02 October 2005

Eleatic Question: 4

Suppose someone wrote a history of Greek mathematics that went like this:

The discovery, by Pythagoreans, of the incommensurability of the hypotenuse of a right triangle with unit sides, had tremendous consequences for the development of mathematical thought. All subsequent mathematicians had to take this result into account. Some simply denied it boldly, insisting that there was "no more reason that a side should be incommensurable than commensurable". Others agreed that, indeed, a right triangle with unit sides had an incommensurable hypotenuse, but they denied that any other triangle did. Still others said that, although incommensurability affected triangles, it was irrelevant to squares (this, even though squares are built out of triangles).
Fairly absurd, isn't it? Yet that's the kind of absurdity we are satisfied with in telling the history or philosophy. Compare:
Parmenides presented what seemed an irrefutable argument that reality is one, continuous, indivisible, unchanging, and perfect. So powerful were his arguments, that all subsequent philosophers regarded themselves as obliged to answer them. The Atomists, for instance, simply denied that it was impossible for nothing to exist: they called nothingness the 'void' and said that there was no more reason for something to exist than nothing. Anaxagoras, although he agreed with Parmenides that nothing could come into existence or perish, nonetheless rejected Parmenides' claim that only one thing exists: he said that there were two fundamental things--Mind and the infinite mixture--and he held, furthermore, that the infinite mixture was not only plural but also infinitely divisible. Empedocles, for his part, simply asserted that four things existed, and that an inexhaustible variety of other things could genuinely come into existence by a kind of mixture of the four elements, caused by a strange force he called 'love'.
Parmenides concludes not-A1, not-A2, not-A3, and not-A4, all on the same grounds. It hardly makes sense to say that someone who, in the face of this, simply asserts A1, is 'replying to Parmenides', no matter what that philosopher does with A2-A4.

Sure, someone might hold that it's obviously false that change doesn't happen (or even that it's obviously false that we can't think of nothing), in just the same way as someone might 'refute' the Dichotomy by simply walking away. But why suppose that anyone, in connection with Parmenides' view, might do this piecemeal? If someone takes Parmenides to be clearly wrong as regards plurality or motion, why ever should we regard such a person to be seriously grappling with Parmenides as regards generation and destruction?