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

§ 15. 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: —