I have said that all abstract knowledge, i.e., all knowledge of reason, is rational knowledge (Wissen), and I have just explained that the concept of feeling is the contradictory opposite of this. But, as reason always brings again before knowledge only what has been received in another way, it does not really extend our knowledge, but merely gives it another form. Thus it enables one to know in the abstract and in general what was known intuitively and in the concrete. But this is far more important than appears at first sight when thus expressed. For all safe preservation, all communicability, all sure and far-reaching application of knowledge to the practical, depend on its having become a rational knowledge (Wissen), an abstract knowledge. Intuitive knowledge is always valid only of the particular case, extends only to what is nearest, and there stops, since sensibility and understanding can really comprehend only one object at a time. Therefore every continuous, coordinated, and planned activity must start from fundamental principles, i.e. from an abstract knowledge, and must be guided in accordance therewith. Thus, for example, knowledge which the understanding has of the relation of cause and effect is in itself much more complete, profound, and exhaustive than what can be thought of it in the abstract. The understanding alone knows from perception, directly and completely, the mode of operation of a lever, a block and tackle, a cog-wheel, the support of an arch, and so on. But on account of the property of intuitive knowledge just referred to, namely that it extends only to what is immediately present, the mere understanding is not sufficient for constructing machines and buildings. On the contrary, reason must put in an appearance here; it must replace intuitions and perceptions with abstract concepts, take those concepts as the guide of action, and, if they are right, success will be attained. In the same way, we know perfectly in pure perception the nature and conformity to law of a parabola, hyperbola, and spiral, but for this knowledge to be reliably applied in real life it must first have become abstract knowledge. Here, of course, it loses its character of intuition or perception, and acquires instead the certainty and definiteness of abstract knowledge. Thus the differential calculus does not really extend our knowledge of curves; it contains nothing more than what was already present in the mere pure perception of them. But it alters the kind of knowledge; it converts the intuitive into an abstract knowledge that is so extremely important for application. Here another peculiarity of our faculty of knowledge comes under discussion, and one that could not be observed previously, until the difference between knowledge of perception and abstract knowledge was made perfectly clear. It is that the relations of space cannot directly and as such be translated into abstract knowledge, but only temporal quantities, that is to say numbers, are capable of this. Numbers alone can be expressed in abstract concepts exactly corresponding to them; spatial quantities cannot. The concept thousand is just as different from the concept ten as are the two temporal quantities in perception. We think of a thousand as a definite multiple of ten into which we can resolve it at will for perception in time, in other words, we can count it. But between the abstract concept of a mile and that of a foot, without any representation from perception of either, and without the help of number, there is no exact distinction at all corresponding to these quantities themselves. In both we think only of a spatial quantity in general, and if they are to be adequately distinguished, we must either avail ourselves of intuition or perception in space, and hence leave the sphere of abstract knowledge, or we must think the difference in numbers. If, therefore, we want to have abstract knowledge of space-relations, we must first translate them into time-relations, that is, numbers. For this reason, arithmetic alone, and not geometry, is the universal theory of quantity, and geometry must be translated into arithmetic if it is to be communicable, precisely definite, and applicable in practice. It is true that a spatial relation as such may also be thought in the abstract, for example “The sine increases with the angle,” but if the quantity of this relation is to be stated, number is required. This necessity for space with its three dimensions to be translated into time with only one dimension, if we wish to have an abstract knowledge (i.e., a rational knowledge, and no mere intuition or perception) of space-relations—this necessity it is that makes mathematics so difficult. This becomes very clear when we compare the perception of curves with their analytical calculation, or even merely the tables of the logarithms of trigonometrical functions with the perception of the changing relations of the parts of a triangle expressed by them. What vast tissues of figures, what laborious calculations, would be required to express in the abstract what perception here apprehends perfectly and with extreme accuracy at a glance, namely how the cosine diminishes while the sine increases, how the cosine of one angle is the sine of another, the inverse relation of the increase and decrease of the two angles, and so on! How time, we might say, with its one dimension must torture itself, in order to reproduce the three dimensions of space! But this was necessary if we wished to possess space-relations expressed in abstract concepts for the purpose of application. They could not go into abstract concepts directly, but only through the medium of the purely temporal quantity, number, which alone is directly connected to abstract knowledge. Yet it is remarkable that, as space is so well adapted to perception, and, by means of its three dimensions, even complicated relations can be taken in at a glance, whereas it defies abstract knowledge, time on the other hand passes easily into abstract concepts, but offers very little to perception. Our perception of numbers in their characteristic element, namely in mere time, without the addition of space, scarcely extends as far as ten. Beyond this we have only abstract concepts, and no longer perceptive knowledge of numbers. On the other hand, we connect with every numeral and with all algebraical signs precise and definite abstract concepts.
Incidentally, it may here be remarked that many minds find complete satisfaction only in what is known through perception. What they look for is reason or ground and consequent of being in space presented in perception. A Euclidean proof, or an arithmetical solution of spatial problems, makes no appeal to them. Other minds, on the contrary, want the abstract concepts of use solely for application and communication. They have patience and memory for abstract principles, formulas, demonstrations by long chains of reasoning, and calculations whose symbols represent the most complicated abstractions. The latter seek preciseness, the former intuitiveness. The difference is characteristic.
Rational or abstract knowledge has its greatest value in its communicability, and in its possibility of being fixed and retained; only through this does it become so invaluable for practice. Of the causal connexion of the changes and motions of natural bodies a man can have an immediate, perceptive knowledge in the mere understanding, and can find complete satisfaction in it, but it is capable of being communicated only after he has fixed it in concepts. Even knowledge of the first kind is sufficient for practice, as soon as a man puts it into execution entirely by himself, in fact when he carries it out in a practical action, while the knowledge from perception is still vivid. But such knowledge is not sufficient if a man requires the help of another, or if he needs to carry out on his own part some action manifested at different times and therefore needing a deliberate plan. Thus, for example, an experienced billiard-player can have a perfect knowledge of the laws of impact of elastic bodies on one another, merely in the understanding, merely for immediate perception, and with this he manages perfectly. Only the man who is versed in the science of mechanics, on the other hand, has a real rational knowledge of those laws, that is to say, a knowledge of them in the abstract. Even for the construction of machines such a merely intuitive knowledge of the understanding is sufficient, when the inventor of the machine himself executes the work, as is often seen in the case of talented workmen without any scientific knowledge. On the other hand, as soon as several men and their coordinated activity occurring at different times are necessary for carrying out a mechanical operation, for completing a machine or a building, then the man controlling it must have drafted the plan in the abstract, and such a cooperative activity is possible only through the assistance of the faculty of reason. But it is remarkable that, in the first kind of activity, where one man alone is supposed to execute something in an uninterrupted course of action, rational knowledge, the application of reason, reflection, may often be even a hindrance to him. For example, in the case of billiards-playing, fencing, tuning an instrument, or singing, knowledge of perception must directly guide activity; passage through reflection makes it uncertain, since it divides the attention, and confuses the executant. Therefore, savages and uneducated persons, not very accustomed to thinking, perform many bodily exercises, fight with animals, shoot with bows and arrows and the like, with a certainty and rapidity never reached by the reflecting European, just because his deliberation makes him hesitate and hang back. For instance, he tries to find the right spot or the right point of time from the mean between two false extremes, while the natural man hits it directly without reflecting on the wrong courses open to him. Likewise, it is of no use for me to be able to state in the abstract in degrees and minutes the angle at which I have to apply my razor, if I do not know it intuitively, in other words, if I do not know how to hold the razor. In like manner, the application of reason is also disturbing to the person who tries to understand physiognomy; this too must occur directly through the understanding. We say that the expression, the meaning of the features, can only be felt, that is to say, it cannot enter into abstract concepts. Every person has his own immediate intuitive method of physiognomy and pathognomy, yet one recognizes that signatura rerum more clearly than does another. But a science of physiognomy in the abstract cannot be brought into existence to be taught and learned, because in this field the shades of difference are so fine that the concept cannot reach them. Hence abstract rational knowledge is related to them as a mosaic is to a picture by a van der Werft or a Denner. However fine the mosaic may be, the edges of the stones always remain, so that no continuous transition from one tint to another is possible. In the same way, concepts, with their rigidity and sharp delineation, however finely they may be split by closer definition, are always incapable of reaching the fine modifications of perception, and this is the very point of the example I have taken here from physiognomy.32
This same property in concepts which makes them similar to the stones of a mosaic, and by virtue of which perception always remains their asymptote, is also the reason why nothing good is achieved through them in art. If the singer or virtuoso wishes to guide his recital by reflection, he remains lifeless. The same is true of the composer, the painter, and the poet. For art the concept always remains unproductive; in art it can guide only technique; its province is science. In the third book we shall inquire more closely into the reason why all genuine art proceeds from knowledge of perception, never from the concept. Even in regard to behaviour, to personal charm in mixing with people, the concept is only of negative value in restraining the uncouth outbursts of egoism and brutality, so that politeness is its commendable work. What is attractive, gracious, prepossessing in behaviour, what is affectionate and friendly, cannot have come from the concept, otherwise “We feel intention and are put out of tune.” All dissimulation is the work of reflection, but it cannot be kept up permanently and without interruption; nemo potest personam diu ferre fictam,33 says Seneca in his book De Clementia; for generally it is recognized, and loses its effect. Reason is necessary in the high stress of life where rapid decisions, bold action, quick and firm comprehension are needed, but if it gains the upper hand, if it confuses and hinders the intuitive, immediate discovery of what is right by the pure understanding, and at the same time prevents this from being grasped, and if it produces irresolution, then it can easily ruin everything.
Finally, virtue and holiness result not from reflection, but from the inner depth of the will, and from its relation to knowledge. This discussion belongs to an entirely different part of this work. Here I may observe only this much, that the dogmas relating to ethics can be the same in the reasoning faculty of whole nations, but the conduct of each individual different, and also the converse. Conduct, as we say, happens in accordance with feelings, that is to say, not precisely according to concepts, but to ethical worth and quality. Dogmas concern idle reason; conduct in the end pursues its own course independently of them, usually in accordance not with abstract, but with unspoken maxims, the expression of which is precisely the whole man himself. Therefore, however different the religious dogmas of nations may be, with all of them the good deed is accompanied by unspeakable satisfaction, and the bad by infinite dread. No mockery shakes the former; no father confessor’s absolution delivers us from the latter. But it cannot be denied that the application of reason is necessary for the pursuit of a virtuous way of living; yet it is not the source of this, but its function is a subordinate one; to preserve resolutions once formed, to provide maxims for withstanding the weakness of the moment, and to give consistency to conduct. Ultimately, it achieves the same thing also in art, where it is not capable of anything in the principal matter, but assists in carrying it out, just because genius is not at a man’s command every hour, and yet the work is to be completed in all its parts and rounded off to a whole.34