CHAPTER VI.

ON THE THIRD CLASS OF OBJECTS FOR THE SUBJECT AND THAT FORM OF THE PRINCIPLE OF SUFFICIENT REASON WHICH PREDOMINATES IN IT.

§ 35. _Explanation of this Class of Objects._

It is the formal part of complete representations--that is to say, the intuitions given us _à priori_ of the forms of the outer and inner sense, _i.e._ of Space and of Time--which constitutes the Third Class of Objects for our representative faculty.

As pure intuitions, these forms are objects for the faculty of representation by themselves and apart from complete representations and from the determinations of being empty or filled which these representations first add to them; since even pure points and pure lines cannot be brought to sensuous perception, but are only _à priori_ intuitions, just as the infinite expansion and the infinite divisibility of Space and of Time are exclusively objects of pure intuition and foreign to empirical perception. That which distinguishes the third class of representations, in which Space and Time are _pure intuitions_, from the first class, in which they are _sensuously_ (and moreover conjointly) _perceived_, is Matter, which I have therefore defined, on the one hand, as the perceptibility of Space and Time, on the other, as objectified Causality.

The form of Causality, on the contrary, which belongs to the Understanding, is not separately and by itself an object for our faculty of representation, nor have we consciousness of it, until it is connected with what is material in our knowledge.

§ 36. _Principle of the Sufficient Reason of Being._

Space and Time are so constituted, that all their parts stand in mutual relation, so that each of them conditions and is conditioned by another. We call this relation in Space, _position_; in Time, _succession_. These relations are peculiar ones, differing entirely from all other possible relations of our representations; neither the Understanding nor the Reason are therefore able to grasp them by means of mere conceptions, and pure intuition _à priori_ alone makes them intelligible to us; for it is impossible by mere conceptions to explain clearly what is meant by above and below, right and left, behind and before, before and after. Kant rightly confirms this by the assertion, that the distinction between our right and left glove cannot be made intelligible in any other way than by intuition. Now, the law by which the divisions of Space and of Time determine one another reciprocally with reference to these relations (position and succession) is what I call the _Principle of the Sufficient Reason of Being, principium rationis sufficientis essendi_. I have already given an example of this relation in § 15, by which I have shown, through the connection between the sides and angles of a triangle, that this relation is not only quite different from that between cause and effect, but also from that between reason of knowledge and consequent; wherefore here the condition may be called _Reason of Being_, _ratio essendi_. The insight into such a _reason of being_ can, of course, become a reason of knowing: just as the insight into the law of causality and its application to a particular case is the reason of knowledge of the effect; but this in no way annuls the complete distinction between Reason of Being, Reason of Becoming, and Reason of Knowing. It often happens, that what according to _one_ form of our principle is _consequence_, is, according to another, _reason_. The rising of the quicksilver in a thermometer, for instance, is the _consequence_ of increased heat according to the law of causality, while according to the principle of the sufficient reason of knowing it is the _reason_, the ground of knowledge, of the increased heat and also of the judgment by which this is asserted.

§ 37. _Reason of Being in Space._

The position of each division of Space towards any other, say of any given line--and this is equally applicable to planes, bodies, and points--determines also absolutely its totally different position with reference to any other possible line; so that the latter position stands to the former in the relation of the consequent to its reason. As the position of this given line towards any other possible line likewise determines its position towards all the others, and as therefore the position of the first two lines is itself determined by all the others, it is immaterial which we consider as being first determined and determining the others, _i.e._ which particular one we regard as _ratio_ and which others as _rationata_. This is so, because in Space there is no succession; for it is precisely by uniting Space and Time to form the collective representation of the complex of experience, that the representation of coexistence arises. Thus an analogue to so-called reciprocity prevails everywhere in the Reason of Being in Space, as we shall see in § 48, where I enter more fully into the reciprocity of reasons. Now, as every line is determined by all the others just as much as it determines them, it is arbitrary to consider any line merely as determining and not as being determined, and the position of each towards any other admits the question as to its position with reference to some other line, which second position necessarily determines the first and makes it that which it is. It is therefore just as impossible to find an end _a parte ante_ in the series of links in the chain of Reasons of Being as in that of Reasons of Becoming, nor can we find any _a parte post_ either, because of the infinity of Space and of the lines possible within Space. All possible relative spaces are figures, because they are limited; and all these figures have their Reason of Being in one another, because they are conterminous. The _series rationum essendi_ in Space therefore, like the _series rationum fiendi_, proceeds _in infinitum_; and moreover not only in a single direction, like the latter, but in all directions.

Nothing of all this can be proved; for the truth of these principles is transcendental, they being directly founded upon the intuition of Space given us _à priori_.

§ 38. _Reason of being in Time. Arithmetic._

Every instant in Time is conditioned by the preceding one. The Sufficient Reason of Being, as the law of consequence, is so simple here, because Time has only one dimension, therefore it admits of no multiplicity of relations. Each instant is conditioned by its predecessor; we can only reach it through that predecessor: only so far as this _was_ and has elapsed, does the present one exist. All counting rests upon this nexus of the divisions of Time, numbers only serving to mark the single steps in the succession; upon it therefore rests all arithmetic likewise, which teaches absolutely nothing but methodical abbreviations of numeration. Each number pre-supposes its predecessors as the reasons of its being: we can only reach the number _ten_ by passing through all the preceding numbers, and it is only in virtue of this insight that I know, that where ten are, there also are eight, six, four.

§ 39. _Geometry._

The whole science of Geometry likewise rests upon the nexus of the position of the divisions of Space. It would, accordingly, be an insight into that nexus; only such an insight being, as we have already said, impossible by means of mere conceptions, or indeed in any other way than by intuition, every geometrical proposition would have to be brought back to sensuous intuition, and the proof would simply consist in making the particular nexus in question clear; nothing more could be done. Nevertheless we find Geometry treated quite differently. Euclid's Twelve Axioms are alone held to be based upon mere intuition, and even of these only the Ninth, Eleventh, and Twelfth are properly speaking admitted to be founded upon different, separate intuitions; while the rest are supposed to be founded upon the knowledge that in science we do not, as in experience, deal with real things existing for themselves side by side, and susceptible of endless variety, but on the contrary with conceptions, and in Mathematics with _normal intuitions_, i.e. figures and numbers, whose laws are binding for all experience, and which therefore combine the comprehensiveness of the conception with the complete definiteness of the single representation. For although, as intuitive representations, they are throughout determined with complete precision--no room being left in _this_ way by anything remaining undetermined--still they are general, because they are the bare forms of all phenomena, and, as such, applicable to all real objects to which such forms belong. What Plato says of his Ideas would therefore, even in Geometry, hold good of these normal intuitions, just as well as of conceptions, _i.e._ that two cannot be exactly similar, for then they would be but one.[148] This would, I say, be applicable also to normal intuitions in Geometry, if it were not that, as exclusively spacial objects, these differ from one another in mere juxtaposition, that is, in place. Plato had long ago remarked this, as we are told by Aristotle:[149] ἔτι δὲ, παρὰ τὰ αἰσθητὰ καὶ τὰ εἴδη, τὰ μαθηματικὰ τῶν πραγμάτων εἶναί φησι μεταξύ, διαφέροντα τῶν μὲν αἰσθητῶν τῷ ἀΐδια καὶ ἀκίνητα εἶναι, τῶν δὲ εἰδῶν τῷ τὰ μὲν πόλλ' ἄττα ὅμοια εἶναι, τὸ δὲ εἶδος αὐτὸ ἓν ἕκαστον μόνον (_item, præter sensibilia et species, mathematica rerum ait media esse, a sensibilibus quidem differentia eo, quod perpetua et immobilia sunt, a speciebus vero eo, quod illorum quidem multa quædam similia sunt, species vero ipsa unaquæque sola_). Now the mere knowledge that such a difference of place does not annul the rest of the identity, might surely, it seems to me, supersede the other nine axioms, and would, I think, be better suited to the nature of science, whose aim is knowledge of the particular through the general, than the statement of nine separate axioms all based upon the same insight. Moreover, what Aristotle says: ἐν τούτοις ἡ ἰσότης ἑνότης (_in illis æqualitas unitas est_)[150] then becomes applicable to geometrical figures.

[148] Platonic ideas may, after all, be described as normal intuitions, which would hold good not only for what is formal, but also for what is material in complete representations--therefore as complete representations which, as such, would be determined throughout, while comprehending many things at once, like conceptions: that is to say, as representatives of conceptions, but which are quite adequate to those conceptions, as I have explained in § 28.

[149] Aristot. "Metaph." i. 6, with which compare x. 1. "Further, says he, besides things sensible and the ideas, there are things mathematical coming in between the two, which differ from the things sensible, inasmuch as they are eternal and immovable, and from the ideas, inasmuch as many of them are like each other; but the idea is absolutely and only one." (Tr.'s Add.)

[150] "In these it is equality that constitutes unity." (Tr.'s Add.)

But with reference to the normal intuitions in Time, _i.e._ to numbers, even this distinction of juxtaposition no longer exists. Here, as with conceptions, absolutely nothing but the _identitas indiscernibilium_ remains: for there is but one five and one seven. And in this we may perhaps also find a reason why 7 + 5 = 12 is a synthetical proposition _à priori_, founded upon intuition, as Kant profoundly discovered, and not an identical one, as it is called by Herder in his "Metakritik". 12 = 12 is an identical proposition.

In Geometry, it is therefore only in dealing with axioms that we appeal to intuition. All the other theorems are demonstrated: that is to say, a reason of knowing is given, the truth of which everyone is bound to acknowledge. The logical truth of the theorem is thus shown, but not its transcendental truth (v. §§ 30 and 32), which, as it lies in the reason of _being_ and not in the reason of _knowing_, never can become evident excepting by means of intuition. This explains _why_ this sort of geometrical demonstration, while it no doubt conveys the conviction that the theorem which has been demonstrated is true, nevertheless gives no insight as to why that which it asserts is what it is. In other words, we have not found its Reason of Being; but the desire to find it is usually then thoroughly roused. For proof by indicating the reason of knowledge only effects conviction (_convictio_), not knowledge (_cognitio_): therefore it might perhaps be more correctly called _elenchus_ than _demonstratio_. This is why, in most cases, therefore, it leaves behind it that disagreeable feeling which is given by all want of insight, when perceived; and here, the want of knowledge _why_ a thing is as it is, makes itself all the more keenly felt, because of the certainty just attained, _that_ it is as it is. This impression is very much like the feeling we have, when something has been conjured into or out of our pocket, and we cannot conceive how. The reason of knowing which, in such demonstrations as these, is given without the reason of being, resembles certain physical theories, which present the phenomenon without being able to indicate its cause: for instance, Leidenfrost's experiment, inasmuch as it succeeds also in a platina crucible; whereas the reason of being of a geometrical proposition which is discovered by intuition, like every knowledge we acquire, produces satisfaction. When once the reason of being is found, we base our conviction of the truth of the theorem upon that reason alone, and no longer upon the reason of knowing given us by the demonstration. Let us, for instance, take the sixth proposition of the first Book of Euclid:--

"If two angles of a triangle are equal, the sides also which subtend, or are opposite to, the equal angles shall be equal to one another." (See fig. 3.)

Which Euclid demonstrates as follows:--

"Let _a b c_ be a triangle having the angle _a b c_ equal to the angle _a c b_, then the side _a c_ must be equal to the side _a b_ also.

"For, if side _a b_ be not equal to side _a c_, one of them is greater than the other. Let _a b_ be greater than _a c_; and from _b a_ cut off _b d_ equal to _c a_, and draw _d c_. Then, in the triangles _d b c_, _a b c_, because _d b_ is equal to _a c_, and _b c_ is common to both triangles, the two sides _d b_ and _b c_ are equal to the two sides _a c_, _a b_, each to each; and the angle _d b c_ is equal to the angle _a c b_, therefore the base _d c_ is equal to the base _a b_, and the triangle _d b c_ is equal to the triangle _a b c_, the less triangle equal to the greater,--which is absurd. Therefore _a b_ is not unequal to _a c_, that is, _a b_ is equal to _a c_."

Now, in this demonstration we have a reason of knowing for the truth of the proposition. But who bases his conviction of that geometrical truth upon this proof? Do we not rather base our conviction upon the reason of being, which we know intuitively, and according to which (by a necessity which admits of no further demonstration, but only of evidence through intuition) two lines drawn from both extreme ends of another line, and inclining equally towards each other, can only meet at a point which is equally distant from both extremities; since the two arising angles are properly but one, to which the oppositeness of position gives the appearance of being two; wherefore there is no reason why the lines should meet at any point nearer to the one end than to the other.

It is the knowledge of the reason of being which shows us the necessary consequence of the conditioned from its condition--in this instance, the lateral equality from the angular equality--that is, it shows their connection; whereas the reason of knowing only shows their coexistence. Nay, we might even maintain that the usual method of proving merely convinces us of their coexistence in the actual figure given us as an example, but by no means that they are always coexistent; for, as the necessary connection is not shown, the conviction we acquire of this truth rests simply upon induction, and is based upon the fact, that we find it is so in every figure we make. The reason of being is certainly not as evident in all cases as it is in simple theorems like this 6th one of Euclid; still I am persuaded that it might be brought to evidence in every theorem, however complicated, and that the proposition can always be reduced to some such simple intuition. Besides, we are all just as conscious _à priori_ of the necessity of such a reason of being for each relation of Space, as we are of the necessity of a cause for each change. In complicated theorems it will, of course, be very difficult to show that reason of being; and this is not the place for difficult geometrical researches. Therefore, to make my meaning somewhat clearer, I will now try to bring back to its reason of being a moderately complicated proposition, in which nevertheless that reason is not immediately evident. Passing over the intermediate theorems, I take the 16th:

"In every triangle in which one side has been produced, the exterior angle is greater than either of the interior opposite angles."

This Euclid demonstrates in the following manner (see fig. 4):--

"Let _a b c_ be a triangle; and let the side _b c_ be produced to _d_; then the exterior angle _a c d_ shall be greater than either of the interior opposite angles _b a c_ or _c b a_. Bisect the side _a c_ at _e_, and join _b e_; produce _b e_ to _f_, making _e f_ equal to _e b_, and join _f c_. Produce _a c_ to _g_. Because _a e_ is equal to _e c_, and _b e_ to _e f_; the two sides _a e_, _e b_, are equal to the two sides _c e_, _e f_, each to each; and the angle _a e b_ is equal to the angle _c e f_, because they are opposite vertical angles; therefore the base _a b_ is equal to the base _c f_, and the triangle _a e b_ is equal to the triangle _c e f_, and the remaining angles of one triangle to the remaining angles of the other, each to each, to which the equal sides are opposite; therefore the angle _b a e_ is equal to the angle _e c f_. But the angle _e c d_ is greater than the angle _e c f_. Therefore the angle _a c d_ is greater than the angle _a b c_."

"In the same manner, if the side _b c_ be bisected, and the side _a c_ be produced to _g_, it may be demonstrated that the angle _b c g_, that is, the opposite vertical angle _a c d_ is greater than the angle _a b c_."

My demonstration of the same proposition would be as follows (see fig. 5):--

For the angle _b a c_ to be even equal to, let alone greater than, the angle _a c d_, the line _b a_ toward _c a_ would have to lie in the same direction as _b d_ (for this is precisely what is meant by equality of the angles), _i.e._, it must be parallel with _b d_; that is to say, _b a_ and _b d_ must never meet; but in order to form a triangle they must meet (reason of being), and must thus do the contrary of that which would be required for the angle _b a c_ to be of the same size as the angle _a c d_.

For the angle _a b c_ to be even equal to, let alone greater than, the angle _a c d_, line _b a_ must lie in the same direction towards _b d_ as _a c_ (for this is what is meant by equality of the angles), _i.e._, it must be parallel with _a c_, that is to say, _b a_ and _a c_ must never meet; but in order to form a triangle _b a_ and _a c_ must meet and must thus do the contrary of that which would be required for the angle _a b c_ to be of the same size as _a c d_.

By all this I do not mean to suggest the introduction of a new method of mathematical demonstration, nor the substitution of my own proof for that of Euclid, for which its whole nature unfits it, as well as the fact that it presupposes the conception of parallel lines, which in Euclid comes much later. I merely wished to show what the reason of being is, and wherein lies the difference between it and the reason of knowing, which latter only effects _convictio_, a thing that differs entirely from insight into the reason of being. The fact that Geometry only aims at effecting _convictio_, and that this, as I have said, leaves behind it a disagreeable impression, but gives no insight into the reason of being--which insight, like all knowledge, is satisfactory and pleasing--may perhaps be one of the reasons for the great dislike which many otherwise eminent heads have for mathematics.

I cannot resist again giving fig. 6, although it has already been presented elsewhere; because the mere sight of it without words conveys ten times more persuasion of the truth of the Pythagorean theorem than Euclid's mouse-trap demonstration.

Those readers for whom this chapter may have a special interest will find the subject of it more fully treated in my chief work, "Die Welt als Wille und Vorstellung," vol. i. § 15; vol. ii. chap. 13.