Then how are the two basic objects of mathematics to be distinguished? μονὰς οὐσία ἄθετος, στιγμὴ δὲ οὐσία θετός· ταύτην ἐκ προσθέσεως (Post. An. I, 27, 87a36).
Both are οὐσία, something that is for itself. The στιγμή, however, over and against the μονάς, is marked by a πρόσθεσις; in the στιγμή there resides a θέσις in a preeminent sense. What is the meaning of this θέσις which characterizes the point in opposition to the μονάς? A thorough elucidation of this nexus would have to take up the question of place and space. Here I can only indicate what is necessary to make understandable the distinction of the ἀκριβές within the disciplines of mathematics.
Θέσις has the same character as ἕξις, διάθεσις. "Ἕξις = to find oneself in a definite situation, to have something in oneself, to retain, and in retaining to be directed toward something. Θέσις = orientation, situation; it has the character of being oriented toward something. ἔστι δὲ καὶ τὰ τοιαῦτα τῶν πρός τι οἷον ἕξις, διάθεσις,... θέσις (Cat. 7, 6b2f.). According to its categorial determination, θέσις is τῶν πρός τι, "it belongs to what is πρός τι." Every θέσις is a θέσις τινός (cf. b6).
α) Τόπος and θέσις (according to Phys. V, 1-5). The absolute
determinateness (φύσει) of τόπος, the relative determinateness
(πρὸς ἡμᾶς) of θέσις. The essence of τόπος: limit (πέρας) and
possibility (δύναμις) of the proper Being of a being.
We need to clarify briefly the distinction between θέσις and τόπος. Aristotle emphasizes that the mathematical objects are οὐκ ἐν τόπῳ (cf. Met. XIV, 5, 1092a19f.), "not anyplace."3 The modern concept of space must not at all be allowed to intrude here. Aristotle determines τόπος at first in an apparently quite naive way. ὅτι μὲν οὖν ἔστιν ὁ τόπος, δοκεῖ δῆλον εἶναι ἐκ τῆς ἀντιμεταστάσεως· ὅπου γὰρ ἔστι νῦν ὕδωρ, ἐνταῦθα ἐξελθόντος ὥσπερ ἐξ ἀγγείου πάλιν ἀὴρ ἐκεῖ ἔνεστιν, ὁτὲ δὲ τὸν αὐτὸν τόπον τοῦτον ἄλλο τι τῶν σωμάτων κατέχει· τοῦτο δὴ τῶν ἐγγιγνομένων καὶ μεταβαλλόντων ἕτερον πάντων εἶναι δοκεῖ· ἐν ᾧ γὰρ ἀήρ ἐστι νῦν, ὕδωρ ἐν τούτῳ πρότερον ἦν, ὥστε δῆλον ὡς ἦν ὁ τόπος τι καὶ ἡ χώρα ἕτερον ἀμφοῖν, εἰς ἣν καὶ ἐξ ἧς μετέβαλον (Phys. IV, 1, 208b1ff.). Τόπος must itself be something. If there formerly was water in a container and if now there is air in it, then the τόπος is something other than that which fills it. The place was already, ἦν, i.e., before specifically water or air was in it. The ἦν does not mean that the τόπος would be something separated, separated from what is in it; the place is simply something other than the two things which have been exchanged in it.
2. Cf. De An. I, 4, 409a6ff.
3. Cf. p. 69.