chiliagon

Consider, for example, the cases of the chiliagon and the ideal state (city), both brought up by Vlastos (1969: 301), the latter also (and earlier) by Maula (1967: 35).

In each case, the excess is to an unworkable extreme: the chiliagon is supposed to have too many sides to imagine, the ideal state, too much justice to realize, the superluminal starship, too great a speed to subsist (retain mass and remain coherent).

For example, it is difficult to place the ideal state in the same category as either the chiliagon or the starship traveling at warp speed.

Yes, the chiliagon seems too intricately nuanced to draw or imagine, but it is not as if it were not a proper mathematical object.

But if I want to think of a chiliagon, although I understand that it is a figure consisting of a thousand sides just as well as I understand the triangle to be a three-sided figure, I do not in the same way imagine the thousand sides or see them as if they were present before me.(23)

In distinguishing between pure intellection and imagination, Descartes uses the example of the chiliagon to illustrate how the mind conceives of those things that are unrepresentable to the imagination.

But once I have found the number, I know the given polygon's nature and properties very well, insofar as they are those of a chiliagon.(46)

There is, perhaps, some intuitive sense in which the heap, and the chiliagon, are more readily understood when one ascertains the number involved, for then, as Leibniz suggests, certain properties can be inferred on the basis of this information.

Thus, the disanalogy noted above stems from the fact that when it comes to our distinct concepts of geometrical figures, such as the one of the chiliagon, the components of these concepts are also distinct, and that enables one to deduce--on the basis of mathematical inference alone--certain properties of the entity.

But once I have found the number, I know the given polygon's nature and properties very well, insofar as they are those of a chiliagon. The upshot is that I have this [distinct] idea of a chiliagon, even though I cannot have the image of one: one's senses and imagination would have to be sharper and more practiced if they were to enable one to distinguish such a figure from a polygon which had one side less"; Leibniz, New Essays, 261.