We know the reasonings of Zeno and Melissus only through scanty fragments, and those fragments transmitted by opponents. But it is plain that both of them, especially Zeno, pressed their adversaries with grave difficulties, which it was more easy to deride than to elucidate. Both took their departure from the ground occupied by Parmenides. They agreed with him in recognising the phenomenal, apparent, or relative world, the world of sense and experience, as a subject of knowledge, though of uncertain and imperfect knowledge. Each of them gave, as Parmenides had done, certain affirmative opinions, or at least probable conjectures, for the purpose of explaining it.[14] But beyond this world of appearances, there lay the real, absolute, ontological, ultra-phenomenal, or Noumenal world, which Parmenides represented as Ens unum continuum, and which his opponents contended to be plural and discontinuous. These opponents deduced absurd and ridiculous consequences from the theory of the One. Herein both Zeno and Melissus defended Parmenides. Zeno, the better dialectician of the two, retorted upon the advocates of absolute plurality and discontinuousness, showing that their doctrine led to consequences not less absurd and contradictory than the Ens unum of Parmenides. He advanced many distinct arguments; some of them antinomies, deducing from the same premisses both the affirmative and the negative of the same conclusion.[15]
[14] Diog. Laert. ix. 24-29.
Zeller (Phil. d. Griech. i. p. 424, note 2) doubts the assertion that Zeno delivered probable opinions and hypotheses, as Parmenides had done before him, respecting phenomenal nature. But I see no adequate ground for such doubt.
[15] Simplikius, in Aristotel. Physic. f. 30. ἐν μέντοι τῷ συγγράμματι αὐτοῦ, πολλὰ ἔχοντι ἐπιχειρήματα, καθ’ ἕκαστον δείκνυσιν, ὅτι τῷ πολλὰ εἶναι λέγοντι συμβαίνει τὰ ἐναντία λέγειν, &c.
Consequences of their assumption of Entia Plura Discontinua. Reductiones ad absurdum.
If things in themselves were many (he said) they must be both infinitely small and infinitely great. Infinitely small, because the many things must consist in a number of units, each essentially indivisible: but that which is indivisible has no magnitude, or is infinitely small if indeed it can be said to have any existence whatever:[16] Infinitely great, because each of the many things, if assumed to exist, must have magnitude. Having magnitude, each thing has parts which also have magnitude: these parts are, by the hypothesis, essentially discontinuous, but this implies that they are kept apart from each other by other intervening parts — and these intervening parts must be again kept apart by others. Each body will thus contain in itself an infinite number of parts, each having magnitude. In other words, it will be infinitely great.[17]
[16] Aristotel. Metaphys. B. 4, p. 1001, b. 7. ἔτι εἰ ἀδιαίρετον αὐτὸ τὸ ἕν, κατὰ μὲν τὸ Ζήνωνος ἀξίωμα, οὐθὲν ἂν εἴη.
ὃ γὰρ μήτε προστιθέμενον μητὲ ἀφαιρούμενον ποιεῖ τι μεῖζον μηδὲ ἕλαττον, οὔ φησιν εἶναι τοῦτο τῶν ὄντων, ὡς δῆλον ὅτι ὄντος μεγέθους τοῦ ὄντος.
Seneca (Epistol. 88) and Alexander of Aphrodisias (see the passages of Themistius and Simplikius cited by Brandis, Handbuch Philos. i. p. 412-416) conceive Zeno as having dissented from Parmenides, and as having denied the existence, not only of τὰ πολλὰ, but also of τὸ ἕν. But Zeno seems to have adhered to Parmenides; and to have denied the existence of τὸ ἕν, only upon the hypothesis opposed to Parmenides — namely, that τὰ πολλὰ existed. Zeno argued thus:—Assuming that the Real or Absolute is essentially divisible and discontinuous, divisibility must be pushed to infinity, so that you never arrive at any ultimatum, or any real unit (ἀκριβῶς ἕν). If you admit τὰ πολλὰ, you renounce τὸ ἕν. The reasoning of Zeno, as far as we know it, is nearly all directed against the hypothesis of Entia plura discontinua. Tennemann (Gesch. Philos. i. 4, p. 205) thinks that the reasoning of Zeno is directed against the world of sense: in which I cannot agree with him.
[17] Scholia ad Aristotel. Physic. p. 334, a. ed. Brandis.