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.