Oxford Studies in Ancient Philosophy, Volume 59 by Victor Caston;

Author:Victor Caston; [Caston, Victor]
Language: eng
Format: epub
ISBN: 9780192602749
Publisher: OUP Premium
Published: 2021-08-20T00:00:00+00:00

3. Infinite by division

I now turn to Aristotle’s claims about the infinity by division of magnitudes.

ἔτι ἄπειρον ἅπαν ἢ κατὰ πρόσθεσιν ἢ κατὰ διαίρεσιν ἢ ἀμφοτέρως. (Physics 3. 4, 204a6–7)

Furthermore, everything is called infinite either in the manner of addition or in the manner of division, or in both ways.

τὸ δὲ μέγεθος ὅτι μὲν κατ’ ἐνέργειαν οὐκ ἔστιν ἄπειρον, εἴρηται, διαιρέσει δ’ ἐστίν· οὐ γὰρ χαλεπὸν ἀνελεῖν τὰς ἀτόμους γραμμάς· λείπεται οὖν δυνάμει εἶναι τὸ ἄπειρον. (Physics 3. 6, 206a16–18)

As for magnitude, we have said that it is not infinite in actuality, but it is infinite by division: for it is not difficult to do away with indivisible lines. It remains, then, that the infinite is potentially.

What sort of infinity is it, precisely, that exists potentially but not actually according to the above passage? According to most commentators, Aristotle’s claim is that no magnitude is actually divided into infinitely many parts.78 But an alternative interpretation is suggested by a remark of Zeller’s, who notes that according to Aristotle a magnitude has no infinitely small part.79 Either interpretation would fit with Aristotle’s use of the notion of division to specify the kind of infinitude he has in mind, since the act of dividing produces both more parts and smaller parts.80

I will argue that Zeller’s interpretation is correct: as Aristotle conceives of it, for a magnitude to be actually infinite by division would be for a magnitude to be actually infinitely small. Correspondingly, for a time to be actually infinite by division would be for a time to be actually infinitely brief; and so on for other kinds of quantity.

To see the difference our choice of interpretation makes, consider the division of a line in the manner of Zeno’s Dichotomy, as follows:

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