, then what was elusive would be brought to a standstill, the curve would be bent straight, and the object would be attained; for as you will see presently, straight lines can always be considered as numbers (real or symbolical), and a fixed relationship between straight lines is therefore at the same time an arithmetical relationship. Thus the curve which is seen, becomes an unseen, logical, arithmetical expression, and can take its place in every arithmetical series by means of various calculations. In this Descartes succeeded. With simple unconsciousness of the magnitude of his achievement the first sentence of his Géométrie tells us: Tous les problèmes de Géométrie se peuvent facilement réduire à tels termes qu'ils n'est besoin par après que de connaître la longueur de quelques lignes droites pour les construire. As coins and watches disappear in the hands of a conjurer, so in the hands of Descartes the visible became invisible, the geometrical, arithmetical. But you will at once remark that with this achievement the inversion was of necessity given at the same time. For it was only necessary to strike into the opposite direction, and at once we were in possession of a form


for every futile arithmetical formula! Here you have the whole Descartes. Goethe declares that man can wrest from Nature nothing more valuable than —

Wenn sie ihm offenbare,
Wie sie das Feste lässt zu Geist verrinnen,
Wie sie das Geisterzeugte fest bewahre —

“When she reveals to him how she lets the substantial lapse into the Spirit, how she preserves as substance that which is the child of the spirit.“

    Since Descartes has pervaded the life of man as teacher, there has been no geometrical form which we have not been able to let “lapse into Spirit,“ that is to say, turn into an arithmetical expression, — into an equation merely thought, — no arithmetical picture “child of the Spirit“ which we have not been able to convert to something seen, something substantial. That is the essence of analytical Geometry.
    Now we may proceed to a closer exposition.
    I hope that you are not scared either by Greek words or by the jargon of mathematics. Both are accessible if you only approach them in the right spirit. Greek was once spoken in a sunny land, — spoken by men who possessed the immeasurable luck not to be forced, as we are, to gag their spiritual life into dead idioms, — men among whom the sage drew his words from the same living well as the shepherd, and so was understood by all: and as regards mathematics this discipline by the application of the right method, was capable of being brought home even to the least gifted, — at any rate in a certain measure, — for mathematical ideas are common to us all, and in their essence elementary. La facilité suprême is what Descartes praises in all true mathematics.
    Analysis comes from αναλύείν, a word which means to unloose and also to set free: it signifies therefore the unloosing of a single perception into simpler component parts, — the setting free of the elements out of a com-


bination. That is why the resolution of any body into materials which are not capable of further disintegration is called “Analysis.“ In mathematics the word implies in the same way the disintegration of a given proposition into its component parts. You will, however, at once meet me with the question, How can one disintegrate figures into component parts? To represent to myself 70 as 10 times 7, or as 58 plus 12, or as 210 divided by 3, is a purely arbitrary proceeding in my brain. The number 70 or 7000 or 7,000,000 is just as simple and just as impossible of disintegration as 7 or 1. Certainly; and yet it is just as capable of disintegration, for the number 1 is capable of disintegration ad infinitum so soon as it pleases me to look upon it as a product. The same holds good of figures; a circle is a circle, a globe a globe, a pyramid a pyramid, each positively a symbol of unity: still I am able to imagine the globe as actually consisting of segments which have grown together, as in the case of the orange, and in accordance with that I am also able to take it to pieces. I can think of the circle as a line rotating round one of its extremities, or as a variety of an ellipse, or as a slice taken out of a cylinder or a cone, or as the place of an endless number of coincident equilateral triangles with the same vertical point, and in fifty other ways besides. In this way the structural unity is at my bidding set free into multitude. I find myself within the domain of pure human will. Here there is no such practical concrete analysis as there is in chemistry, where by mechanical methods of attack I can resolve a combined body into several qualitatively different component parts, nor is there any operation analogous to philosophical “analytics,“ in which complicated ideas and conceptions are reduced to the elements of which they are composed: but mathematical analysis is the autocratic setting free of a given magnitude into several other magnitudes for purely practical reasons, in


order, that is to say, in that way better to calculate, and this end is attained as soon as the original idea in space has been reduced to an expression in which there is neither space, nor possibility of representation, — an expression which is in accord with numbers. In a wider sense the converse process belongs also to mathematical analysis, — the construction of a superficial image or of a solid body out of a combination of numbers. It had already occurred to the Greeks of later times to transfer to the realm of numerical calculation geometrical problems which it was difficult or impossible to solve by direct means. But the next point was one which they did not attain, for it was contrary to the genius of that people to convert the visible into the invisible, and therefore they made no great progress in that direction. In contradistinction to the Greeks the Aryan Indian achieved his best work in the logical calculation of conceptions (arithmetic and algebra): but he lacked that geometrical eye which is dominant in matters of form. It was the Teuton who was the first to possess the right intellectual aptitude for this twofold work, and Descartes was the one and only man who stood so exactly upon the boundary line that, without being a mathematician, and after a short period of study, he by pure instinct forced the door through which hundreds and thousands dashed after him. Car en mathématiques — he says in the last sentence of his Géométrie — lorsqu'on a les deux ou trois premiers termes, il n'est pas malaisé de trouver les autres. Et j'espère que nos neveux me sauront gré, non seulement des choses que j'ai ici expliquées, mais aussi de celles que j'ai omises volontairement, afin de leur laisser le plaisir de les inventer.
    Descartes, as I have said, set to work with the utmost simplicity. He was in the twenties, and an officer; in order to fill the leisure of winter quarters, and because he had remarked that the study of the mathematical sciences is of incomparable methodical value (elles accoutument


l'esprit à se repaître de vérités), 47 he undertook to take a bird's-eye view of this discipline. But he had always abominated numbers, the wading about in a sea of endless calculations; pour ce qui est des nombres je n'ai jamais prétendu d'y rien savoir, he writes to his mathematical confessor Père Mersenne; he belongs to the open-eyed division of mankind; mathematics are for him the science of forms and motions: his repugnance to arithmetic is so strong that it is only geometrically that he establishes all its operations, addition, subtraction, division, multiplication, even the extraction of Roots. Toutes ces opérations doivent être ramenées à l'examen de l'imagination, et il faut les figurer aux yeux, pour ensuite en expliquer l'usage et la pratique. 48 But Descartes soon remarked that mathematics as taught by the professors are a prosy, dull affair, a compound of many parts: he despaired of learning them in this way, and of making them into a living knowledge. And yet, he said to himself, all these branches of mathematics deal with the same thing, the relation of magnitudes to one another: a plague upon all their Geometry and arithmetic! I shall henceforth only fix my eyes directly upon these relative magnitudes: je pensai qu'il valait mieux que j'examinasse seulement ces proportions en général.
    Of course you understand what he means by the word proportions. It may be a matter of comparison of absolute magnitude between similarly formed bodies; that is the simplest case, and always without more ado to be referred to the difference of numbers, in other words to arithmetic; but the comparison may also be the relation to one another of different forms, and this is what is so actively present to Descartes. In this case it is not a question of whether a thing is great or small, but of the forms which are possible to our phantasy, seeing that each is different from the other, and without the absolute magnitude coming under observation. The circle is a form differing


from the square, it is also a form differing from the ellipse or the volute. The same holds good of a globe, a cube, a pyramid. They are different forms. Humanly speaking each one of these forms is subject to a special law, or if you prefer it a special thought, and this thought is a fixed relation to the extension in space towards the various directions. Goethe says of the perception of natural objects, “There is in the Object something of an unknown law which corresponds to the unknown law in the Subject.“ 49 In geometrical forms we supply both object and subject, and thence the idea of law is at once a matter of compelling strictness and of boundless elasticity. A ball of the size of the planet Jupiter is, as a matter of thought, as like a billiard ball as two hairs are to each other, for the reason that the relative proportions are the same: the comparison of the two is exhaustively expressed by a simple arithmetical proportional equation

a : b = 1 : x

a, the billiard ball, is in relation to b (Jupiter) as 1 to x (the requisite number). But, on the contrary, we are dealing with a quite different sort of comparison if I hold the billiard ball and the cue together, and do not wish simply to establish the relative bulk of the two, for which a pair of scales would suffice, but to bring into comparative relation the law of form of the one, and the law of form of the other, that is to say, reduce them to an equation of mutual relation. All this, the interrelation of magnitudes and the interrelation of forms, is what Descartes speaks of as proportions. What he asserts is this, that the different branches of mathematics ultimately ne considèrent autre chose que les diverses proportions qui s'y trouvent. And inasmuch as the monotony of mere arithmetic is repugnant to him, and he is wearied by the calculations incident to the geometry of solids, he just asks himself one question: What will be the simplest


way for me to compare different forms with one another? The answer is, by reducing the problem to one of the relations of various straight lines to one another, à cause que je ne trouvais rien de plus simple ni que je pusse plus distinctement représenter à mon imagination et à mes sens. You see then the first and constant thing postulated is something which can be represented, clearly represented, obvious to the senses. Yes. But supposing that I should analyse many forms in this way, I shall obtain a whole forest of lines. How can I carry them in my memory? And how can I continue my investigations of their reciprocal interrelations? For this purpose the lines must be reduced to an expression in numbers: il fallait que je les expliquasse par quelques chiffres, les plus courts qu'il serait possible. And he ends by saying, je pensais que par ce moyen j'emprunterais tout le meilleur de la géométrie et de l'algèbre, et corrigerais tous les défauts de l'une par l'autre.
    You now know how it was that the idea of analytical geometry arose in the intellect of Descartes, and in what form it floated before him. It contains absolutely nothing abstruse or learned which need scare us laymen. On the contrary, it was in direct opposition to the professorial men of science that Descartes invented his method, and in answer to a friend who communicates to him attacks from all sides on his geometry, he writes, J'aurais mauvaise opinion de mes pensées, si je voyais que les doctes les approuvassent. I imagine that you will already have remarked what is the turning-point of the whole method. It is the establishment of the line as intermediate between form and Numbers. And this means exactly: it is the discovery of that point wherein the doctrine of magnitudes, sense and understanding, perception and thought merge into one another, where the visible becomes invisible, and vice versa. In a late work, which unfortunately remained unfinished, the Règles


pour la direction de l'esprit, Descartes very clearly laid down this mediatory principle of his: par les lignes il faut représenter tantôt des grandeurs continues (i.e. forms) tantôt la pluralité et le nombre; l'industrie humaine ne peut rien trouver de plus simple pour exposer toutes les différences des rapports; so the relation between the straight lines stands exactly in the middle, pointing on one side to the visible form, on the other to the essence of abstract numbers.
    We are now sufficiently equipped to start upon the concrete observation of analytical geometry. But I have to insist that what follows must be treated as a series of ideas without your ever for a moment being contented with thought alone, as apart from ocular demonstration.
    Surely it is plain to the eyes that I can, if I so choose, conceive every straight line as a number? For example, if three straight lines stand in relation to one another as 5, 4, and 3 (it is immaterial whether we are speaking of yards, or feet, or metres, or miles), I can call them simply 5, 4, and 3, and so calculate with them: every builder does that daily, and that is really geometrical analysis, for it is the conversion of a conception in space into a number which has nothing to do with space. But your builder now goes a step further. Supposing that the line 5 represents the one side of the house which is to be built quadrilaterally and that the builder wishes to know the size of the area for his work, there is no need for him to measure it with his measuring tape, nor to set it out on paper: the required area results from the sum 5 times 5. This sum 5 times 5 is what the science of arithmetic calls the square of 5, or 52, or 5 in the second power. And if the house were to be of the same height as the breadth, the builder need only write 53 that is to say, 5 times 5 multiplied by 5, and he would know exactly what would be the space included in the cube-shaped


house. If the sides of the house are unequal, he has to multiply the one by the other as a x b, then by the height, c, and with this abc he has at once at his command all the conjuring tricks known to him by the study of arithmetical logic, without reference to the concrete house which has to be built.

square, rectangle, cube

For as soon as I have written down the numbers instead of the line or the area or the body, the visible form fades away — a2 is simply the number a multiplied by itself, b3 is b multiplied by itself and then again by the product, abc the multiplication of the numbers a, b, and c with one another. 50 The form absolutely disappears, and only the


measurements remain. Up to this point the matter is simplicity itself. But if you examine your power of imagination, it will soon answer that of all forms it is only those that are rectangular that allow themselves to be reduced in this simple way to lines, and consequently again resolved into numbers, whether these rectangles be plain or solid. All these rectangular forms can be imagined as originating in the rectilinear movement of limited straight lines, and this rectilinear movement may, like the straight line itself, be expressed as number without further trouble. If I say a I name the line: if I say a2 I name the superficies which comes into existence when a moves lengthwise along its length; if I say a3 I am naming the cube which arises when this superficies moves upwards to the length of a. The same thing takes place in the case of rectangles with unequal sides; we can represent all of these to ourselves as proceeding from the movement of two or, as the case may be, three lines of different lengths. Thus ab is the movement of a along b, and abc is the movement of the superficies ab along c. The line is therefore comprehensible as a number (explete or symbolical), and what is perceived as the movement of this line is to be understood as the multiplication of the number: a2 is the multiplication of the number by itself, ab the multiplication of the number by another number. Therefore, inasmuch as rectangular figures can without more ado be reduced to single straight lines and to single rectilinear movements, — it is easy to reduce them to an expression in numbers. The numbers 5, 4, 3, or the letters a, b, c, correspond to the length of the component lines, and what we call exponents, that is to say, those smaller cyphers which Descartes taught us to write above and to the right of the larger figures, — for that was his invention, — denote the movement of the lines. As soon as you conceive of the matter as visible, these algebraic figures are shorn of all their abstract


terrors. The small 2, as in a2, points to a simple movement out of which only a superficies arises, therefore a space of two dimensions, hence the 2; the small 3 points to a double movement, and consequently to a solid body, that is to say, a form of three dimensions, hence the 3. When therefore I reduce rectangular figures to the measurement of length and indications of movement, denoting the measurement by ordinary cyphers, and add the movements by small cyphers written in above, as exponents, I have obtained a very simple expression which I can, at will, look upon as a visible form, or as an arithmetical conception. But how am I to deal with forms which are not rectangular? Question your own sound natural power of conception. Unless a man be a second Descartes he will have difficulty in finding the answer.
    Not to extend this mathematical excursus too far, we will only take into consideration one single case, that of the curves in a plane, that is to say, of such curves as you may draw upon a sheet of paper, and which correspond proportionally to the rectangular superficies. How can these curved lines be made capable of a similar solution into arithmetical magnitudes? Without the help of straight lines the transition from curves to numbers is unthinkable. Numbers have no analogy in any shape with visible things, beyond on the one hand the circumstantial analogy with objects exhibited side by side, and on the other hand with straight lines. This second analogy is not, as you might think, drawn from the first, but arises out of the essence of numbers which are to be thought of as a rectilinear continuation. The numbers 5, 6, 7, are essentially identical in their nature, only 6 is longer than 5 and shorter than 7. 51 The curve, on the other hand, is an idea which arithmetical conception can never reach: it lacks the necessary pliability. The essence of the curve is form, the essence of arithmetic is indifference to form. It is therefore only the straight


line that can be of any assistance in the task of converting form into numbers. For we may define the straight line as follows: it is the only line which even if it be produced to infinity creates no form. It is pure magnitude and pure numbers. How then can I bend into pure formless magnitude and pure comprehensible numbers magnitude which is possessed of form and numbers conditioned by form, and at the same time locked in form? Here, too, I can only succeed if I reduce form to movement, but even so the movement must be rectilinear.
    Take a ruler: let a slider with a pencil slide to and fro from one end to the other on this ruler, and let this


pencil be placed perpendicularly to the ruler in a capsule which may be drawn out and compressed at pleasure: if you hold the ruler immovably on a sheet of paper you are able to draw the most complicated curves with the point of the pencil, as you, on the one hand, push along the ruler the sliding capsule which carries the pencil, and on the other hand, by lengthening and shortening the distance between the pencil's point and the ruler. You must now consider the resulting visible curve as being produced by the length of the ruler and the pencil; and as a matter of fact that is what it is. So this curve expresses the varying relation between three straight lines, of which the one, the ruler, has retained its length unaltered, while the two others which express the length


of the pencil and the position of the slider on the ruler, have been changeable. Looked upon purely from the mechanical point of view, that is the proceeding of Descartes in the analytical dissection of a curve. It was with the help of such considerations and instruments that he arrived at his thought. You see how this man is always and everywhere wandering on the boundary line. The problem as a whole deals with the conversion of the visible into the invisible and vice versa. Its solution he arrives at by a perpetual shifting to and fro of the ideas of rest and movement. For the curve which he wishes to “analyse,“ the circle, the ellipse, the spiral, the volute, etc., is in the first place something granted, a symbol of that which is perfected, eternal, immovable. But next he considers how he may regard it as arising out of the movement of straight lines, and thus rest becomes movement. Then there is the return of movement into rest. For these lines in motion serve to attain an immovable arithmetical expression.
    A concrete example will at once show how Descartes obtains the straight lines for a given curve, and out of the lines an arithmetical expression. I choose for the purpose the simplest curve in a plane, the circle.
    With the help of a piece of packthread and a piece of blue chalk I draw a circle on the wall. Our circle is obtained by turning a straight line round one of its ends: that, however, gives us nothing available for analysis, which needs the relation to one another of several lines. This causes Descartes to refer the generative law of this fixed figure to the relation between one immutable straight line of an ascertained length and two other movable lines (cf. our immovable ruler with the movable pencil attached to it). Only you must not for a moment imagine as regards the construction, which is the result, that it possesses any thinkable significance in nature, outside the human brain, or that in practice a circle can


come into existence in that manner; Descartes only delineates it, because it is his pleasure to do so, because the thing can be thought of in that way, and because all sorts of amusing results arise out of it. Well then, how does Descartes set to work? He accepts the circle as given, within it he draws two straight lines perpendicular to one another, and the feat is done. The one line — the most important one — he draws from the circumference to the centre. For the sake of greater clearness I draw the line horizontally, but I might if I chose draw it in any other position. This line is always called R, from the initial letter of the Latin word Radius which signifies the spoke of a wheel, and later was adopted into scientific language as a description of the half of the diameter of the circle. Descartes had all the more reason for retaining the sign R in that the French word for the half-diameter is Rayon, and the Germans have only to think of their own original word Radspeiche (the spoke) for the R to lose all the evil taste of the dust of the schools. This line R is a fixed, immutable, ascertained mathematical magnitude. If the circle is a concrete and present figure I can measure it with a yard measure: if we are only dealing with the form of the circle in general, I cannot represent any length in cyphers, but the line R is none the less a recognised, immutable magnitude, that is to say, in relation to whatever may be the circular line of which it denotes the half-diameter. Well then, upon this immutable line I set up at right angles to it a second line which, inasmuch as it is an unknown magnitude, I call y, and which I represent to myself as movable upon R, that is to say, which I can move to and fro upon R, from one end to the other, exactly as we did just now with our pencil. But this second line is not only movable, but also of variable length. In every single place, that is to say along the whole length of R, its length differs; and indeed the organic relation between its length and its


place is settled by the curve in question, in this case therefore by the line of the circle, inasmuch as we always produce this line y to the periphery of the circle by which we allow it to be cut off. If, therefore, this movable line is raised at the inmost point of R, that is to say, at the centre of the circle, it becomes itself a half-diameter and its length equals that of the line R; that is the maximum length which it can attain; if on the contrary it is set up at the outermost point of R, it is immediately cut off by the circumference line, and its length becomes zero. Between zero and a length equal to R the line y can have every conceivable measurement. And, as a mere glance at the diagram will show, its length will everywhere be determined by its place, and its place by its length. One word more and then we shall have gathered all that we want. The line R is, as we know, immutable: but it now contains a movable element, namely the point at which the movable line y is erected. You need only think of the slider in our mechanical example. I will now make use of the centre of the circle as a starting-point, and from that measure the lines to the point where the line y meets the line R, and this line I will call x. Since y moves along R this line x is manifestly variable and its value, as a single glance at the diagram will show, will always diminish or increase in an inverse ratio to that of y. If y is at the centre, x dwindles away at once to zero. If y is at the outer end of the Radius, x becomes equal to R. x is, as you see, in the same case as y; its length value can take every step between zero and the length of R; but, in addition, its value must always of necessity be conditioned by y. As a result of this construction we have now three values, of which one, R, is immutable and the two others, x and y, are mutable. What unites these three values into one organic relation to one another, is first their fixed reciprocal position in space, secondly their fixed relation to the centre-point and to the peri-


phery of the circle by which they are bounded. Now as these values stand in a relation to the circle, so too does the circle stand in a relation to them, and in this way they will serve us to gain an expression for the curve in lines, and that means in numbers. This relation we can describe in the following very simple fashion:

R2 = x2 + y2


That is to say, expressed in words, the square on R, — no matter whether the circle be great or small, and no matter what may be the position of y, is always equal to the square on y multiplied by *) the square of x. But you must not be led astray if this equation talks of squares:
for R, x, and y, are lines, and the exponent 2 points, as you will remember, to the movement of a line along its own length; every one of these three squares is therefore
    *) Should be: “added to“, Redesdale made a translation mistake here. Chamberlain wrote “vermehrt um“.


resolved by this formula into a line and a movement: but the line as well as the movement can be comprehended without more ado as a number; consequently that representation of a visible geometrical relation is at the same time, if we choose, an algebraical equation, and that means a purely arithmetical expression. As such it belongs to the protean domain of abstract mathematics, “thought without contents“; it is conception without perception, and so gains in pliability and logical multiplicity of significations what it loses in visibility. This algebraical equation (R2 = x2 + y2 ) is the analysis of that flat curve which we call a circle.
    I have no intention of stopping to furnish a proof of the correctness of this affirmation that R2 = x2 + y2. It is very easily demonstrated geometrically, and is to be found in Euclid (as the Pythagorean proposition). Its interest for us only lies in the fundamental idea of Descartes, the idea of the resolution of the visible relations of measurement and form into invisible, abstract arithmetical relations. Any one who is interested in the matter can construct the proof empirically with the help of a circle and a millimetre measure. Nor shall I wait to show that there are other analytical equations which can be made up in behoof of the circle, and that for many other curves, as also for the analysis of bodies of three dimensions, a far more circumstantial process is needed; the principle remains the same. Moreover, my conscience pricks me in that I have not led you precisely on the same way as that which Descartes followed, and because I have been so bold as to exhibit analytical geometry in a manner completely different from that ordinarily adopted. You can read Descartes, if you are able, for his Géométrie is not easy, inasmuch as he wrote it with purposeful obscurity in order to avoid plagiarisms; 52 or you can take up that beautiful monument of German industry, Cantor's Vorlesungen über Geschichte der mathe-


matik, if you wish to make acquaintance with pure mathematics in their historical development. In neither place will you meet with my exposition; if I have failed, this hint may be taken as an apology. 53 If I have taken my own road it has been because I had a goal of my own in view. My peculiar way of looking at the subject grew out of our precedent course of thought, to which it now carries us back. Let me therefore, in closing this mathematical excursus, only say briefly that this algebraical analysis of geometrical perceptions is the foundation of almost the whole immense development of modern mathematics, and with them of all physics. The expert mathematician, it may be said, sees in his mind's eye, in such a scheme as R2 = x2 + y2, things which otherwise he would never have seen in the mere visible symbol of the circle. True, the seen curve has faded away, but in its place that creative law of form, as we have called it, appears perhaps even more distinct, — at any rate as a stimulus to new thoughts. The analytical equation is to the mathematician what the ground plan is to the architect; unintelligible to the layman, such a manner of schematising reveals to the expert things which he would never have been able to see in the concrete: that is to say, it leads him to the discovery of relations between the different forms which no power of perception would have been able to reach — and he has only to discover another, cleverly chosen, algebraical formula for his curve, in order to obtain an elevation in addition to his ground plan. He is now also in a position to investigate the properties of forms which, on account of their great complications, would be beyond the power of the eye to unravel, and perhaps impossible to represent mechanically. Thanks to this method, he has reached a point where he can investigate the properties of figures of four dimensions, as well as of others that are beyond the power of imagination.


    You see that this road leads to the deepest depths of metaphysics, but at the same time and in the same measure to the contemplation of the unseen. For now all equations can be converted into form, and with the help of two lines which exactly correspond to the R and y of our example, dreary, dull rows of cyphers, such, for instance, as statistics, are conjured into curves which at once furnish every layman with intelligible ideas, and allow the mathematician to penetrate the mysterious laws to which Phenomena are obedient. 54
    We need go no further now. Let me add a general survey.
    What Descartes' intervention has signified for mathematics in general may, I think, be summed up in a remark which at the same time points directly to that which we have had in view in this excursus. We might indeed, unless I am mistaken, show that the peculiar duplex character of the infinitesimal calculus, called into life by Descartes and followed up under his instigation by Ferrat, Pascal, Barrow, Newton, Leibniz, the brothers Bernouilli, and others, rests upon the fact that it stands with one foot in perception, with the other in abstraction. To grasp the fundamental conception of the infinitesimal calculus (that is to say, remember, as “thought“), is so infinitely difficult, not to say impossible, that Carnot, one of the most competent of specialists, assures us that very many professional mathematicians have not understood the significance of their own calculations; yet as a consolation he adds: il est certaines idées primitives qui laissent toujours quelque nuage dans l'esprit, mais dont les premières conséquences une fois tirées, ouvrent un champ vaste et facile à parcourir. 55 Historically the infinitesimal calculus grew out of the observation of geometrical problems, and out of the lucky inspiration to consider these as phenomena of motion: far from being an abstraction, this mode of calculation is unthinkable


unless we take the perception of the senses as a starting point. Infinitely small magnitudes are magnitudes that the eye no longer can see, but only the conception can still imagine: the transition from sensibility to understanding takes place here materially: the calculation by letters penetrates like a microscope where the object fades before the naked eye, and communicates “imaginary images“ to the brain (see page 74 seq.). Perception and abstraction are both of them moving in a region near the central line of demarcation. Up to Descartes' time, then, mathematics had, so to say, always been hopping upon one leg, either in perception or in abstraction. He taught them to stride forward vigorously on both feet; the start in mathematics could not long be delayed. We too to-day, within our modest limits shall gain a similar advantage.

* * * * * *
    We have now come to an end of the constructive part of this lecture. It would be delightful to follow Descartes still further; the proud, angular, domineering, and at the same time aristocratically reserved and sensitive nature of this thinker, fills us with respect and sympathy, and there would still be much to bring forward about him and his life in amplification and correction of the known descriptions of him; something of this will perhaps, now that we are quite accurately informed as to the principles of his method of perception, weave itself in automatically in the further course of our studies. But at this moment another duty lies before us, that is to say, to turn to account the sum of our labours of to-day for the recognition of Kant's intellectual aptitudes. 56
    It is the fashion, — wrong as a matter of method, — to start from the simplest point, from that which analysis shows as the simplest component parts. Far rather should that which is best known serve as starting-point

in expositions and explanations, whether it be complicated or simple. This is the only way in which direct perceptions with that power of persuasion of which it has the monopoly, can maintain its rights. That is why I chose as the main theme of my first lecture the conflict between Idea and Experience. In that way we certainly gripped the problem of perception by the most complex and difficult phenomenon that it perhaps ever exhibits. But the advantage was just this, that we at once faced the whole, that is to say, that which is living, true, and sure, as it is common and well known to us all. We all have our experiences and our ideas, and even if we are not accustomed to analyse them, one word is enough, and every one knows what we are talking about; and even though Goethe's perceptions and Goethe's ideas were of an august nature, they none the less spoke directly to our understanding, and that which was perceptible might almost have filled the whole lecture. Next, however, we followed Goethe's advice to “work our way out of the whole into the parts“; in the second lecture we grasped the problem more closely on both sides, when we made the conflict between the pure form of all perception and the empirical material of perception our chief subject of study. Simple, and apparently easy to survey, was the relation between the two in the plastic artist, who kept before his eyes the scheme of his understanding, half pure and half perceptible, in order that he might see more exactly, that is to say, in order to “think“ more clearly that which was seen, to comprehend it more exactly: far more complicated did it become, — harder, that is to say, to expound, and so also harder rightly to grasp, — as soon as the understanding drew the phenomenon over to itself, so that the pure scheme of the senses became the main point, whilst the empirical phenomenon itself, or at any rate its foundation in the perception of the senses, — paled almost to fading away.


Calling to our support the conflict between physical optics and Goethe's doctrine of colours, we tried to gain as clear an explanation as possible of these relations. To-day a new conflict has arisen before us, no longer the one between pure perception and empirical perception, but that between perception as a function of the senses, and conceptible thought as a function of the understanding. Carefully considered, this conflict is far simpler than that between idea and experience, and even in its essence easier to grasp than that between pure and empirical; what makes it difficult to unravel is the complex interlocking of the parts: what I have had before me to-day as my chief aim has been to arrive at a clear conception upon this point; no man in the whole history of the world could render us such conspicuous service in this as Descartes.
    You must know that with the help of Descartes we have become acquainted with a way of seeing, a recognition, a conviction, a view, a method, — call it what you will — which is absolutely fundamental for Kant's philosophy. And this point is just the one of all others which is looked upon as the darkest in his philosophy; it is the pons asinorum before which the great majority of the flighty searchers after knowledge turn tail — and not they alone! I could name a worthy modern student and editor of Kant, who only so far masters the difficulties which are to him insuperable, that he declares roundly that this fundamental thought of Kant's “has no scientific value,“ and therefore that it is not worth while to break one's head over what Kant may have meant: indeed that the whole difficulty was only an “invention“ of Kant's — Guter Mut, halbe Arbeit, says the proverb, and so apparently thought the learned Professor. Still we look at the matter from another point; happily it is not my business to explain the famous and dreaded chapter of the Kritik der reinen Vernunft — Von dem Schematismus der reinen

Verstandesbegriffe — I have only to show in the commonest outline those foundations of perception which later, in the artistic connection of the system, go to the greatest depths, and are therefore the subjects of the most secret exposition. “Unhappy is the speaking man,“ cries Emerson. “If I speak I define, I confine, and am less.“ This “unhappiness of the speaking man“ Kant had to experience: still he would not consent to make his thoughts less, the crabbed genius of truth forbade it; and so they became dark, dark as the powers which rise in the growing life of the golden-cocooned chrysalis; he who does not call eyes and heart to his help will never understand this thinker and will never, freed from the darkness of the pupa, fly aloft with him on the wings of a new knowledge. But to-day, as I said before, our office is far more modest, and I am glad to be able to give the surprising assertion that we have achieved our task, and we now need only recapitulate it briefly, systematically, and with peculiar reference to Kant.
    Descartes is of special value for the understanding of Kant because, with a striking resemblance in his intellectual aptitude in general, he has little capability and still less inclination to busy himself with the nice analysis of abstract comprehensions; that is why with him everything remains so concretely visible. That he insisted upon the critique of the human intellect as an indispensable foundation for all science is proved by a quotation at the very outset of this lecture; moreover the expression “pure reason“ occurs often in his works; 57 yet whatever there is of pure metaphysics in his philosophy is rather symbolical than critical. Masterfully and forcibly he simplifies, and then he places his rough-hewn blocks as landmarks to show that he too has travelled through this domain, and then hurries on further to those scientific investigations which take complete hold of him. Still these somewhat rough-hewn symbols of


metaphysical knowledge have exercised an incomparable power upon later thought, as for example the distinction of every substance into thought and expansion. Like an Alexander among philosophers he thus cut a Gordian Knot which all the desperate attempts of the spiritualists and materialists have never been able to join together again.
    “Reason,“ says Kant, “proves its loftiest duties when it distinguishes between the world of the senses and the world of the understanding.“ Its loftiest duties! So great weight does he attach to this first and elementary direction of critical reflection! The senses and the understanding are in his view “the two extreme ends“ of human knowledge. The shortest formula is as follows: “the business of the senses is to perceive, that of the understanding to think.“ More closely thought out, and more accurately analysed, it runs thus: our knowledge springs from two intellectual sources, of which the first is the reception of notions (receptivity of impressions), the second the power of appreciating a thing through the agency of the notion so received (spontaneity of conceptions); the first gives us the object: by the second the object is conceived in relation to the notion (as a mere diagnosis of the mind), and all this is rather an accessory, a preamble, a preparation, an exercise of the understanding in the intellectual nursery; the true depth of the Kantian method of perception is first attained when the philosopher reaches the certainty that the one “end“ of knowledge (the senses) is incapable of the smallest result without the other “end“ (the understanding). Unless the senses afford notions no thought can arise; and unless thought furnishes its directing power, no perception of an object can take place. Experience, — and in that word we express all that we are, — is therefore always “a product.“ If experience is always a product, then it would be simpler not to think of it as arising

and combined out of two different and separate origins, but rather as an original unity, which is only split into two component parts by analysis. Yet this objection is in reality very superficial, and all that it effects is the reopening of the door to insipid empiricism, according to which the understanding arises out of the senses, and to objectless mysticism, which makes the world of perception arise out of the reason or the will; against which Kant at once admits that “the two stems arise perhaps out of a common, but to us unknown, root,“ while he declines to waste his strength upon this unknown and unknowable thing (unknowable inasmuch as it lies outside experience), 58 but at once declares that we possess no organ or power by which we can ever go beyond experience, and that in all experience the two stems are there, always capable of being proved to be distinct, and always postulated as united. 59
    Kant is the only philosopher of experience — I wish to lay earnest stress upon that — he is the only strict philosopher of experience known to the history of human thought. That makes his greatness, and it is that which makes him so unapproachable to most people. To philosophise with Schopenhauer is a delight, not to say a luxury; Kant, on the contrary, warns us with inexorable earnestness, “That the understanding of which the first duty is to think, should instead of that fall into extravagance, is something not to be forgiven.“ To philosophise with Büchner, Haeckel, and their like seems to comfort many brains that we may presume to be atavistically retrograde; but Kant finds only one predicate, impertinent, adequate to the affirmations of materialism and naturalism, and he exclaims with loathing, “Whoso has once tasted Criticism, is for ever disgusted with all dogmatic nonsense.“ You must not be misled by that much-abused word Idealism. In my further lectures I shall have to offer a few remarks upon Kant's nomen-


clature; here one word will suffice: when a critic of the Reine Vernunft described Kant's teaching as a “system of the higher idealism,“ the sage at once answered wittily and with fine solemnity, “For goodness' sake not higher! High towers and the metaphysically great men who resemble them, both surrounded by much wind, are not for me. My place is the fruitful depth of experience.“ 60
    If you wish to know Kant's method of Seeing, if you wish to overlook no fundamental feature of his intellectual personality, then you must never lose sight of this famous saying, “my place is the fruitful depth of experience.“ This limitation brings into play at the same time a second fundamental characteristic, — that of unconditional truthfulness. The very same truthfulness which finds such crabbedly lofty expression in his moral writings rules here in the philosophic critique of human reason. And even this removes Kant far away from us, makes him inaccessible to most of us. It is not the truth that we long for, but lies: and lies are on the watch for us everywhere; the lie invisible and unnoticed, like the bacteria and microbes, worms its way into our brain in the character of “suggestion,“ nests there and multiplies, until even if we were able to get rid of the intruder and its brood, we should still be unable to destroy its network without ruining our own power of thought. It needs not only extraordinary keenness of thought, but also extraordinary honesty of thought, and incorruptible love of truth, even a whole life of self-discipline, to fit oneself to the fact that our whole thought and being is surrounded by a brazen wall, and that we must resign ourselves to our fate, since we have neither wings to fly over the ramparts, nor the power of reaching the other side by burrowing under the earth.
    From this strict limitation to experience we arrive not only at Kant's peculiar method of perception, but also at the special difficulties which many of his perceptions

present to our understanding. One word more in this or in that direction, — the least little strain exercised upon our thought, — what Kant calls experience, — and all difficulty would disappear. But Kant never makes allowances. “My place is the fruitful depth of experience!“ I believe that I know that Kant may be called the greatest of all thinkers: yet I know with absolute certainty that he is the honestest of all men, and that this loftiness of character means the Sun under whose rays his mental work ripens.
    With all this you must be familiar if you wish to understand Kant's position in regard to the “twofold stem“ of all human knowledge, and actually to know why he looked upon the world in this way and in no other. His iron law binds him to experience alone; he will neither dig for that “common root,“ — always destined to remain hypothetical and perfectly incomprehensible, since we can only understand that which belongs to experience, — nor will he have aught to say to dreaming and Dogma. That is why he considers the twofold sense and understanding as double, and that is why the organism of the practical union of the two inside all experience can only be disentangled by the most painfully exact observation and critique of the facts of experience in the mental life. Kant is not concerned with being easily understood; what he is concerned with is spotless truth, above all with never overstepping the boundary of experience. Great is the reward! Kant is right: the bathos of experience is fruitful. What we learn here is inexhaustible, and it is not only true but useful. Kant's philosophy distinguishes itself in toto from all other methods of philosophy in this, that it watches over itself practically step by step; it is always directing itself towards two goals, natural science and moral doctrine. What can I know? What am I to do? Those are the two great questions which exercised


the sage of Königsberg. And what characterises him alone among all others is that in the answer to both questions he forbids any overstepping of the boundary of experience. Thence we see not only a Goethe but also a Johannes Müller leaning upon Kant, and thence to-day not only our most important and freest professors of philosophy, but in the same way many of our leading investigators of Nature, go back to Kant. Few have been adequately schooled to grasp Kant purely and fully; but merely to touch the banner-bearer of crabbed, and at the same time energetic and richly active truth, suffices to ennoble all thought.
    For to-day we must content ourselves with the standpoint of a Descartes who looked upon the last questions of philosophy rather from a psychological than from a purely metaphysical point of view. That the understanding and the senses are two is a matter known to us clearly and in detail by practical examples from the history of the Sciences. By seeing Descartes at work, first in the domain of the physical sciences, where on the one side the aether and on the other the laws of motion served us as main examples, and secondly within the narrower field of mathematics, we became aware of a tolerably complicated relation, which might otherwise easily have remained unknown to us. We discovered that between those “two extreme ends of human knowledge,“ as Kant called them, there lies a uniting middle land: — outwards the boundaries of this buffer-country are rather indistinct, while, on the contrary, the dividing line which runs through the middle, and separates the two halves of our intellect from one another, remains clear and sharply defined, even to a hair's breadth. We are taught that we must make our comprehensions evident to the senses, otherwise they remain empty: true: but what the understanding sees in making its comprehensions evident to the senses, is not that other “extreme


end,“ not the unadulterated perception given by the senses, but only a schematism of the senses, schemes which at their best reach the divisional middle line. We have to submit our perceptions to comprehensions, otherwise they are blind: to be sure, however, these conceptions must for this purpose be very essentially materialised, and the result is not pure thoughts, but a symbolism. It is then certainly no simple occurrence when we make our conceptions perceptible to the senses, and bring our perceptions under the category of conceptions.


    Here is our ultraschematic diagram, ready to render us further service. If we were to direct our eyes simply towards the general division into understanding and perception by the senses, we should not reach far beyond Aristotle, who also in close connection with Plato distinguished the nature of thought (νοήτικον) from the nature of perception by the senses (ἀισθἡτικον), and who consequently was like Kant and Descartes, antimaterialist and antispiritualist. The matter first gains a living interest, as well psychologically as metaphysically, through the discovery of the intermediary domain, and of the complicated phenomena which take place there. “Never


is a comprehension directly in relation to an object, but only to some different idea of that object.“ That was the one great discovery. It is amplified by the next: even if all perception of the senses brings to us something manifold, “we receive in the first place no objects of empirical knowledge, and therefore no experience“; the rather does experience first come into being by the co-operation of a function of the understanding, since it is comprehensible imagination which brings into combination, “the many-sidedness of perception“ — with that unity which is indispensable to all experience. There you have Scheme and Symbol. And I am convinced that you now accurately understand the whole matter at issue, since you have seen by Descartes' great thoughts of Inertia on the one side and the Aether on the other, what a Scheme is and what a Symbol is, — how each arises, what it means, and what are its limits.
    I should like to remind you once more of Goethe's precious saying, “all thinking is useless for thought.“ True thoughts always come as it were of themselves, their birthplace lies just in that middle land where perception and conception join hands. And it is the same in the case of the understanding of the thoughts of another, where the chief matter of importance is far more the subjection, than any exertion, of the intellect. Kant himself warns us that “Insight cannot be forced and hurried by exertion.“ The man who wrinkles his forehead and draws his eyebrows together, will never make any progress. The expression of the true desire to understand is the widely open eye which shows how inwardly as well the mind greedily sucks up every ray of light in the one endeavour — to See. If you have yourself already seen what the other man saw and how he saw it, then his thoughts will automatically reach you. Hence, now that we are about to take a very decisive step, I repeat the petition which I have already made to you, to think

as little as possible, and see as much as possible. “In order to be comprehensible one must talk for the eye,“ says Herder in his Reisejournal.
    You remember the first steps which we had to take in order to arrive at a comprehension of the boundary-land between the senses and the understanding. Here again Descartes helped us: we only needed to see him at work in the province of pure mathematics. Without the assistance of mathematics we should never have arrived at complete distinctness. But since without perception there can be no thought, and since almost all perceptions are of empirical origin, that is to say, arise through impressions from without taken up by our senses, so in the majority of cases the problem of experience is from the outset very complicated, — as we saw in the case of metamorphosis. You have only to examine our Scheme in order to convince yourselves how difficult it must be to ascertain the precise mental topography, that is to say, the exact place of an idea, which is forced upon us, given by perception, thought by conceptions, — but only thought when it is given, only given when it is thought. Such an idea has generally speaking no fixed place; it is shifted to and fro; the commutator of the middle land suddenly converts the one into the other, and e.g. what was in Goethe a pure intensive Symbol of the senses, in Darwin is converted into a perfectly artificial, abstract, logical Scheme. The advantage of mathematics was that we found there pure schematic thought and pure symbolical perception within our own mind and without any adulteration from outside. Hence the topography was perfectly fixed, and hence with mathematical precision, — as we may well say, — scheme and symbol corresponded to one another. The most important point which we gained from analytical geometry, was this, that the dualism of our intellect, expounded by Plato, Aristotle, and Descartes, but first accurately analysed by


Kant, is no matter of theoretical acceptation, but a mathematically assured fact. All Monism is a lie, not of course a subjective lie, nor a lie for those who are, or think they are, capable of soaring above all experience, still an objective lie, a lie so soon as Monism is to have value inside of experience. A gradual transition from perception to conceptions, or vice versa, from conceptions to perception, — an absorption of any sort, — is something which never takes place. In empirical experience there is still room for doubt; the invisibly complicated relations lead to many a deception: Mathematics, however, teach us something better. We certainly shall never deny that

and R2=x2+y2

are two exactly corresponding expressions; *) but no unprejudiced man will be able to avoid feeling the artificiality and arbitrariness, I might almost say the tyranny, of such a proposition. Logic is powerless against it, for such propositions are outside the pale of logic: perception loses its rights in face of it, for perception is suppressed. The proposition possesses no trace of a meaning beyond the connection which I, as man, assign to it. That I have the courage of such a proposition does not prove that it has any objective sense outside of my own intellect, but only that there is a subject which is capable of uniting into symmetrical relation the two dissimilar parts of its intellectual organism. And it is just this construction of relations, not drawn from experience, but by means of which, uniting the two parts of our mind, we first make experience possible, — just as by the relations between


mental calculation and perception we make the higher mathematics possible, — it is just this that Kant describes by the dreaded, often used and seldom understood word “transcendental.“ 61
    Mathematical analysis has here served us as an example; but I must ask you to make a careful distinction. It was the single man of genius who succeeded in setting up the equation R2=x2+y2, and in endowing it with meaning; another man may introduce another equation with the same object. What is no achievement of ours, but is simply a fundamental law of the human intellect, is the fact that it is only the straight line that has the power to convert form into numbers and vice versa. Thus in our mathematical undertaking we were in reality bound to a transcendental principle, though we were hardly conscious of it. Now we must go a step further and enter upon the field where the arbitrary will of man has no voice, but where inexorable laws of our mind are the informing power — the transcendental laws of our reason.
    In our schematic diagram we have left white the level spaces on either side of the hatching: these were supposed to represent pure sensibility and pure understanding. Now what Kant detects is as follows: the Symbol on the one side and the Scheme on the other, do not originate in the middle region, in transition and in combination, but all perception is at its very outset symbolical, and all thought is at its birth schematic. Although the commutation, although the switches which are to alter our direction may only exist in the middle domain, that is merely a matter of psychological insight: metaphysically, on the other hand, the knowledge that our reason is as a general proposition confined within Scheme and Symbol, is of fundamental importance. That is the transcendental fixed boundary of all that of which we are conscious as experience; Experience is never a pure apperception of what is and takes place outside our


human mind, but it is always a question of an experience which is schematised and symbolised. Almost all men, including our so-called empirical investigators of Nature, maintain that the human understanding possesses capabilities which are independent of scheme; that is to say, that it is at any rate partially set free from the bondage of fixed methods of thought; and they hold that human perception can equally see things as they are, and not as the tyranny of our one-eyed cyclopean form-sense of space transforms them for the benefit of mankind: but the man who maintains this doctrine is defending conceptions which are far beyond all experience, and indeed beyond all possible experience: that man is a dogmatist. Kant refuses to take this aeronautic flight: he remains prosaically, heroically, and recusantly on the terra firma of facts, and says: all human perception happens through the intermediary of a fixed Symbol — this Symbol of all pure perception is Space; all human thought only moves within a perfectly fixed, limited, inevitable Scheme — this Scheme of all pure thought is the table of the Primary Conceptions of pure understanding, also called “Categories.“ 62 The fact that the conceptions of the understanding do not permit of being referred to any single conception (like Perception to Space), is one which, as you will see presently, is founded upon the essence of our intellectual mechanism; but those conceptions do form a simple, strictly united scheme, acting on all sides as condition. We have, therefore, on the one side the one idea Space as an indispensable fundamental form of the senses, and on the other side the single group of the few pure conceptions of the understanding which make up an organic unity.
    I should like briefly to limit and more closely define a saying of which I made use just now in order to serve as a support to your ideas, but which might possibly lead to misunderstandings later. I said: all perception


is at its very outset symbolical, and all thought is at its birth schematic: I should like you to grasp that not strictly, but only as analogy. You have seen how Symbol and Scheme arose out of the reciprocal interpenetration of senses and understanding; briefly, therefore, Symbol and Scheme are not original, but derived; but what space is according to Kant's conviction, that you can first and best imagine by the analogy of a Symbol, — what is the table of the comprehensions of pure understanding you can first and best imagine by the analogy with a Scheme — and now that the above reservation has been made you can fearlessly facilitate your entrance into Kant's world of ideas by the following formula: in the last resort all different symbols may be referred to one symbol, all different schemes of thought may be referred to a many-branched but yet single and united scheme. And these are Space and the Table of the Primary Conceptions.
    This is a point which we may say we have reached with the help of the scope (étendue) and thought (pensée) of Descartes. But a mere lively perception is not enough; it must also be correct. And in order that your view of the world may be the same as that of Kant I will cite two short passages. First as regards Space. “Space is nothing more than the mere form of all phenomena of the outward senses, that is to say, the subjective condition of the power of the senses, by which alone outer perception becomes possible. Now since the liability of the subject to be affected by circumstances necessarily precedes all perception of these objects, we can understand how the form of all phenomena can be given in the mind before all true perceptions, and how they, as a pure perception in which all objects must be fixed, are able to contain principles of the relations of the objects to one another before all experience.“ It is a little more difficult to find words for the pure primary conceptions


of the understanding, — words which, without a previous exegesis of the Kantian system, should be directly intelligible and yet express a great deal, — perhaps the following might serve the purpose. “Just as space implies the condition of perception in a possible experience, so are the Categories nothing more than the Conditions of thought in a possible experience; they are forms of thought which imply the power of uniting into one consciousness the Manifold which is given in perception; and since experience is knowledge by perceptions linked together, so the Categories are conditions of the possibility of experience.“ 63
    If you have paid attention to my request not to cramp your thought in a narrow gangway, but rather to yield yourselves openly and without reserve to a new method of perception, you surely will have succeeded in following me so far. Of the utmost importance are two notions which are easy to retain. Space is the necessary form, the Symbol, of all phenomena: the uniting of that which is manifold in experience into a single consciousness takes place by the intermediary of an immovable Scheme of thought. The little which remains to be said will offer no difficulties if only you never for a moment turn away from the principle of the perceptible incorporation of thought.
    As you will have gathered from his words, Kant believes in a condition of the power of the senses and in a condition of thought: it is the interplay between these two conditions that gives birth to “experience.“ And we may be sure that this conviction of Kant's does not rest upon logical system-mongering, but, quite on the contrary, on precise analytical observation of the functions of the mind; his method is, as he says himself, “imitated from that of the investigator of nature.“ It is here important in the first place to remark that, even if all the power of the senses is subjected to one condition — extension in space

Diagram— it is still the one thing which brings about the manifold in nature, whereas, on the other hand, it is the prime function of the many-branched understanding to bind this multiplicity into unity. Beyond the power of the senses, — it is impossible not to premise this — lies the objective world, a real chaos of multiplicity: on the hither side of the understanding lies — nothing! nothing but a unity, that unity which Kant calls Reason, and which is familiar to us as the true ego. According to Kant, moreover, there is, as you see, a progressive simplification and unification. Here again a schematic diagram will render us preliminary services. Think of the limitless objective world outside a circle.
    It is from that world that the senses take their impressions, and that too under the strictly simplified law that they force it by the compulsion of a single form, that of extension in space, to what, if I might so call it, is a “simple multiplicity.“ Then the understanding reduces this multiplicity to a few primary conceptions standing in relation to one another, and these primary conceptions coalesce in a consciousness of their unity which might be called Reason. In this way the circles are packed the one within the other.
    Here there are two things deserving of special attention: first the peculiar intermediary position and function of the understanding, and next the special relation of the inmost circle (Reason) to the outermost circle, that is to the surrounding world.
    Further in the way of simplification perception cannot go, inasmuch as it brings all the impressions under the


one form of extension; on the other hand, the understanding, if it were to proceed in the same fashion, would bring up no further simplification, but only a reflected image. For simply thinking as a merely formal conception, while it no doubt leads to the idea of oneness, can yet embrace all multiplicity: for example, to the mathematician the number 1 means the “infinitely great,“ and the single “type“ of the zoologist can embrace a boundless wealth of forms. The conception “unity“ gains a living meaning only when used to distinguish the idea of organic combination as opposed to mere formal fusion; as soon as the one conditions the other, by which in its turn it is conditioned, the two together form a true unity. True organic unity can never arise out of singularity, but only out of plurality. That is why the essence of thought is systematic organisation, dissection, conjunction. You cannot think without passing judgment, and you cannot pass the simplest judgment, — for instance “the room is big“ — unless you are in possession of three several conceptions, the subject, the predicate, the copula. Each of these three is derived from a special primary conception, substantia, existentia, multiplicitas. The leading simplification, carried out by the senses, is violent and coarse like the first preparation of some material, like the dressing of yarn; it is only later that the threads are woven into an organic unity. There must, therefore, be numerous primary conceptions, otherwise it would be impossible to introduce order, connection, unity (and that means sense) into the mass of impressions which are afforded by perception. Kant writes, “Combination does not exist in the things, and cannot be in any way borrowed from them, and so, in the first instance, be taken up through perception into the understanding, — but is a function of the understanding which is of itself no more than the power to combine and to bring into unity that which our senses give us as

manifold.“ That is a memorable saying. Our understanding is a power to combine and to reduce that which Is manifold into unity. 64 It will be understood moreover, I hope, that the multiplicity of conceptions in the face of the unity of the form of perception means a progressive unification, and that without this “wholesale conjunction“ there would be no such thing as knowledge, nothing but chaos, or as Kant puts it, a “rhapsody of perceptions.“ (R.V. 495.)
    The second condition which here deserves your attention is the intimate connection between the innermost circle and the outermost. The ego and the world stand in reciprocal interchange: each is necessary to the other: neither can be grasped, seen and dissected except so far as it is reflected in the other. The powers of the senses and understanding hover between two Unknowns: the one immeasurably great, the other without any magnitude, without space; the one imaginably rich in an inexhaustible multitude of forms, the other completely devoid of form, and for that very reason unthinkable. If we consider the relation from the standpoint of perceptibility, then we must say with Kant, “the world is the sum total of all phenomena“ (R.V. 391), the ego, on the contrary, “the poorest of all ideas“ (R.V. 408, 404), indeed “an idea empty of all contents.“ Still, if we pursue the matter conceptively we discover that the “world“ is really only an idea, an image in the focus imaginarius such as we made acquaintance with in the first lecture, an image projected out of the ego into the Inscrutable. 65 And so the two stand over against one another as correlatives: without the world no ego, without the ego no world.
    It is, of course, impossible for me to engage in a more searching discussion of this subject: but it will be worth your while to follow up the lead which I have given. Later on you will find in Kant the most fascinatingly


deep amplifications of the matter. My object for the moment has been above all to show accurately the twofold boundary of experience, because that is so essential to Kant's method of Seeing. I spoke a while ago of a wall, and said that Kant's principle was to confine himself within this boundary of experience. But there are indeed two walls. One wall inwards, and a second wall outwards, and it is the intervening space to which Kant confines himself as the only space of experience. And here again is something of which you must possess an exact and comprehensible idea, otherwise the next thing will be that you will once more fall into the clutches of the all-wise dogmatists, and will lose the moral and intellectual greatness as well as the scientific certainty of Kant's renunciation. It is manifest how empty is the purpose to try and solve the great riddles of existence, out of a nature which is of our own creation, of which the necessary laws are the laws of our own understanding: but Kant will tell you that the opposite proceeding is exactly as deceptive. There is no ego of experience which might serve as a foundation, upon which to raise a dogmatic erection either of the comprehension with Fichte and Hegel, or of the senses with Schopenhauer; the ego lies beyond, or, if you prefer it, on the hither side of, experience. You will see more clearly from the juxtaposition of the two following short formulae than from long arguments, the yawning gulf that separates the schools of philosophy; Schopenhauer teaches us that

    The world IS MY idea.

Kant says —

    MY WORLD is idea.

The difference is immense. For the one is a monstrous, indeed, if you look closely into it, a mad Dogma which presupposes Nature and the ego as peculiar existences, and then sets up categorical conclusions as to the relation


between them: the other is the simple affirmation of the result of critical reflection within the boundaries of experience — a reflection which teaches that whether inwardly or outwardly we are concerned with nothing except Symbols and Schemes, so that we can make no pronouncement about a “world“ beyond the fact that we human beings are compelled to imagine one. Besides this there results from Kant's doctrine the significant inversion, I am an idea of the world, a fact upon which Kant never wearies to descant, since this is really what is contained in the otherwise empty ego. Whereas from Schopenhauer's fundamental doctrine there is no result beyond the necessity of inventing a second dogma in addition to the first, which is what happens with the dictum “I am Will.“ 66
    But now and in conclusion there still remains a question for us to examine, which equally moves entirely within the frame of Descartes' philosophy and of the material for ideas which we have gained in the course of this lecture. With this object let us return to the region of experience and to our old Scheme, which I shall furnish with new terms, for now we shall look upon the matter “objectively“ instead of “subjectively.“ Understanding and the power of the senses have been considered as functions of the human mind; instead of that we will now take into consideration that which corresponds with those functions, so to speak, as object. Where before we wrote “the senses,“ we now write “space“; where we wrote “understanding“ we will now write “the primary conceptions of the understanding.“ But what are we to write in the hatched middle space? We learnt from the history of sciences, and specially from analytical geometry, that no transition takes place from the one side of the middle line of separation to the other except violently, suddenly, and through transcendental encroachment. That will clearly be the case here, for if in mathematics we were


dealing with quite pure, that is to say, quite human, ideas and conceptions, we have now risen a step higher: we have here space as the primitive form of all possible perception and the pure conceptions of the understanding, or categories, as the all-embracing primitive forms of all thought. What is to take the part here which mathematics played in empiricism? Where shall we find a transcendental commutator, a transcendental straight line?
    Before answering the question I must call attention to another trifling matter, because it will help towards the clear setting out of the problem. Everybody without exception finds it much easier to understand Kant's doctrine of space than his doctrine of the pure primitive conceptions of the understanding. Of the readers of the Reine Vernunft, perhaps ninety per cent do not get beyond the first part, which treats of the form of perception (space). And that does not happen, as one might imagine, because it is easier to understand that which is perceptible than that which is abstract. On the contrary, it arises from the fact that it is easier for a man to follow a logical demonstration than to accustom himself to a new and strange method of perception. It is from out of the understanding that space is contemplated, and therefore it is, as our first Scheme shows us, a “perception of thought,“ or if you please a symbolical thought. For that reason the argument can be presented in almost pure logical form with firstly, secondly, thirdly: nothing remains hazy. But the categories, on the contrary must, if we wish to understand them rightly, be approached from the side of the senses, that is to say, we must contemplate them as Schemes, and yet see them. Pure conceptions of the understanding cannot be further analysed and explained in terms of logic, for they themselves are the simplest elements of thought: a subject which can only be grasped by perception, defies all definition. You may define space, but you cannot

define the single pure primary conceptions of the understanding, at any rate not logically. The name “space“ expresses something fixed; whereas all names for primary conceptions are mere helps in need, only, as it stands in the Dissertation, — cognitio symbolica. (D. § 10.) The name substance (or stability), for instance, is nothing but hocus-pocus until you learn to understand that we are here face to face with “perceptible thought,“ a pre-logical thought, a thought lying on that middle territory where conceptions are first born of the union with the senses. The same observation applies to causality, reality, etc. Here, standing as we do upon the two topmost rungs of the two ladders, I can only see the one when I take my place on the other. That is the one great, perhaps the greatest, difficulty against which Kant has to fight, and a chief cause of the much complained of “darkness“ of his philosophy. If the primary conceptions of our human thought were abstract, self-reliant thoughts, we should in any case be able to talk about them; but they are nothing outside their relation to the power of the senses: that at any rate is Kant's view, and that is the reason why nobody by mere thought and without joining with it active ideas, can grasp Kant's real meaning upon the subject of the primary conceptions. He says, “the categories afford us no knowledge of things excepting by their possible application to empirical perception,“ and hence “we cannot really define any single category without having recourse to conditions of the senses, therefore to the form of phenomena to which they must consequently be confined as their only objects, because if this condition is removed all significance, that is to say, relation to the object, falls away, and it is then impossible for us by any example to make ourselves grasp what sort of thing is meant by such-like conceptions.“ (R.V. 147, 300.)
    Here again I am forced to content myself with a mere


suggestion: but you will very soon realise what is the purport of this reminder.
    Supposing then that the primary conceptions of our understanding existed without any relation to the senses, it would be impossible to see how the requisite “application to perception“ could take place, and our knowledge could in no case be a “knowledge of things“: it would be at best a “cloud-cuckoo-town“ * knowledge or, as Kant puts it, a too ambitious knowledge. It would be just the same as if we imagined a mathematical proposition, purely logical, incapable of being turned into anything perceptible, a calculation with an unknown x, y, z, in which everything would naturally fit in correctly without conveying the slightest meaning. And the same holds good, though of course in an inverted sense, in the case of space, as to which we have already seen in Newton, that thought cannot easily grasp it, and yet loves to busy itself with it. And you will understand me when I maintain: that that “transcendental straight line“ of our research, which is to serve us as a commutator between the primary conceptions of thought and the primary form of perception, must not only, like mathematics, be at the same time perception and thought, but must turn its perceptive side to thought, and its thinking side to perception.
    These are conditions which Time alone can fulfil.
    Time is at once conception and perception; Kant introduces it sometimes as the one, sometimes as the other. He calls it “Inner perception,“ or “Form of inner perception,“ but then again “Form of the inner sense.“ 67 However you look at it, it always remains something “inner,“ because like its empirical embodiment, mathematics, it fills the inner or middle domain. And yet it is in so far something “outer“ as it serves to transport each of the two fundamental functions of our

    * See the “Birds“ of Aristophanes.


knowledge out of itself and into the other form. “The conception of change, and with it the conception of motion as change of place, is only intelligible through and in the idea of Time; unless this idea were perception


(inner perception) no conception, whatever it might be, would make the possibility of a change comprehensible.“ 68 It is only in Time therefore that a world can exist for us. That is one side of the question. On the other side it is precisely Time that gives us the idea of stability, and as such the form of the perception of ourselves and of our subjective condition. For, as Kant says, “Time can be no definition of outer phenomena: it belongs neither to form nor position, etc.; on the other hand, it does define the relation of ideas in our subjective condition.“ 69 Without Time, then, no Ego! and not only no Ego, but generally no conception of substance, that is to say, no idea of any object which remains stable in the midst of all change. We have to thank Time for motion and stability, for development and being, for World and Ego.


    The analogy with mathematics must strike you at once: and you will perceive that in our new Scheme we have placed our indications on their proper sides; for Time as stability manifestly corresponds with Geometry which is mathematical form, while Time as fluent motion is the foundation of the conception of numbers. 70 It is true that time as stability is not form, but it furnishes the commutator, the salient point at which the idea of stability arises through the transcendental union of understanding and the power of the senses, that is to say, of space and conception. In the same way the logically unthinkable conception of a change of place in space is impossible without a similar intermediary position of Time.
    Let me go more closely into what I have hinted at. For the function of Time must be made clear to you without any reservation, and it can be made so if you will only take Descartes' analytical geometry as your counsellor step by step. Descartes would have been unable to carry out the critique of the human intellect as Kant did, and yet in the practice of his method of thinking he has shown a manner of Seeing which is in complete harmony with that of Kant. The part played in Descartes' mathematics by the straight line, is played in Kant's analysis of the human intellect by Time. 71 As you will remember, the straight line is not form, inasmuch as it is the only visible thing which gives birth to no form; and it is not a number, inasmuch as every straight line can represent every possible number, and therefore remains entirely indifferent to the conception of numbers: and in spite of that it alone furnishes a footing for the conversion of form into numbers and vice versa. It is just the same with Time. It lies altogether outside of the true conceptions of the understanding; it also lies outside of all perception. Time, as we said before, is at once conception and perception; now we will speak with

greater precision and say: Time is neither perception nor conception, but can never be far removed from either; that is to say, without Time we can neither perceive nor conceive. That is why Kant calls Time “the constant correlatum of all existence.“ (R.V. 226.) Space too is “no real object which can be outwardly perceived,“ but is “the very form of phenomena“ (R.V. 459), and indeed in the famous dispute as to whether it was possible or not that there should exist an empty space, — in which Descartes with unerring genius took the part of the impossibility, Newton with the childish simplicity which was peculiar to him asserted the certainty of this monstrous idea, — our abstract jingle of words is based upon a perception which is at least imagined. Time, on the other hand, offers no handle to perception, for where we conceive it as extension and quasi-visible, that happens by means of an allegory inasmuch as we draw a line in our thoughts and analogically use it to represent Time, which “outward figurative idea of Time“ Kant has discussed in detail. (R.V. 154, seq.) Neither has thought any more power of comprehending Time; it always contrives to elude thought. St. Augustine with a sigh says, et confiteor tibi, Domine, adhuc ignorare me quid sit tempus. 72 The old Greeks had already discovered that Time could only exist for the thinker who measures and counts its hours, and Descartes, who has no objection to predicate “lasting,“ that is “stability,“ of things, gives it as his opinion that le temps n'est rien qu'une façon de penser. 73 And yet the measures of time deduced from various motions, such as those of heavenly bodies, for instance, do not correspond to the actual lived or living life of a man: for him a minute may contain years, years may glide by unobserved like a short autumn morning: he measures Time not by length, but by gradation, that is to say, by the sensations which it contains. Here there is no possibility of bridging over the gulf, our double nature asserts itself too abruptly.


    If ever you study Kant's Reine Vernunft you must not linger too long over the preliminary doctrine of the senses; for here unfortunately Time is at first dealt with as if it belonged to space, and indeed to space alone, and that means, therefore, to the power of the senses. That has given birth to a misunderstanding widely spread and disseminated by Schopenhauer into the remotest strata of the imperfectly cultured, by which Space and Time are represented as two forms of perception of equal value and parallel to one another. To talk of the “ideality of Space and Time“ has become a commonplace platitude; and yet we have to deal with two completely different things: Space is the only form of all pure perception; Time is an intermediary between perception and understanding, which in itself and by itself can neither be perceived nor imagined. That is why I recommend you to hurry on further through Kant's work, till you reach the place where it will be shown to you how the conjunction between understanding and the power of the senses (i.e. between the primary conceptions and the idea of space), takes place. Here you will once again meet Time introduced as a twin-sister of space, but as precisely in the same relationship to understanding, and so “as an intermediary idea, on the one side a matter of the intellect, on the other of the senses“; on the one side “similar to the category, on the other side similar to phenomenon in so far as Time is contained in every empirical idea of the Manifold.“ Then will Kant's perception for the first time really become clear to you. Time is an intermediary idea, on the one side belonging to the intellect, on the other to the senses: what that means you now understand thoroughly and in detail. You need only think of the analogously intermediate part which the straight line plays in mathematics; and you will see clearly how important such an intermediary is for our whole intellectual life, if you remember from the


history of our sciences which we touched upon at the beginning of the lecture, that nothing which has been observed can be thought without the intermediary of a Scheme, and no thought can gain any constructive value without the intermediary of a symbol: Time is the foundation of all these intermediary processes. And now if you look at the title of this chapter of the Reine Vernunft, you will discover with amazement that this chapter so impatiently awaited because it is the most indispensable of all, the one by which at last the goal is reached, this brilliant solution of riddles, the famous, dreaded chapter decried as impenetrably dark, of which I spoke at the beginning of this passage, is the chapter on “the schematisation of the pure conceptions of the understanding.“ And I think you will ask with me, what sort of perception can there be in the brains of a professional, state-paid commentator on Kant, who singles out this chapter to condemn it as valueless? 74
    That, however, is a matter of indifference. If Kant himself in the chapter in question asks, as he does literally, “How is it possible that pure conceptions of the understanding can be applied to phenomena?“ we cannot but think him fully justified. As he says, “It is clear that there must be a third.“ We too see that clearly. And when after proving that this “third“ must be Time — a demonstration of the all-conquering power of conviction — he goes on to show that the combination of the “first“ and the “second“ which takes place inside this “third“ is no fusion, but nothing more and nothing less than a “placing of the two together in relation to one another“ by intermediation; and when he calls this intermediation a Scheme, and consequently makes the relations of the primary conceptions of the understanding to the nature of our impressions of the senses, take place through the intermediary of a schematisation of pure understanding; — then all this is perfectly clear and


natural and according to expectation; we knew it already from empiricism and mathematics: what is new to us is at most the fact that this relation is fundamental in all human knowledge without exception, since it is that which gives birth to what may be called Experience. In my humble opinion there is only one thing lacking to make Kant perfectly clear: that is to say, a chapter upon “the symbolism of the pure power of the senses.“ Not that I should wish to obliterate Kant's distinction between spontaneity and receptivity, between function and inclination, between activity and passivity, 75 as characteristic in the understanding and in the power of the senses, — for mathematics and Time itself teach us, as Kant himself has done, that we not only comprehend our perceptions, but also perceive our thoughts, and that both these processes, not one alone, take place through the intermediary of that “third.“ Because Kant at the beginning of his critique lays stress upon a one-sided view of the relation of Time to Space, the reader is taken by surprise when he finds it brought into an equally close relation to the comprehensions of the understanding, and described as “a third“; and next the first one-sidedness is amplified by a second, since now he only lays stress upon the schematic intermediation of Time, therefore, to use Kant's expression, upon the way in which the power of the senses realises understanding, not on the way in which the understanding realises the power of the senses, that is to say, the symbolising activity of Time. 76 Yet there can be no doubt that we have correctly represented Kant's perception: the whole Kritik der Reinen Vernunft and that of the Urteilskraft (power of judgment) bears witness to that; Kant may have had good metaphysical reasons for his inconsistent exposition — indeed, he certainly did have them, though they have not come under our observation here. We have only been dealing with his method of Seeing, and of the many

proofs which have been laid before you, I need only remind you of one, which furnished the clue to the whole examination of Descartes: “Thoughts without contents are empty, perceptions without conceptions are blind. Therefore it is just as necessary to make our comprehensions obvious to the senses, — i.e. to add to them the object of perception, — as to make our perception comprehensible to ourselves, — i.e. to subject it to comprehensions.“ To bring one's perceptions under conceptions is called Schematising: to make conceptions obvious to the senses is called Symbolising. Neither can take place except by one and the same intermediation, that is to say, through a single element essentially unified, even though it should appear iridised in two colours, — otherwise no unity would exist in the understanding — that transcendental unity, which arises in consciousness by the combination of Scheme and Symbol. Time is that commutator. Precisely where the middle line separates, there the two-sided “commutation“ takes place, the conversion of the one into the other. We saw it in mathematics, we saw it in the empirical sciences, we shall be aware of it in every single one of our thoughts, as soon as we have been attentive and have learnt to appreciate the indispensable intermediation of the Proteus Time.
    One thing must be fixedly borne in mind, that for Kant, Time, like mathematics, is a purely formal principle. For that reason and because its special function is combination, therefore it is present everywhere, in every thought and in every perception. In order to communicate my comprehensions to the senses, I need Time: in order to make my perceptions comprehensible, again I need Time. Two examples: the Aether is little more than the vanishing thought of stability hardly felt by the senses: the observations on motion teach us that the same point may be in two places, which would have


no sense for pure understanding, unless the drawing of a line of Time made it thinkable. Even so Time can nowhere, neither in thought nor in perception, be grasped otherwise than as something existing for itself. Thought and perception cannot exist apart from Time, yet Time is nothing apart from its relation to thought and perception; its essence is to be the fundamental relation of all relations, that through which relations as a general principle arise and have their being. Everywhere the strict analogy with mathematics! and therefore for us the relatively easy mastery of an otherwise so difficult, so incomprehensible subject.
    If you wish briefly to sum up what twofold Time has achieved on behalf of our knowledge you may say: since Time as Stability is the means of subjecting the manifold power of the senses under the yoke of the conceptions of the understanding, it bestows upon that power Unity: inasmuch as Time as Motion combines the unity of the inner sense (i.e. the unity of Reason) with the power of the senses, here it bestows manifoldness. 77 Two examples: Kant has shown how every one of our primary conceptions, magnitude, gradation, causality, reciprocal action, reality, necessity, etc., grasps the matter of perception by the intermediation of a Scheme of Time, and draws it together into unity; 78 on the other hand, Kant has also shown that every affection of the senses can only be perceived as motion, from which he draws the definition, “The fundamental principle of a Something which is an object of the outer senses must be Motion,“ and motion equally demands Time as a correlative, and can only by the intermediation of Time give manifoldness to thought. And now at last the knot is tied fast, since understanding assimilates motion, and produces perfect Scheme, while perception takes causality, reciprocal action, necessity, and other pure primary thoughts, and amalgamates them so completely with


that which has been observed, that it almost fancies that it can see them with its eyes.
    We too must tie a famous knot to end this long and laborious lecture. You remember how as a condition for the understanding of Descartes' special talent we set up the formula that he knew how to make the invisible visible, and the visible invisible. How far this formula is applicable to Kant you now know too accurately for any further explanation to be necessary. The saying, put so simply and abruptly, has but little significance if it is used in connection with Kant's rich world of thought. Still, even so it can render certain services in this direction. That the invisible comprehension is powerless till the actual visible object has been offered to it by perception, and that, on the other hand, this same perception remains blind, unless comprehensions transfer this visibility into the invisibility of the world of thought — this, combined with the doctrine that it is Time which schematically and symbolically cares for the hither and thither of the transformations: this it is which taken together makes up the essence of Kant's perception in regard to human knowledge. Now the limitation to experience, forms, as you will remember, an indispensable part of this perception of Kant's. What we see are only appearances due to the twofold conditions of Form of the Senses and Schematisation of the understanding: and if we add “the third,“ Time, the conditioned phenomena become threefold. As for the things themselves, and what may be their essence, we have neither the disposition nor the possibility to form an opinion, and it is just such a riddle that the individual remains to himself. Yet we cannot prevent two powerful ideas from growing out of this experience of ours, however strictly we may imprison them within their double rampart: — the World and the Ego. We have already spoken of this, and I will only add that World and Ego are as it were the two ends of the knot that I have in


view: the Visible and the Invisible κατ' έξοχήν. The World, the visible end, is nothing else but that threefold conditioned phenomenon in its highest, all-embracing potentiality, the symbol of all symbols; if I remove the Ego to which it appears, nothing remains, nothing, that is to say, that would have any possible meaning for us men; for it is only the Ego that can bring forward the idea of the World. Yet the converse holds equally good. The Ego can be neither thought nor perceived unless it be mirrored by the World; if I remove the World, the Ego, the Inner thing, fades away. What remains is an empty Scheme of all Schemes, that is to say, a Nothing. Here too there is an interchange, and we can tie a knot as we bend the Inner outwards, the Outer inwards. Nothing hinders us from conceiving the Ego as the invisible World, the World as the visible Ego. That we are accustomed to look upon the senses as outer, the understanding as inner, is after all nothing more than a convention, than a superficial analogical deduction from the organs of sense and the brain in the bodies of the vertebrate animals. The diagram on p. 289 might just as well be reversed; reason or the Ego the all-embracing circle, the World in the inmost circle.
    In this way do relations complicate themselves as soon as we cease to limit ourselves to the domain of experience. This limitation, however, is not always possible. We cannot simply go to the order of the day about Ego and World, about Soul and God. And so many an idle chatterer, and also many a noble man, and among the latter none bolder than Giordano Bruno, has soared aloft upon the wings of fictitious knowledge, in order to solve the riddle of the world and the riddle of his own being outside the boundaries of experience. In what a different spirit Kant set to work upon such questions, you will have suspected from our work to-day, and you will guess that he must also have reached different results:


I hope that this will show itself fully as a result of a comparison with Bruno. In his case, as in that of Descartes, we have to deal with a specific thinker, and yet their attitude towards the material at issue is almost directly opposite: for whereas Descartes only exercises the critical function in the domain of knowledge as a discipline of limitation in order to be able to devote himself in safety and freedom to empirical observation, and to the hypothetical and theoretical significance of concrete Nature, Bruno lives only in the empyrean of abstraction and speculation, and accredits human reason and its logical inferences with all knowledge and all power. That is why he has to take up an essentially different relation to Kant. In order to promote the interests of our investigation we gave precedence, whilst dealing with Descartes, to the similarity with Kant, leaving unnoticed the points in which they differed; in Giordano Bruno we shall, on the contrary, gain our brightest illumination from the points of difference. And so we shall let our day of empiricism and the critique of experience be followed by a morrow of dialectics and dreams.


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