On the Standard Interpretation (let's call it) of the Third Man Argument (TMA), that argument is pointing to an infinite and vicious regress that arises from the One Over Many argument. The regress is infinite, because each time a Form is introduced to account for a collection, the argument can be reapplied to the new collection, consisting of that Form and the members of that previous collection, generating a new Form. The regress is vicious, because the Forms are precisely meant to explain the unity or commonality of the many particulars that share in them. If the One Over Many never succeeds in establishing, somewhere at the end of the line, some single, pre-eminent Form for each discernible kind, then it never succeeds in explaining how that kind is unified or somehow one. A single Form is meant to explain the unity of the kind; if the Forms for each kind are infinite, then recourse to a Form cannot explain the unity of the kind.
We have seen (What Does the Text Say?) how it is possible, and natural, to understand the conclusion of the TMA as putting forward exactly this difficulty.
These considerations give rise to various difficulties for the interpretation that Rickless has offered, that there is an additional argument implicit in the text.
1. Rickless concedes that the TMA is meant to establish an infinite regress. I ask him: Does he think the regress itself is also vicious? If so, then, I ask him: Why, on his interpretation, does Parmenides fail to point this out, but rather (as Rickless has it) presses on immediately to another difficulty, putatively based upon this one? Doesn't Parmenides show himself keen to point out every difficulty that affects the Forms? Why would he pass over this difficulty and not give it separate attention--a difficulty which is so striking, that Aristotle exploited it to great effect, and nearly all other readers of the TMA have presumed that this was the sole difficulty raised by the passage?
2. On Rickless' interpretation, we should expect to see an infinity (of some kind or other) mentioned twice in the TMA: the first infinity being the regress of Forms; the second being the multiplicity within each member of that regress. But we find only one mention of an infinity. (Readers may wish to consult once more the translation of the TMA, here.) Isn't it the simpler and more natural interpretation to identify this with the regress--and then say that the second infinity in fact is not in the passage at all?
3. Rickless' interpretation requires that an argument like the following is implicit in the TMA:
(i) There is an infinite regress of Forms.
(ii) Each Form earlier in the sequence participates in all the Forms later in the sequence.
(iii) Thus an infinite number of predicates can be asserted of each Form in the sequence.
(iv) But that of which many predicates can be asserted is itself manifold.
(v) And that of which an infinity of predicates can be asserted is itself infinitely divided.
(vi) Thus, each Form in the sequence is infinitely divided.
But it's implausible on its face to claim that all of this is implicit. Moreover, Parmenides is careful to make all of his steps explicit in every other objection that he raises: the TMA, on Rickless' interpretation, would be a wildly anomalous exception and actually a poor bit of philosophical writing.
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Man, this blogging business is getting more enjoyable by the minute! If I don't quit my day job, I may have to join Bloggers Anonymous...
OK, let's take each objection in order. First, you say that the infinite regress of the TMA is vicious "because the Forms are precisely meant to explain the unity or commonality of the many particulars that share in them," and yet, "if the One Over Many never succeeds in establishing, somewhere at the end of the line, some single, pre-eminent Form for each discernible kind, then it never succeeds in explaining how that kind is unified or somehow one." Now my first response to this is: Where do you find this in the text itself? You fault me (see below) for finding implicit reasoning in the TMA. As I see it, the standard reconstruction, no less than mine, requires the attribution of implicit reasoning. Parmenides pretty clearly assumes that the regress leads to a problem for Socrates' theory of forms. But he never says that the problem is that the infinite hierarchy of largenesses generated by the Third Man Argument is vicious. (We modern analytic philosophers, who have been trained to look for vicious regresses under every rock, are primed to see vicious regresses where there aren't any. Let's try this as a test. Gather a group of analytic philosophers who've never heard of the TMA, and give them an argument homologous to the TMA that generates an infinite hierarchy. I bet you a grande mocha with whipped cream and chocolate chips that they will immediately chant in unison: "vicious regress".) Here's my second response. Socrates supposes, to begin with, that, for any group of large things, each member of that group is large by virtue of partaking of a form of largeness that is "one" (OM). You simply assume, without any explicit indication in the text supporting this assumption, that "one" in OM means "unique". But there is an alternative (defended in some of my previous comments) that you cannot in all fairness dismiss without argument, which is that "one" in OM means "one in relation to something", as in "one among many", or "one over many". This is more than enough to establish that the standard interpretation should not be viewed as the default interpretation. One needs to look at the textual merits of both the standard and non-standard interpretations, and then let the chips fall where they may. (As I've argued in previous comments, there is a significant amount of textual of evidence *from the Parmenides*, much of it having to do with the place of the TMA in the dialogue as a whole, to suggest that the non-standard interpretation is, all things considered, superior.)
Reply to (1): I do not think that the regress is meant to be vicious.
Reply to (2): You say (i) that there is only one mention of infinity in the text of the TMA, (ii) that this infinity is mentioned in conjunction with the regress, but (iii) according to my interpretation, one would expect infinity to be mentioned twice: once in conjunction with the regress itself, and a second time in conjunction with the multiplicity of each form in the regress. I agree with (i), but I deny both (ii) and (iii).
Consider (ii). The TMA mentions infinity once, *but not in conjunction with the regress* (at least not on my interpretation). Infinity is in fact mentioned at the very end of the TMA, *in conjunction with the multiplicity of "each of your forms"*. So let me turn the tables: Why, if the standard interpretation is correct, does Parmenides *fail* to mention infinity in conjunction with the regress?
Now consider (iii). As I see it, the TMA is an enthymeme. This wouldn't be the first enthymeme to be found in the Platonic corpus (or even in the *Parmenides*--see below), so there is no particularly powerful reason, ex ante, to expect that the TMA is non-enthymematic. If it is an enthymeme, there is no particular reason to expect that infinity should be mentioned twice. In fact, what one should expect Plato to mention is the infinity that matters more. And here, as I've just argued in response to (ii), what Plato mentions is the infinity linked to the multiplicity of each form, *not* the infinity linked to the regress. This is a clue that Plato wants us to focus, not on the regress per se, but rather on the fact that the regress leads to the fact that each form of largeness in the hierarchy is infinitely many.
Reply to (3): You say, first, that it is "implausible on its face" to find in the TMA the implicit reasoning I find there. But this is only "implausible on its face" on the assumption that the last line of the TMA is given its standard interpretation. On my interpretation of the last line (for a defense of this interpretation, see my paper and previous comments), what is "implausible on its face" is the standard interpretation of the TMA! As I see it, then, this objection simply begs the question against the non-standard interpretation of the argument.
You say, second, that "Parmenides is careful to make all of his steps explicit in every other objection that he raises". If this were true, it would surely lead us to suppose that the TMA is not an enthymeme. But, as I argue in my paper, this is false. For example, in the Whole-Part Dilemma Parmenides assumes, without explicitly stating, that no form can be both one and many. This is what allows him to infer, from the fact that a form is divisible (or divided), that it is not one (see 131c9-11). And, although I don't make the case for this in my paper, it's clear that each of the four little arguments that appear in quick succession after the Whole-Part Dilemma is enthymematic.
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