§15 [99-101]

Those disciplines are more rigorous and more fundamental which proceed from fewer ἀρχαί, which hence posit fewer original determinations in the beings which are their theme. αἱ γὰρ ἐξ ἐλαττόνων ἀκριβέστεραι τῶν ἐκ προσθέσεως λεγομένων, οἷον ἀριθμητικὴ γεωμετρίας (982a26f.). Arithmetic is in this way distinguished from geometry. Arithmetic has fewer ἀρχαί than geometry. In the case of geometry, a πρόσθεσις, something additional, takes place as regards the ἀρχαί. In order to understand this we need a brief general orientation regarding Aristotle's conception of mathematics. We will provide that in an excursus, which will serve at the same time as a preparation for our interpretation of Plato.

§ 15. Excursus: General orientation regarding the essence of
mathematics according to Aristotle.

We want to proceed so as to present the basic issues: a) in μαθηματική in general and, b) in ἀριθμητική and γεωμετρία.

a) Fundamental issues in mathematics in general
(Phys. II, 2). Χωρίζειν as the basic act of mathematics.
Critique of the χωρισμός in Plato's theory of Ideas.

The μαθηματικοὶ ἐπιστῆμαι have as their theme τὰ ἐξ ἀφαιρέσεως, that which shows itself by being withdrawn from something and specifically from what is immediately given. The μαθηματικά are extracted from the φυσικὰ ὄντα, from what immediately shows itself.1 Hence Aristotle says: ὁ μαθηματικὸς χωρίζει (cf. Phys. II, 2, 193b31ff.). Χωρίζειν, separating, is connected with χώρα, place; place belongs to beings themselves. The μαθηματικός takes something away from its own place. ἄτοπον δὲ καὶ τὸ τόπον ἅμα τοῖς στερεοῖς τοῖς μαθηματικοῖς ποιῆσαι (ὁ μὲν γὰρ τόπος τῶν καθ᾽ ἕκαστον ἴδιος, διὸ χωριστὰ τόπῳ, τὰ δὲ μαθηματικὰ οὐ πού), καὶ τὸ εἰπεῖν μὲν ὅτι ποὺ ἔσται, τί δέ ἐστιν ὁ τόπος μή (Met. XIV, 5, 1092a17ff.). What is peculiar is that the mathematical is not in a place: οὐκ ἐν τόπῳ. Taken in terms of modern concepts, this has the ring of a paradox, especially since τόπος is still translated as "space." But only a σῶμα φυσικόν has a τόπος, a location, a place. This χωρίζειν, which we will encounter in Plato's theory of the χωρισμός of the Ideas, where Plato indeed explicitly assigns to the Ideas a τόπος, namely the οὐρανός, this χωρίζειν is for Aristotle the way in which the mathematical itself becomes objective.

1. Cf. Met. XI, 3, 1061a28f; De Caelo III, 1, 299a15ff; Met. XIII, 3; Met. XII, 8, 1073b6ff.