Infinite Divisibility and Actual Parts in Hume's Treatise

Hume Studies Volume 28, Number 1, April 2002, pp. 3-25 Infinite Divisibility and Actual Parts in Hume's Treatise THOMAS HOLDEN I believe that the smallest portion of matter may be practically divided ad infinitum; that equal qualities taken from equal qualities, an unequal quality will remain; that two and two make seven; that the sun rules the night, the stars the day; and the moon is made of green cheese. Tobias Smollett, The History and Adventures of an Atom (1769)1 Introduction The ferocious controversy in early modern natural philosophy over the structure of continua focuses on a cluster of supposed paradoxes of infinite divisibility. According to the recent commentary, these paradoxes are straightforwardly mathematical in nature: they simply challenge mathematical constructions of infinite divisibility. So interpreted, the paradoxes are then quite easy to disarm. They rest on quaint mathematical mistakes—forgivable in the early modern period, perhaps, but clear errors all the same. But this interpretation of the Enlightenment controversy will not stand scrutiny. The early modern debate depends crucially on a body of metaphysical doctrine concerning the 'filling' or 'stuffing' of actual physical continua—a body of doctrine that dominates the natural philosophy of the period and that sets the background for the debate over infinite divisibility. The controversy is Thomas Holden is Assistant Professor of Philosophy, Syracuse University, Syracuse, NY 13244-1170, USA. e-mail: tholden@syr.edu 4 Thomas Holden not then a purely mathematical debate, exclusively concerned with the mathematically tractable properties of continua. And once we appreciate this, we will see that the paradoxes are not so readily dismissed. In this paper I focus on Hume's main argument against infinite divisibility in Book 1, part 2 of the Treatise.2 Using this argument as an example, I want to embarrass the standard mathematical reading and vindicate my alternative metaphysical interpretation. The point can then be generalized to the wider Enlightenment debate. I focus on Hume's argument in particular, since here there is a wealth of commentary and an exceptionally clear case of a major Enlightenment philosopher charged—quite unjustly, I will argue— with the most grotesque mathematical blundering. I. Hume's Lead Argument against Infinite Divisibility and the Standard Objections Although there are several arguments against infinite divisibility in the notoriously thorny Book 1, part 2 of the Treatise, commentators have—naturally enough—focussed on the one argument that heads his discussion and that Hume clearly sees as the centerpiece of his case.3 In sections 1 and 2, Hume frames this lead argument in terms of the divisibility of our ideas and impressions of extended entities. But in section 4, he makes it clear that similar reasoning would apply to extended entities in the extra-mental physical world.4 And—as has been well pointed out in the literature5—the lead argument is indeed general: it purports to show that no finite thing can be infinitely divisible , and that every finite thing must resolve to a finite number of first elements. So we can follow the commentators in bracketing the fact that Hume introduces his argument in terms of mental representations of extended things. It applies no less to extended entities in the physical world. Hume's lead argument is brisk. It runs as follows: (Hl) "[Wjhatever is capable of being divided in infinitum, must consist of an infinite number of parts"; "Every thing capable of being infinitely divided contains an infinite number of parts" (T 1.2.1.2, 1.2.2.2; SBN 26, 29). (H2) "[T]he idea of an infinite number of parts is individually the same idea with that of an infinite extension;... no finite extension is capable of containing an infinite number of parts" (T 1.2.2.2; SBN 30). Therefore: (H3) "[N]o finite extension is infinitely divisible" (Tl.2.2.2; SBN 30). Hume Studies Infinite Divisibility and Actual Parts 5 In short: (i) whatever is infinitely divisible has an infinite number of parts; (ii) whatever has an infinite number of parts is infinitely large; so (iii) nothing finitely extended is infinitely divisible. I will be maintaining that this argument is essentially incomplete or abbreviated. Were it stated in full...