Achievements and Fallacies in Hume's Account of Infinite Divisibility

Throughout history, almost all mathematicians, physicists, and philosophers have been of the opinion that space and time are infinitely divisible. That is, it is usually believed that space and time do not consist of atoms, but that any piece of space and time of non-zero size, however small, can itself be divided into still smaller parts. This assumption is included in geometry, as in Euclid, and also in the Euclidean and non-Euclidean geometries used in modern physics. Of the few who have denied that space and time are infinitely divisible, the most notable are the ancient atomists, and Berkeley and Hume. All of these assert not only that space and time might be atomic, but that they must be. Infinite divisibility is, they say, impossible on purely conceptual grounds. In the hundred years or so before Hume's Treatise, there were occasional treatments in places such as the Port Royal Logic and Isaac Barrow's mathematical lectures of the 1660s.1 They do not add anything substantial to medieval treatments of the same topic.2 Mathematicians certainly did not take seriously the possibility that space and time might be atomic; Pascal, for example, instances the Chevalier de MA©rA©'s belief in atomic space as proof of his total incompetence in mathematics.3 The problem acquired a more philosophical cast when Bayle, in his Dictionary, tried to show that both the assertion and the denial of the infinite divisibility of space led to contradictions; the problem thus appears as a general challenge to "Reason."4 The problem