The World as Will and Representation

of in words, but for the idea, these words are absolutely meaningless. On the other hand, the law of causality and the treatment and investigation of nature which is based upon it, lead us necessarily to the conclusion that, in time, each more highly organised state of matter has succeeded a cruder state: so that the lower animals existed before men, fishes before land animals, plants before fishes, and the unorganised before all that is organised; that, consequently, the original mass had to pass through a long series of changes before the first eye could be opened. And yet, the existence of this whole world remains ever dependent upon the first eye that opened, even if it were that of an insect. For such an eye is a necessary condition of the possibility of knowledge, and the whole world exists only in and for knowledge, and without it is not even thinkable. The world is entirely idea, and as such demands the knowing subject as the supporter of its existence. This long course of time itself, filled with innumerable changes, through which matter rose from form to form till at last the first percipient creature appeared,—this whole time itself is only thinkable in the identity of a consciousness whose succession of ideas, whose form of knowing it is, and apart

from which, it loses all meaning and is nothing at all. Thus we see, on the one hand, the existence of the whole world necessarily dependent upon the first conscious being, however undeveloped it may be; on the other hand, this conscious being just as necessarily entirely dependent upon a long chain of causes and effects which have preceded it, and in which it itself appears as a small link. These two contradictory points of view, to each of which we are led with the same necessity, we might again call an antinomy in our faculty of knowledge, and set it up as the counterpart of that which we found in the first extreme of natural science. The fourfold antinomy of Kant will be shown, in the criticism of his philosophy appended to this volume, to be a groundless delusion. But the necessary contradiction which at last presents itself to us here, finds its solution in the fact that, to use Kant’s phraseology, time, space, and causality do not belong to the thing-in-itself, but only to its phenomena, of which they are the form; which in my language means this: The objective world, the world as idea, is not the only side of the world, but merely its outward side; and it has an entirely different side—the side of its

inmost nature—its kernel—the thing-in-itself. This we shall consider in the second book, calling it after the most immediate of its objective manifestations—will. But the world as idea, with which alone we are here concerned, only appears with the opening of the first eye. Without this medium of knowledge it cannot be, and therefore it was not before it. But without that eye, that is to say, outside of knowledge, there was also no before, no time. Thus time has no beginning, but all beginning is in time. Since, however, it is the most universal form of the knowable, in which all phenomena are united together through causality, time, with its infinity of past and future, is present in the beginning of knowledge. The phenomenon which fills the first present must at once be known as causally bound up with and dependent upon a sequence of phenomena which stretches infinitely into the past, and this past itself is just as truly conditioned by this first present, as conversely the present is by the past. Accordingly the past out of which the first present arises, is, like it, dependent upon the knowing subject, without which it is nothing. It necessarily happens, however, that this first present does not manifest itself as the first, that is, as having no past for its parent,

but as being the beginning of time. It manifests itself rather as the consequence of the past, according to the principle of existence in time. In the same way, the phenomena which fill this first present appear as the effects of earlier phenomena which filled the past, in accordance with the law of causality. Those who like mythological interpretations may take the birth of Kronos (χρονος), the youngest of the Titans, as a symbol of the moment here referred to at which time appears, though, indeed it has no beginning; for with him, since he ate his father, the crude productions of heaven and earth cease, and the races of gods and men appear upon the scene.

This explanation at which we have arrived by following the most consistent of the philosophical systems which start from the object, materialism, has brought out clearly the inseparable and reciprocal dependence of subject and object, and at the same time the inevitable antithesis between them. And this knowledge leads us to seek for the inner nature of the world, the thing-in-itself, not in either of the two elements of the idea, but in something quite distinct from it, and which is not encumbered with such a fundamental and insoluble antithesis.

Opposed to the system we have explained, which starts from the object in order to derive the subject from it, is the system which starts from the subject and tries to derive the object from it. The first of these has been of frequent and common occurrence throughout the history of philosophy, but of the second we find only one example, and that a very recent one; the “philosophy of appearance” of J. G. Fichte. In this respect, therefore, it must be considered; little real worth or inner meaning as the doctrine itself had. It was indeed for the most part merely a delusion, but it was delivered with an air of the deepest earnestness, with sustained loftiness of tone and zealous ardour, and was defended with eloquent polemic against weak opponents, so that it was able to present a brilliant exterior and seemed to be something. But the genuine earnestness which keeps truth always steadfastly before it as its goal, and is unaffected by any external influences, was entirely wanting to Fichte, as it is to all philosophers who, like him, concern themselves with questions of the day. In his case, indeed, it could not have been otherwise. A man becomes a philosopher by reason of a certain perplexity, from which he seeks to free himself. This is Plato’s

θαυμαξειν, which he calls a μαλα φιλοσοφικον παθος. But what distinguishes the false philosopher from the true is this: the perplexity of the latter arises from the contemplation of the world itself, while that of the former results from some book, some system of philosophy which is before him. Now Fichte belongs to the class of the false philosophers. He was made a philosopher by Kant’s doctrine of the thing-in-itself, and if it had not been for this he would probably have pursued entirely different ends, with far better results, for he certainly possessed remarkable rhetorical talent. If he had only penetrated somewhat deeply into the meaning of the book that made him a philosopher, “The Critique of Pure Reason,” he would have understood that its principal teaching about mind is this. The principle of sufficient reason is not, as all scholastic philosophy maintains, a veritas aeterna—that is to say, it does not possess an unconditioned validity before, outside of, and above the world. It is relative and conditioned, and valid only in the sphere of phenomena, and thus it may appear as the necessary nexus of space and time, or as the law of causality, or as the law of the ground of knowledge. The inner nature of the world, the thing-in-itself can never be found by the guidance

of this principle, for all that it leads to will be found to be dependent and relative and merely phenomenal, not the thing-in-itself. Further, it does not concern the subject, but is only the form of objects, which are therefore not things-in-themselves. The subject must exist along with the object, and the object along with the subject, so that it is impossible that subject and object can stand to each other in a relation of reason and consequent. But Fichte did not take up the smallest fragment of all this. All that interested him about the matter was that the system started from the subject. Now Kant had chosen this procedure in order to show the fallacy of the prevalent systems, which started from the object, and through which the object had come, to be regarded as a thing-in-itself. Fichte, however, took this departure from the subject for the really important matter, and like all imitators, he imagined that in going further than Kant he was surpassing him. Thus he repeated the fallacy with regard to the subject, which all the previous dogmatism had perpetrated with regard to the object, and which had been the occasion of Kant’s “Critique”. Fichte then made no material change, and the fundamental fallacy, the assumption of a relation of reason and consequent between object

and subject, remained after him as it was before him. The principle of sufficient reason possessed as before an unconditioned validity, and the only difference was that the thing-in-itself was now placed in the subject instead of, as formerly, in the object. The entire relativity of both subject and object, which proves that the thing-in-itself, or the inner nature of the world, is not to be sought in them at all, but outside of them, and outside everything else that exists merely relatively, still remained unknown. Just as if Kant had never existed, the principle of sufficient reason is to Fichte precisely what it was to all the schoolmen, a veritas aeterna. As an eternal fate reigned over the gods of old, so these aeternæ veritates, these metaphysical, mathematical and metalogical truths, and in the case of some, the validity of the moral law also, reigned over the God of the schoolmen. These veritates alone were independent of everything, and through their necessity both God and the world existed. According to the principle of sufficient reason, as such a veritas aeterna, the ego is for Fichte the ground of the world, or of the non-ego, the object, which is just its consequent, its creation. He has therefore taken good care to avoid examining further or limiting the principle of sufficient

reason. If, however, it is thought I should specify the form of the principle of sufficient reason under the guidance of which Fichte derives the non-ego from the ego, as a spider spins its web out of itself, I find that it is the principle of sufficient reason of existence in space: for it is only as referred to this that some kind of meaning and sense can be attached to the laboured deductions of the way in which the ego produces and fabricates the non-ego from itself, which form the content of the most senseless, and consequently the most wearisome book that was ever written. This philosophy of Fichte, otherwise not worth mentioning, is interesting to us only as the tardy expression of the converse of the old materialism. For materialism was the most consistent system starting from the object, as this is the most consistent system starting from the subject. Materialism overlooked the fact that, with the simplest object, it assumed the subject also; and Fichte overlooked the fact that with the subject (whatever he may call it) he assumed the object also, for no subject is thinkable without an object. Besides this he forgot that all a priori deduction, indeed all demonstration in general, must rest upon some necessity, and that all necessity is based on the

principle of sufficient reason, because to be necessary, and to follow from given grounds are convertible conceptions.(10) But the principle of sufficient reason is just the universal form of the object as such. Thus it is in the object, but is not valid before and outside of it; it first produces the object and makes it appear in conformity with its regulative principle. We see then that the system which starts from the subject contains the same fallacy as the system, explained above, which starts from the object; it begins by assuming what it proposes to deduce, the necessary correlative of its starting-point.

The method of our own system is toto genere distinct from these two opposite misconceptions, for we start neither from the object nor from the subject, but from the idea, as the first fact of consciousness. Its first essential, fundamental form is the antithesis of subject and object. The form of the object again is the principle of sufficient reason in its various forms. Each of these reigns so absolutely in its own class of ideas that, as we have seen, when the special form of the principle of sufficient reason which governs any class of ideas is known, the nature of the whole class is known also: for the whole class, as idea, is no more than this form of the principle of sufficient reason itself; so that time itself is nothing but the principle of existence in it, i.e., succession; space is nothing but the principle of existence in it, i.e., position; matter is nothing but causality; the concept (as will appear immediately) is nothing but relation to a ground of knowledge. This thorough and consistent relativity of the world as idea, both according to its universal form (subject and object), and according to the form which is subordinate to this (the principle of sufficient reason) warns us, as we said before, to seek the inner nature of the world in an aspect of it which is quite different and quite distinct from the idea; and in the next book we shall find this in a fact which is just as immediate to every living being as the idea.

But we must first consider that class of ideas which belongs to man alone. The matter of these is the concept, and the subjective correlative is reason, just as the subjective correlative of the ideas we have already considered was understanding and sensibility, which are also to be attributed to all the lower animals.(11)

As from the direct light of the sun to the borrowed light of the moon, we pass from the immediate idea of perception, which stands by itself and is its own warrant, to reflection, to the abstract, discursive concepts of the reason, which obtain their whole content from knowledge of perception, and in relation to it. As long as we continue simply to perceive, all is clear, firm, and certain. There are neither questions nor doubts nor errors; we desire to go no further, can go no further; we find rest in perceiving, and satisfaction in the present. Perception suffices for itself, and therefore what springs purely from it, and remains true to it, for example, a genuine work of art, can never be false, nor can it be discredited through the lapse of time, for it does not present an opinion but the thing itself. But with abstract knowledge, with reason, doubt and error appear in the theoretical, care and sorrow in the practical. In the idea of perception, illusion may at moments take the place of the real; but in the sphere of abstract thought, error may reign for a thousand years, impose its yoke upon whole nations, extend to the noblest impulses of humanity, and, by the help of its slaves and its dupes, may chain and

fetter those whom it cannot deceive. It is the enemy against which the wisest men of all times have waged unequal war, and only what they have won from it has become the possession of mankind. Therefore it is well to draw attention to it at once, as we already tread the ground to which its province belongs. It has often been said that we ought to follow truth even although no utility can be seen in it, because it may have indirect utility which may appear when it is least expected; and I would add to this, that we ought to be just as anxious to discover and to root out all error even when no harm is anticipated from it, because its mischief may be very indirect, and may suddenly appear when we do not expect it, for all error has poison at its heart. If it is mind, if it is knowledge, that makes man the lord of creation, there can be no such thing as harmless error, still less venerable and holy error. And for the consolation of those who in any way and at any time may have devoted strength and life to the noble and hard battle against error, I cannot refrain from adding that, so long as truth is absent, error will have

free play, as owls and bats in the night; but sooner would we expect to see the owls and the bats drive back the sun in the eastern heavens, than that any truth which has once been known and distinctly and fully expressed, can ever again be so utterly vanquished and overcome that the old error shall once more reign undisturbed over its wide kingdom. This is the power of truth; its conquest is slow and laborious, but if once the victory be gained it can never be wrested back again.

Besides the ideas we have as yet considered, which, according to their construction, could be referred to time, space, and matter, if we consider them with reference to the object, or to pure sensibility and understanding (i.e., knowledge of causality), if we consider them with reference to the subject, another faculty of knowledge has appeared in man alone of all earthly creatures, an entirely new consciousness, which, with very appropriate and significant exactness, is called reflection. For it is in fact derived from the knowledge of perception, and is a reflected appearance of it. But it has assumed a nature fundamentally different. The forms of perception do not affect it, and even the principle of sufficient reason which reigns over all objects has an entirely different aspect with regard to it. It is just this new, more highly endowed, consciousness, this abstract reflex of all that belongs to perception in that conception of the reason which has nothing to do with perception, that gives to man that thoughtfulness which distinguishes his consciousness so entirely from that of the lower animals, and through which his whole behaviour upon earth is so different from that of his irrational fellow-creatures. He far surpasses them in power and also in suffering. They live in the present alone, he lives also in the future and

the past. They satisfy the needs of the moment, he provides by the most ingenious preparations for the future, yea for days that he shall never see. They are entirely dependent on the impression of the moment, on the effect of the perceptible motive; he is determined by abstract conceptions independent of the present. Therefore he follows predetermined plans, he acts from maxims, without reference to his surroundings or the accidental impression of the moment. Thus, for example, he can make with composure deliberate preparations for his own death, he can dissemble past finding out, and can carry his secret with him to the grave; lastly, he has an actual choice between several motives; for only in the abstract can such motives, present together in consciousness, afford the knowledge with regard to themselves, that the one excludes the other, and can thus measure themselves against each other with reference to their power over the will. The motive that overcomes, in that it decides the question at issue, is the deliberate determinant of the will, and is a sure indication of its character. The brute, on the other hand, is determined by the present impression; only the fear of present compulsion can constrain its desires, until at last this fear has become custom, and as such continues to determine it; this

is called training. The brute feels and perceives; man, in addition to this, thinks and knows: both will. The brute expresses its feelings and dispositions by gestures and sounds; man communicates his thought to others, or, if he wishes, he conceals it, by means of speech. Speech is the first production, and also the necessary organ of his reason. Therefore in Greek and Italian, speech and reason are expressed by the same word; ὁ λογος, il discorso. Vernunft is derived from vernehmen, which is not a synonym for the verb to hear, but signifies the consciousness of the meaning of thoughts communicated in words. It is by the help of language alone that reason accomplishes its most important achievements,—the united action of several individuals, the planned co-operation of many thousands, civilisation, the state; also science, the storing up of experience, the uniting of common properties in one concept, the communication of truth, the spread of error, thoughts and poems, dogmas and superstitions. The brute first knows death when it dies, but man draws consciously nearer to it every hour that he lives; and this makes life at times a questionable good even to him who has not recognised this character of constant annihilation in the whole of life. Principally on this account man has philosophies and religions, though it is

uncertain whether the qualities we admire most in his conduct, voluntary rectitude and nobility of feeling, were ever the fruit of either of them. As results which certainly belong only to them, and as productions of reason in this sphere, we may refer to the marvellous and monstrous opinions of philosophers of various schools, and the extraordinary and sometimes cruel customs of the priests of different religions.

It is the universal opinion of all times and of all nations that these manifold and far-reaching achievements spring from a common principle, from that peculiar intellectual power which belongs distinctively to man and which has been called reason, ὁ λογος, το λογιστικον, το λογιμον, ratio. Besides this, no one finds any difficulty in recognising the manifestations of this faculty, and in saying what is rational and what is irrational, where reason appears as distinguished from the other faculties and qualities of man, or lastly, in pointing out what, on account of the want of reason, we must never expect even from the most sensible brute. The philosophers of all ages may be said to be on the whole at one about this general knowledge of reason, and they have also given prominence to several very important manifestations of it; such as, the control of the emotions and passions, the capacity for drawing conclusions and formulating general principles, even such as are true prior to all experience, and so forth. Still all their explanations of the peculiar nature of reason are wavering, not clearly defined, discursive, without unity and concentration; now laying stress on one manifestation, now on another, and therefore often at variance with each other. Besides this, many start from the opposition between reason and revelation, a distinction

which is unknown to philosophy, and which only increases confusion. It is very remarkable that up till now no philosopher has referred these manifold expressions of reason to one simple function which would be recognised in them all, from which they would all be explained, and which would therefore constitute the real inner nature of reason. It is true that the excellent Locke in the “Essay on the Human Understanding” (Book II., ch. xi., §§ 10 and 11), very rightly refers to general concepts as the characteristic which distinguishes man from the brutes, and Leibnitz quotes this with full approval in the “Nouveaux Essais sur l’Entendement Humaine” (Book II., ch. xi., §§ 10 and 11.) But when Locke (in Book IV., ch. xvii., §§ 2 and 3) comes to the special explanation of reason he entirely loses sight of this simple, primary characteristic, and he also falls into a wavering, undetermined, incomplete account of mangled and derivative manifestations of it. Leibnitz also, in the corresponding part of his work, behaves in a similar manner, only with more confusion and indistinctness. In the Appendix, I have fully considered how Kant confused and falsified the conception of the nature of reason. But whoever will take the trouble to go through in this reference the mass of philosophical writing which has appeared since

Kant, will find out, that just as the faults of princes must be expiated by whole nations, the errors of great minds extend their influence over whole generations, and even over centuries; they grow and propagate themselves, and finally degenerate into monstrosities. All this arises from the fact that, as Berkeley says, “Few men think; yet all will have opinions.”

The understanding has only one function—immediate knowledge of the relation of cause and effect. Yet the perception of the real world, and all common sense, sagacity, and inventiveness, however multifarious their applications may be, are quite clearly seen to be nothing more than manifestations of that one function. So also the reason has one function; and from it all the manifestations of reason we have mentioned, which distinguish the life of man from that of the brutes, may easily be explained. The application or the non-application of this function is all that is meant by what men have everywhere and always called rational and irrational.(12)

Concepts form a distinct class of ideas, existing only in the mind of man, and entirely different from the ideas of perception which we have considered up till now. We can therefore never attain to a sensuous and, properly speaking, evident knowledge of their nature, but only to a knowledge which is abstract and discursive. It would, therefore, be absurd to demand that they should be verified in experience, if by experience is meant the real external world, which consists of ideas of perception, or that they should be brought before the eyes or the imagination like objects of perception. They can only be thought, not perceived, and only the effects which men accomplish through them are properly objects of experience. Such effects are language, preconceived and planned action and science, and all that results from these. Speech, as an object of outer experience, is obviously nothing more than a very complete telegraph, which communicates arbitrary signs with the greatest rapidity and the finest distinctions of difference. But what do these signs mean? How are they interpreted? When some one speaks, do we at once translate his words into pictures of the fancy, which instantaneously flash upon us, arrange and link themselves together, and assume form and colour according to the words that are poured forth, and their grammatical inflections?

What a tumult there would be in our brains while we listened to a speech, or to the reading of a book? But what actually happens is not this at all. The meaning of a speech is, as a rule, immediately grasped, accurately and distinctly taken in, without the imagination being brought into play. It is reason which speaks to reason, keeping within its own province. It communicates and receives abstract conceptions, ideas that cannot be presented in perceptions, which are framed once for all, and are relatively few in number, but which yet encompass, contain, and represent all the innumerable objects of the actual world. This itself is sufficient to prove that the lower animals can never learn to speak or comprehend, although they have the organs of speech and ideas of perception in common with us. But because words represent this perfectly distinct class of ideas, whose subjective correlative is reason, they are without sense and meaning for the brutes. Thus language, like every other manifestation which we ascribe to reason, and like everything which distinguishes man from the brutes, is to be explained from this as its one simple source—conceptions, abstract ideas which cannot be presented in perception, but are general, and have no individual existence in space and time. Only in single cases do we pass

from the conception to the perception, do we construct images as representatives of concepts in perception, to which, however, they are never adequate. These cases are fully discussed in the essay on the principle of sufficient reason, § 28, and therefore I shall not repeat my explanation here. It may be compared, however, with what is said by Hume in the twelfth of his “Philosophical Essays,” p. 244, and by Herder in the “Metacritik,” pt. i. p. 274 (an otherwise worthless book). The Platonic idea, the possibility of which depends upon the union of imagination and reason, is the principal subject of the third book of this work.

Although concepts are fundamentally different from ideas of perception, they stand in a necessary relation to them, without which they would be nothing. This relation therefore constitutes the whole nature and existence of concepts. Reflection is the necessary copy or repetition of the originally presented world of perception, but it is a special kind of copy in an entirely different material. Thus concepts may quite properly be called ideas of ideas. The principle of sufficient reason has here also a special form. Now we have seen that the form under which the principle of sufficient reason appears in a class of ideas always constitutes and exhausts the whole nature of the class, so far as it consists of ideas, so that time is throughout succession, and nothing more; space is throughout position, and nothing more; matter is throughout causation, and nothing more. In the same way the whole nature of concepts, or the class of abstract ideas, consists simply in the relation which the principle of sufficient reason expresses in them; and as this is the relation to the ground of knowledge, the whole nature of the abstract idea is simply and solely its relation to another idea, which is its ground of knowledge. This, indeed, may, in the first instance, be a concept, an abstract idea, and this again may have only a similar abstract ground of knowledge; but the chain of grounds of knowledge does not extend ad infinitum; it must end at last in a concept which has its ground in knowledge of perception; for the whole world of reflection rests on the world of perception as its ground of knowledge. Hence the class of abstract ideas is in this respect distinguished from other classes; in the latter the principle of sufficient reason always demands merely a relation to another idea of the same class, but in the case of abstract ideas, it at last demands a relation to an idea of another class.

Those concepts which, as has just been pointed out, are not immediately related to the world of perception, but only through the medium of one, or it may be several other concepts, have been called by preference abstracta, and those which have their ground immediately in the world of perception have been called concreta. But this last name is only loosely applicable to the concepts denoted by it, for they are always merely abstracta, and not ideas of perception. These names, which have originated in a very dim consciousness of the distinctions they imply, may yet, with this explanation, be retained. As examples of the first kind of concepts, i.e., abstracta in the fullest sense, we may take “relation,” “virtue,” “investigation,” “beginning,” and so on. As examples of the second kind, loosely called concreta, we may take such concepts as “man,” “stone,” “horse,” &c. If it were not a somewhat too pictorial and therefore absurd simile, we might very appropriately call the latter the ground floor, and the former the upper stories of the building of reflection.(13)

It is not, as is commonly supposed, an essential characteristic of a concept that it should contain much under it, that is to say, that many ideas of perception, or it may be other abstract ideas, should stand to it in the relation of its ground of knowledge, i.e., be thought through it. This is merely a derived and secondary characteristic, and, as a matter of fact, does not always exist, though it must always exist potentially. This characteristic arises from the fact that a concept is an idea of an idea, i.e., its whole nature consists in its relation to another idea; but as it is not this idea itself, which is generally an idea of perception and therefore belongs to quite a different class, the latter may have temporal, spacial, and other determinations, and in general many relations which are not thought along with it in the concept. Thus we see that several ideas which are different in unessential particulars may be thought by means of one concept, i.e., may be brought under it. Yet this power of embracing several things is not an essential but merely an accidental characteristic of the concept. There may be concepts through which only one real object is thought, but which are nevertheless abstract and general, by no means capable of presentation individually and as perceptions. Such, for example, is the conception which any one may have of a particular town which he only knows from geography; although only this one town is thought under it, it might yet be applied to several towns differing in certain respects. We see then that a concept is not general because of being abstracted from several objects; but conversely, because generality, that is to say, non-determination of the particular, belongs to the concept as an abstract idea of the reason, different things can be thought by means of the same one.

It follows from what has been said that every concept, just because it is abstract and incapable of presentation in perception, and is therefore not a completely determined idea, has what is called extension or sphere, even in the case in which only one real object exists that corresponds to it. Now we always find that the sphere of one concept has something in common with the sphere of other concepts. That is to say, part of what is thought under one concept is the same as what is thought under other concepts; and conversely, part of what is thought under these concepts is the same as what is thought under the first; although, if they are really different concepts, each of them, or at least one of them, contains something which the other does not contain; this is the relation in which every subject stands to its predicate. The recognition of this relation is called judgment. The representation of these spheres by means of figures in space, is an exceedingly happy idea. It first occurred to Gottfried Plouquet, who used squares for the purpose. Lambert, although later than him, used only lines, which he placed under each other. Euler carried out the idea completely with circles. Upon what this complete analogy between the relations of concepts, and those of figures in space, ultimately rests, I am unable to say. It is, however, a very fortunate circumstance for logic that all the relations of concepts, according to their possibility, i.e., a priori, may be made plain in perception by the use of such figures, in the following way:—

(1.) The spheres of two concepts coincide: for example the concept of necessity and the concept of following from given grounds, in the same way the concepts of Ruminantia and Bisulca (ruminating and cloven-hoofed animals), also those of vertebrate and red-blooded animals (although there might be some doubt about this on account of the annelida): they are convertible concepts. Such concepts are represented by a single circle which stands for either of them. (2.) The sphere of one concept includes that of the other. [Illustration: Category "horse" within category "animal".] (3.) A sphere includes two or more spheres which exclude each other and fill it. [Illustration: Circle divided into thirds "right", "acute", and "obtuse".] (4.) Two spheres include each a part of the other. [Illustration: Two overlapping circles, one "flower" and one "red".] (5.) Two spheres lie in a third, but do not fill it. [Illustration: A large circle, "matter", within which are two other circles, "water" and "earth".] This last case applies to all concepts whose spheres have nothing immediately in common, for there is always a third sphere, often a much wider one, which includes both.

To these cases all combinations of concepts may be referred, and from them the entire doctrine of the judgment, its conversion, contraposition, equipollence, disjunction (this according to the third figure) may be deduced. From these also may be derived the properties of the judgment, upon which Kant based his pretended categories of the understanding, with the exception however of the hypothetical form, which is not a combination of concepts, but of judgments. A full account is given in the Appendix of “Modality,” and indeed of every property of judgments on which the categories are founded.

With regard to the possible combinations of concepts which we have given, it has only further to be remarked that they may also be combined with each other in many ways. For example, the fourth figure with the second. Only if one sphere, which partly or wholly contains another, is itself contained in a third sphere, do these together exemplify the syllogism in the first figure, i.e., that combination of judgments, by means of which it is known that a concept which is partly or wholly contained in another concept, is also contained in a third concept, which again contains the first: and also, conversely, the negation; the pictorial representation of which can, of course, only be two connected spheres which do not lie within a third sphere. If many spheres are brought together in this way we get a long train of syllogisms. This schematism of concepts, which has already been fairly well explained in more than one textbook, may be used as the foundation of the doctrine of the judgment, and indeed of the whole syllogistic theory, and in this way the treatment of both becomes very easy and simple. Because, through it, all syllogistic rules may be seen in their origin, and may be deduced and explained. It is not necessary, however, to load the memory with

these rules, as logic is never of practical use, but has only a theoretical interest for philosophy. For although it may be said that logic is related to rational thinking as thorough-bass is to music, or less exactly, as ethics is to virtue, or æsthetics to art; we must yet remember that no one ever became an artist by the study of æsthetics; that a noble character was never formed by the study of ethics; that long before Rameau, men composed correctly and beautifully, and that we do not need to know thorough-bass in order to detect discords: and just as little do we need to know logic in order to avoid being misled by fallacies. Yet it must be conceded that thorough-bass is of the greatest use in the practice of musical composition, although it may not be necessary for the understanding of it; and indeed æsthetics and even ethics, though in a much less degree, and for the most part negatively, may be of some use in practice, so that we cannot deny them all practical worth, but of logic even this much cannot be conceded. It is nothing more than the knowledge in the abstract of what every one knows in the concrete. Therefore we call in the aid of logical rules, just as little to enable

us to construct a correct argument as to prevent us from consenting to a false one, and the most learned logician lays aside the rules of logic altogether in his actual thought. This may be explained in the following way. Every science is a system of general and therefore abstract truths, laws, and rules with reference to a special class of objects. The individual case coming under these laws is determined in accordance with this general knowledge, which is valid once for all; because such application of the general principle is far easier than the exhaustive investigation of the particular case; for the general abstract knowledge which has once been obtained is always more within our reach than the empirical investigation of the particular case. With logic, however, it is just the other way. It is the general knowledge of the mode of procedure of the reason expressed in the form of rules. It is reached by the introspection of reason, and by abstraction from all content. But this mode of procedure is necessary and essential to reason, so that it will never depart from it if left to itself. It is, therefore, easier and surer to let it proceed itself according to its nature in each particular case, than to present to it the knowledge abstracted from this procedure

in the form of a foreign and externally given law. It is easier, because, while in the case of all other sciences, the general rule is more within our reach than the investigation of the particular case taken by itself; with the use of reason, on the contrary, its necessary procedure in a given case is always more within our reach than the general rule abstracted from it; for that which thinks in us is reason itself. It is surer, because a mistake may more easily occur in such abstract knowledge, or in its application, than that a process of reason should take place which would run contrary to its essence and nature. Hence arises the remarkable fact, that while in other sciences the particular case is always proved by the rule, in logic, on the contrary, the rule must always be proved from the particular case; and even the most practised logician, if he remark that in some particular case he concludes otherwise than the rule prescribes, will always expect to find a mistake in the rule rather than in his own conclusion. To desire to make practical use of logic means, therefore, to desire to derive with unspeakable trouble, from general rules, that which is immediately known with the greatest certainty in the particular case. It is just

as if a man were to consult mechanics as to the motion of his body, and physiology as to his digestion; and whoever has learnt logic for practical purposes is like him who would teach a beaver to make its own dam. Logic is, therefore, without practical utility; but it must nevertheless be retained, because it has philosophical interest as the special knowledge of the organisation and action of reason. It is rightly regarded as a definite, self-subsisting, self-contained, complete, and thoroughly safe discipline; to be treated scientifically for itself alone and independently of everything else, and therefore to be studied at the universities. But it has its real value, in relation to philosophy as a whole, in the inquiry into the nature of knowledge, and indeed of rational and abstract knowledge. Therefore the exposition of logic should not have so much the form of a practical science, should not contain merely naked arbitrary rules for the correct formation of the judgment, the syllogism, &c., but should rather be directed to the knowledge of the nature of reason and the concept, and to the detailed investigation of the principle of sufficient reason of knowing. For logic is only a paraphrase of this principle, and, more exactly, only of that exemplification of it in which the ground that gives truth to

the judgment is neither empirical nor metaphysical, but logical or metalogical. Besides the principle of sufficient reason of knowing, it is necessary to take account of the three remaining fundamental laws of thought, or judgments of metalogical truth, so nearly related to it; and out of these the whole science of reason grows. The nature of thought proper, that is to say, of the judgment and the syllogism, must be exhibited in the combination of the spheres of concepts, according to the analogy of the special schema, in the way shown above; and from all this the rules of the judgment and the syllogism are to be deduced by construction. The only practical use we can make of logic is in a debate, when we can convict our antagonist of his intentional fallacies, rather than of his actual mistakes, by giving them their technical names. By thus throwing into the background the practical aim of logic, and bringing out its connection with the whole scheme of philosophy as one of its chapters, we do not think that we shall make the study of it less prevalent than it is just now. For at the present day every one who does not wish to remain uncultured, and to be numbered with the ignorant and incompetent multitude, must study speculative philosophy. For

the nineteenth century is a philosophical age, though by this we do not mean either that it has philosophy, or that philosophy governs it, but rather that it is ripe for philosophy, and, therefore, stands in need of it. This is a sign of a high degree of civilisation, and indeed, is a definite stage in the culture of the ages.(14)

Though logic is of so little practical use, it cannot be denied that it was invented for practical purposes. It appears to me to have originated in the following way:—As the love of debating developed among the Eleatics, the Megarics, and the Sophists, and by degrees became almost a passion, the confusion in which nearly every debate ended must have made them feel the necessity of a method of procedure as a guide; and for this a scientific dialectic had to be sought. The first thing which would have to be observed would be that both the disputing parties should always be agreed on some one proposition, to which the disputed points might be referred. The beginning of the methodical procedure consisted in this, that the propositions admitted on both sides were formally stated to be so, and placed at the head of the inquiry. But these propositions were at first concerned only with the material of the inquiry. It was soon observed that in the process of going back to the truth admitted on both sides, and of deducing their assertions from it, each party followed certain forms and laws about which, without any express agreement, there was no difference of opinion. And from this it became evident that these must constitute the peculiar and natural procedure of reason

itself, the form of investigation. Although this was not exposed to any doubt or difference of opinion, some pedantically systematic philosopher hit upon the idea that it would look well, and be the completion of the method of dialectic, if this formal part of all discussion, this regular procedure of reason itself, were to be expressed in abstract propositions, just like the substantial propositions admitted on both sides, and placed at the beginning of every investigation, as the fixed canon of debate to which reference and appeal must always be made. In this way what had formerly been followed only by tacit agreement, and instinctively, would be consciously recognised and formally expressed. By degrees, more or less perfect expressions were found for the fundamental principles of logic, such as the principles of contradiction, sufficient reason, excluded middle, the dictum de omni et nullo, as well as the special rules of the syllogism, as for example, ex meris particularibus aut negativis nihil sequitur, a rationato ad rationem non valet consequentia, and so on. That all this was only brought about slowly, and with great pains, and up till the time of Aristotle remained very incomplete, is evident from the awkward and tedious way in which logical truths are brought out in many of the Platonic dialogues, and still more from what

Sextus Empiricus tells us of the controversies of the Megarics, about the easiest and simplest logical rules, and the laborious way in which they were brought into a definite form (Sext. Emp. adv. Math. l. 8, p. 112). But Aristotle collected, arranged, and corrected all that had been discovered before his time, and brought it to an incomparably greater state of perfection. If we thus observe how the course of Greek culture had prepared the way for, and led up to the work of Aristotle, we shall be little inclined to believe the assertion of the Persian author, quoted by Sir William Jones with much approval, that Kallisthenes found a complete system of logic among the Indians, and sent it to his uncle Aristotle (Asiatic Researches, vol. iv. p. 163). It is easy to understand that in the dreary middle ages the Aristotelian logic would be very acceptable to the controversial spirit of the schoolmen, which, in the absence of all real knowledge, spent its energy upon mere formulas and words, and that it would be eagerly adopted even in its mutilated Arabian form, and presently established as the centre of all knowledge. Though its authority has since declined, yet up to our own time logic has retained the credit of a self-contained, practical, and highly important science. Indeed, in

our own day, the Kantian philosophy, the foundation-stone of which is taken from logic, has excited a new interest in it; which, in this respect, at any rate, that is, as the means of the knowledge of the nature of reason, it deserves.

Correct and accurate conclusions may be arrived at if we carefully observe the relation of the spheres of concepts, and only conclude that one sphere is contained in a third sphere, when we have clearly seen that this first sphere is contained in a second, which in its turn is contained in the third. On the other hand, the art of sophistry lies in casting only a superficial glance at the relations of the spheres of the concepts, and then manipulating these relations to suit our purposes, generally in the following way:—When the sphere of an observed concept lies partly within that of another concept, and partly within a third altogether different sphere, we treat it as if it lay entirely within the one or the other, as may suit our purpose. For example, in speaking of passion, we may subsume it under the concept of the greatest force, the mightiest agency in the world, or under the concept of the irrational, and this again under the concept of impotency or weakness. We may then repeat the process, and start anew with each concept to which the argument leads us. A concept has almost always several others, which partially come under it, and each of these contains part of the sphere of the first, but also includes in its own

sphere something more, which is not in the first. But we draw attention only to that one of these latter concepts, under which we wish to subsume the first, and let the others remain unobserved, or keep them concealed. On the possession of this skill depends the whole art of sophistry and all finer fallacies; for logical fallacies such as mentiens, velatus, cornatus, &c., are clearly too clumsy for actual use. I am not aware that hitherto any one has traced the nature of all sophistry and persuasion back to this last possible ground of its existence, and referred it to the peculiar character of concepts, i.e., to the procedure of reason itself. Therefore, as my exposition has led me to it, though it is very easily understood, I will illustrate it in the following table by means of a schema. This table is intended to show how the spheres of concepts overlap each other at many points, and so leave room for a passage from each concept to whichever one we please of several other concepts. I hope, however, that no one will be led by this table to attach more importance to this little explanation, which I have merely given in passing, than ought to belong to it, from the nature of the subject. I have chosen as

an illustration the concept of travelling. Its sphere partially includes four others, to any of which the sophist may pass at will; these again partly include other spheres, several of them two or more at once, and through these the sophist takes whichever way he chooses, always as if it were the only way, till at last he reaches, in good or evil, whatever end he may have in view. In passing from one sphere to another, it is only necessary always to follow the direction from the centre (the given chief concept) to the circumference, and never to reverse this process. Such a piece of sophistry may be either an unbroken speech, or it may assume the strict syllogistic form, according to what is the weak side of the hearer. Most scientific arguments, and especially philosophical demonstrations, are at bottom not much more than this, for how else would it be possible, that so much, in different ages, has not only been falsely apprehended (for error itself has a different source), but demonstrated and proved, and has yet afterwards been found to be fundamentally wrong, for example, the Leibnitz-Wolfian Philosophy, Ptolemaic Astronomy, Stahl’s Chemistry, Newton’s Theory of Colours, &c. &c.(15)

Through all this, the question presses ever more upon us, how certainty is to be attained, how _judgments __ are to be established_, what constitutes rational knowledge, (wissen), and science, which we rank with language and deliberate action as the third great benefit conferred by reason.

Reason is feminine in nature; it can only give after it has received. Of itself it has nothing but the empty forms of its operation. There is no absolutely pure rational knowledge except the four principles to which I have attributed metalogical truth; the principles of identity, contradiction, excluded middle, and sufficient reason of knowledge. For even the rest of logic is not absolutely pure rational knowledge. It presupposes the relations and the combinations of the spheres of concepts. But concepts in general only exist after experience of ideas of perception, and as their whole nature consists in their relation to these, it is clear that they presuppose them. No special content, however, is presupposed, but merely the existence of a content generally, and so logic as a whole may fairly pass for pure rational science. In all other sciences reason has received its content from ideas of perception; in mathematics from the relations of space and time, presented in intuition or perception prior to all experience; in pure natural science, that is, in what we know of the course of nature prior to any experience, the content of the science proceeds from the pure understanding, i.e., from the a priori knowledge of the law of causality and its connection with those pure intuitions or perceptions of space and time. In all other sciences everything that is not derived from the sources we have just referred to belongs to experience. Speaking generally, to know rationally (wissen) means to have in the power of the mind, and capable of being reproduced at will, such judgments as have their sufficient ground of knowledge in something outside themselves, i.e., are true. Thus only abstract cognition is rational knowledge (wissen), which is therefore the result of reason, so that we cannot accurately say of the lower animals that they _rationally __ know_ (wissen) anything, although they have apprehension of what is presented in perception, and memory of this, and consequently imagination, which is further proved by the circumstance that they dream. We attribute consciousness to them, and therefore although the word (bewusstsein) is derived from the verb to know rationally (wissen), the conception of consciousness corresponds generally with that of idea of whatever kind it may be. Thus we attribute life to plants, but not consciousness. Rational knowledge (wissen) is therefore abstract consciousness, the permanent possession in concepts of the reason, of what has become known in another way.

In this regard the direct opposite of rational knowledge is feeling, and therefore we must insert the explanation of feeling here. The concept which the word feeling denotes has merely a negative content, which is this, that something which is present in consciousness, is not a concept, is not abstract rational knowledge. Except this, whatever it may be, it comes under the concept of feeling. Thus the immeasurably wide sphere of the concept of feeling includes the most different kinds of objects, and no one can ever understand how they come together until he has recognised that they all agree in this negative respect, that they are not abstract concepts. For the most diverse and even antagonistic elements lie quietly side by side in this concept; for example, religious feeling, feeling of sensual pleasure, moral feeling, bodily feeling, as touch, pain, sense of colour, of sounds and their harmonies and discords, feeling of hate, of disgust, of self-satisfaction, of honour, of disgrace, of right, of wrong, sense of truth, æsthetic feeling, feeling of power, weakness, health, friendship, love, &c. &c. There is absolutely nothing in common among them except the negative quality that they are not abstract rational knowledge. But this diversity becomes more striking when the apprehension of space relations presented a priori in perception, and also the knowledge

of the pure understanding is brought under this concept, and when we say of all knowledge and all truth, of which we are first conscious only intuitively, and have not yet formulated in abstract concepts, we feel it. I should like, for the sake of illustration, to give some examples of this taken from recent books, as they are striking proofs of my theory. I remember reading in the introduction to a German translation of Euclid, that we ought to make beginners in geometry draw the figures before proceeding to demonstrate, for in this way they would already feel geometrical truth before the demonstration brought them complete knowledge. In the same way Schleiermacher speaks in his “Critique of Ethics” of logical and mathematical feeling (p. 339), and also of the feeling of the sameness or difference of two formulas (p. 342). Again Tennemann in his “History of Philosophy” (vol. I., p. 361) says, “One felt that the fallacies were not right, but could not point out the mistakes.” Now, so long as we do not regard this concept “feeling” from the right point of view, and do not recognise that one negative characteristic which alone is essential to it, it must constantly give occasion for misunderstanding and controversy, on account of the excessive wideness of its sphere, and its entirely

negative and very limited content which is determined in a purely one-sided manner. Since then we have in German the nearly synonymous word empfindung (sensation), it would be convenient to make use of it for bodily feeling, as a sub-species. This concept “feeling,” which is quite out of proportion to all others, doubtless originated in the following manner. All concepts, and concepts alone, are denoted by words; they exist only for the reason, and proceed from it. With concepts, therefore, we are already at a one-sided point of view; but from such a point of view what is near appears distinct and is set down as positive, what is farther off becomes mixed up and is soon regarded as merely negative. Thus each nation calls all others foreign: to the Greek all others are barbarians; to the Englishman all that is not England or English is continent or continental; to the believer all others are heretics, or heathens; to the noble all others are roturiers; to the student all others are Philistines, and so forth. Now, reason itself, strange as it may seem, is guilty of the same one-sidedness, indeed one might say of the same crude ignorance arising from vanity, for it classes under the one concept, “feeling,” every modification of consciousness which does not immediately belong to its

own mode of apprehension, that is to say, which is not an abstract concept. It has had to pay the penalty of this hitherto in misunderstanding and confusion in its own province, because its own procedure had not become clear to it through thorough self-knowledge, for a special faculty of feeling has been set up, and new theories of it are constructed.

Rational knowledge (wissen) is then all abstract knowledge,—that is, the knowledge which is peculiar to the reason as distinguished from the understanding. Its contradictory opposite has just been explained to be the concept “feeling.” Now, as reason only reproduces, for knowledge, what has been received in another way, it does not actually extend our knowledge, but only gives it another form. It enables us to know in the abstract and generally, what first became known in sense-perception, in the concrete. But this is much more important than it appears at first sight when so expressed. For it depends entirely upon the fact that knowledge has become rational or abstract knowledge (wissen), that it can be safely preserved, that it is communicable and susceptible of certain and wide-reaching application to practice. Knowledge in the form of sense-perception is valid only of the particular case, extends only to what is nearest, and ends with it, for sensibility and understanding can only comprehend one object at a time. Every enduring, arranged, and planned activity must therefore proceed from principles,—that is, from abstract knowledge, and it must be conducted in accordance with them. Thus, for example, the knowledge of the relation of cause and effect arrived at by the understanding, is in itself far completer, deeper and more exhaustive than anything that can be

thought about it in the abstract; the understanding alone knows in perception directly and completely the nature of the effect of a lever, of a pulley, or a cog-wheel, the stability of an arch, and so forth. But on account of the peculiarity of the knowledge of perception just referred to, that it only extends to what is immediately present, the mere understanding can never enable us to construct machines and buildings. Here reason must come in; it must substitute abstract concepts for ideas of perception, and take them as the guide of action; and if they are right, the anticipated result will happen. In the same way we have perfect knowledge in pure perception of the nature and constitution of the parabola, hyperbola, and spiral; but if we are to make trustworthy application of this knowledge to the real, it must first become abstract knowledge, and by this it certainly loses its character of intuition or perception, but on the other hand it gains the certainty and preciseness of abstract knowledge. The differential calculus does not really extend our knowledge of the curve, it contains nothing that was not already in the mere pure perception of the curve; but it alters the kind of knowledge, it changes the intuitive into an abstract knowledge, which is so valuable for application.

But here we must refer to another peculiarity of our faculty of knowledge, which could not be observed until the distinction between the knowledge of the senses and understanding and abstract knowledge had been made quite clear. It is this, that relations of space cannot as such be directly translated into abstract knowledge, but only temporal quantities,—that is, numbers, are suitable for this. Numbers alone can be expressed in abstract concepts which accurately correspond to them, not spacial quantities. The concept “thousand” is just as different from the concept “ten,” as both these temporal quantities are in perception. We think of a thousand as a distinct multiple of ten, into which we can resolve it at pleasure for perception in time,—that is to say, we can count it. But between the abstract concept of a mile and that of a foot, apart from any concrete perception of either, and without the help of number, there is no accurate distinction corresponding to the quantities themselves. In both we only think of a spacial quantity in general, and if they must be completely distinguished we are compelled either to call in the assistance of intuition or perception in space, which would be a departure from abstract knowledge, or we must think the difference in numbers. If then we wish to have abstract

knowledge of space-relations we must first translate them into time-relations,—that is, into numbers; therefore only arithmetic, and not geometry, is the universal science of quantity, and geometry must be translated into arithmetic if it is to be communicable, accurately precise and applicable in practice. It is true that a space-relation as such may also be thought in the abstract; for example, “the sine increases as the angle,” but if the quantity of this relation is to be given, it requires number for its expression. This necessity, that if we wish to have abstract knowledge of space-relations (i.e., rational knowledge, not mere intuition or perception), space with its three dimensions must be translated into time which has only one dimension, this necessity it is, which makes mathematics so difficult. This becomes very clear if we compare the perception of curves with their analytical calculation, or the table of logarithms of the trigonometrical functions with the perception of the changing relations of the parts of a triangle, which are expressed by them. What vast mazes of figures, what laborious calculations it would require to express in the abstract what perception here apprehends at a glance completely and with perfect accuracy, namely, how the co-sine diminishes as the sine increases, how the co-sine of one angle is the sine of another, the inverse

relation of the increase and decrease of the two angles, and so forth. How time, we might say, must complain, that with its one dimension it should be compelled to express the three dimensions of space! Yet this is necessary if we wish to possess, for application, an expression, in abstract concepts, of space-relations. They could not be translated directly into abstract concepts, but only through the medium of the pure temporal quantity, number, which alone is directly related to abstract knowledge. Yet it is worthy of remark, that as space adapts itself so well to perception, and by means of its three dimensions, even its complicated relations are easily apprehended, while it eludes the grasp of abstract knowledge; time, on the contrary, passes easily into abstract knowledge, but gives very little to perception. Our perceptions of numbers in their proper element, mere time, without the help of space, scarcely extends as far as ten, and beyond that we have only abstract concepts of numbers, no knowledge of them which can be presented in perception. On the other hand, we connect with every numeral, and with all algebraical symbols, accurately defined abstract concepts.

We may further remark here that some minds only find full satisfaction in what is known through perception. What they seek is the reason and consequent of being in space, sensuously expressed; a demonstration after the manner of Euclid, or an arithmetical solution of spacial problems, does not please them. Other minds, on the contrary, seek merely the abstract concepts which are needful for applying and communicating knowledge. They have patience and memory for abstract principles, formulas, demonstrations in long trains of reasoning, and calculations, in which the symbols represent the most complicated abstractions. The latter seek preciseness, the former sensible perception. The difference is characteristic.

The greatest value of rational or abstract knowledge is that it can be communicated and permanently retained. It is principally on this account that it is so inestimably important for practice. Any one may have a direct perceptive knowledge through the understanding alone, of the causal connection, of the changes and motions of natural bodies, and he may find entire satisfaction in it; but he cannot communicate this knowledge to others until it has been made permanent for thought in concepts. Knowledge of the first kind is even sufficient for practice, if a man puts his knowledge into practice himself, in an action which can be accomplished while the perception is still vivid; but it is not sufficient if the help of others is required, or even if the action is his own but must be carried out at different times, and therefore requires a pre-conceived plan. Thus, for example, a practised billiard-player may have a perfect knowledge of the laws of the impact of elastic bodies upon each other, merely in the understanding, merely for direct perception; and for him it is quite sufficient; but on the other hand it is only the man who has studied the science of mechanics, who has, properly speaking, a rational knowledge of these laws, that is, a knowledge of them in the

abstract. Such knowledge of the understanding in perception is sufficient even for the construction of machines, when the inventor of the machine executes the work himself; as we often see in the case of talented workmen, who have no scientific knowledge. But whenever a number of men, and their united action taking place at different times, is required for the completion of a mechanical work, of a machine, or a building, then he who conducts it must have thought out the plan in the abstract, and such co-operative activity is only possible through the assistance of reason. It is, however, remarkable that in the first kind of activity, in which we have supposed that one man alone, in an uninterrupted course of action, accomplishes something, abstract knowledge, the application of reason or reflection, may often be a hindrance to him; for example, in the case of billiard-playing, of fighting, of tuning an instrument, or in the case of singing. Here perceptive knowledge must directly guide action; its passage through reflection makes it uncertain, for it divides the attention and confuses the man. Thus savages and untaught men, who are little accustomed to think, perform certain physical exercises, fight with beasts, shoot with bows and arrows and the like, with a certainty and rapidity which the reflecting European never attains to,

just because his deliberation makes him hesitate and delay. For he tries, for example, to hit the right position or the right point of time, by finding out the mean between two false extremes; while the savage hits it directly without thinking of the false courses open to him. In the same way it is of no use to me to know in the abstract the exact angle, in degrees and minutes, at which I must apply a razor, if I do not know it intuitively, that is, if I have not got it in my touch. The knowledge of physiognomy also, is interfered with by the application of reason. This knowledge must be gained directly through the understanding. We say that the expression, the meaning of the features, can only be felt, that is, it cannot be put into abstract concepts. Every man has his direct intuitive method of physiognomy and pathognomy, yet one man understands more clearly than another these signatura rerum. But an abstract science of physiognomy to be taught and learned is not possible; for the distinctions of difference are here so fine that concepts cannot reach them; therefore abstract knowledge is related to them as a mosaic is to a painting by a Van der Werft or a Denner. In mosaics, however fine they may

be, the limits of the stones are always there, and therefore no continuous passage from one colour to another is possible, and this is also the case with regard to concepts, with their rigidity and sharp delineation; however finely we may divide them by exact definition, they are still incapable of reaching the finer modifications of the perceptible, and this is just what happens in the example we have taken, knowledge of physiognomy.(16)

This quality of concepts by which they resemble the stones of a mosaic, and on account of which perception always remains their asymptote, is also the reason why nothing good is produced in art by their means. If the singer or the virtuoso attempts to guide his execution by reflection he remains silent. And this is equally true of the composer, the painter, and the poet. The concept always remains unfruitful in art; it can only direct the technical part of it, its sphere is science. We shall consider more fully in the third book, why all true art proceeds from sensuous knowledge, never from the concept. Indeed, with regard to behaviour also, and personal agreeableness in society, the concept has only a negative value in restraining the grosser manifestations of egotism and brutality; so that a polished manner is its commendable production. But all that is attractive, gracious, charming in behaviour, all affectionateness and friendliness, must not proceed from the concepts, for if it does, “we feel intention, and are put out of tune.” All dissimulation is the work of reflection; but it cannot be maintained constantly and without interruption: “_nemo __ potest personam diu ferre fictum_,” says Seneca in his book de clementia; and so it is generally found out and loses its effect. Reason is needed in the full stress of life, where quick conclusions, bold action, rapid and sure comprehension are required, but it may easily spoil all if it gains the upper hand, and by perplexing hinders the intuitive, direct discovery, and grasp of the right by simple understanding, and thus induces irresolution.

Lastly, virtue and holiness do not proceed from reflection, but from the inner depths of the will, and its relation to knowledge. The exposition of this belongs to another part of our work; this, however, I may remark here, that the dogmas relating to ethics may be the same in the reason of whole nations, but the action of every individual different; and the converse also holds good; action, we say, is guided by feelings,—that is, simply not by concepts, but as a matter of fact by the ethical character. Dogmas occupy the idle reason; but action in the end pursues its own course independently of them, generally not according to abstract rules, but according to unspoken maxims, the expression of which is the whole man himself. Therefore, however different the religious dogmas of nations may be, yet in the case of all of them, a good action is accompanied by unspeakable satisfaction, and a bad action by endless remorse. No mockery can shake the former; no priest’s absolution can deliver from the latter. Notwithstanding this, we must allow, that for the pursuit of a virtuous life, the application of reason is needful; only it is not its source, but has the subordinate function of preserving resolutions which have been made, of providing maxims to withstand the weakness of the moment, and give consistency to action. It plays the same part ultimately in art also, where it has just as little to do with the essential matter, but assists in carrying it out, for genius is not always at call, and yet the work must be completed in all its parts and rounded off to a whole.(17)

All these discussions of the advantages and disadvantages of the application of reason are intended to show, that although abstract rational knowledge is the reflex of ideas of perception, and is founded on them, it is by no means in such entire congruity with them that it could everywhere take their place: indeed it never corresponds to them quite accurately. And thus, as we have seen, many human actions can only be performed by the help of reason and deliberation, and yet there are some which are better performed without its assistance. This very incongruity of sensuous and abstract knowledge, on account of which the latter always merely approximates to the former, as mosaic approximates to painting, is the cause of a very remarkable phenomenon which, like reason itself, is peculiar to human nature, and of which the explanations that have ever anew been attempted, are insufficient: I mean laughter. On account of the source of this phenomenon, we cannot avoid giving the explanation of it here, though it again interrupts the course of our work to do so. The cause of laughter in every case is simply the sudden perception of the incongruity between a concept and the real objects which have been thought through it in some relation, and laughter itself is just the expression of this incongruity. It often occurs in this way: two or more real objects are thought through one concept, and the identity of the concept is transferred to the objects; it then becomes strikingly apparent from the entire difference of the objects in other respects, that the concept was only applicable to them from a one-sided point of view. It occurs just as often, however, that the incongruity between a single real object and the concept under which, from one point of view, it has rightly been subsumed, is suddenly felt. Now the more correct the subsumption of such objects under a concept may be from one point of view, and the greater and more glaring their incongruity with it, from another point of view, the greater is the ludicrous effect which is produced by this contrast. All laughter then is occasioned by a paradox, and therefore by unexpected subsumption, whether this is expressed in words or in actions. This, briefly stated, is the true explanation of the ludicrous.

I shall not pause here to relate anecdotes as examples to illustrate my theory; for it is so simple and comprehensible that it does not require them, and everything ludicrous which the reader may remember is equally valuable as a proof of it. But the theory is confirmed and illustrated by distinguishing two species into which the ludicrous is divided, and which result from the theory. Either, we have previously known two or more very different real objects, ideas of sense-perception, and have intentionally identified them through the unity of a concept which comprehends them both; this species of the ludicrous is called wit. Or, conversely, the concept is first present in knowledge, and we pass from it to reality, and to operation upon it, to action: objects which in other respects are fundamentally different, but which are all thought in that one concept, are now regarded and treated in the same way, till, to the surprise and astonishment of the person acting, the great difference of their other aspects appears: this species of the ludicrous is called folly. Therefore everything ludicrous is either a flash of wit or a foolish action, according as the procedure has been from the discrepancy of the objects to the identity of the concept, or the converse; the former always intentional, the latter always unintentional, and from without. To seem to reverse the starting-point, and to conceal wit with the mask of folly, is the art of the jester and the clown. Being quite aware of the diversity of the objects, the jester unites them, with secret wit, under one concept, and then starting from this concept he receives from the subsequently discovered diversity of the objects the surprise which he himself prepared. It follows from this short but sufficient theory of the ludicrous, that, if we set aside the last case, that of the jester, wit must always show itself in words, folly generally in actions, though also in words, when it only expresses an intention and does not actually carry it out, or when it shows itself merely in judgments and opinions.

Pedantry is a form of folly. It arises in this way: a man lacks confidence in his own understanding, and, therefore, does not wish to trust to it, to recognise what is right directly in the particular case. He, therefore, puts it entirely under the control of the reason, and seeks to be guided by reason in everything; that is to say, he tries always to proceed from general concepts, rules, and maxims, and to confine himself strictly to them in life, in art, and even in moral conduct. Hence that clinging to the form, to the manner, to the expression and word which is characteristic of pedantry, and which with it takes the place of the real nature of the matter. The incongruity then between the concept and reality soon shows itself here, and it becomes evident that the former never condescends to the particular case, and that with its generality and rigid definiteness it can never accurately apply to the fine distinctions of difference and innumerable modifications of the actual. Therefore, the pedant, with his general maxims, almost always misses the mark in life, shows himself to be foolish, awkward, useless. In art, in which the concept is unfruitful, he produces lifeless, stiff, abortive mannerisms. Even with regard to ethics, the purpose to act rightly or nobly cannot always be carried out in accordance with abstract maxims; for in many cases the excessively nice distinctions in the nature of the circumstances necessitate a choice of the right proceeding directly from the character; for the application of mere abstract maxims sometimes gives false results, because the maxims only half apply; and sometimes cannot be carried out, because they are foreign to the individual character of the actor, and this never allows itself to be entirely discovered; therefore, inconsistencies arise. Since then Kant makes it a condition of the moral worth of an action, that it shall proceed from pure rational abstract maxims, without any inclination or momentary emotion, we cannot entirely absolve him from the reproach of encouraging moral pedantry. This reproach is the significance of Schiller’s epigram, entitled “Scruples of Conscience.” When we speak, especially in connection with politics, of doctrinaires, theorists, savants, and so forth, we mean pedants, that is, persons who know the things well in the abstract, but not in the concrete. Abstraction consists in thinking away the less general predicates; but it is precisely upon these that so much depends in practice.

To complete our theory it remains for us to mention a spurious kind of wit, the play upon words, the calembourg, the pun, to which may be added the equivocation, the double entendre, the chief use of which is the expression of what is obscene. Just as the witticism brings two very different real objects under one concept, the pun brings two different concepts, by the assistance of accident, under one word. The same contrast appears, only familiar and more superficial, because it does not spring from the nature of things, but merely from the accident of nomenclature. In the case of the witticism the identity is in the concept, the difference in the reality, but in the case of the pun the difference is in the concepts and the identity in the reality, for the terminology is here the reality. It would only be a somewhat far-fetched comparison if we were to say that the pun is related to the witticism as the parabola (sic) of the upper inverted cone to that of the lower. The misunderstanding of the word or the quid pro quo is the unintentional pun, and is related to it exactly as folly is to wit. Thus the deaf man often affords occasion for laughter, just as much as the fool, and inferior writers of comedy often use the former for the latter to raise a laugh.

I have treated laughter here only from the psychical side; with regard to the physical side, I refer to what is said on the subject in the “Parerga,” vol. II. ch. vi., § 98.(18)

By means of these various discussions it is hoped that both the difference and the relation between the process of knowledge that belongs to the reason, rational knowledge, the concept on the one hand, and the direct knowledge in purely sensuous, mathematical intuition or perception, and apprehension by the understanding on the other hand, has been clearly brought out. This remarkable relation of our kinds of knowledge led us almost inevitably to give, in passing, explanations of feeling and of laughter, but from all this we now turn back to the further consideration of science as the third great benefit which reason confers on man, the other two being speech and deliberate action. The general discussion of science which now devolves upon us, will be concerned partly with its form, partly with the foundation of its judgments, and lastly with its content.

We have seen that, with the exception of the basis of pure logic, rational knowledge in general has not its source in the reason itself; but having been otherwise obtained as knowledge of perception, it is stored up in the reason, for through reason it has entirely changed its character, and has become abstract knowledge. All rational knowledge, that is, knowledge that has been raised to consciousness in the abstract, is related to science strictly so called, as a fragment to the whole. Every one has gained a rational knowledge of many different things through experience, through consideration of the individual objects presented to him, but only he who sets himself the task of acquiring a complete knowledge in the abstract of a particular class of objects, strives after science. This class can only be marked off by means of a concept; therefore, at the beginning of every science there stands a concept, and by means of it the class of objects concerning which this science promises a complete knowledge in the abstract, is separated in thought from the whole world of things. For example, the concept of space-relations, or of the action of unorganised bodies upon each other, or of the nature of plants, or of animals, or of the successive changes of the surface of the globe, or

of the changes of the human race as a whole, or of the construction of a language, and so forth. If science sought to obtain the knowledge of its object, by investigating each individual thing that is thought through the concept, till by degrees it had learned the whole, no human memory would be equal to the task, and no certainty of completeness would be obtainable. Therefore, it makes use of that property of concept-spheres explained above, that they include each other, and it concerns itself mainly with the wider spheres which lie within the concept of its object in general. When the relations of these spheres to each other have been determined, all that is thought in them is also generally determined, and can now be more and more accurately determined by the separation of smaller and smaller concept-spheres. In this way it is possible for a science to comprehend its object completely. This path which it follows to knowledge, the path from the general to the particular, distinguishes it from ordinary rational knowledge; therefore, systematic form is an essential and characteristic feature of science. The combination of the most general concept-spheres of every science, that is, the knowledge of its first principles, is the indispensable condition of mastering it; how far we advance from these to the more

special propositions is a matter of choice, and does not increase the thoroughness but only the extent of our knowledge of the science. The number of the first principles to which all the rest are subordinated, varies greatly in the different sciences, so that in some there is more subordination, in others more co-ordination; and in this respect, the former make greater claims upon the judgment, the latter upon the memory. It was known to the schoolmen,(19) that, as the syllogism requires two premises, no science can proceed from a single first principle which cannot be the subject of further deduction, but must have several, at least two. The specially classifying sciences: Zoology, Botany, and also Physics and Chemistry, inasmuch as they refer all inorganic action to a few fundamental forces, have most subordination; history, on the other hand, has really none at all; for the general in it consists merely in the survey of the principal periods, from which, however, the particular events cannot be deduced, and are only subordinated to them according to time, but according to the concept are co-ordinate with them. Therefore, history, strictly speaking, is certainly rational knowledge, but is not science. In mathematics, according to Euclid’s treatment, the axioms alone are indemonstrable first principles, and all demonstrations are in gradation strictly subordinated to them.

But this method of treatment is not essential to mathematics, and in fact each proposition introduces quite a new space construction, which in itself is independent of those which precede it, and indeed can be completely comprehended from itself, quite independently of them, in the pure intuition or perception of space, in which the most complicated construction is just as directly evident as the axiom; but of this more fully hereafter. Meanwhile every mathematical proposition remains always a universal truth, which is valid for innumerable particular cases; and a graduated process from the simple to the complicated propositions which are to be deduced from them, is also essential to mathematics; therefore, in every respect mathematics is a science. The completeness of a science as such, that is, in respect of form, consists in there being as much subordination and as little co-ordination of the principles as possible. Scientific talent in general is, therefore, the faculty of subordinating the concept-spheres according to their different determinations, so that, as Plato repeatedly counsels, a science shall not be constituted by a general concept and an indefinite multiplicity immediately under it, but that knowledge shall descend by degrees from the general to the particular, through intermediate concepts and divisions, according to closer and closer definitions. In Kantian language this is called satisfying equally the

law of homogeneity and that of specification. It arises from this peculiar nature of scientific completeness, that the aim of science is not greater certainty—for certainty may be possessed in just as high a degree by the most disconnected particular knowledge—but its aim is rather the facilitating of rational knowledge by means of its form, and the possibility of the completeness of rational knowledge which this form affords. It is therefore a very prevalent but perverted opinion that the scientific character of knowledge consists in its greater certainty, and just as false is the conclusion following from this, that, strictly speaking, the only sciences are mathematics and logic, because only in them, on account of their purely a priori character, is there unassailable certainty of knowledge. This advantage cannot be denied them, but it gives them no special claim to be regarded as sciences; for the special characteristic of science does not lie in certainty but in the systematic form of knowledge, based on the gradual descent from the general to the particular. The process of knowledge from the general to the particular, which is peculiar to the sciences, involves the necessity that in the sciences much should be established by deduction from preceding propositions, that is to say, by demonstration; and this has given rise to the old mistake

that only what has been demonstrated is absolutely true, and that every truth requires a demonstration; whereas, on the contrary, every demonstration requires an undemonstrated truth, which ultimately supports it, or it may be, its own demonstration. Therefore a directly established truth is as much to be preferred to a truth established by demonstration as water from the spring is to water from the aqueduct. Perception, partly pure a priori, as it forms the basis of mathematics, partly empirical a posteriori, as it forms the basis of all the other sciences, is the source of all truth and the foundation of all science. (Logic alone is to be excepted, which is not founded upon perception but yet upon direct knowledge by the reason of its own laws.) Not the demonstrated judgments nor their demonstrations, but judgments which are created directly out of perception, and founded upon it rather than on any demonstrations, are to science what the sun is to the world; for all light proceeds from them, and lighted by their light the others give light also. To establish the truth of such primary judgments directly from perception, to raise such strongholds of science from the innumerable multitude of real objects, that is the work of the faculty of judgment, which consists in the power of rightly and accurately

carrying over into abstract consciousness what is known in perception, and judgment is consequently the mediator between understanding and reason. Only extraordinary and exceptional strength of judgment in the individual can actually advance science; but every one who is possessed of a healthy reason is able to deduce propositions from propositions, to demonstrate, to draw conclusions. To lay down and make permanent for reflection, in suitable concepts, what is known through perception, so that, on the one hand, what is common to many real objects is thought through one concept, and, on the other hand, their points of difference are each thought through one concept, so that the different shall be known and thought as different in spite of a partial agreement, and the identical shall be known and thought as identical in spite of a partial difference, all in accordance with the end and intention which in each case is in view; all this is done by the faculty of judgment. Deficiency in judgment is silliness. The silly man fails to grasp, now the partial or relative difference of concepts which in one aspect are identical, now the identity of concepts which are relatively or partially different. To this explanation of the faculty of judgment, moreover, Kant’s division of it into reflecting and subsuming judgment may be applied, according

as it passes from the perceived objects to the concepts, or from the latter to the former; in both cases always mediating between empirical knowledge of the understanding and the reflective knowledge of the reason. There can be no truth which could be brought out by means of syllogisms alone; and the necessity of establishing truth by means of syllogisms is merely relative, indeed subjective. Since all demonstration is syllogistic, in the case of a new truth we must first seek, not for a demonstration, but for direct evidence, and only in the absence of such evidence is a demonstration to be temporarily made use of. No science is susceptible of demonstration throughout any more than a building can stand in the air; all its demonstrations must ultimately rest upon what is perceived, and consequently cannot be demonstrated, for the whole world of reflection rests upon and is rooted in the world of perception. All primal, that is, original, evidence is a perception, as the word itself indicates. Therefore it is either empirical or founded upon the perception a priori of the conditions of possible experience. In both cases it affords only immanent, not transcendent knowledge. Every concept has its worth and its existence only in its relation, sometimes very indirect, to an idea of perception; what is true of

the concepts is also true of the judgments constructed out of them, and of all science. Therefore it must in some way be possible to know directly without demonstrations or syllogisms every truth that is arrived at through syllogisms and communicated by demonstrations. This is most difficult in the case of certain complicated mathematical propositions at which we only arrive by chains of syllogisms; for example, the calculation of the chords and tangents to all arcs by deduction from the proposition of Pythagoras. But even such a truth as this cannot essentially and solely rest upon abstract principles, and the space-relations which lie at its foundation also must be capable of being so presented a priori in pure intuition or perception that the truth of their abstract expression is directly established. But of mathematical demonstration we shall speak more fully shortly.

It is true we often hear men speak in a lofty strain of sciences which rest entirely upon correct conclusions drawn from sure premises, and which are consequently unassailable. But through pure logical reasoning, however true the premises may be, we shall never receive more than an articulate expression and exposition of what lies already complete in the premises; thus we shall only explicitly expound what was already implicitly understood. The esteemed sciences referred to are, however, specially the mathematical sciences, particularly astronomy. But the certainty of astronomy arises from the fact that it has for its basis the intuition or perception of space, which is given a priori, and is therefore infallible. All space-relations, however, follow from each other with a necessity (ground of being) which affords a priori certainty, and they can therefore be safely deduced from each other. To these mathematical properties we have only to add one force of nature, gravity, which acts precisely in relation to the masses and the square of the distance; and, lastly, the law of inertia, which follows from the law of causality and is therefore true a priori, and with it the empirical datum of the motion impressed, once for all, upon each of these masses. This is the whole material of astronomy, which both by its simplicity and its

certainty leads to definite results, which are highly interesting on account of the vastness and importance of the objects. For example, if I know the mass of a planet and the distance of its satellite from it, I can tell with certainty the period of the revolution of the latter according to Kepler’s second law. But the ground of this law is, that with this distance only this velocity will both chain the satellite to the planet and prevent it from falling into it. Thus it is only upon such a geometrical basis, that is, by means of an intuition or perception a priori, and also under the application of a law of nature, that much can be arrived at by means of syllogisms, for here they are merely like bridges from one sensuous apprehension to others; but it is not so with mere pure syllogistic reasoning in the exclusively logical method. The source of the first fundamental truths of astronomy is, however, properly induction, that is, the comprehension of what is given in many perceptions in one true and directly founded judgment. From this, hypotheses are afterwards constructed, and their confirmation by experience, as induction approaching to completeness, affords the proof of the first judgment. For example, the apparent motion of the planets is known empirically; after many false

hypotheses with regard to the spacial connection of this motion (planetary course) the right one was at last found, then the laws which it obeyed (the laws of Kepler), and, lastly, the cause of these laws (universal gravitation), and the empirically known agreement of all observed cases with the whole of the hypotheses, and with their consequences, that is to say, induction, established them with complete certainty. The invention of the hypotheses was the work of the judgment, which rightly comprehended the given facts and expressed them accordingly; but induction, that is, a multitude of perceptions, confirmed their truth. But their truth could also be known directly, and by a single empirical perception, if we could pass freely through space and had telescopic eyes. Therefore, here also syllogisms are not the essential and only source of knowledge, but really only a makeshift.

As a third example taken from a different sphere we may mention that the so-called metaphysical truths, that is, such truths as those to which Kant assigns the position of the metaphysical first principles of natural science, do not owe their evidence to demonstration. What is a priori certain we know directly; as the form of all knowledge, it is known to us with the most complete necessity. For example, that matter is permanent, that is, can neither come into being nor pass away, we know directly as negative truth; for our pure intuition or perception of space and time gives the possibility of motion; in the law of causality the understanding affords us the possibility of change of form and quality, but we lack powers of the imagination for conceiving the coming into being or passing away of matter. Therefore that truth has at all times been evident to all men everywhere, nor has it ever been seriously doubted; and this could not be the case if it had no other ground of knowledge than the abstruse and exceedingly subtle proof of Kant. But besides this, I have found Kant’s proof to be false (as is explained in the Appendix), and have shown above that the permanence of matter is to be deduced, not from the share which time

has in the possibility of experience, but from the share which belongs to space. The true foundation of all truths which in this sense are called metaphysical, that is, abstract expressions of the necessary and universal forms of knowledge, cannot itself lie in abstract principles; but only in the immediate consciousness of the forms of the idea communicating itself in apodictic assertions a priori, and fearing no refutation. But if we yet desire to give a proof of them, it can only consist in showing that what is to be proved is contained in some truth about which there is no doubt, either as a part of it or as a presupposition. Thus, for example, I have shown that all empirical perception implies the application of the law of causality, the knowledge of which is hence a condition of all experience, and therefore cannot be first given and conditioned through experience as Hume thought. Demonstrations in general are not so much for those who wish to learn as for those who wish to dispute. Such persons stubbornly deny directly established insight; now only the truth can be consistent in all directions, and therefore we must show such persons that they admit under one form and indirectly, what they deny under another form and directly; that is, the logically necessary connection

between what is denied and what is admitted.

It is also a consequence of the scientific form, the subordination of everything particular under a general, and so on always to what is more general, that the truth of many propositions is only logically proved,—that is, through their dependence upon other propositions, through syllogisms, which at the same time appear as proofs. But we must never forget that this whole form of science is merely a means of rendering knowledge more easy, not a means to greater certainty. It is easier to discover the nature of an animal, by means of the species to which it belongs, and so on through the genus, family, order, and class, than to examine on every occasion the animal presented to us: but the truth of all propositions arrived at syllogistically is always conditioned by and ultimately dependent upon some truth which rests not upon reasoning but upon perception. If this perception were always as much within our reach as a deduction through syllogisms, then it would be in every respect preferable. For every deduction from concepts is exposed to great danger of error, on account of the fact we have considered above, that so many spheres lie partly within each other, and that their content is often vague or uncertain. This is illustrated by a multitude of demonstrations of false doctrines and sophisms of every kind. Syllogisms are indeed perfectly certain as regards form, but they are very uncertain on account of their matter, the concepts. For, on the one hand, the spheres of these are not sufficiently sharply defined, and, on the other hand, they intersect each other in so many ways that one sphere is in part contained in many others, and we may pass at will from it to one or another of these, and from this sphere again to others, as we have already shown. Or, in other words, the minor term and also the middle can always be subordinated to different concepts, from which we may choose at will the major and the middle, and the nature of the conclusion depends on this choice. Consequently immediate evidence is always much to be preferred to reasoned truth, and the latter is only to be accepted when the former is too remote, and not when it is as near or indeed nearer than the latter. Accordingly we saw above that, as a matter of fact, in the case of logic, in which the immediate knowledge in each individual case lies nearer to hand than deduced scientific knowledge, we always conduct our thought according to our immediate knowledge of the laws of thought, and leave logic unused.(20)

If now with our conviction that perception is the primary source of all evidence, and that only direct or indirect connection with it is absolute truth; and further, that the shortest way to this is always the surest, as every interposition of concepts means exposure to many deceptions; if, I say, we now turn with this conviction to mathematics, as it was established as a science by Euclid, and has remained as a whole to our own day, we cannot help regarding the method it adopts, as strange and indeed perverted. We ask that every logical proof shall be traced back to an origin in perception; but mathematics, on the contrary, is at great pains deliberately to throw away the evidence of perception which is peculiar to it, and always at hand, that it may substitute for it a logical demonstration. This must seem to us like the action of a man who cuts off his legs in order to go on crutches, or like that of the prince in the “Triumph der Empfindsamkeit” who flees from the beautiful reality of nature, to delight in a stage scene that imitates it. I must here refer to what I have said in the sixth chapter of the essay on the principle of sufficient reason, and take for granted that it is

fresh and present in the memory of the reader; so that I may link my observations on to it without explaining again the difference between the mere ground of knowledge of a mathematical truth, which can be given logically, and the ground of being, which is the immediate connection of the parts of space and time, known only in perception. It is only insight into the ground of being that secures satisfaction and thorough knowledge. The mere ground of knowledge must always remain superficial; it can afford us indeed rational knowledge that a thing is as it is, but it cannot tell why it is so. Euclid chose the latter way to the obvious detriment of the science. For just at the beginning, for example, when he ought to show once for all how in a triangle the angles and sides reciprocally determine each other, and stand to each other in the relation of reason and consequent, in accordance with the form which the principle of sufficient reason has in pure space, and which there, as in every other sphere, always affords the necessity that a thing is as it is, because something quite different from it, is as it is; instead of in this way giving a thorough insight into the nature of the triangle, he sets up certain

disconnected arbitrarily chosen propositions concerning the triangle, and gives a logical ground of knowledge of them, through a laborious logical demonstration, based upon the principle of contradiction. Instead of an exhaustive knowledge of these space-relations we therefore receive merely certain results of them, imparted to us at pleasure, and in fact we are very much in the position of a man to whom the different effects of an ingenious machine are shown, but from whom its inner connection and construction are withheld. We are compelled by the principle of contradiction to admit that what Euclid demonstrates is true, but we do not comprehend why it is so. We have therefore almost the same uncomfortable feeling that we experience after a juggling trick, and, in fact, most of Euclid’s demonstrations are remarkably like such feats. The truth almost always enters by the back door, for it manifests itself per accidens through some contingent circumstance. Often a reductio ad absurdum shuts all the doors one after another, until only one is left through which we are therefore compelled to enter. Often, as in the proposition of Pythagoras, lines are drawn, we don’t know why, and it afterwards appears that they were traps which close unexpectedly and take prisoner the assent of the astonished learner, who must now admit what remains wholly inconceivable

in its inner connection, so much so, that he may study the whole of Euclid through and through without gaining a real insight into the laws of space-relations, but instead of them he only learns by heart certain results which follow from them. This specially empirical and unscientific knowledge is like that of the doctor who knows both the disease and the cure for it, but does not know the connection between them. But all this is the necessary consequence if we capriciously reject the special kind of proof and evidence of one species of knowledge, and forcibly introduce in its stead a kind which is quite foreign to its nature. However, in other respects the manner in which this has been accomplished by Euclid deserves all the praise which has been bestowed on him through so many centuries, and which has been carried so far that his method of treating mathematics has been set up as the pattern of all scientific exposition. Men tried indeed to model all the sciences after it, but later they gave up the attempt without quite knowing why. Yet in our eyes this method of Euclid in mathematics can appear only as a very brilliant piece of perversity. But when a great error in life or in science has been intentionally and methodically carried

out with universal applause, it is always possible to discover its source in the philosophy which prevailed at the time. The Eleatics first brought out the difference, and indeed often the conflict, that exists between what is perceived, φαινομενον,(21) and what is thought, νουμενον, and used it in many ways in their philosophical epigrams, and also in sophisms. They were followed later by the Megarics, the Dialecticians, the Sophists, the New-Academy, and the Sceptics; these drew attention to the illusion, that is to say, to the deception of the senses, or rather of the understanding which transforms the data of the senses into perception, and which often causes us to see things to which the reason unhesitatingly denies reality; for example, a stick broken in water, and such like. It came to be known that sense-perception was not to be trusted unconditionally, and it was therefore hastily concluded that only rational, logical thought could establish truth; although Plato (in the Parmenides), the Megarics, Pyrrho, and the New-Academy, showed by examples (in the manner which was afterwards adopted by Sextus Empiricus) how syllogisms and concepts were also sometimes misleading, and indeed produced paralogisms and sophisms which arise much more easily and are far harder to explain than the illusion of sense-perception. However, this rationalism, which arose in opposition to empiricism, kept

the upper hand, and Euclid constructed the science of mathematics in accordance with it. He was compelled by necessity to found the axioms upon evidence of perception (φαινομενον), but all the rest he based upon reasoning (νουμενον). His method reigned supreme through all the succeeding centuries, and it could not but do so as long as pure intuition or perception, a priori, was not distinguished from empirical perception. Certain passages from the works of Proclus, the commentator of Euclid, which Kepler translated into Latin in his book, “De Harmonia Mundi,” seem to show that he fully recognised this distinction. But Proclus did not attach enough importance to the matter; he merely mentioned it by the way, so that he remained unnoticed and accomplished nothing. Therefore, not till two thousand years later will the doctrine of Kant, which is destined to make such great changes in all the knowledge, thought, and action of European nations, produce this change in mathematics also. For it is only after we have learned from this great man that the intuitions or perceptions of space and time are quite different from empirical perceptions, entirely independent of any impression of the senses, conditioning it, not conditioned by it, i.e., are a priori, and therefore are not exposed to the illusions of sense; only after we have learned

this, I say, can we comprehend that Euclid’s logical method of treating mathematics is a useless precaution, a crutch for sound legs, that it is like a wanderer who during the night mistakes a bright, firm road for water, and carefully avoiding it, toils over the broken ground beside it, content to keep from point to point along the edge of the supposed water. Only now can we affirm with certainty that what presents itself to us as necessary in the perception of a figure, does not come from the figure on the paper, which is perhaps very defectively drawn, nor from the abstract concept under which we think it, but immediately from the form of all knowledge of which we are conscious a priori. This is always the principle of sufficient reason; here as the form of perception, i.e., space, it is the principle of the ground of being, the evidence and validity of which is, however, just as great and as immediate as that of the principle of the ground of knowing, i.e., logical certainty. Thus we need not and ought not to leave the peculiar province of mathematics in order to put our trust only in logical proof, and seek to authenticate mathematics in a sphere which is quite foreign to it, that of concepts. If we

confine ourselves to the ground peculiar to mathematics, we gain the great advantage that in it the rational knowledge that something is, is one with the knowledge why it is so, whereas the method of Euclid entirely separates these two, and lets us know only the first, not the second. Aristotle says admirably in the Analyt., post. i. 27: “Ακριβεστερα δ᾽ επιστημη επιστημης και προτερα, ἡτε του ὁτι και του διοτι ἡ αυτη, αλλα μη χωρις του ὁτι, της του διοτι” (Subtilior autem et praestantior ea est scientia, quâ QUOD aliquid sit, et CUR sit una simulque intelligimus non separatim QUOD, et CUR sit). In physics we are only satisfied when the knowledge that a thing is as it is is combined with the knowledge why it is so. To know that the mercury in the Torricellian tube stands thirty inches high is not really rational knowledge if we do not know that it is sustained at this height by the counterbalancing weight of the atmosphere. Shall we then be satisfied in mathematics with the qualitas occulta of the circle that the segments of any two intersecting chords always contain equal rectangles? That it is so Euclid certainly demonstrates in the 35th Prop. of the Third Book; why it is so remains doubtful. In the same way the proposition of

Pythagoras teaches us a qualitas occulta of the right-angled triangle; the stilted and indeed fallacious demonstration of Euclid forsakes us at the why, and a simple figure, which we already know, and which is present to us, gives at a glance far more insight into the matter, and firm inner conviction of that necessity, and of the dependence of that quality upon the right angle:—

In order to improve the method of mathematics, it is especially necessary to overcome the prejudice that demonstrated truth has any superiority over what is known through perception, or that logical truth founded upon the principle of contradiction has any superiority over metaphysical truth, which is immediately evident, and to which belongs the pure intuition or perception of space.

That which is most certain, and yet always inexplicable, is what is involved in the principle of sufficient reason, for this principle, in its different aspects, expresses the universal form of all our ideas and knowledge. All explanation consists of reduction to it, exemplification in the particular case of the connection of ideas expressed generally through it. It is thus the principle of all explanation, and therefore it is neither susceptible of an explanation itself, nor does it stand in need of it; for every explanation presupposes it, and only obtains meaning through it. Now, none of its forms are superior to the rest; it is equally certain and incapable of demonstration as the principle of the ground of being, or of change, or of action, or of knowing. The relation of reason and consequent is a necessity in all its forms, and indeed it is, in general, the source of the concept of necessity, for necessity has no other meaning. If the reason is given there is no other necessity than that of the consequent, and there is no reason that does not involve the necessity of the consequent. Just as surely then as the consequent expressed in the conclusion follows from the ground of knowledge given in the premises, does the ground of being in space determine its

consequent in space: if I know through perception the relation of these two, this certainty is just as great as any logical certainty. But every geometrical proposition is just as good an expression of such a relation as one of the twelve axioms; it is a metaphysical truth, and as such, just as certain as the principle of contradiction itself, which is a metalogical truth, and the common foundation of all logical demonstration. Whoever denies the necessity, exhibited for intuition or perception, of the space-relations expressed in any proposition, may just as well deny the axioms, or that the conclusion follows from the premises, or, indeed, he may as well deny the principle of contradiction itself, for all these relations are equally undemonstrable, immediately evident and known a priori. For any one to wish to derive the necessity of space-relations, known in intuition or perception, from the principle of contradiction by means of a logical demonstration is just the same as for the feudal superior of an estate to wish to hold it as the vassal of another. Yet this is what Euclid has done. His axioms only, he is compelled to leave resting upon immediate evidence; all the geometrical truths which follow are demonstrated logically, that is to say, from the agreement of the assumptions made in the proposition

with the axioms which are presupposed, or with some earlier proposition; or from the contradiction between the opposite of the proposition and the assumptions made in it, or the axioms, or earlier propositions, or even itself. But the axioms themselves have no more immediate evidence than any other geometrical problem, but only more simplicity on account of their smaller content.