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5 the Case for Infinite Divisibility I. Introduction

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5 the Case for Infinite Divisibility I. Introduction The Architecture of Matter, Galileo to Kant Holden, Thomas, Department of Philosophy, Syracuse University Print publication date: 2004, Published to Oxford Scholarship Online: January 2005 Print ISBN-13: 978-0-19-926326-4, doi:10.1093/0199263264.001.0001 5 The Case for Infinite Divisibility Thomas Holden I shall not believe that you are entirely cured of mathematics so long as you maintain that these tiny bodies, about which we were disputing the other day, can be divided in infinitum. Chevalier de Meré, in a letter to Pascal I. Introduction If the interpretation advanced in Chapter 1 is correct, the fundamental challenge facing Enlightenment matter theory was the apparent conflict between the geometrization of nature on the one hand, and the actual parts metaphysic on the other. Geometry seemed to mandate matter's infinite divisibility; the actual parts doctrine to preclude it. Chapters 2 to 4 dealt with the one side of this conflict: the actual parts doctrine and its various corollaries (genuine and otherwise). In the current chapter I look at the other side of the conflict: the geometrization of nature and the case for infinite divisibility. This chapter focuses on the two central arguments for infinite divisibility. Of the various arguments for infinite divisibility presented in early modern period, these two are the most philosophically interesting and also the most popular: they can justly be considered the classic arguments on this side of the debate. First there is (i) the argument from geometry, which purports to establish infinite divisibility by way of certain constructions in classical geometry. Although this was by far the most popular argument for infinite divisibility during the Enlightenment, it is given short shrift nowadays: it clearly presupposes that physical space and matter conform to the axioms of classical geometry, and thus begs the question. (The objection is certainly lethal. However, as we shall see, the more sophisticated early modern advocates of the argument from geometry were well aware of this possible petitio charge and attempted to answer it in various end p.206 philosophically intriguing ways.) The second classic argument for infinite divisibility was slightly less popular in the Enlightenment literature, but is more direct and can seem more compelling. This is (ii) the conceptual argument , which asserts that, as a conceptual truth, whatever is extended has parts that are themselves extended; that the concept of a minimal extension is self-contradictory; and thus briskly concludes that whatever is extended is divisible ad infinitum. Once I have covered these two classic arguments in the body of the chapter, I then briefly survey the other less popular and less interesting arguments for infinite divisibility in a short appendix. First a word about the status of these arguments. This book addresses the question of whether or not matter is infinitely m- divisible—that is, the question of whether or not it has logically separable parts within parts to infinity. But the arguments in this chapter do not deal ab initio with the m-divisibility of matter. Rather, they deal with f-divisibility of space—that is, with the question of whether or not any region of space contains subregions ad infinitum. So even if these arguments prove successful, they will not immediately prove that matter is infinitely m-divisible. To reach that further claim, two additional steps are required. First, one would have to infer from the infinite f-divisibility of a region of space to the infinite f-divisibility of any material body that occupies that space. This step looks uncontroversial: if a given region of space has spatially distinct parts ad infinitum, then surely a body that occupies that space likewise has spatially distinct parts ad infinitum, each subregion of space playing host to one such spatially differentiable subpart of the body. The second step would then be to infer from the infinite f-divisibility of the body to its infinite m-divisibility: from the claim that it has spatially differentiable parts within parts ad infinitum to the conclusion that it has logically separable parts within parts ad infinitum. Now, I think that we can grant this second step at least some initial plausibility. In the case of matter, doesn't f-divisibility entail m-divisibility? If a body has spatially distinct parts, won't those same parts each be logically separable from one another? Couldn't God rupture and distance any two such parts? So (it seems to me) the burden of proof is at least initially on those who would resist this second step, to show how they can block the inference from the f-divisibility of a body to its m-divisibility. A few philosophers have, however, attempted to meet this challenge. (This, in essence, is the strategy of Boscovich and the early Kant with their metaphysic of f-divisible but m-indivisible force-shell atoms.) So—if these thinkers are correct—this second step may prove more problematic. end p.207 To sum up the dialectical situation and set the scope of the current chapter. This chapter examines the classic arguments PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2003 - 2010. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use (for details see http://www.oxfordscholarship.com/oso/public/privacy_policy.html). Subscriber: Chelyabinsk State University; date: 11 November 2010 The Architecture of Matter, Galileo to Kant Holden, Thomas, Department of Philosophy, Syracuse University Print publication date: 2004, Published to Oxford Scholarship Online: January 2005 Print ISBN-13: 978-0-19-926326-4, doi:10.1093/0199263264.001.0001 presented on this side of the early modern debate: (i) the argument from geometry and (ii) the conceptual argument . Each of these arguments purports to establish the infinite f-divisibility of any region of space and ipso facto the infinite f-divisibility of any body (since each body occupies a region of space and, if a body occupies an infinitely f-divisible region of space, it is itself infinitely f-divisible). So these arguments, if they are successful, prove the infinite f-divisibility of any body. It was also widely thought, and it is certainly initially plausible, that the infinite f-divisibility of matter would entail its infinite m-divisibility. But this is a further step that I will examine in the next chapter. In this chapter, the focus will simply be the classic arguments for the infinite f-divisibility of space and, ipso facto, the infinite f-divisibility of body. The chapter proceeds as follows. Sections II–IV are devoted to the argument from geometry. Section II reconstructs this argument and presents examples of the sort of geometrical proof that were thought to establish infinite f-divisibility. In section III I present what I take to be the fatal objection to this argument, giving due credit to the period's so-called ‘ungeometrical philosophers’ responsible for this objection. Then in section IV I examine possible attempts to salvage the argument from geometry and show why they each must fail. This last section will take us into a brief survey of the rival Enlightenment accounts of the epistemological status of geometry. Sections V and VI are then given over to the conceptual argument. In section V I reconstruct this argument and document its popularity with both early modern thinkers and various present-day philosophers. In section VII I then attack the argument: its initial plausibility, I think, proves specious. Neither the argument from geometry nor the conceptual argument, the two classic a priori arguments for the infinite f-divisibility of space, then, is a success. This appears to leave the question an empirical matter: for all logic mandates, space may or may not be infinitely f-divisible. Finally, an appendix documents two other (less interesting and less popular) types of argument for infinite f-divisibility also found in the period literature. II. The Argument from Geometry The doctrine of the infinite f-divisibility of space was orthodox in all the main traditions of Enlightenment natural philosophy. Notwithstanding end p.208 their famous differences over the ontological status of space, the central factions—Cartesian, classical Newtonian, and Aristotelian–scholastic—were all in agreement on this question. Along with these traditions we could add individual luminaries such as Galileo, Hobbes, Barrow, Spinoza, Locke, Leibniz, Euler, and Kant. Even Gassendi, the great champion of p-indivisible material atoms, renounces Epicurus on this question: contra the Greek atomist, space and matter are both endlessly f- 1 divisible. 1 See Gassendi in the Syntagma, in Opera Omnia (Stuttgart-Bad: Canstatt, 1964), liber III, sectio I, caput V, 256–6. Compare Epicurus' rejection of space's infinite f-divisibility in the Letter to Herodotus—for discussion, see Richard Sorabji, ‘Atoms and Time Atoms’, in Norman Kretzmann (ed.), Infinity and Continuity in Ancient and Medieval Thought (Ithaca, NY: Cornell University Press, 1982), 37–82. (This leaves only a few dissenting figures whose rejection of infinite f-divisibility, however just, can only be described as idiosyncratic when set against the main traditions of natural philosophy. These dissenters include the hardcore Epicureans Charleton and the younger Newton, the ever-sceptical Bayle, and the champions of minima sensibilia Berkeley and Hume.) By far the most popular form of argument for the doctrine of infinite f-divisibility involved the invocation of geometrical constructions. These constructions underwrote clear mathematical proofs, advancing from elementary geometrical axioms through lock-step theorems to demonstrate the doctrine. And no doubt because of geometry's celebrated status in the Enlightenment as a paradigm of scientia, offering self-evident step-by-step deductions, these elegant, clear, and a priori demonstrations seemed to most period philosophers to present incontrovertible proof of infinite f-divisibility.
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