First Aspect. The Idea Subordinated To The Principle Of Sufficient Reason: The Object Of Experience And Science.

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

§ 14. 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 [pg 081] 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, [pg 082] 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 [pg 083] 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 [pg 084] 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 [pg 085] 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 [pg 086] 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 [pg 087] 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 [pg 088] 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 [pg 089] 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 [pg 090] 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 [pg 091] 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 [pg 092] 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 [pg 093] 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, [pg 094] 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 [pg 095] 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:—

[pg 096]

In the case of unequal catheti also, and indeed generally in the case of every possible geometrical truth, it is quite possible to obtain such a conviction based on perception, because these truths were always discovered by such an empirically known necessity, and their demonstration was only thought out afterwards in addition. Thus we only require an analysis of the process of thought in the first discovery of a geometrical truth in order to know its necessity empirically. It is the analytical method in general that I wish for the exposition of mathematics, instead of the synthetical method which Euclid made use of. Yet this would have very great, though not insuperable, difficulties in the case of complicated mathematical truths. Here and there in Germany men are beginning to alter the exposition of mathematics, and to proceed more in this analytical way. The greatest effort in this direction has been made by Herr Kosack, teacher of mathematics and physics in the Gymnasium at Nordhausen, who added a thorough attempt to teach geometry according to my principles to the programme of the school examination on the 6th of April 1852.

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 [pg 097] 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 [pg 098] 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.

When a criminal is examined, a procès-verbal is made of his statement in order that we may judge of its truth from its consistency. But this is only a makeshift, and we are not satisfied with it if it is possible to investigate the truth of each of his answers for itself; especially as he might lie consistently from the beginning. But Euclid investigated space according to this first method. He set about it, indeed, under the correct assumption that nature must everywhere be consistent, and that therefore it must also be so in space, its fundamental form. Since then the parts of space stand to each other in a relation of reason and consequent, no single property of space can be different from what it is without being in contradiction with all the others. But this is a very troublesome, unsatisfactory, and roundabout way to follow. It prefers indirect knowledge to direct, which is just as certain, and it separates the knowledge that a thing is from the knowledge why it is, to the great disadvantage of the science; and lastly, it entirely withholds from the beginner insight into the laws of space, and indeed renders him unaccustomed to the special investigation of the ground and inner connection of things, inclining him to be satisfied with a mere historical knowledge that a thing is as it is. The exercise of acuteness which this method is unceasingly extolled as affording consists merely in this, that the pupil practises drawing conclusions, i.e., he practises applying the principle of contradiction, but specially he exerts his memory to retain all those data whose agreement is to be tested. [pg 099] Moreover, it is worth noticing that this method of proof was applied only to geometry and not to arithmetic. In arithmetic the truth is really allowed to come home to us through perception alone, which in it consists simply in counting. As the perception of numbers is in time alone, and therefore cannot be represented by a sensuous schema like the geometrical figure, the suspicion that perception is merely empirical, and possibly illusive, disappeared in arithmetic, and the introduction of the logical method of proof into geometry was entirely due to this suspicion. As time has only one dimension, counting is the only arithmetical operation, to which all others may be reduced; and yet counting is just intuition or perception a priori, to which there is no hesitation in appealing here, and through which alone everything else, every sum and every equation, is ultimately proved. We prove, for example, not that (7 + 9 × 8 - 2)/3 = 42; but we refer to the pure perception in time, counting thus makes each individual problem an axiom. Instead of the demonstrations that fill geometry, the whole content of arithmetic and algebra is thus simply a method of abbreviating counting. We mentioned above that our immediate perception of numbers in time extends only to about ten. Beyond this an abstract concept of the numbers, fixed by a word, must take the place of the perception; which does not therefore actually occur any longer, but is only indicated in a thoroughly definite manner. Yet even so, by the important assistance of the system of figures which enables us to represent all larger numbers by the same small ones, intuitive or perceptive evidence of every sum is made possible, even where we make such use of abstraction that not only the numbers, but indefinite quantities and whole operations are thought only in the abstract and indicated as so thought, as [sqrt](r^b) so that we do not perform them, but merely symbolise them.

We might establish truth in geometry also, through [pg 100] pure a priori perception, with the same right and certainty as in arithmetic. It is in fact always this necessity, known through perception in accordance with the principle of sufficient reason of being, which gives to geometry its principal evidence, and upon which in the consciousness of every one, the certainty of its propositions rests. The stilted logical demonstration is always foreign to the matter, and is generally soon forgotten, without weakening our conviction. It might indeed be dispensed with altogether without diminishing the evidence of geometry, for this is always quite independent of such demonstration, which never proves anything we are not convinced of already, through another kind of knowledge. So far then it is like a cowardly soldier, who adds a wound to an enemy slain by another, and then boasts that he slew him himself.22

After all this we hope there will be no doubt that the evidence of mathematics, which has become the pattern and symbol of all evidence, rests essentially not upon demonstration, but upon immediate perception, which is thus here, as everywhere else, the ultimate ground and source of truth. Yet the perception which lies at the basis of mathematics has a great advantage over all other perception, and therefore over empirical perception. It is a priori, and therefore independent of experience, which is always given only in successive parts; therefore everything is equally near to it, and we can start either from the reason or from the consequent, as we please. Now this makes it absolutely reliable, [pg 101] for in it the consequent is known from the reason, and this is the only kind of knowledge that has necessity; for example, the equality of the sides is known as established by the equality of the angles. All empirical perception, on the other hand, and the greater part of experience, proceeds conversely from the consequent to the reason, and this kind of knowledge is not infallible, for necessity only attaches to the consequent on account of the reason being given, and no necessity attaches to the knowledge of the reason from the consequent, for the same consequent may follow from different reasons. The latter kind of knowledge is simply induction, i.e., from many consequents which point to one reason, the reason is accepted as certain; but as the cases can never be all before us, the truth here is not unconditionally certain. But all knowledge through sense-perception, and the great bulk of experience, has only this kind of truth. The affection of one of the senses induces the understanding to infer a cause of the effect, but, as a conclusion from the consequent to the reason is never certain, illusion, which is deception of the senses, is possible, and indeed often occurs, as was pointed out above. Only when several of the senses, or it may be all the five, receive impressions which point to the same cause, the possibility of illusion is reduced to a minimum; but yet it still exists, for there are cases, for example, the case of counterfeit money, in which all the senses are deceived. All empirical knowledge, and consequently the whole of natural science, is in the same position, except only the pure, or as Kant calls it, metaphysical part of it. Here also the causes are known from the effects, consequently all natural philosophy rests upon hypotheses, which are often false, and must then gradually give place to more correct ones. Only in the case of purposely arranged experiments, knowledge proceeds from the cause to the effect, that is, it follows the method that affords certainty; but these experiments [pg 102] themselves are undertaken in consequence of hypotheses. Therefore, no branch of natural science, such as physics, or astronomy, or physiology could be discovered all at once, as was the case with mathematics and logic, but required and requires the collected and compared experiences of many centuries. In the first place, repeated confirmation in experience brings the induction, upon which the hypothesis rests, so near completeness that in practice it takes the place of certainty, and is regarded as diminishing the value of the hypothesis, its source, just as little as the incommensurability of straight and curved lines diminishes the value of the application of geometry, or that perfect exactness of the logarithm, which is not attainable, diminishes the value of arithmetic. For as the logarithm, or the squaring of the circle, approaches infinitely near to correctness through infinite fractions, so, through manifold experience, the induction, i.e., the knowledge of the cause from the effects, approaches, not infinitely indeed, but yet so near mathematical evidence, i.e., knowledge of the effects from the cause, that the possibility of mistake is small enough to be neglected, but yet the possibility exists; for example, a conclusion from an indefinite number of cases to all cases, i.e., to the unknown ground on which all depend, is an induction. What conclusion of this kind seems more certain than that all men have the heart on the left side? Yet there are extremely rare and quite isolated exceptions of men who have the heart upon the right side. Sense-perception and empirical science have, therefore, the same kind of evidence. The advantage which mathematics, pure natural science, and logic have over them, as a priori knowledge, rests merely upon this, that the formal element in knowledge upon which all that is a priori is based, is given as a whole and at once, and therefore in it we can always proceed from the cause to the effect, while in the former kind of knowledge we are generally obliged to proceed from the effect to the [pg 103] cause. In other respects, the law of causality, or the principle of sufficient reason of change, which guides empirical knowledge, is in itself just as certain as the other forms of the principle of sufficient reason which are followed by the a priori sciences referred to above. Logical demonstrations from concepts or syllogisms have the advantage of proceeding from the reason to the consequent, just as much as knowledge through perception a priori, and therefore in themselves, i.e., according to their form, they are infallible. This has greatly assisted to bring demonstration in general into such esteem. But this infallibility is merely relative; the demonstration merely subsumes under the first principles of the science, and it is these which contain the whole material truth of science, and they must not themselves be demonstrated, but must be founded on perception. In the few a priori sciences we have named above, this perception is pure, but everywhere else it is empirical, and is only raised to universality through induction. If, then, in the empirical sciences also, the particular is proved from the general, yet the general, on the other hand, has received its truth from the particular; it is only a store of collected material, not a self-constituted foundation.

So much for the foundation of truth. Of the source and possibility of error many explanations have been tried since Plato's metaphorical solution of the dove-cot where the wrong pigeons are caught, &c. (Theætetus, p. 167, et seq.) Kant's vague, indefinite explanation of the source of error by means of the diagram of diagonal motion, will be found in the “Critique of Pure Reason,” p. 294 of the first edition, and p. 350 of the fifth. As truth is the relation of a judgment to its ground of knowledge, it is always a problem how the person judging can believe that he has such a ground of knowledge and yet not have it; that is to say, how error, the deception of reason, is possible. I find this possibility quite analogous to that of illusion, or the deception of the understanding, [pg 104] which has been explained above. My opinion is (and this is what gives this explanation its proper place here) that every error is an inference from the consequent to the reason, which indeed is valid when we know that the consequent has that reason and can have no other; but otherwise is not valid. The person who falls into error, either attributes to a consequent a reason which it cannot have, in which case he shows actual deficiency of understanding, i.e., deficiency in the capacity for immediate knowledge of the connection between the cause and the effect, or, as more frequently happens, he attributes to the effect a cause which is possible, but he adds to the major proposition of the syllogism, in which he infers the cause from the effect, that this effect always results only from this cause. Now he could only be assured of this by a complete induction, which, however, he assumes without having made it. This “always” is therefore too wide a concept, and instead of it he ought to have used “sometimes” or “generally.” The conclusion would then be problematical, and therefore not erroneous. That the man who errs should proceed in this way is due either to haste, or to insufficient knowledge of what is possible, on account of which he does not know the necessity of the induction that ought to be made. Error then is quite analogous to illusion. Both are inferences from the effect to the cause; the illusion brought about always in accordance with the law of causality, and by the understanding alone, thus directly, in perception itself; the error in accordance with all the forms of the principle of sufficient reason, and by the reason, thus in thought itself; yet most commonly in accordance with the law of causality, as will appear from the three following examples, which may be taken as types or representatives of the three kinds of error. (1.) The illusion of the senses (deception of the understanding) induces error (deception of the reason); for example, if one mistakes a painting for an alto-relief, and actually [pg 105] takes it for such; the error results from a conclusion from the following major premise: “If dark grey passes regularly through all shades to white; the cause is always the light, which strikes differently upon projections and depressions, ergo—.” (2.) “If there is no money in my safe, the cause is always that my servant has got a key for it: ergo—.” (3.) “If a ray of sunlight, broken through a prism, i.e., bent up or down, appears as a coloured band instead of round and white as before, the cause must always be that light consists of homogeneous rays, differently coloured and refrangible to different degrees, which, when forced asunder on account of the difference of their refrangibility, give an elongated and variously-coloured spectrum: ergo—bibamus!”—It must be possible to trace every error to such a conclusion, drawn from a major premise which is often only falsely generalised, hypothetical, and founded on the assumption that some particular cause is that of a certain effect. Only certain mistakes in counting are to be excepted, and they are not really errors, but merely mistakes. The operation prescribed by the concepts of the numbers has not been carried out in pure intuition or perception, in counting, but some other operation instead of it.