Descartes and Newton on the (In)Divisibility of Space



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Descartes and Newton on the (In)Divisibility of Space
Both Newton and Descartes postulate the existence of an infinitely extended entity and, although the respective entities they posit differ in dramatic ways, they both argue for its infinitude in the same way. For Descartes, what is infinitely extended is the plenum – the fusion of all the material bodies that exist:

What is more we recognize that this world, that is, the whole universe of corporeal substance, has no limits to its extension. For no matter where we imagine the boundaries to be, there are always some indefinitely extended spaces beyond them, which we not only imagine but also perceive to be imaginable in a true fashion, that is, real. (p. 232)

Newton argues in an extraordinarily similar vein about his absolute space: “Space is extended infinitely in all directions. For, we cannot imagine any limit anywhere without at the same time imagining that there is space beyond it” (p. 23). Notice that both philosophers are arguing that given the existence of some part of the infinitely extended thing the remainder is guaranteed to exist. In other words, there is no way to separate out one part of the extended entity from the rest. This seems to suggest that both Descartes and Newton would view the entire extended thing as a metaphysical atom – that is, a thing which cannot be divided even by God. Surprisingly, such a view is only half right.

Here is what Descartes claims in the section just prior to the one quoted above: “We also know that it is impossible that there should exist atoms, that is, pieces of matter that are by their very nature indivisible” (p. 231). Descartes appears to be denying that there are indivisible material things. Since the whole of extended substance is surely a material thing, he must therefore be denying that it is indivisible. Newton on the other hand seems to welcome the view I suggested earlier. “Moreover, lest anyone should for this reason imagine God to be like a body, extended and made of divisible parts, it should be known that spaces themselves are not actually divisible…” (p. 26). In his famous correspondence with Leibniz, Newton’s mouthpiece Samuel Clarke speaks even more directly of the indivisibility of space: “For infinite space is one, absolutely and essentially indivisible, and to suppose it parted is a contradiction in terms…” (p. 19). How can both Newton and Descartes argue for the infinitude of their respective extended entities and yet disagree so completely when it comes to the entity’s indivisibility?

First and foremost, we need to get a handle on the relevant notion of indivisibility. We can find a useful analysis in Descartes’s Principles. As such, the first section deals with Descartes’s plenum and how Descartes can argue that it does not count as a metaphysical atom. In the second section I turn to an examination of Newton’s view and how he argues that his space is a metaphysical atom. I argue that Newton’s view is a reasonable one. And, this implies that one might accept the claim that all extended things have parts and avoid the inference from that claim to the claim that all extended things are metaphysically divisible.
1. Descartes’s Divisible Plenum

Although I investigate Descartes’s view in this section, it will be instructive to turn to a remarkable exchange in the Leibniz-Clarke correspondence. The culmination of a debate about God’s relation to space is found in Leibniz’s fifth letter:

I objected that space cannot be in God because it has parts. Hereupon the author seeks another subterfuge by departing from the received sense of words, maintaining that space has no parts because its parts are not separable and cannot be removed from one another by being plucked out. But it is sufficient that space has parts, whether those parts are separable or not… (pp. 48-9)

Surely Leibniz is exactly right. The question of separability is different, though possibly closely related, to the question of complexity. Despite Clarke’s protests to the contrary, surely Newton and Clarke hold that space has parts; what they deny is that those parts are separable from one another. Being separable is the notion of divisibility that I want to investigate, and the notion that Descartes must think fails to apply to his plenum in order to claim that it is not a metaphysical atom.

First, let me establish that the notion of divisibility in play in Descartes is more than merely the claim that a particular thing has proper parts.

Even if we imagine that God has chosen to bring it about that some particle of matter is incapable of being divided into smaller particles, it will still not be correct, strictly speaking, to call this particle indivisible. For, by making it indivisible by any of his creatures, God certainly could not thereby take away his own power of dividing it, since it is quite impossible for him to diminish his own power… (pp. 231-2)

If the notion of indivisibility that Descartes was speaking about merely amounted to the claim that a body lacked parts, it is very difficult to imagine the state of affairs he is suggesting that God might bring about. Specifically, Descartes would be suggesting that God could make it such that all created beings could not think of a certain particle as having distinct parts. But surely this is not the state of affairs Descartes is imagining God actualizing. Indeed, Descartes must be thinking of God making a certain particle physically indivisible but then claiming that this would not amount to true or metaphysical indivisibility. Hence, the divisibility that Descartes is thinking of is the separating of some proper parts of a corporeal body, not merely the existence of the proper parts.

But, this invoking of the powers of God brings us back to our entry into the discussion. Descartes holds that given the existence of one finite corporeal substance, we are ensured that there are distinct corporeal substances that together with the hypothesized corporeal substance make up an infinitely large plenum. If this is the case, not even God could make only one finite corporeal substance. So, should we then infer that the material plenum is indivisible? Have we found an inconsistency in Descartes? I believe that there are multiple reasons to think that the answer to these questions is no. First, it is useful to reflect on Descartes’s discussion of the requirements that are necessary for a real distinction between substances.

We can perceive that two substances are really distinct simply from the fact that we can clearly and distinctly understand one apart from the other... and things which God has the power to separate, or to keep in being separately, are really distinct. (p. 213).

There is something of an ambiguity here. The power that God is claimed to have over distinct substances might be read in two different ways:



Ontological Independence: Two things x and y are really distinct iff x can exist without y and vice versa.

Mere Separability: Two things x and y are really distinct iff x and y can exist separated from one another.

I suggest that this discussion of real distinction relates to the problem at hand because the question of material divisibility reduces to the question of whether the proper parts of a material thing are really distinct. Recall the last bit of the quotation from Descartes’s discussion of the impossibility of atoms: “…it is quite impossible for [God] to diminish his own power…” (p. 232). The power of divisibility is the power to keep the parts “in being separately”.

So, to return to the central issue, in order for Descartes to claim that the plenum is not a metaphysical atom1 he must show that the parts of the plenum are really distinct. That is the parts of the plenum must meet either the ontological independence criterion or must meet the mere separability criterion. I actually think that the parts of the plenum meet both criteria, but in distinct ways.

First, consider the ontological independence criterion. As the passage at the very beginning of this chapter explicitly claims, any finite part of the plenum is not a completely ontologically independent substance; that is, there must be infinitely many other finite corporeal bodies if one exists. So, we might conclude that the parts of the plenum are not ontologically independent and hence not divisible. I think we should resist this temptation. Notice that any two finite corporeal substances could exist without the other. As far as I can tell, Descartes has no qualms about the annihilation of any particular finite corporeal substance. When discussing the possibility of a vacuum, Descartes discusses what would happen if God were to annihilate a particular body.

Hence, if someone asks what would happen if God were to take away every single body contained in a vessel, without allowing any other body to take the place of what had been removed, the answer must be that the sides of the vessel would, in that case, have to be in contact. (p. 231).

As far as Descartes is concerned, the body inside the vessel could be annihilated by God. It is just that the sides of the vessel would come to be in contact if the intervening substance was annihilated. So, any finite corporeal substance is really distinct from any other. Is this enough to establish that the plenum is divisible? Well, call some finite corporeal substance x, and the plenum minus x, y. Now it is clear that y is ontologically independent from x. x could be annihilated without y’s ceasing to exist – hence, y does not depend for its existence on x. Some parts of y that were not in contact prior to the annihilation of x will be in contact post annihilation, but nothing else is required to change; indeed, y is still infinitely extended so there is no need to postulate any further corporeal substances that are not proper parts of y. But, could x exist without y? I think x can exist without y, but this is more controversial. The question we need to answer is whether y is mereologically fragile or not. If y is mereologically fragile, then x can indeed exist without y. Indeed, just consider some other finite corporeal substance z that does not overlap x (that is, has no proper parts in common with x). By the same reasoning as above, z could be annihilated. But, if y is mereologically fragile, then once z is annihilated, y ceases to exist. Instead, a proper part of y exists. But, this suggests that x could exist without y existing. Another way to think about this issue is as follows: Descartes claims that x cannot exist without infinitely many other finite corporeal substances. But, surely this does not imply that x cannot exist without the particular finite substances that do in fact exist – those finite substances might be replaced by other finite substances. If this is correct, the parts of the plenum are ontologically independent and hence the plenum is divisible.

But, is y mereologically fragile? Can y survive the loss of some of its parts; indeed, given the latter argument, can y survive the loss of all of its parts? Since this is an historical project, the question bifurcates. On the one hand we might ask if y is actually mereologically fragile. On the other, we should ask if Descartes thinks that y is mereologically fragile. In answer to the first question, I have very strong temptations to claim that y is mereologically fragile. y seems to me to be a mere aggregate, and as an aggregate would therefore be mereologically fragile. The more relevant question though, is the second. Remember that we are asking how Descartes can claim that the plenum is not a metaphysical atom. So, does Descartes think that the plenum minus a finite part is mereologically fragile? This question is not at all easy to answer. We are here asking what the individuation conditions for corporeal substances are. There are places where Descartes explicitly claims that the human body, for instance, can undergo a change of many parts and yet remain the same.2 Of course, this doesn’t settle matters as one might well claim that Descartes thinks that it is the same human but not the same body. In the end I just do not think that it is clear whether or not Descartes would think that y is mereologically fragile or not.

[Since this is a work in progress, there is another consideration that I think speaks in favor of the ontological independence of a finite portion of the plenum and the entire plenum. Specifically, in terms of contemporary metaphysics, I see no problem with the finite portion expanding in size. Indeed, I think it possible that a finite piece of gunk (atomless matter) might expand to infinite proportion. Whether or not that view is consistent with Descartes I do not know.]

We can avoid all of these controversies. If we turn to the mere separability criterion, we can establish that the parts of the plenum are distinct substances and hence that the plenum is indivisible without making any assumptions about the identity conditions of the infinite material complement of any particular body. Again, a cursory analysis of the criterion might seem to tell against the divisibility of the plenum. Consider some material body x, and the material body identical to the plenum minus x, y. We are assuming for the sake of argument that x and y do not meet the ontological independence criterion. But, if this is the case, one might think that x and y are not separable. Given Descartes’s view that a vacuum is impossible, it does not look like x could cease to be contiguous with y. Indeed, since we are not assuming anything about the identity conditions of y, suppose that any bodies distinct from x that are posited to exist are parts of y. Surely this suggests that x could not be separated from y and hence that the plenum is indivisible.

I think that the above reasoning depends upon a flawed view of what is required for x to be separated from y. One way for x to be separated from y is for x to be located some distance away from y. We are supposing this to be impossible. Another way of x being separated from y is for the parts of y to change their distance relations to x. And this can certainly happen. All we need is for x to move relative to parts of y which is surely possible. Consider Descartes’s discussion of matter and motion: “If the division into parts occurs simply in our thought, there is no resulting change; any variation in matter or diversity in its many forms depends on motion.” (p. 232). I think the general idea is something like the following. Imagine some big homogeneous corporeal substance. When considering such an entity one might very well be tempted to think that the corporeal substance has no parts,3 or that the parts we attribute to the substance are merely the work of our minds conceptually carving up the world but not grabbing any real metaphysical distinction. But, now imagine that in that homogeneous substance, a proper part of it is moving with respect to all the other parts of the substance (perhaps by spinning). In such a case, I am pulled to say that the motion of the proper part with respect to the whole separates it from the rest of the homogeneous substance. So, I think that such relative motion is a mark of something being distinct from another. The proper parts of Descartes’s plenum certainly move with respect to one another. Hence, his plenum does not count as a metaphysical atom.

We have seen that there are two distinct ways in which to argue that Descartes’s plenum is divisible. We might either claim that the proper parts of the plenum can exist without one another, or we might point to the fact that the parts of the plenum can move with respect to one another. The parts of Newtonian space on the other hand, do not meet either of these criteria for divisibility. Hence, as I argue below, even though there are strong affinities in the ways that Newton and Descartes assert that their respective infinite substances are infinite in extent, Descartes’s plenum is not a metaphysically indivisible atom while Newtonian space is.
2. Newton’s Indivisible Space

In Isaac Newton's most philosophical work, De Gravitatione (date unknown), his main target for criticism is Descartes. Indeed, the lion's share of the text is occupied with refuting the view of motion Descartes sets out in his Principles of Philosophy:

...motion is 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. (p. 233)

Surprisingly, in arguing against this view of motion, Newton ends up by asserting the existence of (i) a space that is distinct from matter and (ii) an extended thing that is indivisible. As we saw above, Descartes holds that all extended things are divisible including his infinitely extended plenum. Descartes argues that no matter how small (or big!) something is, if it is extended it has distinct proper parts. But, he then concludes that those parts could exist separate from one another – either because the one could exist without the other or the one could move with respect to the other. In contrast to this reasoning, Newton appears to hold that the parts of space are inseparable from one another. In contrast to Descartes’s plenum, Newton claims that the parts of space cannot move with respect to one another. But the most natural way to understand how he argues for the immobility of space will also guarantee the inseparability of the parts of space. Hence, the parts of Newtonian space meet neither the ontological independence criterion nor the mere seperability criterion. Therefore, Newtonian space is an atom – an extended indivisible thing.


2.1. Newton’s Two Theses

As stated above, Newton’s main target for criticism in his De Gravitatione is Descartes’s principles of motion. Famously, Descartes equates space and matter. As such, according to Descartes, the motion of a particular body is analyzed in terms of the distinct bodies that immediately surround the body in question. Newton argues that such a view of motion is untenable. Although Newton gives many different criticisms, most importantly for our purposes here, Newton argues that bodies cannot have determinate motions on Descartes's view. Newton assumes that for the motion of a body to be determinate one must be able to identify the place from which the body started to move after it has completed some part of the motion. But, Newton claims, such identification is impossible for Descartes because the place that the body previously occupied, being determined only by the bodies that surrounded it initially, no longer exists (as the bodies that initially surrounded the body in question have themselves changed position). Since the place where the motion began no longer exists and it is necessary to identify that place in order for a body to have a determinate motion, Newton concludes that it is impossible for bodies to have determinate motions. In order to avoid this difficulty, Newton postulates the existence of a space that is distinct from bodies. Bodies move by occupying different parts of space at different times.

Since Newton's primary complaint against Descartes is the latter's seeming commitment to the continual flux of locations, Newton is motivated to ensure that, on his theory of motion, locations are not the sorts of things that change with respect to one another. In explicating his notion of space, Newton discusses the immobility of the parts of space:

Moreover, the immobility of space will be best exemplified by duration. For just as the parts of duration are individuated by their order, so that (for example) if yesterday could change places with today and become the later of the two, it would lose its individuality and would no longer be yesterday but today; so the parts of space are individuated by their positions, so that if any two could change their positions, they would change their individuality at the same time and each would be converted numerically into the other. The parts of duration and space are understood to be the same as they really are only because of their mutual order and position; nor do they have any principle of individuation apart from that order and position, which consequently cannot be altered. (p. 25)

This theme is repeated in the Principia, Newton's masterpiece:

Just as the order of the parts of time is unchangeable, so, too, is the order of the parts of space. Let the parts of space move from their places, and they will move (so to speak) from themselves. For times and spaces are, as it were, the places of themselves and of all things. All things are placed in time with reference to order of succession and in space with reference to order of position. It is of the essence of spaces to be places, and for primary places to move is absurd. (p. 66)

Neither of these passages is entirely obvious. It is clear that in both passages Newton concludes that the parts of space cannot move; but, what is the argument? In the De Gravitatione passage, Newton seems to think that the way that the parts of space are individuated ensures that the parts of space are immobile. Our first job then is to determine what Newton means by a principle of individuation. Newton claims that the parts of space are individuated by their position. At the very least, this means that position is an essential property of each part of space.

Essentiality: (x)(Sx → Px)

That is, for all x, if x is a part of space, then x has a position. But, merely having a position will hardly distinguish one part of space from another. Instead, each part of space must have a unique position.



Uniqueness: (x)(y){(Sx & Sy) → [(P1x & P1y) → (x=y)]}

In other words, for any two things x and y, if x and y are parts of space, then if x and y both have some particular position P1, then x and y are identical. However, notice that uniqueness is not nearly a strong enough condition for Newton's purposes. Indeed, in the De Gravitatione passage Newton claims that if two parts of space were to swap positions they would change their individuality. Our uniqueness condition above only ensures that at most one part of space has a particular position at any given time. Newton makes the much stronger claim that the identity of the part of space is determined by its position. This suggests Newton’s individuation requirement:



Individuation: (x)(y){(Sx & Syx) → [(P1x & P1y) → (x=y)]}

In other words, for any two things x and y, if x and y are parts of space, then if x has some position P1 and possibly y has position P1, then x and y are identical. Only one part of space can have any particular position because parts of space are the parts of space they are in virtue of having a particular position.

But, what is the nature of the positions that the parts of space have? “...positions, properly speaking, do not have quantity and are not so much places as attributes of places.” (p. 65). As we saw in the Principia passage, the parts of space are themselves places. Hence, positions are attributes of parts of space. But, what type of attributes are positions? The way that Newton speaks of positions, it looks as though for a thing x to have some position, there must be some distinct other thing y it has that position in relation to. For instance:

But since the parts of space cannot be seen and cannot be distinguished from one another by our senses, we use sensible measures in their stead. For we define all places on the basis of the positions and distances of things from some body that we regard as immovable, and then we reckon all motions with respect to these places, insofar as we conceive of bodies as being changed in position with respect to them. (p. 66)

And also:

Moreover, the only places that are unmoving are those that all keep given positions in relation to one another from infinity to infinity and therefore always remain immovable and constitute the space that I call immovable. (p. 67)

There are a couple of things to learn from these passages. In the second passage notice that Newton explicitly claims that some places keep certain positions in relation to one another for all times, and space is constituted by such places. That is, parts of space are places and have positions but are not identical to their positions. The more interesting question is how to understand the position of those places. There are, I think, two options to take. Either (1) places have certain positions essentially and in virtue of these positions their relations to one another do not change, or (2) places are essentially related to one another in a particular way and as the positions of the places just are their relations to one another, the positions of the places do not change. I think that Newton holds the latter of these two views for a couple of reasons. First, as stated above, in both of these passages Newton talks of position as a relation between one thing and another. Look especially at the second passage where Newton speaks of the positions of regions of space. They are “positions in relation to one another.” Indeed, had Newton held position (1) above he ought to have just made the more straightforward claim that places keep their positions and hence are immovable. Second, how do positions determine the relations between things? Say that one place is at position x and another is at position y; why does that make them a particular distance d away from one another? On the later view this question does not arise. To have position x just is to be d away from position y. Finally, if we look back to the initial passages from the Principia and De Gravitatione, we see that Newton illuminates the case of the positions of places by reference to the order of times. But, notice that order is a relational property. If Newton thought that positions were not relational, it is curious why he would use his notion of duration to explicate his notion of space.

Hence, we might attribute the following sort of view to Newton: Bodies occupy parts of space – the parts of space are the places that all things occupy. Bodies have absolute positions in virtue of the positions of the parts of space they occupy. The parts of space themselves have positions and these positions individuate the parts of space. In other words, every part of space has a position that is both essential and necessarily unique to it. Finally, the position of a part of space is some relation between it and distinct parts of space. So, parts of space are individuated by their relations to other parts of space. Of course, this means that the individuation conditions of the parts of space are extrinsic. Hence we should attribute something like the following to Newton:


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