## Thursday, March 26, 2009

### The Phaedo equality argument

I've never quite got the Phaedo 75 "equality" argument. The point is made that whenever we have two equal things in the physical world, they are never simply equal, but are always only equal in some respect. From this we are supposed to infer that we do not get the concept of equality from the two things. Here are two readings that build arguments out of the text. Whether they're faithful to what Plato is saying is a different question.

Reading 1: Take two sticks. They are related in many respects. In one respect, they are equal. In another, they are not. Their may be equal length, but not in their width. Moreover, the length of the one is certainly not equal to the width of the other. (I include that remark in case one is tempted to say: "Why not just consider the same stick twice over, and then it'll surely be equal to itself?" But no, it, too, will only be equal to itself secundum quid—its length will be equal to its length, but not to its width, say.) The two sticks are related in all kinds of ways other than equality. Among these many relations that they stand in (such as inequality in width, difference in color, similarity in value, etc.), there is equality, in repect of length. To recognize the equality, in respect of length, among the many relations that they stand in, requires that we already have the concept of equality so that seeing it in the crowd of relations will pick it out from that crowd.

Objection: We can't do it just with two sticks, but if we have enough items, we can abstract equality from them. For it might be that a1 and b1 stand in a multitudinous set M1 of relations including equality, and a2 and b2 stand in a multitudinous set M2 of relations, and so on. But maybe the intersection of M1,M2,...,Mn contains only equality.

Response: There are so many relations that things stand in, that it is very unlikely that the intersection will be a mere singleton.

Objection: We can get to equality as long as we specify "in respect of length". So we do get the concept of equality from the sticks—"the relation in which their lengths stand to one another."

Response: First, the lengths of the sticks stand in infinitely many relations, equality being but one of those relations. (To give a non-Platonic example, the two lengths stand in the relation of being equal or the same color. Or the two lengths stand in the relation of being observed by the same observer.) So the problem reappears. Second, "length" must be defined in some respect—from which exact point on one end of the stick do we measure to which exact point on the other end do we measure? And, note, that almost surely we cannot really exactly specify points—the Cartesian coordinates are triples of real numbers, and almost no real numbers can be exactly specified (there are uncountably many real numbers, but only countably many can be exactly specified by us), so almost surely the ones here cannot be.

Reading 2: The two sticks are only equal in some respect R. But even the claim "the two sticks are equal in respect R" only holds in some further respect. And so on. Hence, we never get to equality itself. Concretely, let's start with: they are equal in respect of length. But that only holds in respect of one time—at some later time, one of the sticks will slightly oscillate and they won't be equal. So, they're equal in respect of length in respect of some time. But now, their length has one value in respect of one way of defining lengths, and another value in respect of another way of defining lengths. (There are probably little whiskers of wood fiber sticking out both ends. Do we measure them, or not? Which ones do we measure? Where in the atoms do we start measuring? And of course we have the uncertainty principles to contend with.) Moreover, in what way do we compare the lengths? Do we take a measuring stick to the one, and then to the other? But equality then only holds in respect of measuring sticks that don't change their lengths. And how do we define the measurement with the measuring sticks? Let's say they have tick marks. Where in the tick mark is the relevant point? It is not impossible that these questions go on ad infinitum. But even if they don't, they go further than we can answer them—and so we didn't get the concept of equality from the sticks.

#### 1 comment:

N. N. said...

Whenever we have two equal things in the physical world, they are never simply equal, but are always only equal in some respect.

This, it seems to me, is where your reading goes astray. At 74b, Socrates says, "do not equal stones and sticks sometimes, while remaining the same, appear to one person to be equal and to another to be unequal" (Grube translation). The point, I take it, is that 'equal' sticks appear both equal and unequal, i.e., they appear to be and to not be equal (cf. Republic, 523c on sensations that "at the same time go over to the opposite sensation").

Sticks that appear to be equal and unequal are imperfectly equal. However, the recognition of the sticks as imperfectly equal requires knowledge of perfect equality (otherwise, in virtue of what are they being recognized as imperfectly equal): "Whenever someone, on seeing something, realizes that that which he now sees wants to be like some other reality but falls short and cannot be like that other since it is inferior, do we agree that the one who thinks this must have prior knowledge of that to which he says it is like, but deficiently so?" (74d)

This knowledge must be acquired before the recognition of the sticks as imperfectly equal, i.e., before sense perception; therefore, it is acquired before birth.