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Hume Studies Volume XX, Number 1, April 1994, pp. 85-101 Achievements and Fallacies in Hume's Account of Infinite Divisibility JAMES FRANKLIN 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 Méré'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 James Franklin is at the Department of Pure Mathematics, School of Mathematics, University of New South Wales, P.O. Box 1, Kensington, New South Wales, Australia 2033. e-mail: 86 James Franklin was still a live one for Kant, whose Second Antinomy includes the infinite divisibility of space as a premise.5 The eighteenth century also felt a certain tension, largely unacknowledged, between the corpuscular hypothesis of matter and the infinite divisibility of space. Newton and most scientists supposed matter and light to be atomic, but unambiguous scientific evidence remained tantalizingly unavailable until Dalton's work after 1800; and while the atomic hypothesis remained essentially a philosophical one, there was an uncomfortable tension between the atomicity of matter and the continuity of space. Thus, Lord Stair in 1685 (in a scientific work reviewed by Bayle) defended the atomicity of matter against mathematical objections concerning infinite divisibility and arrived at a position close to Hume's.6 Nevertheless, it is obviously hard to explain why space and time should be infinitely divisible, and how this could be known if it were true: surely knowing it requires that measurement should be able to follow nature into the infinitely small? The details of the argument in this period are not very relevant to what Hume says and so will not be discussed here. (Nor are the details interesting mathematically, since they just consist in extracting from Euclid the implicit assumption of infinite divisibility.) Suffice it to say that by 1739, the problem of the infinite divisibility of space and time had the status of an old chestnut, not unlike the problem of interpreting quantum mechanics today . Problems of this kind attract the attention of two kinds of philosopher: the technical expert, who follows the scientists into the intricacies, and the Young Turk, eager to rush in where others fear to tread and cut the Gordian knot with his brilliant new insight. No further explanation seems necessary as to why Hume should have written on the question nor why he should have given it such prominence at the beginning of the Treatise. To solve a long-running problem with his "experimental method of reasoning" would have been a simple demonstration of its value. But in omitting his treatment of space and time almost entirely from the later Enquiry, Hume seems to admit tacitly that it was not a success with its intended audience. It has had no better reception since. Almost all commentators, even ones who...


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