Plato's Sophist [110-111]

Instead, through the θέσις, they acquire an autonomy over and against the physical body. The geometrical objects are indeed not in a place; nevertheless, I can determine in them an above and a below, a right and a left. In a square, e.g., I can determine the sides: above, below, right, left. I still have here the possibility of a determination of the θέσις, the possibility of an analysis situs, i.e., of drawing out differentiations in the sites as such, although the geometrical objects themselves, in what they are, do not possess these determinations. Geometrical objects can always be oriented in accord with a θέσις. Every geometrical point, every element, line, and surface is fixed through a θέσις. Every geometrical object is an οὐσία θετός.5 This θέσις does not have to be a determination, but it pertains to one. On the other hand, the unit, the μονάς, does not bear in itself this orientation; it is οὐσία ἄθετος. In mathematics, the θέσις survives only in geometry, because geometry has a greater proximity to the αἰσθητόν than does arithmetic.

The geometrical consists of a manifold of basic elements—point, line, etc.—which are the πέρατα for the higher geometrical figures. But it is not the case that the higher figures are put together out of such limits. Aristotle emphasizes that a line will never arise out of points (Phys. VI, 1, 231a24ff.), a surface will never arise out of a line, nor a body out of a surface. For between any two points there is again and again a γραμμή, etc. This sets Aristotle in the sharpest opposition to Plato. Indeed, the points are the ἀρχαί of the geometrical, yet not in such a way that the higher geometrical figures would be constructed out of their summation. One cannot proceed from the στιγμή to the σῶμα. One cannot put a line together out of- points. For in each case there is something lying in between, something that cannot itself be constituted out of the preceding elements. This betrays the fact that in the οὐσία θετός there is certainly posited a manifold of elements, but, beyond that, a determinate kind of connection is required, a determinate kind of unity of the manifold. In the realm of arithmetic the same holds. For Aristotle, the μονάς, the unit, is itself not yet number; instead, the first number is the number two.6 Since the μονάς, in distinction to the elements of geometry, does not bear a θέσις, the mode of connection in each realm of objectivities is very different. The mode of connection of an arithmetical whole, of a number, is different than that of a geometrical whole, than a connection of points. Number and geometrical figures are in themselves in each case a manifold. The "fold" is the mode of connection of the manifold. We will understand the distinction between στιγμή and μονάς only if we grasp in each of these the respective essence of the structure of that mode of manifoldness.

5. Cf. p. 71f.

6. Ct. Met. V, 6, 1016b18, 1016b15, and 1021a13; Phys. IV, 12, 220a17ff.