§15 [111-113]

What is the essence of the mode of manifoldness of points, lines, etc.? What is the essence of the mode of manifoldness of number?

γ) The structure of the connection of the manifold in geometry
and arithmetic; συνεχές and ἐφεξῆς.

Our consideration will set forth from the point. We have indicated that geometrical objects still have a certain kinship with what is in αἰσθάνεσθαι. Everything in αἰσθάνεσθαι possesses μέγεθος; everything perceivable has extension. Extension, as understood here, will come to be known as continuousness. Since everything perceivable has extension, μέγεθος, it is an οὐκ ἀδιαίρετον. τὸ αἰσθητόν πᾶν ἐστι μέγεθος καὶ οὐκ ἔστιν ἀδιαίρετον αἰσθητόν (De Sensu VII, 449a20). This peculiar structure of the αἰσθητόν is preserved in the geometrical, insofar as the geometrical, too, is continuous, συνεχές. The point presents only the ultimate and most extreme limit of the continuous. For τὸ δὲ πάντῃ <ἀδιαίρετον,> καὶ θέσιν ἔχον στιγμή, (Met. V, 6, 1016b25f.), "That which cannot be resolved further, in any regard, and specifically that which has a θέσις, an orientation as to site, is the point." Conversely, the γραμμή is μοναχῇ διαιρετὸν (cf. b26f.), that which is resolvable as to one dimension; the surface, ἐπίπεδον, is διχῇ διαιρετὸν (cf. b27), that which is doubly resolvable; and the body, σῶμα, is πάντῃ καὶ τριχῇ διαιρετὸν (b27), that which is divisible trebly, i.e., in each dimension. The question is what Aristotle understands by this peculiar form of connection we call the continuous. Characteristically, Aristotle acquires the determination of continuousness not, as one might suppose, within the compass of his reflections on geometry but within those on physics. It is there that he faces the task of explicating the primary phenomena of copresence, and specifically of worldly co-presence, that of the φύσει ὄντα: Physics, V, chapter 3. I will present, quite succinctly, the definitions of the phenomena of co-presence in order that you may see how the συνεχές is constituted and how the mode of manifoldness within number is related to it. You will then also see to what extent the geometrical has a πρόσθεσις, i.e., to what extent there is more co-posited in it than in number.

αα) The phenomena of co-presence as regards φύσει ὄντα
(Phys. V, 3).

1.) Aristotle lists, as the first phenomenon of co-presence, i.e., of objects being with and being related to one another, specifically as regards the φύσει ὄντα, the ἅμα, the "concurrent," which is not to be understood here in a temporal sense, but which rather concerns place. What is concurrent is what is in one place. We must be on our guard not to take these determinations as self-evident and primitive. The fundamental value of these analyses resides in the fact that Aristotle, in opposition to every sort of theoretical construction, took as his point of departure what is immediately visible.