On the Fourfold Root of the Principle of Sufficient Reason

§ 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.

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.