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.
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.)
