Jul
24
Space and Concepts of Space
July 24, 2007 | Leave a Comment
In his Physics, Aristotle, famously, denies that there can be such a thing as an actually infinite magnitude. That seems plausible enough, but he goes on to deny that anything can be actually infinite, which is a slightly more problematic claim. The most notorious example of the infinite, the continuum, was only potentially infinite for Aristotle. He held that a potentiality is to be understood in terms of the activity that would be required in order for the thing to become the case, so to say that there are infinite number of points on a line segment AB is only to say that the line AB may be arbitrarily bisected at any point. Does the mathematician need the concept of the infinite to do the work of mathematics? Not according to Aristotle:
This reasoning does not deprive the mathematicians of their study, either, in refuting the existence in actual operation of an intransversable infinite in extent. Even as it is they do not need the infinite, for they make no use of it; they need only that there should be a finite line of any size they wish (Physics III.7).
This last claim is clearly false in actual practice, because Euclid, writing not long after Aristotle, made no secret of the fact that he thought that it was necessary to postulate infinitely long lines in certain proofs, and it is not clear how the postulates he had in mind could be proved without making an appeal to a possibly infinite line.
Aristotle’s mistake is interesting because it raises an extremely important question in the history of mathematics: how did Aristotle conceive of the relation between space and the science of geometry? Kant, notoriously, held geometry to be the science of space, only to be proven wrong with the advent of non-Euclidean geometries in the 19th century.
Because he thought the cosmos itself to be finite (because there are no infinitely large magnitudes), Aristotle could not countenance the idea that a line might extend beyond the boundary of the cosmos, but why couldn’t one imagine a line that theoretically extended beyond the cosmos just for the purposes of constructing a proof of some postulate? Perhaps because such an assumption would raise the question of what sort of “space” there might be “outside” the cosmos, a possibility rather like division by zero as far as Aristotle was concerned, or would have been concerned if he had had a concept of zero.
