§15 [102-104]

Those who speak of the Ideas are not themselves clear about which possibilities χωρισμός harbors. Χωρισμός has a justifiable sense in mathematics, but not where it is a matter of determining the ἀρχαί of beings. τὰ γὰρ φυσικὰ χωρίζουσιν ἧττον ὄντα χωριστὰ τῶν μαθηματικῶν (193b36f.). Such a one "posits the φύσει ὄντα (i.e., the ἀρχαί pertaining to these as such) for themselves, in a separate place, but they are even less to be removed from their place." For the φύσει ὄντα are κινούμενα; in every category of physical beings there resides a determinate relation to motion. In his Ideas, as ἀρχαί, however, the man in question leaves out precisely the κίνησις which is the basic character of the φύσει ὄντα, with the result that he makes of these ἀρχαί genuine beings, among which finally even κίνησις itself becomes one. Yet it is possible to determine the ἀρχαί of the moving φύσει ὄντα in such a way that the ἀρχαί are not taken as divorced from motion and, furthermore, such that κίνησις itself is not taken as an Idea and hence as χωριστόν. In the ἀρχαί the κινούμενον ή κινούμενον must be co-perceived and hence must basically be something else as well, namely the τόπος itself whereby Being and presence are determined.

Let this suffice as an initial orientation concerning the mathematician in opposition to the physicist and at the same time as an indication of the connection of the mathematical χωρίζειν with the one Plato himself promulgates as the determination of the method of grasping the Ideas. We will see later why the Ideas were brought into connection with mathematics. Let us now ask how, within mathematics, geometry differs from arithmetic.

b) The distinction between geometry and arithmetic. The
increasing "abstraction" from the φύσει ὄν: στιγμή = οὐσία
θετός; μονάς = οὐσία ἄθετος.

Geometry has more ἀρχαί than does arithmetic. The objects of geometry are λαμβανόμενα ἐκ προσθέσεως (cf. Post. An. I, 27, 87a35f.), "they are gained from what is determined additionally, through θέσις." Πρόσθεσις does not simply mean "supplement. " What is the character of this πρόσθεσις in geometry? λέγω δ' ἐκ προσθέσεως, οἷον μονὰς οὐσία ἄθετος, στιγμὴ δὲ οὐσία θετός· ταύτην ἐκ προσθέσεως (87a35ff.). Aristotle distinguishes the basic elements of geometry from those of arithmetic. The basic element of arithmetic is μονάς, the unit; the basic element of geometry is στιγμή, the point. Μονάς, the unit—related to μόνον, "unique," "alone"—is what simply remains, μένειν, what is "alone," "for itself." In the case of the point, a θέσις is added. τὸ δὲ μηδαμῇ διαιρετὸν κατὰ τὸ ποσὸν στιγμὴ καὶ μονάς, ἡ μὲν ἄθετος μονὰς ἡ δὲ θετὸς στιγμή (Met. V, 6, 1016b29f.). "What is in no way divisible according to quantity are the point and the μονάς; the latter, however, is without θέσις, the point with θέσις."2