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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.173) Appendix D: Leibniz and Modal Relationalism

(p.173) Appendix D: Leibniz and Modal Relationalism

Geometric Possibility
Oxford University Press

In so far as Leibniz is willing to grant a sort of reality to the phenomenal world, he takes space to have a determinate structure (more on this in a moment). And he emphatically denies that space can profitably be thought of as composed of geometrically related parts.1 This makes him some sort of relationalist.

But what sort? Should we view Leibniz as a conservative relationalist, aiming to explicate claims concerning the structure of space in terms closely parallel to those offered by substantivalists (but with material points playing the role of points of space)? Or is Leibniz a sort of modal relationalist, employing a notion of geometric possibility in giving content to claims about the structure of space?

The orthodox view has it that Leibniz is a modal relationalist. Indeed, modal relationalism is traditionally introduced and motivated through the quotation of some suggestive and cryptic remarks that Leibniz makes in his correspondence with Clarke. For example:

As for my own opinion, I have said more than once that I hold space to be something purely relative, as time is—that I hold it to be an order of coexistences, as time is an order of successions. For space denotes, in terms of possibility, an order of things that exist at the same time, considered as existing together, without entering into their particular manners of existing.2

Leibniz makes similar remarks elsewhere—e.g., in his correspondence with Des Bosses:

(p.174) space, like time, is a certain order, namely (in the case of space) that of coexisting, which includes not only actual things but also possibles. It follows that it is something indefinite, like every continuum whose parts are not actual but can be taken at will, just like the parts or fractions of a unity.3

Now, such remarks on their own do not establish that Leibniz is a modal relationalist. After all, Leibniz, like Descartes, holds that matter forms a plenum with the structure of Euclidean three‐space. So we could understand his remarks about possibilia not as intended to suggest that possibilia be employed to probe the structure of empty parts of space, but as intended merely to draw our attention to the fact that there are many ways that Euclidean space can be filled with matter.4

I believe, however, that a pretty good case can be made for the orthodox reading of Leibniz as being a modal rather than a conservative relationalist. My case comes in two parts. In the first I argue that there are pretty conclusive reasons for denying that Leibniz is a conservative relationalist. In the second I argue that the texts strongly suggest that he is indeed a modal relationalist. I conclude by considering some worries one might have about these arguments.

Leibniz not a Conservative Relationalist

A good place to begin is with Leibniz's views about void space. On Descartes's view, void space is thoroughly impossible: speaking of extension void of matter involves a conceptual incoherence in much the same way as would speaking of a chain of mountains without any valleys; indeed, if God were to annihilate the contents of a full vessel, the result would be that the walls of the vessel would then be in contact.5 Leibniz explicitly rejects the Cartesian view:

although I deny that there is any vacuum, I distinguish matter from extension, and I grant that if there were a vacuum inside a sphere the opposite poles within the hollow would still not touch. But I believe that divine perfection does not permit such a situation to occur.6

(p.175) In another passage, Leibniz indicates his reasons for denying that the poles inside a void sphere would touch.

If there were a vacuum in space (for instance, if a sphere were empty inside), one could establish its size.…It follows from this that we can refute someone who says that if there is a vacuum between two bodies then they touch, since two opposite poles within an empty sphere cannot touch—geometry forbids it.7

Clearly there is no conceptual incoherence involved in the notion of void space, since we are able to reason about situations involving vacuum. And if there is no conceptual incoherence in the notion of the situation considered, then, one would think, it follows that there exist possible worlds in which this situation occurs.8

So Leibniz certainly allows the possibility of worlds in which matter does not (always) fill all of Euclidean space because there are (at least sometimes) bubbles of void within material extension. In his correspondence with Clarke, he also allows that there are possible worlds in which material extension is of finite extent: “[a]bsolutely speaking, it appears that God can make the material universe finite in extension…”9

Now, the recognition of the possibility of bubbles of void within matter and of the possibility of a finite material universe can be consistently combined with conservative relationalism.10 But anyone who goes in for such a combination must deny that space is three‐dimensional and Euclidean at every world—e.g., no conservative relationalist could consistently maintain that space was infinite if the material world permanently had the structure of Aristotle's spherical cosmos.

However, as Bertrand Russell notes in passing, Leibniz did take the structure of space and time to be the same in every possible world.11

Consider, for instance, Leibniz's assertion that “space and time taken together constitute the order of possibilities of the one entire universe, so (p.176) that these orders—space and time, that is—relate not only to what actually is but also to anything that could be put in its place…”12 In light of the discussion above, it is natural to read this as telling us that the fact that space is Euclidean in structure at our world determines what sort of configurations of matter are possible. Thus, there is some possible world whose matter forms a finite spherical cosmos of Aristotelian type—but no world whose matter forms a Klein bottle, or any other configuration that could not be embedded in a Euclidean space of three dimensions. But what should we think about the structure of space at an Aristotelian world—is space infinite there (with the Euclidean geometry of our own world providing the order of possibilities) or is it finite there (with the limited extent of matter at that world determining that relative to that world only finite cosmoi are possible)? Leibniz's stance is, I think, unequivocal. He tells us that “time and space indicate possibilities beyond any that might be supposed to be actual. Time and space are of the nature of eternal truths, which equally concern the possible and the actual.”13 And for Leibniz the eternal truths are of course genuinely necessary.14 So it would appear that, for Leibniz, from the fact that the structure of space at our world is Euclidean, it follows that every world has Euclidean spatial geometry. And from that it follows in turn that Leibniz was not a conservative relationalist, since he is committed to taking space to be infinite even at worlds of Aristotelian structure.

It would seem that the only way to evade this conclusion would be to show that distinct notions of possibility are in play in the passages in which Leibniz allows that worlds of finite material extent are possible and in the passages in which he seems to imply that space and time have a fixed structure across possible worlds—perhaps the infinitude of matter is functioning as a tacit presupposition in the latter sort of passage. But this suggestion will not work. Consider an analogy that Leibniz develops in his discussion in “On the Ultimate Origination of Things” for the optimization problem that God faces in creating a world:

in this context, time, place, or in a word, the receptivity or capacity of the world can be taken for the cost or the plot of ground on which the most pleasing building (p.177) possible is to be built, and the variety of shapes corresponds to the pleasingness of the building and the number and elegance of the rooms. And the situation is like that in certain games, in which all places on the board are supposed to be filled in accordance with certain rules, where at the end, blocked by certain spaces, you will be forced to leave more places empty than you could have wanted to, unless you used some trick.15

It seems that God is to consider the possible worlds that result from variant ways of filling in space and time with matter—with space and time themselves possessing their structure independently of their material contents. From this it follows that “there would be as much as there possibly can be, given the capacity of time and space (that is, the capacity of the order of possible existence); in a word, it is just like tiles laid down so as to contain as many as possible in a given area.”16 It seems clear here that space has the same structure at worlds in which matter is sparse as it does at worlds in which it forms a plenum with the structure of Euclidean space—and that the infinitude of matter is a consequence, rather than a presupposition, of the thesis that the structure of space is invariant across worlds.

That puts an end to the interpretation of Leibniz as a conservative relationalist: there would appear to be no evading the conclusion that he countenances possible worlds in which space is Euclidean even though the extent of matter is permanently limited to some fixed finite size.

Leibniz a Modal Relationalist

But what does it mean to say that space is infinite in a world in which material extension is bounded? The answer of modal relationalists turns on the notion of geometric possibility—space is infinite if and only if a linear, unbounded array of material points is geometrically possible. Is this Leibniz's answer as well?

Certainly, at various points in the New Essays on Human Understanding it appears that Leibniz is up to something very like this. After rejecting the view that space is a substance, Leibniz asserts that space is rather: “a relationship: an order, not only among existents, but also among possibles as though they existed.”17 In a nearby passage, Leibniz considers the temporal analogue of a vacuum and remarks that this “vacuum which (p.178) can be conceived in time indicates, along with that in space, that time and space pertain as much to possibles as to existents.”18 Why does the possible existence of void space indicate that space pertains to merely possible existents as well as to actual ones? Why think of space as a relationship among possible existents as if they were actual? Here it seems that Leibniz has motivations for speaking of possibilia in the same breath as space which far outstrip the tame observation that Euclidean space may be filled by matter in many ways. These motivations are most explicit in a passage in which he is commenting on Locke's insistence that we should distinguish extension from material extension:

there is no need to postulate two extensions, one abstract (for space) and the other concrete (for body). For the concrete one is as it is only by virtue of the abstract one…In fact time and space are only kinds of order; and an empty place within one of these orders (called ‘vacuum’ in the case of space), if it occurred, would indicate the mere possibility of the missing item and how it relates to the actual.19

This seems to suggest that in worlds in which matter does not fill all of Euclidean space, there is nonetheless some sense in which the complete pattern of Euclidean space exists and makes possible the pattern of extension instantiated by matter—and that the gap between the full Euclidean pattern and the pattern materially instantiated somehow directs us towards possible ways of filling out material extension so that it would instantiate the full Euclidean pattern of spatial relations.

There is no knockdown argument here. But it does seem to me that in these passages we have modal relationalism all but made explicit.


The discussion above invites a number of worries. I discuss five such: three that concern the thesis that Leibniz recognizes the possibility of void space; two that concern the thesis that Leibniz takes the structure of space and time to be the same at every possible world. In each case, the question is whether the textual support adduced in favour of these theses above is (p.179) significantly undermined by Leibnizean considerations that pull in the opposite direction.

Worry I. Vacuum Inconsistent with Divine Nature

One of Leibniz's favourite arguments against the existence of void space appears to be to establish its impossibility: Leibniz argues that space must be supposed to be full of matter, since to do otherwise would be to detract from God's perfection.20 This argument can be found in many forms throughout Leibniz's work.21 The central point is made very succinctly in section 2 of Leibniz's second letter to Clarke: “the more matter there is, the more God has occasion to exercise his wisdom and power.”22 But since God exists necessarily and has his perfections necessarily, it would seem that anything implied by these perfections must itself be necessary. So Leibniz has no more room to recognize the possibility of void space than do Aristotle and Descartes—and hence Leibniz has no motive at all to embrace modal relationalism.

But of course this argument really only serves to draw our attention to a very general problem for Leibniz (and Leibniz scholars). Parallel reasoning would suggest that there are no contingent truths: God actualizes this world because it is the best of all possible worlds; but surely, whichever world is best is necessarily best; and, since it follows from divine perfection, surely it is necessarily true that God actualizes the best world. Leibniz himself seems to have been tempted at one time to accept the necessitarian conclusion, but his considered view appears to have been that it should be vigourously rejected.23 So in attempting to make out Leibniz's views, it seems only fair to allow him to take void space to be genuinely possible despite the fact that its non‐actuality follows from divine nature alone.

Worry II. Space and Matter Inseparable for Leibniz

A second objection to the claim that Leibniz allows void space is based on the following.

(p.180) I do not say that matter and space are the same thing. I only say that there is no space where there is no matter and that space in itself is not an absolute reality. Space and matter differ as time and motion. However, these things, though different, are inseparable.24

On one reading of this passage, Leibniz is telling us that space and material extension are necessarily coextensive—that there is no time at any world at which there is empty space. If this is right, then we have a powerful counterweight to the texts in which Leibniz appears to allow void space.

But there is another natural reading, on which Leibniz is telling us here merely that there is no time or world which is completely devoid of matter but at which space exists. That this second reading is to be preferred is strongly suggested by the context of the passage under consideration. Leibniz is engaging with Clarke's off‐target gibe that anyone who takes matter and space to be the same must regard the material world as necessarily infinite in extent and eternal in duration (because space and time are). The passage above serves to set up Leibniz's assertion that:

it does not follow that matter is eternal and necessary, unless we suppose space to be eternal and necessary—a supposition ill‐grounded in all respects.25

In these passages Leibniz is concerned not with whether there might be empty space within or outside the material world but with the question of worlds and times devoid of matter.

(Note that it is important for Leibniz to insist that while God's existence is necessary, that of space is not.26 To this end, he asserts that there is a possible situation in which God creates nothing and (hence) in which nothing other than God exists—not even space and time.27 Since for Leibniz space has the same structure in every possible world, there is no sense in allowing that God could create distinct empty worlds, and so Leibniz has no use for a distinction between a situation in which God creates an empty world and one in which no world is created.)

(p.181) Worry III. Void Space Imaginary for Leibniz

Consider next passages in which Leibniz speaks of void space (whether outside or within the cosmos) as being imaginary.

The same reason which shows that extramundane space is imaginary proves that all empty space is an imaginary thing, for they differ only as greater and less.28

Since space itself is an ideal thing like time, space out of the world must necessarily be imaginary, as the schoolmen themselves have acknowledged. The case is the same with empty space within the world, which I take to be imaginary…29

What does Leibniz intend to communicate by telling us that void space is imaginary? One possibility that may come to mind is that he is telling us that void space is not merely non‐existent but impossible—for Leibniz does in fact sometimes employ the term ‘imaginary’ with something like this force.

The whole difficulty here has therefore only come from a wrong idea of contingency and of freedom, which was thought to have need of a complete indifference or equipoise, an imaginary thing, of which neither a notion nor an example exists, nor ever can exist.30

But caution is required here: it is far from obvious that in speaking of void space as imaginary in his letters to Clarke Leibniz meant to indicate that it was impossible. (i) As Leibniz indicates in one of the passages under discussion, ‘imaginary space’ was a Scholastic term of art—one which had, and was known to have, a dizzying array of established meanings by the time Leibniz was writing.31 For example, for some prominent authors, such as Suárez, the distinction between real and imaginary space was the distinction between space occupied by body and empty space capable of being occupied by body.32 (ii) Leibniz himself sometimes uses ‘imaginary’ (p.182) to describe entities that are non‐actual but possible.33 (iii) Elsewhere Leibniz speaks of the imagination as an internal sense concerned with the objects that are the concern of arithmetic and geometry.34

Worry IV. Dimension of Space Varies Across Worlds

Now we turn to a couple of worries concerning the claim that Leibniz takes space and time to have the same structure in each possible world.

In a famous passage in the New Essays Concerning Human Understanding, Leibniz discusses the epistemology of spatial and temporal vacua. In the portion of this passage dealing with the spatial case (quoted above on p. 175), Leibniz asserts that bubbles of void within material extension are not only possible but in principle measurable. Things are quite different in the temporal case: “if there were a vacuum in time, i.e., a duration without change, it would be impossible to establish its length.”35 Why is this? The reason appears to be that in the case of a ball‐shaped void, we can determine that opposite points on the boundary do not touch because “geometry forbids it.” Whereas in the temporal case, “we could not refute anyone who said that two successive worlds are contiguous in time, with no possible interval between them. We could not refute him, I say, because that interval is indeterminable.” This claim raises some interesting questions.36 But for present purposes, the important point comes in the final remark that Leibniz makes in this discussion: “If space were only a line, and if bodies were immobile, it would also be impossible to establish the length of the vacuum between two bodies.”

(p.183) Applying the same standards as were used above in arguing that Leibniz is committed to the possibility of void space, it appears that we must conclude that Leibniz here commits himself to the possibility of worlds in which space has only a single dimension. And of course if this is accepted then we must abandon the thesis that Leibniz takes the structure of space to be the same across possible worlds.

A couple of possible responses suggest themselves. One option is to read Leibniz's remarks concerning the situation in which space is a line and bodies are immobile as referring to a world in which the matter distribution is a one‐dimensional continuum while space, as usual, has the structure of Euclidean three‐space.

Another possibility is to note that Leibniz is perfectly capable in the course of technical discussions of getting carried away and saying things that he ought not to—things that are inconsistent with principles that he holds dear.37 So perhaps here we have a case in which he says something carelessly which deeper consideration would lead him to retract—the remark in question is, after all, a fairly casual one.

This suggestion can be bolstered by considering a passage in which Leibniz very conveniently addresses the question why our world is three‐dimensional. The context is provided by a puzzle raised by Bayle—why should matter have three dimensions rather than, say, two or four?38 Leibniz notes that Bayle himself appears to expect that the answer should lie in God's will. Leibniz denies this and makes much of his denial. On his view, that matter should have three dimensions follows not from considerations of what sort of world is best but rather from a “geometrical and blind necessity.”39

This provides very strong evidence for the verdict that on Leibniz's considered view, the dimension of space is invariant across possible worlds. Unfortunately, the case is not quite conclusive—there are at least two sorts of worry that one might consider grounds for appeal.

  1. (1) Consider the reason that Leibniz offers for taking matter to be necessarily three‐dimensional:

    (p.184) the ternary number is determined for it not by the reason of the best, but by a geometrical necessity, because the geometricians have been able to prove that there are only three straight lines perpendicular to one another which can intersect at one and the same point.40

This would appear to be baldly question‐begging.41 Worse, for present purposes, it appears at best to head off the possibility that space could have more than three dimensions while leaving untouched the possibility that space could be one‐ or two‐dimensional.

  1. (2) In a discussion of the foundations of the calculus, Leibniz remarks that “even if someone refuses to admit infinite and infinitesimal lines in a rigourous and metaphysical sense and as real things, he can still use them with confidence as ideal concepts which shorten his reasoning.”42 Higher‐dimensional spaces have the same status: “we can also conceive of dimensions beyond three, and even of powers whose exponents are not ordinary numbers—all in order to establish ideas fitting to shorten our reasoning and founded on realities.” It may appear that Leibniz here comes perilously close to conceding that higher‐dimensional spaces are genuinely conceivable. If this is so, it may be difficult for him to show that the notion of a world in which matter has more than three dimensions is incoherent, as would seem to be required in order to show that in creating the world, God had no choice but to make it three‐dimensional.

These last two points certainly raise real worries about whether Leibniz ought to take matter to be necessarily three‐dimensional. But I do not think that they manage to raise real worries about the claim that he in fact did so, given his unequivocal pronouncement on the question in his discussion of Bayle's question about the dimensionality of space. (In any case, so long as it is granted that space is Euclidean at every world, Leibniz's acceptance of the possibility of a finite material world forces him towards something like modal relationalism—whether or not he takes the number of spatial dimensions to vary from world to world.)

Worry V. Extent of Time Varies Across Worlds

A somewhat similar worry can be raised about the extent of time. Above we considered a passage in which Leibniz noted that space is not eternal if (p.185) the duration of the created world is finite. In a similar vein, Leibniz remarks: “If there were no creatures, there would be neither time nor place, and consequently no actual place.”43 These texts suggest but do not mandate an interpretation of Leibniz as believing that the extent of time in a given world is equal to the duration of material extension at that world.

Curiously, there is a text in which Leibniz appears to come down on the other side on this issue. Locke suggests that having fixed the length of the year in terms of the motion of the Sun, it makes sense to speak of the durations of various epochs lying prior the creation of the material universe.44 In his commentary on this passage, Leibniz does not object that this notion makes no sense, but appears rather to approve of it (aside from a quibble about biblical dates).45

So it is far from clear that consideration of this issue should move us away from the claim that Leibniz takes space and time to have the same structure in every possible world.46 (In any case, a reasonable fallback position is available, under which space is Euclidean at every world while the structure of time differs from world to world in virtue of differing in extent. This would suffice for the present purpose of arguing the Leibniz should be classified as a modal relationalist rather than as a conservative one.)


(1) Lamarra, “Leibniz on Locke on Infinity.”

(2) §4 of Leibniz's third letter to Clarke. Translation of Ariew (ed.), Leibniz and Clarke. For similar language, see §41 of Leibniz's fourth letter and §§47, 104, and 106 of his fifth letter.

(3) Look and Rutherford (eds.), Leibniz–Des Bosses141

(4) For suggestions along these lines, see Earman, World Enough and Space‐Time, §6.12 and Futch, Leibniz's Metaphysics of Time and Space, ch. 2.

(5) Principles of Philosophy,Garber, Descartes' Metaphysical Physics,127–55

(6) New Essays on Human UnderstandingRemnant and Bennett (eds.), Leibniz

(7) Ibid. §II.xv.11. Translation of Remnant and Bennett (eds.), Leibniz.

(8) 170Ariew and Garber (eds.), Leibniz

(9) §30 of Leibniz's fifth letter to Clarke (see also §73 of the same letter). Translation of Ariew (ed.), Leibniz and Clarke.

(10) Difficulties arise when one considers worlds more exotic than those Leibniz likely had in mind. See the discussion of §2 of Ch. II above.

(11) A Critical Exposition of the Philosophy of Leibniz,Rescher, “The Plurality of Space‐Time Frameworks,”

(12) Loemker (ed.), Leibniz

(13) New EssaysRemnant and Bennett (eds.), Leibniz

(14) Adams, Leibniz,

(15) Ariew and Garber (eds.), Leibniz150

(16) Ibid.

(17) Remnant and Bennett (eds.), Leibniz

(18) Remnant and Bennett (eds.), Leibniz

(19) §II.iv.5. Translation of Remnant and Bennett (eds.), Leibniz. The suggestion that the merely ideal in some sense governs the real can also be found in Leibniz's letter to Varignon of 2 Feb. 1702; see Loemker (ed.), Leibniz, p. 544.

(20) Leibniz's other favourite argument against the void is driven by the principle of sufficient reason. To the extent that that principle is taken to be necessary, the considerations engaged below arise for that argument as well.

(21) Garber, “Leibniz: Physics and Philosophy,”

(22) Leibniz and Clarkeibid.27

(23) Adams, Leibniz,

(24) §62 of Leibniz's fifth letter to Clarke. Translation of Ariew (ed.), Leibniz and Clarke.

(25) ibid.

(26) TheodicyFarrer (ed.), Leibniz

(27) Robinet (ed.), Correspondance118

(28) §7 of Leibniz's fourth letter to Clarke. Translation of Ariew (ed.), Leibniz and Clarke.

(29) §33 of Leibniz's fifth letter to Clarke; translation, ibid. See also Leibniz's letters to Rémond of 27 March 1716 (Robinet (ed.), Correspondance, pp. 61 f.) and to Des Bosses of 29 May 1716 (Ariew and Garber (eds.), Leibniz, pp. 201–6).

(30) TheodicyFarrer (ed.), Leibniz

(31) Grant, Much Ado About Nothing,120

(32) Grant, Much Ado, §7.2. Bayle appears to have taken this for the standard use: see remark G of the article on Leucippus in the Historical and Critical Dictionary. Gassendi likewise follows this use—although that does not stop him from going on to speak of imaginary space as a chimera (see the passages quoted on pp. 110 and 121 f. in Lolordo, Pierre Gassendi and the Birth of Early Modern Philosophy). Further, Leibniz himself appears to have followed this use in notes written in 1676 (Parkinson (ed.), De Summa Rerum, p. 77). I mention all of this in order to drive home the point that one should not jump to conclusions about the sense of ‘imaginary space.’ It is of course clear that Leibniz himself was not following this use in the passages cited in fnn. 28 and 29 above—cf. esp. §29 of his fifth letter to Clarke.

(33) Loemker (ed.), Leibniz

(34) Letter to Queen Sophie Charlotte of Prussia; Ariew and Garber (eds.), Leibniz, pp. 186–92. For discussion and references concerning this theme, see McRae, “The Theory of Knowledge,” pp. 178–86. Note that Leibniz also speaks of space as imaginary in his memorandum on Copernicanism and relativity of motion (see Ariew and Garber (eds.), Leibniz, p. 91). It seems plausible that this use has its roots in Leibniz's account of mathematics.

(35) Remnant and Bennett (eds.), Leibniz

(36) (1) How should we understand the claim that geometry forbids us from taking the poles of the empty sphere to touch? (2) If Leibniz is allowing, as he seems to be, that there could be worlds that differ only as to the empirically inaccessible length of a certain period of changelessness, how can this be reconciled with the sort of verificationist sentiment that he gives vent to in his correspondence with Clarke (see esp. §52 of Leibniz's fifth letter)?

(37) Loemker (ed.), Leibniz

(38) Theodicy, §351.

(39) Farrer (ed.), Leibniz

(40) Farrer (ed.), Leibniz

(41) Russell, Critical Exposition,

(42) Loemker (ed.), Leibniz

(43) §106 of his fifth letter to Clarke. Translation of Ariew (ed.), Leibniz and Clarke.

(44) An Essay Concerning Human Understanding, §II.xiv.24.

(45) New Essays, §II.xiv.24.

(46) Futch, Leibniz's Metaphysics,