By all these various considerations it is hoped that the difference and the relation between the cognitive method of reason, rational knowledge, the concept, on the one hand, and the immediate knowledge in purely sensuous, mathematical perception or intuition and in apprehension by the understanding on the other, has been brought out quite clearly. Further, there have been also the incidental discussions on feeling and laughter, to which we were almost inevitably led by a consideration of that remarkable relation of our modes of cognition. From all this I now return to a further discussion of science as being, together with speech and deliberate action, the third advantage which the faculty of reason confers on man. The general consideration of science which here devolves upon us will be concerned partly with its form, partly with the foundation of its judgements, and finally with its content.
We have seen that, with the exception of the basis of pure logic, all rational knowledge has its origin not in reason itself, but, having been otherwise gained as knowledge of perception, it is deposited in reason, since in this way it has passed into quite a different method of cognition, namely the abstract. All rational knowledge, that is to say, knowledge raised to consciousness in the abstract, is related to science proper as a part to the whole. Every person has obtained a rational knowledge about many different things through experience, through a consideration of the individual things presented to him; but only the person who sets himself the task of obtaining a complete knowledge in the abstract about some species of objects aspires to science. Only by a concept can he single out this species; therefore at the head of every science there is a concept through which the part is thought from the sum-total of all things, and of which that science promises a complete knowledge in the abstract. For example, the concept of spatial relations, or of the action of inorganic bodies on one another, or of the nature of plants and 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 structure of a language, and so on. If science wished to obtain the knowledge of its theme by investigating every individual thing thought through the concept, till it had thus gradually learnt the whole, no human memory would suffice, and no certainty of completeness would be obtainable. It therefore makes use of that previously discussed property of concept-spheres of including one another, and it goes mainly to the wider spheres lying generally within the concept of its theme. When it has determined the relations of these spheres to one another, all that is thought in them is also determined in general, and can now be more and more accurately determined by separating out smaller and smaller concept-spheres. It thus becomes possible for a science to embrace its theme completely. This path to knowledge which it follows, namely that from the general to the particular, distinguishes it from ordinary rational knowledge. Systematic form is therefore an essential and characteristic feature of science. The combination of the most general concept-spheres of every science, in other words, the knowledge of its main principles, is the indispensable condition for mastering it. How far we want to go from these to the more special propositions is a matter of choice; it does not increase the thoroughness but the extent of learning. The number of the main principles to which all the rest are subordinated varies greatly as between the different sciences, so that in some there is more subordination, in others more coordination; and in this respect the former make greater claims on the power of judgement, the latter on memory. It was known even to the scholastics36 that, because the syllogism requires two premisses, no science can start from a single main principle that cannot be deduced further; on the contrary, it must have several, at least two, of these. The strictly classificatory sciences, such as zoology, botany, even physics and chemistry, in so far as these latter refer all inorganic action to a few fundamental forces, have the most subordination. History, on the other hand, has really none at all, for the universal in it consists merely in the survey of the principal periods. From these, however, the particular events cannot be deduced; they are subordinate to them only according to time, and are coordinate with them according to the concept. Therefore history, strictly speaking, is rational knowledge certainly, but not a science. In mathematics, according to Euclid’s treatment, the axioms are the only indemonstrable first principles, and all demonstrations are in gradation strictly subordinate to them. This method of treatment, however, is not essential to mathematics, and in fact every proposition again begins a new spatial construction. In itself, this is independent of the previous constructions, and can actually be known from itself, quite independently of them, in the pure intuition of space, in which even the most complicated construction is just as directly evident as the axiom is. But this will be discussed in more detail later. Meanwhile, every mathematical proposition always remains a universal truth, valid for innumerable particular cases. A graduated process from the simple to the complicated propositions that are to be referred to them is also essential to mathematics; hence mathematics is in every respect a science. The completeness of a science as such, that is to say, according to form, consists in there being as much subordination and as little coordination of the principles as possible. Scientific talent in general, therefore, is the ability to subordinate the concept-spheres according to their different determinations, so that, as Plato repeatedly recommends, science may not be formed merely by something universal and an immense variety of things placed side by side directly under it, but that knowledge may step down gradually from the most universal to the particular through intermediate concepts and divisions, made according to closer and closer definitions. According to Kant’s expressions, this means complying equally with the law of homogeneity and with the law of specification. From the fact that this constitutes real scientific completeness, it follows that the aim of science is not greater certainty, for even the most disconnected single piece of knowledge can have just as much certainty; its aim is rather facility of rational knowledge through its form and the possibility, thus given, of completing such knowledge. It is for this reason a prevalent but perverted opinion that the scientific character of knowledge consists in greater certainty; and just as false is the assertion, following from this, that mathematics and logic alone are sciences in the proper sense, because only in them, on account of their wholly a priori nature, is there irrefutable certainty of knowledge. This last advantage cannot be denied them, but it does not give them a special claim to the nature of science. For that is to be found not in certainty, but in the systematic form of knowledge, established by the gradual descent from the universal to the particular. This way of knowledge from the universal to the particular, peculiar to the sciences, makes it necessary that in them much is established by deduction from previous propositions, that is by proofs. This has given rise to the old error that only what is demonstrated is perfectly true, and that every truth requires a proof. On the contrary, every proof or demonstration requires an undemonstrated truth, and this ultimately supports it or again its own proofs. Therefore a directly established truth is as preferable to a truth established by a proof as spring water is to piped water. Perception, partly pure a priori, as establishing mathematics, partly empirical a posteriori, as establishing all the other sciences, is the source of all truth and the basis of all science. (Logic alone is to be excepted, which is based not on knowledge of perception, but on reason’s direct knowledge of its own laws.) Not the demonstrated judgements or their proofs, but judgements drawn directly from perception and founded thereon instead of on any proof, are in science what the sun is to the world. All light proceeds from them, and, illuminated thereby, the others in turn give light. To establish the truth of such primary judgements directly from perception, to raise such foundations of science from the immense number of real things, is the work of the power of judgement. This consists in the ability to carry over into abstract consciousness correctly and exactly what is known in perception; and judgement accordingly is the mediator between understanding and reason. Only outstanding and extraordinary strength of judgement in an individual can actually advance the sciences, but anyone who has merely a healthy faculty of reason is able to deduce propositions from propositions, to demonstrate, to draw conclusions. On the other hand, to lay down and fix in appropriate concepts for reflection what is known through perception, so that, firstly, what is common to many real objects is thought through one concept, and secondly, their points of difference are thought through just as many concepts; this is done by the power of judgement. From this what is different is known and thought as different, in spite of a partial agreement; and what is identical is known and thought as identical, in spite of a partial difference, all according to the purpose and consideration that actually exist in each case. This too is the work of judgement. Want of judgement is silliness. The silly person fails to recognize, now the partial or relative difference of what is in one respect identical, now the identity of what is relatively or partially different. Moreover, to this explanation of the power of judgement Kant’s division of it into reflecting and subsuming judgement can be applied, according as it passes from the objects of perception to the concept, or from the concept to the objects of perception, in both cases always mediating between knowledge of the understanding through perception and reflective knowledge of reason. There can be no truth that could be brought out absolutely through syllogisms alone, but the necessity of establishing truth merely through syllogisms is always only relative, indeed subjective. As all proofs are syllogisms, we must first seek for a new truth not a proof, but direct evidence, and only so long as this is wanting is the proof to be furnished for the time being. No science can be capable of demonstration throughout any more than a building can stand in the air. All its proofs must refer to something perceived, and hence no longer capable of proof, for the whole world of reflection rests on, and is rooted in, the world of perception. All ultimate, i.e., original, evidence is one of intuitive perception, as the word already discloses. Accordingly, it is either empirical or based on the perception a priori of the conditions of possible experience. In both cases, therefore, it affords only immanent, not transcendent knowledge. Every concept has its value and its existence only in reference to a representation from perception, although such reference may be very indirect. What holds good of the concepts holds good also of the judgements constructed from them, and of all the sciences. Therefore it must be possible in some way to know directly, even without proofs and syllogisms, every truth that is found through syllogisms and communicated by proofs. This is most difficult certainly in the case of many complicated mathematical propositions which we reach only by chains of syllogisms; for example, the calculation of the chords and tangents to all arcs by means of deductions from the theorem of Pythagoras. But even such a truth cannot rest essentially and solely on abstract principles, and the spatial relations at the root of it must also be capable of being so displayed for pure intuition a priori, that their abstract expression is directly established. But shortly we shall discuss demonstration in mathematics in detail.
It may be that people often speak in a lofty tone about sciences which rest entirely on correct conclusions from sure premisses, and are therefore incontestably true. But through purely logical chains of reasoning, however true the premisses may be, we shall never obtain more than an elucidation and exposition of what already lies complete in the premisses; thus we shall only explicitly expound what was already implicitly understood therein. By these esteemed sciences are meant especially the mathematical, in particular astronomy. But the certainty of astronomy arises from the fact that it has for its basis the intuition or perception of space, given a priori, and hence infallible. All spatial relations, however, follow from one another with a necessity (ground of being) that affords a priori certainty, and they can with safety be derived from one another. To these mathematical provisions is added only a single force of nature, namely gravity, operating exactly in proportion to the masses and to the square of the distance; and finally we have the law of inertia, a priori certain, because it follows from the law of causality, together with the empirical datum of the motion impressed on each of these masses once for all. This is the whole material of astronomy, which, by both its simplicity and its certainty, leads to definite results that are very interesting by virtue of the magnitude and importance of the objects. For example, if I know the mass of a planet and the distance from it of its satellite, I can infer with certainty the latter’s period of revolution according to Kepler’s second law. But the basis of this law is that at this distance only this velocity simultaneously chains the satellite to the planet, and prevents it from falling into it. Hence only on such a geometrical basis, that is to say, by means of an intuition or perception a priori, and moreover under the application of a law of nature, can we get very far with syllogisms, since here they are, so to speak, merely bridges from one perceptive apprehension to another. But it is not so with merely plain syllogisms on the exclusively logical path. The origin of the first fundamental truths of astronomy is really induction, in other words, the summarizing into one correct and directly founded judgement of what is given in many perceptions. From this judgement hypotheses are afterwards formed, and the confirmation of these by experience, as induction approaching completeness, gives the proof for that first judgement. For example, the apparent motion of the planets is known empirically; after many false hypotheses about the spatial connexion of this motion (planetary orbit), the correct one was at last found, then the laws followed by it (Kepler’s laws), and finally the cause of these laws (universal gravitation). The empirically known agreement of all observed cases with the whole of the hypotheses and with their consequences, hence induction, gave them complete certainty. The discovery of the hypothesis was the business of the power of judgement which rightly comprehended the given fact, and expressed it accordingly; but induction, in other words perception of many kinds, confirmed its truth. But this truth could be established even directly through a single empirical perception, if we could freely pass through universal space, and had telescopic eyes. Consequently, even here syllogisms are not the essential and only source of knowledge, but are always in fact only a makeshift.
Finally, in order to furnish a third example from a different sphere, we will observe that even the so-called metaphysical truths, that is, such as are laid down by Kant in the Metaphysical Rudiments of Natural Science, do not owe their evidence to proofs. We know immediately what is a priori certain; this, as the form of all knowledge, is known to us with the greatest necessity. For instance, we know immediately as negative truth that matter persists, in other words, that it can neither come into being nor pass away. Our pure intuition or perception of space and time gives the possibility of motion; the understanding gives in the law of causality the possibility of change of form and quality, but we lack the forms for conceiving an origin or disappearance of matter. Therefore this truth has at all times been evident to all men everywhere, and has never been seriously doubted; and this could not be the case if its ground of knowledge were none other than the very difficult and hair-splitting proof of Kant. But in addition, I have found Kant’s proof to be false (as explained in the Appendix), and I have shown above that the permanence of matter is to be deduced not from the share that time has in the possibility of experience, but from that which space has. The real foundation of all truths which in this sense are called metaphysical, that is, of abstract expressions of the necessary and universal forms of knowledge, can be found not in abstract principles, but only in the immediate consciousness of the forms of representation, manifesting itself through statements a priori that are apodictic and in fear of no refutation. But if we still want to furnish a proof of them, this can consist only in our showing that what is to be proved is already contained in some undoubted truth as a part or a presupposition of it. Thus, for example, I have shown that all empirical perception implies the application of the law of causality. Hence knowledge of this is a condition of all experience, and therefore cannot be given and conditioned through experience, as Hume asserted. Proofs are generally less for those who want to learn than for those who want to dispute. These latter obstinately deny directly established insight. Truth alone can be consistent in all directions; we must therefore show such persons that they admit under one form and indirectly what under another form and directly they deny, i.e. the logically necessary connexion between what is denied and what is admitted.
Moreover, it is a consequence of the scientific form, namely subordination of everything particular under something general, and then under something more and more general, that the truth of many propositions is established only logically, namely through their dependence on other propositions, and hence through syllogisms which appear simultaneously as proofs. But we should never forget that this entire form is a means only to facilitating knowledge, not to greater certainty. It is easier to know the nature of an animal from the species to which it belongs, and so on upwards from the genus, family, order, and class, than to examine the animal itself which is given to us on each occasion. But the truth of all propositions deduced by syllogisms is always only conditioned by, and ultimately dependent on, a truth that rests not on syllogisms, but on perception or intuition. If this perception were always as much within our reach as deduction through a syllogism is, it would be in every way preferable. For every deduction from concepts is exposed to many deceptions on account of the fact, previously demonstrated, that many different spheres are linked and interlocked, and again because their content is often ill-defined and uncertain. Examples of this are the many proofs of false doctrines and sophisms of every kind. Syllogisms are indeed perfectly certain as regards form, but very uncertain through their matter, namely the concepts. For on the one hand the spheres of these are often not defined with sufficient sharpness, and on the other they intersect one another in so many different ways, that one sphere is partly contained in many others, and therefore we can pass arbitrarily from it to one or another of these, and again to others, as we have already shown. Or, in other words, the minor and also the middle term can always be subordinated to different concepts, from which we choose at will the major term and the middle, whereupon the conclusion turns out differently. Consequently, immediate evidence is everywhere far preferable to demonstrated truth, and the latter is to be accepted only when the former is too remote, and not when it is just as near as, or even nearer than, the latter. Therefore we saw above that actually with logic, where in each individual case immediate knowledge lies nearer at hand than derived scientific knowledge, we always conduct our thinking only in accordance with immediate knowledge of the laws of thought, and leave logic unused.37