Bayle's trilemma

In his famous article on Zeno of Elea in his Dictionnaire historique et critique, Bayle represents a latter-day Zeno arguing against motion by arguing against the existence of spatial extension. There are three possible theories of spatial extension: (i) space is made of mathematical points, (ii) space is ‘granular’, made of finite atoms, (iii) space is made of parts that are divisible in infinitum . These three exhaust the possible theories, but none of them is tenable. The first demands that a quantity of things with no extension eventually makes up an extension, which is absurd. The second forbids us from recognizing the atoms as having parts, such as left-hand and right-hand edges, but this we must do. The third, Aristotelian, theory improperly shelters behind the notion of a purely potential infinity, forgetting that spatial (and temporal) extension is actual; it also forgets that space is continuous, i.e. its parts touch each other, whereas between any two elements of an infinite series of the kind proposed there is an infinite number of other elements. No two elements touch, any more than any two fractions are ‘next to’ one another. A modern mathematical treatment of the paradoxes (e.g. Adolf Grünbaum, Philosophical Problems of Space and Time, 1963) may be used to show how the first option is tenable since, as it is defined set-theoretically, the sum of any finite or infinite number of dimensionless points need not necessarily be zero. We can assign zero length to unit point sets, and differing finite lengths to the unions or sums of those sets that make up a finite interval. However, the solution requires that we are happy co-ordinating the fundamental points of space with the point sets of the theory, and the way in which space becomes describable in these terms, or in other words the ontological problem that Bayle (or Zeno) is posing, may be felt not to have been addressed. Kant makes the divisibility of space the subject of the second antinomy, and a crucial part of his idealist argument that it is we who organize experience spatially, rather than the world which is constituted of a spatial manifold. See also Zeno's paradoxes.