I've been reading David Foster Williams Everything and More: A Compact History of ∞ which contains some discussion of Zeno's paradoxes. Here I'd like to say a word about the Dichotomy, and hopefully explain how a pluralist can believe in motion, that is, move and be immersed in world of motion without being paralyzed by doubt. I'd also like to say why I feel that Diogenes the Cynic, whose answer to the paradox was to silently walk, has been given short shrift.
The Dichotomy states that "that which is in locomotion must arrive at the half-way stage before it arrives at its goal." That's incorrect. Motion, or any abstract continuum expressly meant to represent something like motion, does not have an intrinsic halfway stage or halfway point. Only a second step of abstraction can add a halfway point to a motion. Taking that second step creates two problems, the first being that one is likely to forget one is dealing with two distinct levels of abstraction, the second being that the abstractions involved may be incommensurable. Motions and divisions are separate realities. Lines and points are separate realities, though both may be brought together under the umbrella of geometry. A certain kind of infinity, Zeno's infinitiy but not all infinite sets, may be considered an artefact of incommensurability. A pluralist is not somebody who believes in the infinite divisibility of the world, but rather in there being multiple incommensurable worlds. There's no hidden monism behind pluralism, as Zeno wrongly suggests with his infinite divisibility. There's just more than one reality.
By remaining silent while he walked, Diogenes gave a brilliant answer to the Dichotomy. He demonstrated the existence of at least two realities: the logos and locomotion. The logos isn't the only reality that philosophy might be concerned with, though it might be said that the logos is the way philosophy concerns itself with various realties. Diogenes faults the way Zeno has deployed the logos. As easily as the logos can contain, it can be contained. Diogenes contained the reality of the logos with his silence. With his feet he not so much refuted or disproved the Dichotomy according to its own implicit terms (logic) as he demonstrated an alternative mode of being that one could, if one were so inclined, believe in. The existence of an alternative to the logos having been demonstrated, the Dichotomy is shown to be inadequate to the critical task it was meant to accomplish, viz., a critique of pluralism. This is not a case of common sense versus philosophy, but of philosophical error and correction. Diogenes was undoubtedly correct.
(There might be a paradox or inconsistency in the way I've argued this. Is Diogenes' walking merely an alternative to the logos, or, as a response to a riddle, is it therefore part of the of the logos? We could see the logos as expansive, perhaps bending its own rules as it encompasses more and more activity. That might suggest a kind of monism, a crumpled monism, if one were inclined to go looking for monisms (and if one weren't so inclined one might also speak of logoi). If the logos isn't so expansive, however, Diogenes' walking takes the measure of the logos in the sense that it marks a limit. That might suggest that different realities aren't so incommensurable. I don't think that's quite right, though, because marking a limit doesn't measure a reality so much as it signifies the existence of another reality. Can one criticize monism from outside the logos? Etymologically, "critical" (κρῐτῐκός) means able to discern, able to separate. (In fact the meaning of the logos may be a mirror of criticism, of sifting and separating.) It would be imperious to say that criticism can only happen within the logos. If I say, then, that Diogenes criticized Zeno at the limit of the logos, as I'm tempted to think, am I therefore accepting of Zeno's infinity?)
Labels: criticism, Diogenes the Cynic, infinity, paradox, Wallace, Zeno