1. Descartes’s Theory

1.4 The Indefinite as Syncategorematic

The scholastic distinction between actual and potential infinities ultimately had its roots in Aristotle’s Physics III.4–8. But so too did another scholastic distinction that I feel is more helpful in coming to grips with Descartes’s notion of the indefinite: namely the distinction between categorematic and syncategorematic infinities.

I say this while acknowledging that one should always be cautious in applying Aristotelian/scholastic jargon to Descartes, and for two reasons. First, Descartes himself explicitly disavowed it, explaining that “it would be very difficult for me to employ the same terminology [as in the schools], when my own views are profoundly different.”^{Footnote 29} Second, even among the schoolmen themselves, there was scant consistency in how terms like these actually got used. Aristotle himself had blended elements of both distinctions together into one, as many set theorists still do today, and as indeed did many of the schoolmen themselves. Ariew, for instance, quotes an explicit identification of the syncategorematic infinite with the potential infinite from Goclenius’s 1613 Lexicon philosophicum.^{Footnote 30} And many scholars of medieval Aristotelianism will still equate them now: Grant, for instance, straightforwardly contrasts “an actual, or categorematic, infinite” with a “syncategorematic, that is, a potential infinite.”^{Footnote 31} However, there were several medieval Aristotelians who thought that these two distinctions could be usefully pried apart. Grant’s identification of the syncategorematic/categorematic with the potential/actual might be compared with an equally clear-cut declaration from Pierre Duhem that the “distinction between the categorematic and the syncategorematic senses of infinite is completely independent from the distinction between potentiality and actuality.”^{Footnote 32} For my part, I agree with Duhem. In particular, and to avoid any potentially misleading ambiguities, I maintain that Descartes’s indefinite universe should be understood as actual but syncategorematic, as I shall now define that term.

The syncategorematic/categorematic distinction boils down to the question of whether the items in question—irrespective of whether these should be actual or potential—can be considered collectively or only distributively. Can we affirm anything of the whole lot of them together as one, or only affirm it piecemeal of each one individually? In the case at hand for Descartes, the question is whether there is a place infinitely/indefinitely far from some given place—say, the present location—or merely an infinite/indefinite series of places, each one at some finite distance from here.

The distinction is perhaps best exemplified by the set of natural or real numbers. There are infinitely many numbers, because each of them is such that there is another greater than it. That is to say,

(1)$$\left( {\forall n} \right)\left( {\exists m} \right)\left( {m > n} \right)$$

This is the syncategorematic infinite. And yet each of these numbers individually is still finite. Whatever else it might be, ‘infinity’ is not a number greater than all the rest: which would amount to a categorematic infinite. It is not the case that^{Footnote 33}

(2)$$\left( {\exists m} \right)\left( {\forall n} \right)\left( {m > n} \right)$$

Descartes’s extended universe was such that every number could represent the distance from here of a place; indeed, of a body. To apply formula (1) to the case at hand, then, we simply need to read the ‘>’ symbol to mean ‘more distant than.’ And, for each such place or body, there either does or at least could exist a further one. I have suggested that Descartes believed that this indefinite sequence of bodies was entirely actual; but the notion of the syncategorematic as such, just like Descartes’s concept of the indefinite, would be just as applicable to their merely potential existence. To capture the latter, we simply need to insert a modal operator into (1):

(1ʹ)$$\left( {\forall n} \right)\diamondsuit \left( {\exists m} \right)\left( {m > n} \right)$$

As we saw, (1ʹ) is all that Descartes was asserting in Principles I.26, with the stronger (1) then being argued for in II.21. But what Descartes never suggested was that a place or body, further away than every other body in this whole indefinite sequence, did or even could exist. That is, it is not even the case that

(2ʹ)$$\diamondsuit \left( {\exists m} \right)\left( {\forall n} \right)\left( {m > n} \right)$$

Descartes’s general distaste for the jargon of the schools notwithstanding, he was certainly familiar with it, and in particular with their notion of the syncategorematic infinite. For it is a term that he did in fact use: albeit only on one solitary occasion, in a letter to Jean-Baptiste Morin of 1638. He was there discussing the various movements to which a body might be subjected, and he wrote: “each body can have several movements, and be pushed by an infinity of diverse forces at the same time; always taking the word ‘infinity’ syncategorematically [sincategorematice (sic)], so that those in the schools should have nothing therein to find fault with.”^{Footnote 34} I have not come across any instances in Descartes’s writings of the opposing term, ‘categorematic’; but his use of ‘always’ in this remark does suggest that he would never have been willing to apply that term—or, at any rate, never to do so when the context was restricted, as here, to created things alone—any more than those schoolmen who surely would have found fault with him had he done so.

When we come to look in detail, in Section 2 below, at Descartes’s various presentations of his argument for an indefinite universe, it should become abundantly clear that he was pushing for such indefiniteness in precisely this syncategorematic sense. The principle that a place/body should exist beyond each other place/body was the very thing that his argument was designed to demonstrate. For instance, and to anticipate, he would tell Henry More: “I cannot but conceive a space beyond whatever bounds you assign to the universe; and on my view such a space is a genuine body.”^{Footnote 35} To anyone who might suggest that the universe should come to an end at some particular finite distance from here, Descartes would insist upon the actual or at least possible existence of an additional body one place further out. And yet still only one place further out—that is, at a slightly larger but still finite distance. Or, if the same considerations should then be reapplied to this additional body, Descartes would offer yet another body, one place further out still.^{Footnote 36}

2. Descartes’s Argument

If a limited universe is as contradictory as that the sum of 1 and 2 is not 3, as Descartes eventually came to maintain, then precisely where does the contradiction lie? Descartes came at the issue via a conceptual analysis of the notion of a limit, with a view to showing that such a notion was inconsistent with the notion of extension.

But the argument he adopted was an old one. For example, the ancient atomists had believed that the universe comprised infinite worlds in an infinite void, and this was one of the approaches they took in establishing that conclusion. If the universe was finite, argued Lucretius, it would certainly have a limit somewhere. However, he continued, “clearly a thing cannot have a limit unless there is something outside to limit it.”^{Footnote 37} But, of course, there cannot be anything outside the universe, that being by definition the totality of everything that exists. The idea was that it is in the nature of a limit, a boundary, or (three-dimensionally) a surface that it should separate one thing from another thing. A finitely extended universe will therefore be unthinkable, because the very act of imagining it as terminating in a boundary will require us to suppose that there is more of it on the other side.

The Stoics also adopted a similar argument, but with a twist. Unlike the atomists (or, for that matter, Descartes), the Stoics did believe that the corporeal cosmos was indeed only finitely large. However, they also believed that this corporeal world was surrounded by an infinite void: “it is itself necessarily limited, whereas what is outside it is a void that extends without limit in every direction.”^{Footnote 38} But, again, it was precisely by contemplating the nature of a limit to the corporeal world that they were led to recognise the real existence of further extension beyond it, albeit void extension in their case. The Stoics agreed with the atomists that anything that was limited would need to be limited by something outside itself. What differentiated them was that they then introduced an additional premise: they further claimed that the limiting thing would need to be of a different nature from what it limited. Hence their two kinds of extension, corporeal on this side of the boundary and void on the other.

Descartes, of course, rejected all talk of void, whether beyond a finite corporeal cosmos or even just in gaps between the corporeal constituents of an indefinite one. But the more fundamental principle, that a boundary cannot be one-sided, was precisely that on which his argument depended. There are five passages in his writings where he offered such an argument, the first couple of which we have already seen:

1. (1) There is, for example, no imaginable extension which is so great that we cannot understand [intelligamus] the possibility of an even greater one; and so we shall describe the size of possible things as indefinite.^{Footnote 39}

2. (2) For no matter where we imagine the boundaries to be, there are always some indefinitely extended spaces beyond them, which we not only imagine [non modo imaginamur] but also perceive [percipimus] to be imaginable in a true fashion [verè imaginabilia], that is, real.^{Footnote 40}

3. (3) Now if we suppose the world to be finite, we are imagining [on imagine] that beyond its bounds there are some spaces which are three-dimensional and so not purely imaginary [pas purement imaginaires], as the philosophers’ jargon has it. These spaces contain matter; and this matter cannot be anywhere but in the world, and this shows that the world extends beyond the bounds we had tried to assign to it.^{Footnote 41}

4. (4) [N]o limit to the world can be imagined without its being understood [intelligam] that there is extension beyond it.^{Footnote 42}

5. (5) I think it involves a contradiction, that the world should be finite or bounded; because I cannot but conceive [concipere] a space beyond whatever bounds you assign to the universe; and on my view such a space is a genuine body.^{Footnote 43}

As noted in the last section, passage (1) does not capture the full force of Descartes’s position: it suggests that an unlimited sequence of greater and greater extensions is at least possible—albeit only in a syncategorematic sense, such that each is possible—but it remains silent on whether these extensions are actual. And perhaps extension can already be said to be indefinite on this basis alone, any claims of actuality being additional to the basic notion of indefiniteness as such. But passages (2) to (5) all make it clear that Descartes was indeed committed to the stronger claim. These spaces beyond the alleged limit are not merely possible: they are truly as we imagine them to be, they contain matter, and are genuine bodies.

It will be observed that Descartes used a variety of epistemological terms in these passages. Passages (2) and (3) both suggest that what we are doing in this thought experiment, at least in the first instance, is imagining further extension beyond any supposed boundary. Now, to some degree, Descartes was simply addressing ‘the philosophers’—those great scholastic champions of extra-mundane imaginary spaces—in their own terms, to show how his own opinion differed from theirs. Still, there does seem to be more going on than just that alone. It appears from these passages that Descartes wanted us actually to form a mental image of this additional extension beyond any purported limit, to picture it in the mind’s eye.^{Footnote 44}

But this does make sense. Earlier in his career—especially in the Rules for the Direction of the Mind—Descartes had gone so far as to suggest that such images were essential to our knowledge of extended things.^{Footnote 45} Admittedly, by the time he was presenting the argument at hand, he had softened that position and come to think that such images were dispensable. In the Second Meditation, for instance, he argued that our knowledge of the wax was not really achieved through imagination at all, but through “purely mental scrutiny.”^{Footnote 46} However, he also made the point to Elisabeth of Bohemia that “body (i.e. extension, shapes and motions) can likewise be known by the intellect alone, but much better by the intellect aided by the imagination.”^{Footnote 47} So perhaps that is what is going on in passages (2) and (3). The images we form of these spaces beyond the alleged limit, even if not absolutely necessary, can still prove a valuable aid, a springboard to launch us towards the ultimate conclusion.

But the fact that Descartes did not think that such images were absolutely necessary is then backed up by passages (4) and (5): for these simply do not mention imagination, and yet Descartes still had no trouble arriving at the same conclusion. Furthermore, (2) and (3) themselves already make it clear that imagination is not only unnecessary; it is also insufficient. We do not only imagine these spaces, and they are not purely imaginary. However helpful the imagination might be as a starting point, something else is going to need to get in on the act, to enable us to go beyond the mere image and discover the real existence of the thing we are imagining. As Descartes was the first to admit, the unassisted faculty of imagination was a decidedly unreliable resource in the search after truth. It is not at all clear that he would have allowed that mere imagination, obscure and confused as it frequently was, could be used to establish even so much as the possibility of things, let alone their real existence.

However, it is clear that Descartes believed that there was indeed another epistemological faculty available to us here, one with the power to legitimise the move from such a mental image to a recognition of real existence, or even to establish the latter conclusion all by itself. Passage (4) states that we ‘understand’ that such bodies really exist; and passage (1), notwithstanding the weakness of what it is actually saying, also uses the same term. Neither of these passages mentions the imagination: for this was precisely the role of the understanding in Descartes’s epistemology, to give us a direct insight into the eternal truths and essences of things, one that would often not involve images at all, or, even when they did play an assisting role in prompting it into activity, certainly did not depend on them.

In contrast to imagination, Descartes felt that a higher faculty like understanding could at least give us a reliable insight into possibility. He regarded it as axiomatic that possible existence was “contained in the concept of a limited thing,”^{Footnote 48} and he wrote: “I know that everything which I clearly and distinctly understand is capable of being created by God so as to correspond exactly with my understanding of it” (emphasis added).^{Footnote 49} And yet merely understanding concepts was still going to be of only limited value when it came to establishing the real existence of anything other than the self (via the cogito) and God (from the infinite objective reality of our idea of him). So what might warrant the stronger claim, that these bodies beyond the purported boundary not only do not exist solely in the mind, nor even exist outside the mind solely as possibilia, but are genuinely real?

Descartes certainly felt that the real existence of any extension was entirely contingent on God’s voluntary decision to create it. God could have decided not to create anything at all. Or he could have created a universe containing nothing but unextended thinking beings, and just left it at that. Moreover, even allowing that God did create extended things, Descartes did not think that our knowledge of their existence could arise directly out of their concept alone, but would depend on God’s veracity, coupled with the propensity he gave us to believe in bodies when presented with sensual perceptions, as outlined in the Sixth Meditation. So, if Descartes’s argument for the indefiniteness of the universe was going to get off the ground, it could not start from nothing, but would need to take the existence of some bodies for granted. And this was indeed the approach that Descartes took. His argument takes the existence of bodies within the alleged boundary as a starting point—established in advance, presumably on the basis of that Sixth Meditation argument—and then seeks to show that these bodies will need to be surrounded by others, and those by further others, and so on indefinitely. For Descartes, it is all or nothing. Either nothing extended exists, which he regarded as a false but nevertheless intelligible and non-contradictory hypothesis. Or an indefinite quantity of extension exists. The one option he rejected as contradictory was that only a finite quantity of extension should exist.

But it follows that bodies cannot be ontologically independent of one another. The actual existence of any matter must entail the actual existence of all possible matter. And Descartes did indeed assert an ontological dependence among the parts of space, in the 1647 letter to Chanut. The letter had been prompted by certain theological concerns, expressed by Queen Christina of Sweden and reported to Descartes by Chanut, among which was the fear that, if one admits a world that is infinite in matter and substance, this will undermine its creation in finite time.^{Footnote 50} If matter must exist beyond any imagined spatial boundary, then should it not, by the same token, exist both before any imagined temporal creation and after any imagined temporal destruction, thereby making matter necessarily existent and independent of God? The same concern about Descartes’s position was frequently expressed by others over the next few decades.^{Footnote 51} Indeed, it became especially acute after 1671, when the Cartesian physicist, Jacques Rohault, actually came unnervingly close to accepting that conclusion, that matter was uncreated and indestructible.^{Footnote 52}

However, Descartes attempted to reassure Christina on this point, writing to Chanut as follows:

Here, Descartes was explicitly contrasting the merely possible existence of the world before any given moment of time, and its actual existence beyond any given point of space. And his grounds for this distinction lay in a contention that there was a necessary connection between the actual existence of the spaces surrounding a body such as a globe and that of the globe itself, a connection that did not hold across temporal boundaries.

On the face of it, it might seem—and it has been argued in the secondary literature, for instance by Jean Laporte^{Footnote 54}—that this is flying in the face of what Descartes said elsewhere about the real distinction between different regions of the extended world. For instance, in Descartes’s principal discussion of the nature of a real distinction, he wrote of extended or corporeal substance that, “if it exists, each and every part of it, as delimited by us in our thought, is really distinct from the other parts of the same substance.” And he spelled out what a real distinction amounted to: it “exists only between two or more substances,” such that God will always retain the power of “keeping one in being without the other.”^{Footnote 55} Again, Descartes made much the same point in a 1642 letter to Guillaume Gibieuf: “I consider the two halves of a part of matter, however small it may be, as two complete substances.”^{Footnote 56} As he explained the expression in the Fourth Replies, Descartes meant by ‘complete substance’ that each of these halves, considered as a body in its own right, would require nothing else (apart from God) to sustain it in existence: it could subsist on its own.^{Footnote 57} There does appear, at least prima facie, to be a challenge in reconciling such remarks with Descartes’s claims about a necessary connection between one body and other, surrounding bodies.

But the crucial thing to observe is that Descartes never suggested that there was a necessary connection between a body and any particular surroundings. All bodies were, after all, supposed to be mobile: but motion, for Descartes, meant precisely that the body would relinquish its spatial relations to its immediate surroundings, and become surrounded by some other set of bodies instead.^{Footnote 58} This body certainly did not need to be connected to that body, and the one could perfectly well be conserved in existence without the other. But it might yet need to be connected to some body or other. The latter is all that Descartes actually suggested to Chanut (and elsewhere) when rejecting the notion of a bounded universe, and it is not so clear after all that there is any conflict between this weaker claim and the real distinction between bodies. Descartes did not care which bodies might turn out to surround a supposedly finite world, but merely insisted that some should.

And yet one might still wonder what—if anything—would entitle Descartes even just to the weaker conclusion. After all, many people in his own era, even when writing directly in response to his claims, simply rejected it. They instead fell back on the traditional scholastic line that, although it was perfectly possible that further extension should exist beyond the boundary of the finite world, none actually did so.

Let us take just two examples. First, Isaac Barrow laid out a theory of space in the 10^th of his Mathematical Lectures, delivered in 1665. Most of the first half of the lecture was directed against Descartes’s position, following which Barrow proceeded to offer his own alternative proposal. “Space,” he wrote, “is nothing else but the mere Power, Capacity, Ponibility, or (begging pardon for the Expressions) Interponibility of Magnitude.”^{Footnote 59} By ‘magnitude,’ he meant specifically corporeal bulk, and by his own neologism ‘ponibility’ (from the Latin ‘ponere,’ to put), he meant simply that it was possible that such bulk should be put into such space. As he explained: “There lies no Body, there is found no actual Dimension beyond the Mass of the Universe; but it is possible for a Body to be constituted and a real Dimension to be extended beyond that itself, i.e. there is an Ultramundane Space.”^{Footnote 60} And the important thing to observe here is that Barrow was not only saying that there lies no body beyond the finite universe, but that there are no actual dimensions there at all, not even those of a void. As we noted in the last section, Barrow was taking the same line as figures like Oresme or the Coimbra commentator had done. His ultramundane space was quite unlike the void of the Stoics which, as we noted at the start of this section, extended without limit in every direction. Barrow’s space was not actually extended: “it has no actual but only potential Figures, Dimensions and Parts consentaneous to its Nature.”^{Footnote 61} Although quantities, measurable in feet or yards or miles, might be assigned to such an ultramundane space, it was not to the space itself that these quantities properly belonged, but only to the bodies that could (but did not actually, or did not yet) exist therein.

Second, a few years later, Jean-Baptiste de La Grange argued against Descartes in much the same way, in Chapter 29 of his own Principes de la philosophie, a chapter entitled “Place and Space are nothing positive.” La Grange recognised the force of the arguments that seemed to be pressing for the existence of further spaces beyond the boundary of a world imagined as finite, and he was comfortable in embracing that conclusion, just as long as it was rightly understood. But the trick was, just as with Barrow, to regard space as nothing positive, and as possessing no real extension of its own. “Space, properly speaking, is a certain capacity of receiving a body […]. That being granted, I say that there are true spaces beyond the heavens. That is evident, since there are capacities there, proper for receiving the bodies that God could create there, and God could produce beyond the heavens several other worlds similar to this one.”^{Footnote 62} However, although it could be truly said that there were spaces out there, and these spaces were indeed real in their own manner, what they did not constitute was a “positive extension.”^{Footnote 63} La Grange accepted that space was unbounded, just as (he observed) the Cartesians themselves were well persuaded. He also believed that space could not be produced or destroyed. However, he could not accept either point as applicable to matter, and this was his solution: far from treating space as material, he treated it as a “nothing.”^{Footnote 64} It was merely the possibility of matter, rather than an actually extended—or even actually existing—thing in its own right. It was unlimited, but only in the sense that God possessed an unlimited creative power. It could not be produced, but only in the sense that God did not produce his own power to produce things. And it could not be destroyed, but only in the sense that God did not have the power to take away his own power.

These, then, were the two positions. On the one hand, traditionalists were perfectly happy to allow the possibility of further extension on the other side of the boundary of a limited universe, just as long as this was acknowledged as a mere possibility. On the other hand, Descartes insisted that there should actually be further extension—and corporeal extension at that—beyond any supposed boundary (entailing, of course, that it was not a genuine boundary after all). Neither side ever managed to persuade the other, and ultimately one might wonder whether this just came down to a battle of intuitions. Descartes felt that, if an extended substance was going to be limited, it would need to be limited by another extended substance, really existing beyond itself in order to demarcate its boundary. But opponents like Barrow or La Grange simply did not accept that premise. They did not think that a limited extended substance would need to be limited by anything beyond it at all, neither by further matter, nor even by an actually extended void. Unless some further consideration can be brought to bear, to decide between these two attitudes, we seem to be in a stand-off.

3. Descartes and Malebranche on the Ontology of Modes

Descartes believed that the actual indefiniteness of the extended world was something we understood. Whether or not the imagination might also happen to play an assisting role along the way, it was the understanding that was doing all the real work. So what was it that the understanding was latching onto? Precisely why did that principle, that a limit could not be one-sided but would need to separate one thing from another thing, so appeal to Descartes? I suggest that the answer lies in Descartes’s conception of a mode of extension.

To illustrate and explain this, and in hopes of resolving the stand-off between those who accepted the principle and those who simply did not, let us turn to another figure who disagreed with Descartes on this point: namely, Nicolas Malebranche. The reason why this case is especially worthy of attention is because, whereas Barrow and La Grange were avowed opponents of Descartes, Malebranche was in many ways a Cartesian. He drew heavily on Descartes’s ideas and arguments, employing them to his own ends in a wide range of different areas of philosophy, and only deviating from the orthodox Cartesian party line when he felt the force of argument required it. But this was one such occasion.

The first difference between Malebranche and Descartes lay in the fact that the former did not think it could be demonstrated that any material extension existed at all. Perhaps faith might warrant such a conclusion, but it could not be established on philosophical grounds alone.^{Footnote 65} Moreover, even allowing that some kind of extended world had indeed been created, Malebranche was still far from persuaded that it would need to be either infinite or indefinite, even just with respect to its extension alone. He was satisfied that the idea of extension was infinite, both in itself (because he regarded it as consubstantial with an infinitely perfect God) and representatively (insofar as it could represent infinitely many possible bodies). But he wrote in his Méditations chrétiennes et métaphysiques of 1683: “The idea you have of extension represents it to you as divisible, mobile, impenetrable: judge without fear that it has these properties essentially. But do not judge that it should be either immense or eternal. It might not exist at all, or it might have very narrow boundaries.”^{Footnote 66} By saying only that created extension might have narrow boundaries, it seems that Malebranche was not closing the door altogether on Descartes’s theory of an unbounded universe. But it is also abundantly clear that he was far from committed to the truth of such a theory, and still less to its necessary truth.

Some 30 years after making that remark in his Méditations, Malebranche would have occasion to echo it the course of an epistolary discussion of Spinoza’s philosophy with Jean-Jacques Dortous de Mairan. At one point, Mairan happened to argue in a Spinozistic way that extension could not be finite: either it did not exist at all, or it existed as infinite. To this end, he established the following lemma: “To be finite in its kind, it is necessary to be delimited by something of the same kind or the same nature.” And he illustrated this by reference to bodies: “Let there be A, a finite body: it is evident that it is bounded and delimited by all the surrounding bodies B, C, D, etc., which are extended like it or which have extension in common with it, and beyond which it does not extend; and if there were no body, nor anything extended, around A, I could not avoid affirming of body A that it is infinite in its kind; for to be delimited by nothing, not to be delimited, is to be infinite.”^{Footnote 67} The underlying principle here was coming from Spinoza’s Ethics (Part One, Definition 2): but, as we have seen, it was also the principle at the heart of Descartes’s argument for an indefinite universe. For one extended thing to be limited, it would need to be limited by another extended thing.

Malebranche, however, disagreed. In response to Mairan, he wrote:

The contrast between this 1714 letter from Malebranche to Mairan and the 1647 letter from Descartes to Chanut, quoted above, should be clear. Both were using the example of a solitary sphere; but Descartes was arguing that it could not be so solitary after all, because its own extension would be necessarily connected with the actual existence of further extension beyond it, whereas Malebranche was arguing that, on the contrary, it did not need anything else to delimit it. To be defined by a round boundary—and a fortiori to have a boundary at all, that is, to be finite—was simply the way it was. Jean Laporte would draw on Malebranche to argue that Descartes had not been entitled to claim that there was a necessary connection between a finite world supposed spherical, and the real existence of further bodies beyond it. “In taking it for limited,” wrote Laporte, “we are not supposing that it is contained in another thing, but on the contrary that it leaves no space outside itself, either empty or full. So is it limited by nothing? It is limited by the deficiencies of its own substance, which extends as far as it extends and no further.”^{Footnote 69}

What it really comes down to is this: does a mode or modification (such as roundness) belong to one individual body, simply in itself, or does it more properly belong to multiple bodies, to the one that interests us only in relation to others?

Malebranche took the former line. In his opinion, the boundary of an extended thing, together with its particular figure (“since figure is nothing but the boundary of extension”^{Footnote 70}), was in fact identical with the thing itself, existing in a certain way: “the actual roundness and motion of a body are but that body shaped and moved in this or that way.”^{Footnote 71} And so, although a modification/mode could not be perceived without a perception of the substance to which it belonged, it could perfectly well be perceived without a perception of anything else. As early as the opening chapter of Malebranche’s first published work, The Search after Truth, he was already claiming that “a figure is round when all the exterior parts of a body are equally distant from one of its parts called its center, independently of any external body” (emphasis added).^{Footnote 72} And so likewise at the end of his life, we find him telling Mairan that figures and other such modifications of extended things belong to those things, and are inconceivable without them, in contrast to the things themselves, which, together with their modifications, can be conceived without—and can exist without—any other extended things to surround them.^{Footnote 73} What impediment, therefore, could there be to a finite universe, confined within a spherical boundary?

Descartes, however, followed the latter path, and thought that a spherical boundary would need to be demarcated by things on both sides of it; for, strictly speaking, this one mode was shared between those things. It was, to borrow an expression from Paul Hoffman, a “straddling mode.”^{Footnote 74} In a discussion ostensibly unrelated to the size of the universe, instead expounding how he thought Transubstantiation might work, Descartes happened to address the ontological status of a surface. He observed that a surface was not a substance in its own right, but “only a mode or manner of being, which cannot be changed without a change in that in which or through which it exists.”^{Footnote 75} But in or through what did this surface exist? Descartes explained: “This surface intermediate between the air and the bread does not differ in reality from the surface of the bread, or from the surface of the air touching the bread; these three surfaces are in fact a single thing and differ only in relation to our thought.”^{Footnote 76} For some purposes, perhaps when reflecting on the fact that it could be surrounded by things other than air, we might choose to focus our attention on the outer surface of the bread. On other occasions, when reflecting on the fact that the same air might find itself surrounding something other than bread (e.g., the body of Christ), we might consider the inner surface of the air. And sometimes we might focus on the interface as such between the two things. But the object of our thought would be just the same in all three cases, and this one common surface belonged properly neither to the air alone, nor to the bread alone, but to both together.

What holds for surfaces in general will also hold for their shapes, shape being merely “a function of the boundaries of this extension.”^{Footnote 77} When arguing against the possibility of a vacuum, Descartes claimed that, if God was to remove the contents of a vessel without replacement, the sides of the vessel would have to touch. In the course of that discussion, he wrote that, “although there is no connection between a vessel and this or that particular body contained in it, there is a very strong and wholly necessary connection between the concave shape of the vessel and the extension, taken in a general sense, which must be contained in the concave shape.”^{Footnote 78} The similarity between this comment, concerning the necessary connection between the concave inner surface of the vessel and the extension within, and the necessary connection that Descartes described to Chanut between the convex outer surface of a world supposed finite and the extension without, should be clear. Again, a finite world would not need to be surrounded by any particular bodies, but it would need to be surrounded by some bodies or other: for otherwise its spherical boundary—as a mode that should belong equally to the objects on both sides, that being the nature of surfaces in general—would be missing one of its requisite relata.

And this was not specific to surfaces and their shapes. The same point applies, I contend, to all the other modes of extension too: they were all straddling. Descartes did not countenance many such things, but only those that were required by his mechanical physics, so we can go through them one by one.

In the case of external place, for instance, Descartes defined this simply as “the surface immediately surrounding what is in the place.”^{Footnote 79} As he acknowledged, this was effectively the same as the old Aristotelian definition of the place of a thing as “the innermost motionless boundary of what contains it.”^{Footnote 80} But, as we have just seen, Descartes believed that such a boundary would properly belong to what was contained as much as to what contained it. He reiterated the same point he had elsewhere made for surfaces considered more generally, but now with specific application to the role that such surfaces would play as external places. The surface that constituted the place, he wrote, was not “any part of the surrounding body but merely the boundary between the surrounding and surrounded bodies, which is no more than a mode. Or rather what is meant is simply the common surface, which is not a part of one body rather than the other but is always reckoned to be the same, provided it keeps the same size and shape.”^{Footnote 81}

As for external place, so too for motion, that being simply a change of place. For Descartes, the motion of a body would always need to make references to its surroundings. He defined motion as “the transfer of one piece of matter, or one body, from the vicinity of the other bodies which are in immediate contact with it, and which are regarded as being at rest, to the vicinity of other bodies.”^{Footnote 82} But then it followed that motion, our customary ways of describing it notwithstanding, should be entirely reciprocal. Strictly speaking, it was not so much that a moving body would be leaving its stationary surroundings, but rather that the body and its surroundings would both be equally moving in relation to one another. And Descartes was perfectly happy to embrace this conclusion: “For transfer is in itself a reciprocal process: we cannot understand that a body AB is transferred from the vicinity of a body CD without simultaneously understanding that CD is transferred from the vicinity of AB. Exactly the same force and action is needed on both sides.”^{Footnote 83} Or again, in the very next paragraph: “whatever is real and positive in moving bodies—that in virtue of which they are said to move—is also to be found in the other bodies which are contiguous with them, even though these are regarded merely as being at rest.”^{Footnote 84}

Of course, the same point also holds for rest itself. One might think that this would go without saying; but we should remember that Descartes did not regard rest as simply the limiting case of motion (i.e., a motion where the speed happens to be zero), but as a distinct mode in its own right, wholly opposed to it. So, for the record, just as motion was the reciprocal transfer of a body away from its surroundings, rest was the absence of such a transfer.^{Footnote 85} That is to say, the definition of rest for one body would still make an ineliminable reference to other bodies.

The size or quantity of a body might look like an exception here, for perhaps that can, after all, be understood solely in terms of the three-dimensional extension that constitutes the body itself, without any reference to surrounding bodies. But then, strictly speaking, size was not a mode of extension. Descartes acknowledged this in Principles II.8: “There is no real difference between quantity and the extended substance; the difference is merely a conceptual one.”^{Footnote 86} And he elaborated in a letter: whereas shape and motion were true modes of corporeal substance, size—together with existence, duration, number and all universals—were rather to be taken as “attributes, or modes of thinking, because […] the thing itself cannot be outside our thought without its existence, or without its duration or size, and so on.”^{Footnote 87} Far from being a mode of a corporeal substance, the size of a body just was that corporeal substance itself. The two things could be considered apart, but not by a modal distinction—and still less by a real distinction—but only by a distinction of reason.

The same, however, was not the case for internal place. Much as Descartes’s notion of external place had linked it directly to the two-dimensional (i.e., depthless) surface intermediate between the body and its surroundings, his notion of internal place linked that directly to the three-dimensional extension that constituted the body. And yet, even despite this, the notion still managed to involve a reference to the surroundings. Descartes identified the internal place of a body with the body’s own extension, but only insofar as it was being considered generically, rather than as something individual. And how, precisely, was that genus defined? According to Descartes, the internal place of a body will remain one and the same, even if the individual body in question should happen to leave it, just as long as the extension that constitutes whatever comes to replace that body “retains the same size and shape and keeps the same position relative to certain external bodies which we use to determine the space in question” (emphasis added).^{Footnote 88}

In short, every single one of the true modes of corporeal substance—surfaces together with their shapes, motion and rest, and both internal and external place—needs to be referred not only to the individual body to which we might be accustomed to ascribing them, but to other bodies too. Where Malebranche believed that a mode of a thing was just that thing itself, existing in a certain way, independently of all other things, Descartes believed that modes—at any rate, the corporeal ones—were relational in nature. Any property that was not thus relational but inhered in the substance itself (“substantiae inesse”) would, properly speaking, qualify not as a mode thereof but an attribute.^{Footnote 89} And that did include size, but it did not include any of these others.

But Aristotle had long since recognised that his own relational definition of place, as the innermost boundary of what contained a thing, entailed that the cosmos as a whole could not have a place, on the grounds that nothing contained it.^{Footnote 90} Had Descartes followed Aristotle in treating the universe as finite, the same entailment would have held for him too, and not only for place but right across the board. Maybe a single finite body, all alone in the world, could somehow—paradoxically—have a size. But it could not have a place; it could neither move nor be at rest; it could not have a shape; and, most fundamentally of all, it could not have a boundary.

It is this, I contend, that was driving Descartes’s argument for the actual indefiniteness of the universe. Indefiniteness as such might only require the possibility of increase beyond any given point: that is, a syncategorematic sequence of expansions, this potential body beyond that one, and that beyond another, though without any suggestion of anything beyond the whole sequence together. But Descartes went a lot further than this, and made it clear that each of these more and more distant bodies should be not only possible but actual. To recall what he told Chanut, the actual existence of spaces surrounding a supposedly finite world is necessarily connected with the actual existence of that world. We can now see why Descartes was so confident in this, and precisely what he thought the understanding was latching onto here, when we recognised a contradiction in the supposition of a world that was actually only finitely large. To be finite just is to be bounded. But the notion of a boundary to the whole extended world will be self-defeating, because the very act of supposing such a boundary will reveal further extension beyond it, actually existing to provide the second relatum that every boundary requires.

Abbreviations

AT

Descartes, René. 1996. Oeuvres de Descartes, edited by Charles Adam and Paul Tannery. 11 vols. Paris: J. Vrin.

CSM

Descartes, René. 1984–1985. The Philosophical Writings of Descartes, translated by John Cottingham, Robert Stoothoff, Dugald Murdoch. 2 vols. Cambridge: Cambridge University Press.

CSMK

Descartes, René. 1991. The Philosophical Writings of Descartes, vol. III: The Correspondence, translated by John Cottingham, Robert Stoothoff, Dugald Murdoch, Anthony Kenny. Cambridge: Cambridge University Press.