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Geometric Possibility$

Gordon Belot

Print publication date: 2011

Print ISBN-13: 9780199595327

Published to Oxford Scholarship Online: September 2011

DOI: 10.1093/acprof:oso/9780199595327.001.0001

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(p.157) Appendix C: Some Background to the Absolute‐Relational Debate

(p.157) Appendix C: Some Background to the Absolute‐Relational Debate

Source:
Geometric Possibility
Publisher:
Oxford University Press

It is traditional among philosophers of space and time to approach the absolute‐relational debate about the nature of space and motion via the views of Descartes, Newton, and Leibniz. In briefest outline, the story runs as follows.1 (1) Descartes defined the place of a body via its relations to its immediate neighbours and motion as change of place so conceived. He attempted to base upon this notion a mechanistic physics governed by rules of impact and the principle of inertia. The result was a disappointing mess. (2) Newton defined the state of motion of a body as its motion relative to absolute space—the latter he thought of as a non‐material existent, neither substance nor accident, consisting of parts that maintain their identity and geometric relations to one another over time. This provided the conceptual underpinnings for his laws of motion. The result was a resounding success. (3) Leibniz's criticisms of Descartes's and Newton's accounts of space and motion have exercised a considerable influence on natural philosophical discussions down to the present day. But his attempts at constructing a systematic and credible competitor to the physics of his rivals never came to fruition—in part, it seems, because Leibniz was pulled in several incompatible directions by his critical intuitions. In particular, it is hard to see how to fit together his relational account of space (which would seem to undercut any notion of absolute motion) with his views about force (which would appear to ground absolute notions of motion).2

(p.158) Of course, philosophical debates about the nature of motion and space began long before the seventeenth century. Indeed, the views of Descartes and Newton are closely related to certain ancient views. And in antiquity, through the middle ages, and down to the seventeenth century, one finds many arguments for and against these views, including precursors of some of those arguments from Newton's writings and from the Leibniz–Clarke correspondence that continue to drive much of the philosophy of space and time. All of this is of course well known to historians of these periods. The following is a sort of bibliographical essay that concludes with a few remarks about what seems to have been genuinely novel to the seventeenth century context.

***

A natural place to begin is with Aristotle's account of the cosmos, place, and motion. The finite material world is organized into a spherical cosmos. The Earth sits at rest at the centre of this cosmos; above the atmosphere are several nested spheres rotating about the Earth; the Moon, Sun, planets, and stars are fixed to these spheres; and the composition of the circular motions of these spheres gives rise to the motions of the heavenly bodies through the sky. The matter of the Aristotelian cosmos forms a plenum. Thus every body is surrounded by matter; this allows Aristotle to take the place of a body to be “the boundary of the containing body at which it is in contact with the contained body.”3 Or at least, this works for bodies in the interior of the cosmos: “the heaven…is not anywhere as a whole, nor in any place, if at least, as we must suppose, no body contains it.”4 We nonetheless say that the outermost sphere of the heavens—the sphere of the fixed stars—completes one revolution each day.5

Aristotle had to contend with atomists, who claimed that movement would be impossible in a plenum and posited a cosmology involving infinitely many atoms moving through an infinite void.6 To the atomists' claim about the impossibility of motion in a plenum, Aristotle retorted, reasonably enough, that “not even movement in respect of place involves a void; for (p.159) bodies may simultaneously make room for one another, though there is no interval separate and apart from the bodies that are in movement. And this is plain even in the rotation of continuous things, as in that of liquids.”7

Aristotle also provides positive arguments against the possibility of a void—not only is there no space empty of matter within the cosmos, but the cosmos itself is not to be thought of as immersed in a larger void space.

  1. (1) In De Caelo, void is characterized as “that in which the presence of body, though not actual, is possible…”8 But it is not possible for there to be matter beyond the cosmos: such matter could not be there naturally, for the natural place of earth, water, air, fire, and the heavenly material is within the cosmos; nor could it have gotten there by violence, for in that case it would have to be located in the natural place of some other matter—and there is none such. So an extra‐cosmic void is impossible.9 The same argument is supposed to show that there can be no other cosmoi located outside of our own.

  2. (2) In the Physics, Aristotle tells us that the partisans of the void regard “it as a sort of place or vessel which is supposed to be ‘full’ when it holds the bulk which it is capable of containing, ‘void’ when it is deprived of that—as if ‘void’ and ‘full’ and ‘place’ denoted the same thing, though the essence of the three is different.”10 Aristotle offers a series of objections to the void in Book IV, chapter 8, showing that a body immersed in a void would be both motionless and move with an infinite velocity, etc. These arguments turn upon the details of the Aristotelian account of natural place, motion through resisting media, etc., and they exercised a considerable influence on medieval discussions of the possibility and nature of motion in a void.11 To these we can add the following remark that occurs in the preamble to Aristotle's discussion of place: “place cannot be body; for if it were there would be two bodies in the same place.”12 Many of Aristotle's medieval successors saw here a powerful consideration against the possibility of the void. For if the void is conceived of as a sort of (p.160) three‐dimensional entity capable of being filled by body, then we must accept that when it is so filled, we have two things existing in the same place—an absurdity.13

***

These views of Aristotle were subject to sustained criticism throughout antiquity, among the Scholastics, and in the early modern period. Let me begin by noting three types of anti‐Aristotelian argument that are of special interest for present purposes. (The following treatment is of course a highly selective one.)

  1. (1) Paradoxes of Aristotelian Motion. If, as is natural, (local) movement is understood as change of place, then the Aristotelian definition of place leads to counter‐intuitive consequences: a body such as a tower moves (because air surrounding it constantly circulates); similarly, bodies can approach one another even if neither moves. These observations provide an argument against the Aristotelian accounts of place and local motion.

  2. (2) Arguments from Cosmic Size and Shape. Many find it plausible that the cosmos does or could change shape, or that it could have had a different size or shape from its actual one—and this seems to suggest that there must be void outside of the cosmos.

  3. (3) Arguments from Possible Motions. Our intuitions recognize the possible states of motions which must be understood as motion relative to the parts of a separately existing void—no account of motion in terms of the relations between material parts will suffice. This provides another sort of argument in favour of the void.

***

I will make some remarks about the history of each of these families of objections, beginning with the Paradoxes of Aristotelian Motion.

Under this heading we find arguments directed against the Aristotelian definition of place. They appear to have first emerged in the writings of Aristotle's immediate successor, Theophrastus.14 They seem to have (p.161) played some role in the rejection by the Aristotelian majority in antiquity of Aristotle's conception of the place of a body as the boundary of the surrounding bodies.15

During the middle ages, Aristotle's account of place was again widely accepted.16 And the paradoxes of motion were then rediscovered and widely discussed.17 In the seventeenth century, they were available even to non‐Scholastic philosophers in, e.g., the Physiologia Epicuro‐Gassendo‐Charltoniana of Walter Charleton.18

It is clear that these arguments create difficulties for Aristotle. He affirms in his Physics that: “It is always with respect to substance or to quantity or to quality or to place that what changes changes.”19 Later he is quite specific in identifying locomotion with change of place.20 And so it seems clear that he regards a body as moving (in our sense) if and only if there is a change along its immediate boundary. And so, prima facie, it seems that a boat moored in a strong current will count as moving, while one drifting downstream along with the current may count as at rest (depending on whether we require the individual parts of water along its surface to be at relative rest, and whether they in fact are).

Now, it isn't clear whether the charge in this form will stick. At one point Aristotle departs from his original characterization of the motion of a body in terms of what is happening at the immediate boundary of the body and maintains instead that:

when what is within a thing which is moved, is moved and changes its place, as a boat on a river, what contains plays the part of a vessel rather than that of place. Place on the other hand is rather what is motionless: so it is rather the whole river that is place, because as a whole it is motionless. Hence we conclude that the innermost motionless boundary of what contains is place.21

(p.162) Here is Myles Burnyeat's influential reading of this passage:

The point of the refinement is this: the place of X was to be the boundary of Y enclosing X, but if Y is moving, this specifies a carrier or vessel of X rather than X's place…The solution is to find Z such that Z is static and Z encloses X at the same boundary as Y does. Example: X=a boat, Y=the body of water flowing in the Cayster, Z=the river Cayster as a geographical entity.22

There is a question of coherence here. Our system of judgements about place and motion will be founded upon an initial choice of a body that counts as motionless. Considering a different body as motionless at the beginning would result in different judgements about place and motion. Now in Aristotle's scheme, it is clearly safe to count the Earth as motionless, and to work outwards from there. But then place is specified by position relative to the surface of the Earth, and motion by change of distance with respect to reference points on the surface of the Earth. So Aristotle's attempt to shore up his definitions of place and motion lead quickly to their supersession by quite different ones. Indeed, one strand of Scholastic thought followed this course, referring motion ultimately to change of a body's relation to the immobile centre and poles of the cosmic sphere.23

***

Now we turn to the positive arguments offered by proponents of the void—the arguments from cosmic size and shape and the arguments from cosmic motion. Here it is convenient to discuss both arguments together, moving from one group of commentators to the next.

THE ATOMISTS. The atomist cosmology featured an infinite number of indivisible particles moving in an infinite void. Our cosmos formed by chance, and will eventually decay—it is one of an infinite number of cosmoi.24

Against the finite spherical universe of Aristotle, Lucretius deploys an argument attributed by ancient authors to Archytas (contemporary of Plato and teacher of Eudoxus): if you are situated at the edge of the (p.163) cosmos, what happens if you extend your staff (or spear, or sword,…) beyond the edge? If there is something there to prevent its extension, then you are not yet at the edge—there is further matter. On the other hand, if you are successful, then there must be receptive void. Repeating the argument whenever a new putative boundary is reached shows that there is infinite extension—of either matter or void.25

Lucretius also gives two detailed arguments in favour of the void, defined as “intangible empty space.”26 The first rests upon the traditional atomist contention that motion would be impossible in a plenum.27 The second, cleaned up and amplified, proceeds thus: suppose two bodies in contact along a surface move away from one another; then air must fill the space between the surfaces initially in contact; but if it moves with only finite velocity, there will be void immediately after the separation of the bodies.28

So far, the arguments given allow us to think of the void of the atomists either as being something like the space of modal relationalists or as being something like Newton's absolute space. But following the arguments just discussed, Lucretius remarks that:

  • If there were no place and space, which we call void,
  • Bodies could not be situated anywhere
  • And they would totally lack the power of movement,
  • As I explained a little while ago.29

Now, Lucretius has earlier told us that bodies could not move if there were no void. But that they would be situated nowhere appears to be a new (p.164) thought—and one on which he never really elaborates. I do not think it much of a stretch to think of Lucretius as taking for granted the absolutist conception here: his void, as an infinite three‐dimensional non‐corporeal entity provides a standard of place and movement for bodies—a body changes place if it occupies a new portion of void and the state of motion of a body is referred to its change of place in the void.30

Indeed, it is not easy to see how we can otherwise make sense of certain characteristic atomist theses. At least from Epicurus onward, atomists held that the void has a natural distinguished direction, downwards, and that the natural motion of atoms is downwards, with atoms of all sizes moving at the same speed.31 In Lucretius we find that this natural motion is sporadically interrupted by the mysterious swerve which puts atoms on collision courses.32 These collisions are ultimately responsible for the formation of cosmic vortices. As Lucretius notes, without the swerve, the atoms

  • Would fall like drops of rain through the void.
  • There would be no collisions, no impacts
  • Of atoms upon atom, so that nature
  • Would never have created anything.33

The most obvious way for us to make sense of this is to refer the motion of atoms to the parts of the void, conceived of as retaining their identity and relations to one another over time. For if in the swerveless atomist universe we look at the relations just between the atoms, we find them utterly static—and we would have no reason to maintain that the atoms were falling down like drops of rain rather than sitting motionless.

THE STOICS. The Stoics, while accepting a spherical and void‐free cosmos, explicitly located it within an infinite void.34 Now, the Stoics more or less (p.165) accept Aristotle's terms—void is that which is capable of being occupied by matter, but is not so occupied. But in favour of the void they offer arguments of the sort that we are interested in.

  1. (1) The cosmos will be or could have been a different shape—so there must be receptive void. The Stoic cosmos is subject to periodic destruction by conflagration, during which the volume of matter is increased manyfold. Thus, Cleomedes in The Heavens:

    If, according to the doctrine of the most accomplished natural philosophers, the whole substance [of the cosmos] is also reduced to fire, it must occupy an immensely larger place, as do solid bodies that are vaporized into fumes. Therefore the place occupied in the conflagration by the substance [of the cosmos] when it expands is currently void, since no body fills it.35

So there exists at least some void outside of the cosmos. According to Simplicius, some Stoics employed Archytas' argument to show that the void must in fact be infinite.36

  1. (2) The possibility of motion of the entire world shows that there must be an infinite void. This argument appears in Cleomedes:

    We can also conceive of the cosmos itself moving from the place that it currently happens to occupy, and together with this displacement of it we shall also at the same time conceive of the place abandoned by the cosmos as void, and the place into which it is transferred as taken over and occupied by it. The latter [place] must be filled void.37

(p.166) Presumably we should add: But there is no limit to the direction, velocity, or duration of this movement, so we must conceive of the void as being infinitely extended in all directions.

In Archytas' argument and in the argument from the conflagration, the existence of the void functions only as a sort of place‐holder for possible deformations or expansions of the cosmos—and so is compatible with an understanding of the void as something like the empty space of a modern modal relationalist and with a broadly Aristotelian account of motion. But with Cleomedes' thought experiment regarding the possible motion of the cosmos as a whole this is no longer possible—the thought experiment is only coherent if the void itself plays a role in defining place and motion. Cleomedes wants us to judge that in the situation described the cosmos is moving through the void—because it successively occupies different parts of the void, rather than because of any characteristic relative motion between its parts. This suggests that, for some Stoics at least, the void ought to be viewed as an infinite three‐dimensional entity, whose parts maintain their identity over time and provide the ultimate grounding for the notions of place and motion.38

THE SCHOLASTICS. No brief summary can do justice to the full range of Scholastic mutations of Aristotelianism. From Edward Grant I take the following points.

  1. (1) The Aristotelian account of place remained essentially unchallenged throughout the medieval period.39 There was, however, active discussion of the paradoxes of motion and problems regarding the motion of cosmic sphere.

  2. (2) There was a unanimous consensus among medieval Scholastics that the cosmos could not be thought of as immersed in an extended, three‐dimensional void.40 Grant identifies a theological basis for this consensus, in Scholastic reluctance to recognize any infinite being in addition to (p.167) God.41 In this context, Aristotle's complaint that if void could be occupied by body, then two things would be in the same place was widely accepted as a decisive argument.42

  3. (3) This position seems entirely compatible with Scholastic use of arguments showing that the world could have been larger that it was, or differently shaped. Archytas' argument was communicated to the Scholastics in works of Simplicius, and was afterwards widely discussed.43 It was also widely accepted that God could have chosen to create a larger world than he had.44 But, of course, this is consistent with the insistence that the extra‐cosmic void is not an extended entity.

  4. (4) In 1277, Parisian theologians, fighting a rearguard action against Aristotelians in the faculty of arts, managed to have a number of propositions condemned by the Bishop of Paris. For a time, the teaching of these propositions was punishable by excommunication. Even after this penalty was lifted, the condemnation continued to have an effect: the condemned propositions continued to be eschewed by conscientious writers. Among the propositions condemned, we find the following.45

    • That there is no more excellent state than to study philosophy.

    • That the only wise men in the world are philosophers.

    • That one should not hold anything unless it is self‐evident or can be manifested from self‐evident principles.

    • That if the heaven stood still, fire would not burn flax because God would not exist.

    • That a sphere is the immediate efficient cause of all forms.

    • That it pertains to the dignity of the higher cause to be able to commit errors and produce monsters unintentionally, since nature is able to do this.

    • That the intellect of the dead Socrates does not have the science of those things of which it once had science.

    • (p.168) That by certain signs one knows men's intentions and changes of intention, and whether these intentions are to be carried out, and that by means of these prefigurations one knows the arrival of strangers, the enslavement of men, the release of captives, and whether those who are coming are acquaintances of thieves.

    • That one should not confess, except for the sake of appearance.

    • That simple fornication, namely that of an unmarried man with an unmarried woman, is not a sin.

    • That God could not move the heaven in a straight line, the reason being that He would then leave a vacuum.

It has been argued that the inclusion of this last proposition had momentous consequences for the development of the concept of space—for in the fourteenth century one finds a number of Scholastics happy to say that God could move the cosmos through the void, or that God was faced with a choice about where in the void to create the cosmos.46 It is difficult to see how the possibility of the translation of the world as a whole along a straight line can be underwritten by anything short of an extended void whose parts maintain their identity through time and (thus) provide a standard of place and motion independent of body.

THE EARLY MODERN ATOMISTS. Spurred in part by a flood of ancient texts previously unavailable in Europe, the sixteenth and seventeenth centuries saw the discussion of a wide variety of non‐Aristotelian accounts of place, space, void, motion, matter and the structure of the cosmos.47 Here we note one particular strand of development which prefigured Newton's absolutist accounts of space and motion: Gassendi's attempt to revive and Christianize ancient atomism.

Gassendi self‐consciously mines ancient and Scholastic authors for arguments. His cosmology features a single material world, created by God and composed of atoms, immersed in an infinite three‐dimensional void space, itself neither substance nor accident but suffused with the omnipresence of (p.169) God.48 In Gassendi and/or his English disciple Charleton, we find the following arguments and claims. (1) The paradoxes of motion cause difficulties for any Aristotelian account of motion.49 (2) The argument of Archytas for the existence of an infinite void.50 (3) God could have created the universe larger than it is or could repeatedly annihilate the universe and create a larger version—so the void must be infinite.51 (4) We can conceive God moving the material world from one location to another.52

***

Obviously this is only the tip of the iceberg. But I hope to have given some feeling for the wealth of interesting arguments and theses salient to the absolute‐relational debate that pre‐date Descartes, Newton, and Leibniz, but which reverberate through, e.g., Newton's De Gravitatione and the Leibniz–Clarke correspondence. In closing this discussion I would like to make a few remarks about what does appear to have been new in discussion of space and motion in the seventeenth century.

The mathematical physics of the seventeenth century took over from astronomy the practice of representing the motions of bodies by curves in Euclidean space, parameterized by time.53 The course of the century saw a progressive widening of the scope and ambitions of this new physics, with its dynamical treatment of the motion of bodies: from its first specimens in Galileo's treatment of free fall and projectile motion near the Earth; to Descartes's qualitative modelling of the celestial motions via vortices; to the competing quantitative accounts of the system of the world offered by Newton and the later vortex theorists (including Leibniz). The first half of the seventeenth century also saw the decisive rejection by astronomers and natural philosophers of Ptolemaic astronomy and the Aristotelian cosmology in which it was set. Of course, these two sets of developments were (p.170) related to one another in many ways. I would like to emphasize just one aspect by claiming that the transition from Aristotelian cosmology to the new cosmologies of the seventeenth century undermined the most straightforward route to interpreting curves in Euclidean space as representing the motions of bodies; and that the competing accounts of the nature of space, the nature of motion, and the relation between the two that one finds in Descartes, Newton, and Leibniz can be viewed as aspects of the process of recognition and resolution of this problem.

From Galileo onwards, the new mechanics was based on one form or another of the principle of inertia, according to which bodies free from interference naturally tend to trace out a certain sort of curve in space. The interpretation of curves in Euclidean space as representing the motion of bodies is unproblematic in contexts in which the motion of all bodies can be understood as motions relative to a natural reference body. For then (to speak anachronistically) one can regard the curves as describing motion in the space picked out by coordinate axes attached to the reference body. The location of a moving body relative to the fixed body is determined at each moment of time by the parameterization of the geometric curve associated with the moving body.

In the mainstream cosmological tradition deriving from Aristotle and Ptolemy, the Earth is at rest at the centre of a finite series of rotating material spheres which exhaust the contents of the universe. In this context the Earth provides a geometrically privileged, fixed body—the natural reference body to which the complicated trajectories of Ptolemaic astronomy can be referred.

For Copernicus and Kepler, the cosmos is still spherical, and both the central sun and the outer surface which encloses the fixed stars are immobile, and are suitable to serve as reference bodies.54 According to Copernicus, the stars are fixed to the surface of the outermost sphere; according to Kepler they are scattered throughout a shell within the outermost sphere, with the shell enclosing a void in which the solar system is located. Copernicus is quite explicit: “the first and supreme of all is the sphere of the fixed stars which contains everything and itself and which, therefore, is at rest. Indeed, it is the place of the world to which are referred the motion and the position of all other stars.”55

(p.171) Galileo, on the other hand, is able to understand the curves that terrestrial bodies trace out in his mechanics as curves relative to the Earth, treated as fixed. But of course, he is also a partisan of the Copernican system, and maintains against Tycho Brahe and Ptolemy that the Earth rotates daily and moves through the heavens annually. And he can make sense of these claims, if he wishes—for like Copernicus and Kepler he maintains that “the fixed stars (which are so many suns) agree with our sun in enjoying perpetual rest.”56

But as the century progressed, new cosmologies emerged in which the Earth orbits the Sun along with the other planets, the Sun itself is just another star, and the stars are scattered haphazardly through space, each being constantly jostled by the fluid or ether in which it is immersed.

In this new context, neither the Earth, nor the Sun, nor the set of “fixed” stars any longer provides a natural reference body for the interpretation of the motion of bodies in terms of geometric curves—the cosmos has no centre, and there is no body that could naturally be taken to be at rest. What is needed is an account of motion that refers motions to something other than body, or one which grapples directly with the fact that only some reference bodies are suitable to refer motions to (in the sense that the law of inertia does not hold if all motion is referred to a body in an arbitrary state of motion)—and that such bodies, if they exist, need not be of any particular astronomical interest.

It was of course Newton who first saw clearly the difficulties involved. He showed that Descartes's analysis of motion in terms of the separation of contiguous bodies was unable to provide the conceptual scaffolding required to make sense of the principle of inertia and concluded that absolute space provided the best foundation for the new mathematical natural philosophy.57 But Kepler seems to have already sensed the difficulties (p.172) that lay ahead—in rejecting the notion of an infinite material universe, he remarks that that notion “carries with it I don't know what secret, hidden horror; indeed, one finds oneself wandering in this immensity, to which are denied limits and center and therefore also all determinate places.”58

Notes:

(1) The canonical sources for this tale are Stein, “Newtonian Space‐Time” and “Some Philosophical Prehistory of General Relativity;” and Earman, World Enough and Space–Time.

(2) For an interesting attempt to resolve this tension, see Jauernig, “Leibniz on Motion and the Equivalence of Hypotheses.”

(3) Physics, IV.4 212a5–7. Translation of McKeon (ed.), Aristotle.

(4) Ibid. 212b8–10. Translation of McKeon (ed.), Aristotle.

(5) I believe that we are supposed to reach this conclusion by regarding the Earth as fixed, then examining the relative motion between each of the surrounding spheres (see below). For a survey of ancient reactions, see Sorabji, Matter, Space and Motion, pp. 193–6. For further discussion, see Morison, On Location, pp. 166–9.

(6) On the evolution of the atomist' notion of void, see Sedley, “Two Conceptions of Vacuum.”

(7) Physics, IV.7 214a28–32. Translation of McKeon (ed.), Aristotle.

(8) Physics, I.9 279a14–15. Translation of McKeon (ed.), Aristotle.

(9) There is some reason to think that that Aristotle here assumes that something is possible only if it happens at some time or other; see Hahm, The Origins of Stoic Cosmology, p. 103 esp. fn. 32.

(10) Physics, IV.6 213a15–20. Translation of McKeon (ed.), Aristotle.

(11) See Grant, Much Ado About Nothing, ch. 3.

(12) Physics, IV.1 209a6–7; translation of McKeon (ed.), Aristotle. See also IV.8 216a34–b10.

(13) For the medieval influence of this argument, see Grant, Much Ado, pp. 32 ff.

(14) See Sorabji, Matter, Space and Motion, ch. 11.

(15) On this, see Sorabji, Matter Spaceam Motica pp. 199–201.

(16) Grant, “Place and Space in Medieval Physical Thought,” p. 154.

(17) See Grant, “The Medieval Doctrine of Place,” § 2 and Much Ado, p. 125.

(18) See p. 69. This work is an eccentrically augmented free translation of a work by Gassendi—and it played a pivotal role in making available in English Gassendi's attempts to Christianize and modernize atomism. Newton is known to have read this work carefully as an undergraduate; see Westfall “The Foundations of Newton's Philosophy of Nature,” p. 172esp. fn. 5.

(19) Physics, III.1 200b33–4. Translation of McKeon (ed.), Aristotle.

(20) Ibid. VIII.6 260a27–8.

(21) Ibid. IV.4 212a15–19.

(22) “The Sceptic in His Place and Time,” 102 n. 15. For further discussion, see Morison, On Location, ch. 5.

(23) Grant, “Medieval Doctrine of Place,” § 3. Presumably it went unnoticed that a body moving along the equator of the cosmic sphere would count as immobile according to this criterion.

(24) It has been argued that one should not attribute to the early atomists the account of void of Epicurus and Lucretius discussed below; see Sedley, “Two Conceptions.”

(25) De Rerum Natura, I.968–83. For discussion of origins of this argument and of Aristotelian responses in antiquity, see Sorabji, Matter, Space and Motion, pp. 125–8. For Scholastic responses, see Grant, Much Ado, pp. 106–8. The argument also appears in More and Gassendi; see Koyré, From the Closed World to the Infinite Universe, p. 123 and Grant, op. cit., 389 n. 168. Here is another popular atomist argument: that which is limited must be limited by something. For this latter argument, see Epicurus, Letter to Herodotus, §41 and Lucretius, De Rerum Natura, I.957–65. For discussion, see Sorabji, op. cit., 136–8.

(26) De Rerum Natura, I.334. Translation of Fowler and Fowler (eds.), Lucretius.

(27) Ibid. I.335–45 and I.370–83. See also Epicurus, Letter to Herodotus, §40. Note Aristotle's response, Physics, IV.7 214a28–32. See Charleton, Physiologia, p. 19. Note: the undergraduate Newton appears to have accepted this argument; Westfall, “The Foundations of Newton's Philosophy of Nature,” p. 174.

(28) De Rerum Natura, I.384–9. See Grant, Much Ado, §4.E for Scholastic responses to this sort of challenge.

(29) Ibid. I.426–9. Translation of Fowler and Fowler (eds.), Lucretius. See also Epicurus, Letter to Herodotus, §40: “And if there did not exist that which we call void and space and intangible nature, bodies would have no place to be in or move through, as they obviously do move;” translation of Inwood and Gerson (eds.), Epicurus, p. 6.

(30) One might worry about this interpretation, on the grounds that the atomists spoke of void as non‐being. But then, they also seem to have thought that the existence of void shows that the non‐existent is just as real as the existent. For discussion and references, see Barnes, The Presocratic Philosophers, §XIX(b).

(31) For discussion of pre‐Epicurean atomism, see chs. 9 and 10 of Furley, The Greek Cosmologists, vol. 1.

(32) According to ancient authorities, Epicurus also held this view; see Inwood and Gerson (eds.), Epicurus, p. 47.

(33) De Rerum Natura, II.222–5; translation of Fowler and Fowler (eds.), Lucretius.

(34) For discussion and references, see e.g. Hahm Origins, ch. IV and Sambursky, Physics of the Stoics, ch. IV. For deflationary readings of the infinitude of the Stoic void, see Todd, “Cleomedes and the Stoic Concept of the Void” and Inwood “Chrysippus on Extension and the Void.” Posidonius appears to have held a heterodox view, according to which the extra‐cosmic void was only just large enough to hold the cosmos at its time of maximum expansion; see Algra, “Posidonius' Conception of the Extra‐Cosmic Void.”

(35) Bowen and Todd (eds.), Cleomedes' Lectures, p. 24. For further discussion and references, see Sorabji, Matter, Space and Motion, p. 129 and Hahm, Origins, p. 106. Note that Cleomedes continues: “But if anyone claims that a conflagration does not occur, such a claim would not confute the existence of the void. For even if we merely conceived of the substance [of the cosmos] expanding, that is, being further extended (granted that there is no possible obstacle to such extension), then this very thing into which it would be conceived as entering in its extension would be void.” Here it is important that the Stoics employed a relatively liberal notion of possibility. See Hahm, Origins, p. 103 for references and discussion.

(36) See Sorabji, Matter, Space and Motion, pp. 126 ff. and Hahm, Origins, p. 106.

(37) Bowen and Todd (eds.), Cleomedes' Lectures, p. 24. (1) Cleomedes himself denied that the cosmos was in fact in motion (see ibid. 26)—but this would seem to be perfectly consistent with taking the possibility of such motion to establish the infinitude of the void, in analogy with the way that the mere possibility of conflagration establishes the existence of the void (see fn. 35 above). (2) Achilles the Grammarian records the following Stoic argument: “If the cosmos were moving down in an infinite void, rain would not overtake the earth. But it does. Therefore the cosmos does not move but stands still;” quoted at pp. 109 f. of Hahm, Origins.

(38) Chrysippus, on the other hand, seems to have denied, on broadly relationalist grounds, that the hypothesis of cosmic motion was a sensible one. For discussion and references, see Hahm, Origins, p. 122.

(39) Grant, “Place and Space,” p. 154.

(40) Grant, Much Ado, p. 180.

(41) This was not, however, viewed as problematic by early Christians; see Sambursky, The Concept of Place in Late Neoplatonism, pp. 14–17.

(42) See Grant, “Medieval and Seventeenth‐Century Conceptions of Infinite Void Space Beyond the Cosmos” and chs. 1 and 2 of Much Ado.

(43) Grant, Much Ado, pp. 106 f.

(44) Ibid. 137. See also Sorabji Matter, Space and Motion, p. 129.

(45) These are propositions 1, 2, 4, 79, 81, 97, 149, 178, 203, 205, and 66 in the numbering and translation found in Lerner and Mahdi (eds.), Medieval Political Philosophy, selection 18.

(46) For discussion and references, see Grant, “The Condemnation of 1277, God's Absolute Power, and Physical Thought in the Late Middle Ages” and Lindberg, The Beginnings of Western Science, pp. 233–44.

(47) For helpful discussion, see Koyré, From the Closed World and Grant, Much Ado, chs. 7 and 8.

(48) For discussion, see Lolordo, Pierre Gassendi and the Birth of Early Modern Philosophy, pp. 106–8. For references to others who rejected the substance‐attribute dichotomy in the case of space, see Grant, Much Ado, pp. 187, 199, 204, 217, 240, and 392 nn. 182 and 185.

(49) Charleton, Physiologia, p. 69.

(50) For references and discussion, see Grant, Much Ado, p. 389 n. 168.

(51) See Charleton, Physiologia, p. 11 and Brush (ed.), Gassendi, p. 387. For discussion, see Lolordo, Gassendi, pp. 109 ff.

(52) Brush (ed.), Gassendi, 388; Charleton, Physiologia, 67 f. Gassendi also says that God faces a choice in deciding where in space to create the world; see the passage quoted on p. 110 of Lolordo, Gassendi.

(53) Leibniz is of course no exception. See e.g. the passages at Loemker (ed.), Leibniz, pp. 577 and 668.

(54) On this point, see Koyré, From the Closed World, pp. 29–34 and 76–87.

(55) Quoted at Koyré, From the Closed World, p. 33. Of course here the planets, including the earth, are numbered among the “other stars.”

(56) Drake (ed.), Galileo, p. 327. Galileo, while content to grant for the sake of argument that the cosmos is spherical in shape, makes a point of noting that there is little evidence that the material universe is finite in extent; ibid. 319 f. And indeed, he appears to have been genuinely undecided on—and quite likely, not especially interested in—questions concerning the finitude, infinitude, or deployment of the stars; see Koyré, From the Closed World, pp. 95–9.

(57) Newton's treatment of Descartes's analysis of motion can be found in De Gravitatione; see esp. the passage at pp. 19–21 of Janiak (ed.), Newton. Spinoza appears to make a similar point in corollary 3 to proposition 22 in pt. 2 of The Principles of Cartesian Philosophy.

(58) Quoted at Koyré, From the Closed World, p. 61.