It has seemed to some people that Zeno’s argument of the race course is directed against the kind of conception of space as infinitely divisible that Anaxagoras is proposing; so they conclude that Zeno is responding to Anaxagoras.

This is chronologically possible. But it seems more likely that Anaxagoras is proposing a way out of Zeno’s paradox of plurality. (Cf. McKirihan, Philosophy Before Socrates, pp. 217-18)

In his Paradox of Plurality, Zeno had argued that a thing’s ultimate parts (of which there are infinitely many) would have to be either (a) of finite size or (b) of no size at all. But if (a), the thing would be infinitely large (for no matter how small the smallest part is, an infinite number of such parts would add up to an infinitely large size); and if (b), the thing would be of no size at all (since the sum of an infinite number of zeroes is still zero).

Anaxagoras can be seen as arguing that this is a false dilemma, for it falsely assumes that there are ultimate parts of the thing we are considering. But if there are no such parts (“of the small there is no smallest”) the argument for the first horn (a) of the dilemma breaks down. This is what we pointed out earlier in criticizing the Infinite Sum Principle we found implicit in the paradox of plurality.

So it seems likelier that Anaxagoras is replying to Zeno than the other way around. Zeno tried to show that if there are many, they would have to be both infinitely large and infinitely small. Anaxagoras replies that they would have to be infinitely large only if there is a smallest of the small. But, he says, there is not a smallest of the small, so Zeno’s argument fails.

Go to the lecture on Zeno 's Argument against Plurality.

Go back to the lecture on Anaxagoras

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Copyright © 2002, S. Marc Cohen