Leibniz argues against Descartes’s conception of material substance based on considerations of unity. I examine a key premise of Leibniz’s argument, what I call the Plurality Thesis—the claim that matter (i.e., extension alone) is a plurality of parts. More specifically, I engage an objection to the Plurality Thesis stemming from what I call Material Monism—the claim that the physical world is a single material substance. I argue that Leibniz can productively engage this objection based on his view that matter is discrete. The discreteness of matter provides two aspects of support for the Plurality Thesis. First, it indicates that the parts of matter do not share boundaries and are, therefore, independent in an important sense. Second, it indicates that the parts of matter are determinate and are, therefore, ontologically prior to the wholes they compose.
Leibniz argues against the Cartesian conception of material substance, that is, the view that matter consists in extension alone, based on considerations of unity. Leibniz’s Argument from Unity, as I will call it, can be rendered as follows:
Matter (i.e., extension alone) is a plurality of parts to infinity. A plurality of parts cannot be a true unity. A substance must be a true unity.
Therefore,
Matter (i.e., extension alone) is not a substance.
Leibniz’s use of the term “matter” in this argument refers both to
This argument, or, at least, nearly-identical arguments, can be found in Leibniz’s texts from as early as the 1670s until as late as 1714; it is a staple of his metaphysical program.
Versions of monism have been around since Parmenides and continue to pop up in present-day discussions.
The picture of the physical world that Material Monism provides can be described as a
This is all to say that a great deal of Leibniz’s theory of substance depends on his ability to defend his position against challenges of this type. Although Leibniz himself does not encounter the challenge stemming from Material Monism until late in his life, his metaphysical system has the resources to engage it productively. As I will argue, Leibniz’s view that matter is (and must be) discrete provides support for the bottom-up conception of the physical world needed to get his Argument from Unity off the ground.
Discreteness is a structural feature of matter: it concerns the relationships between the parts of a material thing. There are two important features of discreteness that I will develop. First, to say that matter is discrete means that the parts of a material thing do not share boundaries with one another, that each part has its own distinct boundary (even though there is no empty space between them), and that each part is itself made up of parts with their own boundaries. This supports the idea that the parts of matter are
In Section
Intuitively, something is discrete if it is broken apart or choppy, as opposed to smooth or without gaps, which would make it continuous. This intuitive rendering can be seen in the traditional distinction between discrete and continuous quantity, which is Aristotelian in origin. I will begin by showing that Leibniz understands the distinction between continuous and discrete in Aristotelian terms, that is, in terms of the parts of a thing sharing or not sharing boundaries, and that this distinction shapes Leibniz’s conception of the nature of materiality.
According to Aristotle, the distinction between discrete and continuous is an exhaustive and mutually exclusive distinction within the category of quantity. It applies to wholes, which is to say, anything with parts or capable of having parts, and sorts them according to the relations that hold between their parts—in particular, the structural relations (or perhaps, more anachronistically, the topological relations). Whether something is continuous or discrete is a function of whether or not its parts
Aristotle presents the distinction as follows:
Of quantities some are discrete, others continuous. … Discrete are number and language; continuous are lines, surfaces, bodies, and also, besides these, time and place. For the parts of a number have no common boundary at which they join together. For example, if five is a part of ten the two fives do not join together at any common boundary but are separate; nor do the three and seven join together at any common boundary. Nor could you ever in the case of a number find a common boundary of its parts, but they are always separate. Hence number is one of the discrete quantities. A line on the other hand, is a continuous quantity. For it is possible to find a common boundary at which its parts join together, a point. And for a surface, a line. For the parts of a plane join together at some common boundary. Similarly in the case of a body one could find a common boundary—a line or a surface—at which the parts of the body join together. (
From this text, we can extract basic characterizations of continuity and discreteness:
In an early letter to Jacob Thomasius, Leibniz develops an account of the discreteness of matter that relies on an explicitly Aristotelian understanding of discreteness. Although many of the views Leibniz expresses in this letter are aspects of a rather grand reconciliation project between Aristotelian philosophy and mechanistic natural philosophy, a project that Leibniz does not continue to develop in these terms, many of the views that Leibniz expresses about matter in this letter persist.
The task of Leibniz’s stated reconciliation project is to explain Aristotle’s principles in mechanical terms, that is, by appeal to only magnitude, figure, and motion.
One especially difficult part of Leibniz’s stated reconciliation project is accounting for Aristotelian forms in merely mechanical terms. Leibniz’s approach to this problem is simply to reduce forms to figures or shapes. But for there to be figures or shapes in matter, Leibniz argues, matter must have boundaries. He writes,
since figure is the boundary [
This passage contains important claims about the relationship between boundaries and discontinuity. Discontinuity, in particular “a discontinuity of parts”— just is the presence of separate boundaries—
The most important part of this early letter concerns Leibniz’s account of how discontinuity is introduced into primary matter:
discontinuity can be introduced into the formerly continuous mass in two ways—first, in such a way that contiguity is at the same time destroyed, when the parts are so pulled apart from each other that a vacuum is left; or in such a way that contiguity remains. This happens when the parts are left together but moved in different directions. For example, two spheres, one included in the other, can be moved in different directions and yet remain contiguous, though they cease to be continuous. (A 2.1, 27 = L 96)
Leibniz identifies two ways that discontinuity can be introduced, both of which involve motion.
The conclusion, then, of Leibniz’s attempt to reduce forms to figures is as follows: “division comes from motion, the bounding [
Although the stated reconciliation project does not figure among Leibniz’s lasting commitments, the connection between motion and division, and between motion and bounded parts is a persistent feature of Leibniz’s account of matter.
While Leibniz’s conception of discreteness initially appears in the context of his physics and is, as I have described it, a structural feature of matter, as his thought develops it takes a decidedly metaphysical turn. Leibniz’s views about matter become closely connected to his engagement with questions related to the composition of the continuum, which Leibniz himself characterizes as one of the two great labyrinths of human reason.
Terms Indicating Discreteness
Terms Indicating Continuity
Bounded Parts
Unbounded Parts
Assignable Parts
Unassignable Parts
Determinate Parts
Indeterminate Parts
Distinguished Parts
Actual Parts
Potential Parts
This table provides pairs of terms that Leibniz uses to contrast the parts of continuous quantities with the parts of discrete quantities.
In some notes from 1676, Leibniz again asserts the discreteness of matter. In the following passage, Leibniz’s tone is tentative, and he flirts with the conclusion that, since matter is discrete, it is ultimately composed of points:
Matter alone can be explained by a plurality without continuity. And matter seems in fact to be a discrete being. For though it is assumed to be solid, matter taken without a cement, through the motion of another body, for example, will be reduced to a state of liquidity or divisibility. Hence it follows that it is composed of points. This I prove as follows: every perfect liquid is composed of points, because it can be dissolved into points, namely, by the motion of a solid within it. Matter therefore is discrete being, not continuous. It is merely contiguous and is united by motion or by some mind. (A 6.3, 473 = L 158)
Note, in particular, the connection Leibniz suggests in the opening of this passage between plurality and discreteness. Matter is a plurality without continuity; matter is discrete. Furthermore, the sense in which matter is a plurality is made very explicit: matter is a collection of points.
An important shift takes place later in the same year, a shift that introduces the second main component of discreteness noted above: the priority of material parts to material wholes. If the discreteness of matter is conceived as a resolution into points, it may follow that the parts of matter do not share boundaries (points, of course, are incapable of overlap), but it is difficult to see how points can be prior to wholes. After all, as Leibniz understands points, they are themselves mere boundaries or extrema of material things.
Later in 1676, Leibniz develops a model for discrete matter that does not entail its resolution into points. In the dialogue It is just as if we suppose a tunic to be scored with folds multiplied to infinity in such a way that there is no fold so small that it is not subdivided by a new fold: and yet in this way no point in the tunic will be assignable [
The same relationship between motion and division is present here as in the Thomasius letter. More importantly, though, Leibniz formulates his commitment to the actually infinite division of matter in a way that avoids the conclusion that matter is composed of points. He claims that “there is no fold so small that it is not subdivided by a new fold”. Consequently, every part of the tunic is assignable—
The tunic model is paired with another important result that Leibniz reaches in the 1670s concerning the different part-whole priority relations that obtain in continuous versus discrete wholes. Here, the discussion appears in a slightly different context (i.e., not primarily a discussion of material structure):
There can be no such thing as a fastest motion, nor a greatest number. For number is something discrete, where the whole is not prior to the parts, but the converse. There cannot be a fastest motion, since motion is a modification, and is the translation of a certain thing in a certain time—in short, just as there cannot be a greatest shape. There cannot be one motion of the whole; but there can be a kind of thought of everything. Whenever the whole is prior to the parts, then it is a maximum, as for example in space and in the continuum. If matter, like shape, is that which makes a modification, then it seems that there is not a whole of matter, either. (A 6.4, 520 = Ar 121)
Leibniz claims that in discrete things the parts are prior to the whole, while in continuous things it is the reverse. This means that the parts of matter (since matter is discrete) have to be, in some sense, independent of the wholes they make up and in fact prior to them. If the parts of matter were points, as Leibniz had previously argued, then the parts could not be prior to the whole because points are, as Leibniz explicitly says in
In a May 1702 study, given the title “On Body and Force, Against the Cartesians”, Leibniz uses the term “distinguished” to express the same commitment as expressed by the terms “assignable” and “bounded”. every repetition (or collection [
The contrast here is between parts that are distinguished—
The emerging picture is further filled in in a 31 October 1705 letter to Princess Sophie. In this case, Leibniz highlights the fact that the parts of matter are we must say that space is not at all composed of points, nor time of instants, nor mathematical motion of moments, nor intensity of extreme degrees. That is, that matter, that the course of things, that finally all actual composites, is a discrete quantity, but that space, time, mathematical motion, intensity or the continual increase one conceives in speed and in other qualities, and finally all that which gives an estimate which ranges over possibilities, is a continuous quantity which is indeterminate [
Here the contrast between continuous and discrete quantity is elaborated in the context of distinguishing between actual and ideal things (here, “possibilities”). The reason Leibniz gives for attributing discrete quantity to actual things is that “the mass of bodies is actually divided in a determinate manner”. In contrast, an ideal thing, which is a continuous quantity, “represents nothing but an indeterminate possibility of dividing it however one likes”. So the contrast is between determinate parts on the one hand and indeterminate parts on the other.
A mere twenty days earlier, Leibniz has formulated a similar distinction in a draft letter to De Volder. The material in this draft can help to clarify what Leibniz means by determinate and indeterminate parts. He writes,
In fact, matter is not continuous but discrete and actually divided to infinity, even if no assignable part of space is devoid of matter. Yet space, like time, is not something substantial, but something ideal, and consists in possibilities, i.e. the order of possible coexistents at any given time. And so, there are no divisions in it, except those that the mind makes, and the part is posterior to the whole. (GP II, 278 = Lo 327)
According to Leibniz, a continuum can be divided in infinitely many ways. Take, for example, a unit line segment. This can be divided into two half-unit segments; or into four quarter-unit segments; or into one half-unit segment and two quarter-unit segments; or into three third-units; and so on, indefinitely. Leibniz’s point is that there is nothing about the unit line segment (and by extension any continuous thing) that dictates how it is to be divided—that is up to some observer to determine. Not so with discrete things. In discrete things everything is actually divided in a determinate manner. In other words, at any time there is an already-given structure of divisions, a way in which the parts are already assigned.
The contrast between indeterminate and determinate parts, then, has to do with whether or not there is an already-given structure of divisions or not. If there is, then what we have is a discrete whole, in which an observer can merely perceive the divisions that are already there. If there is no such structure, then what we have is a continuous whole, in which an observer can “create” divisions (at least,
This idea is drawn out clearly in a very well-known passage from a 19 January 1706 letter to De Volder. In this passage, Leibniz draws a distinction between actual and ideal things:
In actual things there is nothing but a discrete quantity, namely the multitude of monads, i.e., simple substances, which in any sensible aggregate, i.e. any aggregate corresponding to the phenomena, is, indeed, greater than any number however large. But continuous quantity is something ideal that pertains to possible things and to actual things in so far as they are possible things. Of course, the continuum involves indeterminate [
This text represents the culmination of Leibniz’s thought over the preceding decades. Actual and ideal are correlated with discrete and continuous, respectively. Actual things (i.e., discrete things) are exhaustively divided into parts in a determinate way. There are parts already given, and prior to any wholes composed of them. Consequently, actual things (in this sense) are always aggregates—that is, collections. They only exist insofar as a multitude of actual parts exists. Ideal things (i.e., continuous things), by contrast, are not divided into parts and, as such, are indefinite in a certain respect. There are no parts already given, and the whole is prior to any parts that are subsequently divided out (division here being division
The central point for the present investigation is this: continuity involves indeterminacy, while discreteness does not. This indeterminacy presents a problem for concrete existence, on Leibniz’s view, because if the parts of a continuous entity are not already specified, if there is not already some built-in structure of divisions, then we are forced to conclude that the continuous entity is composed from points. But this leads to paradox, as Leibniz’s sustained engagement with the Problem of the Composition of the Continuum makes clear.
Leibniz is, therefore, committed to the following general claims: (1) for matter to actually exist is for it to be discrete, (2) to be discrete is to be a plurality, and (3) to be discrete is to have parts that are prior to the wholes they compose. To be sure, the metaphysical commitments in the background of this passage are very different from those operating in Leibniz’s letter to Thomasius. But given the trajectory that I have followed from 1669, I think that the connection is clear: the physically bounded parts of the Thomasius letter are consonant with the metaphysically actual parts of the letter to De Volder; the latter is just a development of the former.
The developmental story just recited can be shown to support the two aspects of discreteness needed to support the Plurality Thesis. In this section, I will address the following two sets of questions. First, how does the discreteness of matter establish that matter is a plurality? In other words, how does the claim that material things have either bounded, assignable, distinguished, determinate, or actual parts give Leibniz license to conclude that material things are, properly speaking, pluralities? And second, how does the discreteness of matter support a bottom-up conception of the physical world? In other words, how does the claim that material things have bounded parts allow Leibniz to conclude that the parts of material things pre-exist the wholes they compose? I will address these questions in order.
Aside from the developmental story presented above, there are conceptual or philosophical reasons that discreteness and plurality are connected. Based on the account of matter’s discreteness developed above, it is apparent that the following claims from the Thomasius letter are persistent features of Leibniz’s metaphysics of matter in some form or other:
To more clearly see the connection between discreteness and the Plurality Thesis, more subtle versions of these claims are needed. In some sense, what I offer here is an important correction to Leibniz’s own articulation of these commitments.
In the early letter to Thomasius, Leibniz outlined a temporally ordered process in which
The fact that matter is always already subdivided slightly changes the meanings of
Consider the example of the two spheres that Leibniz presented to Thomasius.
It follows from the understanding of discreteness outlined so far that discreteness grounds a certain type of independence among the parts of material things and any wholes they might compose. Call the type of independence “independence with respect to motion”:
As I have suggested, though not yet in these particular words, having distinct boundaries is a necessary condition of independence with respect to motion. That is, unless
Thus, as I see it, the discreteness of matter undergirds the Plurality Thesis by supporting the contention that matter is a plurality, but also by specifying the sense in which this is so. Now what about the connection between the discreteness of matter and the bottom-up conception of the physical world required by the Plurality Thesis? To address this question, I will turn to an objection to Leibniz’s Argument from Unity raised by Burcher de Volder. De Volder objects to the Plurality Thesis by suggesting an alternative model of the physical world, one on which the world is a single, material substance. On De Volder’s view the “parts” of matter are merely modes of the single, extended substance, not distinct things in their own right.
While granting Leibniz the point that material parts are independent with respect to motion, De Volder presents the following objection:
For if there is indeed no empty space, as you submit, it will not be possible for one part, which anyone might imagine for themselves, to be conceived without the others. From this it seems to follow that there is no real distinction [
De Volder is arguing from a global dependence among material parts to the conclusion that the distinction between them is not strong enough to establish the type of plurality that Leibniz wants.
Though I cannot fully engage De Volder’s objection and Leibniz’s attempts to reply here, I want to develop one key idea as it relates to the discussion of discreteness above. The viability of De Volder’s objection hinges on the possibility that a material whole (namely, the entire material world) can be prior to its parts, that is, that a top-down conception of the physical world is viable. As I have developed it above, Leibniz’s characterization of matter as discrete explicitly rejects this possibility. To exist and be material requires, on Leibniz’s analysis, that material parts are prior to material wholes. Why is this? If things were otherwise, then material things would be indeterminate, their structures would not be entirely specified or determined at any given time. But this, according to Leibniz, is inconsistent with concrete material existence. Thus, even though material parts are not absolutely fundamental, they do enjoy a relative fundamentality when considered in relation to wholes they compose. In fact, they must, if the material world is to meet a crucial requirement of existence: to exist is to be
Consideration of De Volder’s objection shows, therefore, that both aspects of discreteness are crucial to its role in undergirding the Plurality Thesis: matter must have parts that are independent of one another (i.e., that have separate boundaries), and matter must have parts that are prior to the wholes they compose. If either of these aspects of discreteness is neglected, it cannot play the role it needs to in providing support for the Plurality Thesis, and, in turn, cannot play the role it needs to in Leibniz’s otherwise powerful Argument from Unity.
One objection that might arise to the account I have provided so far is that I take one controversial claim, that is, the Plurality Thesis, and rest it squarely on another (perhaps more) controversial claim, what I will call the
There are two sets of questions pertaining to Leibniz’s commitment to the Determinateness Thesis that I will discuss here. First, in what sense must existing things be determinate for Leibniz? What are Leibniz’s reasons for thinking that to exist is to be determinate in this sense? And second, is this commitment consistent with other views Leibniz holds? In particular, is the Determinateness Thesis consistent with Leibniz’s analysis of bodies, including the claim that bodies have no precise shapes? I will claim that the sense in which existing things must be determinate is that they must have
The sense in which existing things are determinate for Leibniz is that they have all features specified. I have already motivated this idea as it pertains to the structural features of material things: to leave any indeterminacy in the structural features of material things is to induce paradoxes of the composition of the continuum. But Leibniz’s commitment to the Determinateness Thesis can be found in other contexts within Leibniz’s metaphysics as well. Consider, for example, Leibniz’s distinction between complete and incomplete notions. In order to exist, the nature of a substance must be Thus the concept of the sphere in general is incomplete or abstract, that is to say one considers in it only the essence of the sphere in general or in theory, disregarding singular circumstances, and consequently it in no way contains what is required for the existence of a certain sphere; but the concept of the sphere Archimedes had put on his tomb is complete and must contain everything that belongs to the subject of this form. This is why in individual or practical considerations, which revolve around singular things, beyond the form of the sphere there enters the matter of which it is made, the place, the time, and the other circumstances which by a continual concatenation would finally embrace the entire succession of the universe, if one could pursue everything these concepts contain. For the concept of this particle of matter of which this sphere is made embraces all the changes it has undergone and will one day undergo. And according to me each individual substance always contains traces of what has ever happened to it and marks of what will ever happen to it. (A 2.2, 45 = Vo 61–63)
Leibniz’s claim is that a
The ultimate basis for Leibniz’s requirement that the notions of existing things must be complete is his commitment to the Principle of Sufficient Reason (PSR), but more specifically, his application of the PSR to God’s creative activity. Consider Leibniz’s remarks in the
To see how Leibniz connects this to his conception of materiality, consider his rejection of material atoms on the grounds that material atoms are intrinsically indiscernible and thus unsuitable objects of God’s creative choice. Writing to Clarke, Leibniz makes his view clear:
This supposition of two indiscernibles, such as two pieces of matter perfectly alike, seems indeed to be possible in abstract terms, but it is not consistent with the order of things, nor with the divine wisdom by which nothing is admitted without reason. The vulgar fancy of such things because they content themselves with incomplete notions. And this is one of the faults of the atomists. (LC 40)
Thus, even though it might seem plausible to suggest that a material atom is
One loose end along these lines is that more recent types of monism, such as the version developed by
Of course, Leibniz is not engaged with a view exactly like Horgan and Potrč’s. However, the version of Material Monism suggested by De Volder (and outlined above) is sufficiently similar that some remarks are in order. Recall that according to De Volder, the entire physical world is a single substance with by derivative force, namely, that by which bodies actually act on one another or are acted upon by one another, I understand, in this context, only that which is connected to motion (local motion, of course), and which, in turn tends further to produce local motion. For we acknowledge that all other material phenomena can be explained by local motion. (GM 6, 237 = AG 120)
Leibniz can, therefore, respond to views like Horgan and Potrč’s by giving just such a mechanical analysis of observable qualities.
However, there is one complication to consider. Even De Volder acknowledges that material parts have differential motions and that it is these motions that explain the observable qualities of matter. Still, De Volder denies that the material world has parts, strictly speaking: matter may have modally distinct regions with differential motions, but these are not independent parts. To address this point, Leibniz needs to leave the mechanical philosophy behind, or, at least, he needs to be clear about its limitations. For Leibniz, the motion of material parts requires the existence of active substances underlying matter. Matter alone is not sufficient to explain of force (e.g., motion, but also resistance to motion) in bodies. Writing to Johann Bernoulli in 1698, Leibniz makes this point clearly:
I have often said … that all phenomena in bodies, even the force of elasticity, can be explained mechanically. But the principles of mechanism or of the laws of motion cannot be derived from the consideration of extension and impenetrability alone; and so there must be something else in bodies from whose modification conatus and impetus arise, as shapes arise from the modification of extension. (Lo 9 = AG 167)
Thus, motion requires something more fundamental “in” bodies that gives rise to it.
It remains to consider whether the Determinateness Thesis stands in tension with any of Leibniz’s other metaphysical commitments. Though the centrality of the Determinateness Thesis to Leibniz’s metaphysics makes this unlikely, there is at least one case that warrants attention, since it concerns certain features of the material world. Leibniz clearly asserts that bodies do not have precise shapes. Sometimes the way Leibniz expresses this view makes it seem as though the shapes of bodies are somehow indeterminate, or as Leibniz says “imaginary”. Take, for example, the following passage from It is even possible to demonstrate that the notions of size, shape, and motion are not as distinct as is imagined and that they contain something imaginary and relative to our perception, as do (though to a greater extent) color, heat, and other similar qualities, qualities about which one can doubt whether they are truly found in the nature of things outside ourselves. (A 6.4, 1545 = AG 44)
Passages like this one make it sound as though Leibniz thinks that bodies ultimately lack features such as size, shape, and motion. If this is the case, doesn’t Leibniz’s rejection of precise shapes in bodies entail that there is some indeterminacy in the material world?
On my view, the absence of precise shapes in bodies is ultimately consistent with the Determinateness Thesis. Leibniz’s rejection of precise shapes in bodies is best understood not as the claim that bodies have
First, the fact that bodies lack precise shapes is, according to Leibniz, a consequence of the actually infinite division of matter. Consider the following passage from Indeed, even though this may seem paradoxical, it must be realized that the notion of extension is not as transparent as is commonly believed. For from the fact that no body is so very small that it is not actually divided into parts excited by different motions, it follows that no determinate shape can be assigned to any body, nor is a precisely straight line, or circle, or any other assignable shape of any body, found in the nature of things, although certain rules are observed by nature even in its deviation from an infinite series. Thus shape involves something imaginary, and no other sword can sever the knots that we tie for ourselves by misunderstanding of the composition of the continuum. (A 6.4, 1622 = Ar 315)
In this passage, Leibniz explicitly concludes that bodies lack precise shapes
Second, on Leibniz’s account, we attribute precise shapes to bodies in virtue of our less-than-perfect senses in concert with our imagination. Importantly, Leibniz is explicit that, in fact, indeterminacy would arise if bodies were conceived as having precise, that is, geometrical, shapes. The failure of bodies to have such shapes is, by contrast, clearly connected to the fact that bodies are divided to infinity:
It is the imperfection and fault of our senses that makes us conceive of physical things as Mathematical Beings, in which there is indeterminacy. It can be demonstrated that there is no line or shape in nature that gives exactly and keeps uniformly for the least space and time the properties of a straight or circular line, or of any other line whose definition a finite mind can comprehend. (GP 7, 563; trans.
Far from being a basis for attributing indeterminacy to bodies, then, the lack of precise shapes is both consistent with, and ultimately follows from the fact that bodies are actually infinitely divided.
Finally, in Leibniz’s considered analysis, the difference between precise, geometrical shapes and the infinitely divided and determinate structure of actually existing bodies is, as he puts it, “less than any given amount that can be specified”, that is, it is an unassignable difference. As Leibniz writes to De Volder,
However, the science of continua, that is, the science of possible things, contains eternal truths, truths which are never violated by actual phenomena, since the difference is always less than any given amount that can be specified. (Lo 333 = AG 186)
Here again, the fact that the difference between the “actual phenomena”, that is, bodies, and true continua, that is, geometrical shapes, is less than any assignable quantity is based on the fact that the division of bodies is actually infinite.
Putting all of this together, the best way to understand Leibniz’s rejection of precise shapes in bodies is to say that, strictly speaking, bodies have
To sum up, then, I believe that even if Leibniz’s commitment to the Determinateness Thesis has not been fully vindicated, I have adequately articulated its character and motivation within Leibniz’s metaphysics, both as it relates to his analysis of materiality and more broadly. I have also ruled out potential concerns about its alignment with certain features of Leibniz’s considered analysis of bodies, in particular his claim that bodies do not have precise shapes. In the end, the Determinateness Thesis is a central commitment of Leibniz’s metaphysical project. Thus, it is not altogether surprising to find it playing an important role in the background of Leibniz’s Argument from Unity, which is one of Leibniz’s major lines of argument against the Cartesian conception of extended substance.
Leibniz’s view that matter is discrete is a view about the character of material things and their parts. According to the characterization given, material things are, strictly speaking, pluralities, that is, collections of pre-existing parts. Without this characterization, Leibniz’s commitment to the Plurality Thesis would be a weak point in his familiar Argument from Unity against material substances.
As I noted above, it was fairly common in the 17th & 18th century to claim that material things have actual parts.
These commitments are connected: the discreteness of matter supports a key premise—the Plurality Thesis—in Leibniz’s Argument from Unity, which is a central argument in the development of Leibniz’s monadological metaphysics. The parts of matter, although actual, are also vanishing; the very boundaries that serve to distinguish material parts are themselves vanishing. Thus, discrete matter itself calls out for a foundation in non-material substances. This thought is at the center of Leibniz’s rejection of the substantiality of material objects. But it only comes to the fore through an examination of Leibniz’s commitment to the discreteness of matter. Since material objects are discrete all the way down, they are inherently pluralities, and therefore, they cannot be ontologically fundamental. The infinitely descending structure of matter points towards the further conclusion that Leibniz wants to draw from the line of thought detailed above: if matter is to exist at all, discrete matter in particular, it requires a foundation in something non-material. Therefore, Leibniz’s commitment to discreteness drives his argument (or, at least, one of his arguments) for non-material substances: first substantial forms and, ultimately, monads.
One text that supports this particular rendering is from Leibniz’s
See, e.g., A 6.4, 1464 = Ar 257–59, GP IV, 478–79 = AG 139, A 2.3, 546 = Lo 73. This argument has received lots of attention in the literature, and I cannot reference all discussion here. For a representative sample, see, e.g.,
For further discussion of the role of the Plurality Thesis in Leibniz’s Argument from Unity, including how Leibniz’s Argument from Unity is vulnerable to objections stemming from certain types of monism, see, e.g.,
For present-day versions, see, e.g.,
I use the term “region” as distinct from “part” in order to emphasize the dependence on the entire physical world. “Part” often brings a sense of independence or priority along with it.
This way of expressing the view shows up in Leibniz’s correspondence with Burcher de Volder. See A 2.3, 530 = Lo 61. See also Section
The distinction I am making here is similar, though not identical to the distinction Thomas Holden draws between the so-called “actual parts doctrine”, according to which each part of a whole is a distinct existent in its own right (
This point is also made by
See, e.g.,
It is worth noting certain features of the dialectical situation as it pertains to Leibniz’s Argument from Unity in general and the Plurality Thesis in particular. If Material Monism is not Descartes’s view, but instead, Descartes accepts Leibniz’s claim that material parts are ontologically prior to material wholes—a view for which there is certainly textual evidence and which has been defended in the literature on this topic (see, e.g.,
My discussion of discreteness is largely indebted to
See
In light of this passage from Aristotle, it appears that Leibniz is not only departing from the Cartesians by claiming that matter is discrete rather than continuous; he is departing from a longer philosophical tradition. Though the reason for why matter is a continuous quantity for Aristotle and the scholastic Aristotelians might differ from that of Descartes, insofar as quantity for Aristotelians is considered a proper accident of body—inalienable but not but not itself part of the essence. For further discussion, see
Of course, Leibniz does continue to employ Aristotelian notions such as
For a detailed and insightful discussion of the conception of discreteness, but especially its counterpart, continuity, in the letters to Thomasius, see
This is Leibniz’s list. See A 2.1, 25 = L 94.
Leibniz will later reject the notion of primary matter, since, as he will later argue, continuous mass, being purely passive, cannot exist. See, e.g., A 3.7, 885 = Lo 9.
Leibniz will later argue that matter has no precise shape, another consideration which he uses against the existence of material substance. See, e.g., A 2.2, 250 = Vo 253. For discussion of Leibniz’s view that bodies have no precise shape, see, e.g.,
Leibniz considers two scenarios because in 1699 at least he holds that “neither a vacuum nor a plenum is necessary; the nature of things can be explained in either way” (A 2.1, 25 = L 94). So he is covering either case. Of course, since Leibniz will later uphold the plenum, the second scenario is more likely to have connections with his later views.
Aristotle characterizes contiguity as follows: “a thing that is in succession and touches is contiguous” (
See, e.g.,
See, e.g., Grua 326 = AG 95: “there are two great labyrinths of the human mind, one concerning the composition of the continuum, and the other concerning the nature of freedom, and they arise from the same source, infinity.” See,
In certain cases, I have resorted to cognates of Leibniz’s actual terminology for ease of expression. However, when I discuss the texts in which each pair of terms is found, I will note the exact words themselves.
Note that the bounded parts that Leibniz is developing in these and later texts are not non-material substances, but material things. Thus the sense in which matter is an aggregate (as this relates to matter’s discreteness) is that it is an aggregate of material parts, not an aggregate of substances. The latter notion comes to be how Leibniz articulates what he calls “second matter”. See, e.g., Leibniz’s 1698 letter to Bernoulli: “secondary matter, i.e. mass, is not a substance, but substances” (A 3.7, 885 = Lo 9). For further discussion of how matter is a plurality in the sense of
Although this text is some 25 years from the 1676 notes, I think there is a clear connection, which is highlighted by the terminological distinctions introduced above.
I will engage this very powerful, but also somewhat elusive idea at more length in Section
One discussion of the various paradoxes engendered by taking a line to be composed from points spans many pages of Leibniz’s dialogue
My claims here also provide a correction to the literature on this topic, which often describes motion as
In later texts, Leibniz often mentions the actually infinite division of matter alongside the claim that each part has different motion from its neighbors, without explicitly stating that the motion is what causes the division. See, e.g.,
See GM 7, 19 = L 667, where Leibniz characterizes the notion of “ingredient [
I believe this is consistent with Leibniz’s later texts, though not stated by Leibniz explicitly, since in many texts Leibniz does not explicitly claim that motion causes divisions. See Footnote
I add the parenthetical “and to exist” because there remains a conception of
Recall: “For example, two spheres, one included in the other, can be moved in different directions and yet remain contiguous, though they cease to be continuous” (A 2.1, 27 = L 96).
Leibniz returns to the example of the spheres in a 1698 text
This result supports views previously developed by
De Volder’s suggestion has a great deal in common with Schaffer’s
De Volder provides different formulations of this argument. See, e.g., A 2.3, 562 = LDV 91. Also, De Volder’s reasoning here is reminiscent of Spinoza in E1P15S, though De Volder does not mention Spinoza: “For if corporeal substance could be so divided that its parts were really distinct [
Of course, Leibniz’s God does not create material parts, but immaterial monads, whose existence and perceptions provide a basis from which bodies result.
For a more detailed discussion of the sense in which possible worlds and possible substances must be determinate for Leibniz, see
For other views that assert the possibility of qualitative heterogeneity without mereological structure, see
It is important to note that the relation between monads and bodies is not a mereological one, since monads or “substantial unities are not parts, but the foundations of phenomena” (Lo 303). Still, the notion of “foundation” employed by Leibniz here suggests a picture in opposition to the type of monism I am considering.
The Argument from Force is roughly the idea that matter cannot be a substance because it is merely passive and thus cannot explain the presence of forces in bodies. See, for example, “On the Nature of Body and the Laws of Motion” (A 6.4, 1976–1980 = AG 245–250). For discussion of this line of argument, see, for example,
For a detailed development of this orientation towards Leibniz’s rejection of precise shapes in the physical world, see
See also Footnote
For a detailed discussion of the so-called “actual parts doctrine” among the early moderns, see
Gottfried Wilhelm Leibniz (1923–). |
|
Gottfried Wilhelm Leibniz (1875–90). |
|
Gottfried Wilhelm Leibniz (1849–63). |
|
Gottfried Wilhelm Leibniz (1948). |
|
René Descartes (1965–76). |
Gottfried Wilhelm Leibniz (1989). |
|
Gottfried Wilhelm Leibniz (2001). |
|
René Descartes (1985–1991). |
|
Gottfried Wilhelm Leibniz (1969). |
|