GA19:69-70 – mathematika (μαθηματικά)

The μαθηματικαὶ ἐπιστήμαι have as their theme τὰ ἐξ ἀφαιρέσεως, that which shows itself by being withdrawn from something and specifically from what is immediately given.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 [Cf. Met. XI, 3,1061a28f; De Caelo III, 1,299a15ff; Met. XIII, 3; Met. XII, 8,1073b6ff]. Hence Aristotle   says: ὁ μαθηματικός   χωρίζει (cf. Phys. II, 2, 193b31ff.). Χωρίζειν, separating, is connected with χώρα, place [Platz  ]; place belongs to beings themselves. The μαθηματικός takes something away from its own place, […] What is peculiar is that the mathematical is not   in a place: οὐκ ἕν   τόπῳ (Met. XIV, 5, 1092a17ff.). 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.

[…] Insofar as the φυσικά ὄντα are κινούμενα, i.e., insofar as motility is a basic determination of their Being, the mathematical can be considered initially as appertaining to beings that move. The mathematical as such is removed from things characterized by motion, […], the mathematical, e.g., a point, is “extracted from beings insofar as they move,” (b34) i.e., insofar as they change, turn around, increase and decrease. And specifically the mathematical is χωριστά τῇ νοήσει, “discerned,” extracted simply in a particular mode of consideration. Κίνησις itself, however, is initially and for the most part κίνησις   κατά τόπον, change of location […] [GA19EN:69-70]