Plato's Sophist [116-117]

To points there pertains the ἅπτεσθαι, touching, and indeed the ἐχόμενον in the preeminent sense of the συνεχές. To the μονάδες, the units, there pertains, however, only the ἐφεξῆς. The mode of connection of the geometrical, of points, is characterized by the συνεχές, the series of numbers by the ἐφεξῆς, where no touching is necessary. The structure of the connection is in the latter case more simple, as compared to the continuum. With points there can always be something in between; between two points there is always an extension which is more or less large. But that is not necessary in the case of the ἐφεξῆς. Here, therefore, another connection obtains. For there is nothing between unity and twoness. Hence it is clear that the being together of the basic elements in the geometrical has the character of the ἅπτεσθαι or of the συνεχές; the being together of numbers has the character of the ἐφεξῆς, of the one after another. Thus in considering geometrical figures we must add something which according to its structure co-posits more elements than ἐφεξῆς does. Such elements, which are constitutive for the συνεχές, are μέγεθος, πρός τι, θέσις, τόπος, ἄμα, ὑπομένον. The ὑπομένον, "from the very outset to be permanently there," pertains to that which is determined by θέσις.9 Therefore the geometrical is not as original as the arithmetical.

Note here that for Aristotle the primary determination of number, insofar as it goes back to the μονάς as the ἀρχή, has a still more original connection with the constitution of beings themselves, insofar as it pertains equally to the determination of the Being of every being that it "is" and that it is "one": every ὃν is a ἔν. With this, the ἀριθμός in the largest sense (ἀριθμός stands here for the ἔν ) acquires for the structure of beings in general a more fundamental significance as an ontological determination. At the same time it enters into a connection with λόγος, insofar as beings in their ultimate determinations become accessible only in a preeminent λόγος, in νόησις, whereas the geometrical structures are grasped in mere αἴσθησις. Αἴσθησις is where geometrical considerations must stop, στήσεται, where they rest. In arithmetic, on the other hand, λόγος, νοεῖν, is operative, which refrains from every θέσις, from every intuitable dimension and orientation.

Contemporary mathematics is broaching once again the question of the continuum. This is a return to Aristotelian thoughts, insofar as mathematicians are learning to understand that the continuum is not resolvable analytically but that one has to come to understand it as something pregiven, prior to the question of an analytic penetration. The mathematician Hermann Weyl has done work in this direction, and it has been fruitful above all for the fundamental problems of mathematical physics.10

9. Cf. Cat., chapter 6, 5a27f.

10. H. Weyl, Raum—Zeit—Materie. Vorlesungen über allgemeine Relativitätstheorie. Berlin 1918; 5., umgearb. Aufl., Berlin, 1923.