Determinate/Indeterminate
We talked last time about the emphasis Leibniz places in his 1705 letter to Sophie on the distinction between the indeterminate character of space and the determinate character of matter. Here are some relevant passages:
[A]
The mass of bodies is actually divided in a determined way, and nothing in it is exactly continuous; but space or the perfect continuity which exists in the idea only signals an indeterminate possibility of dividing as one would like. In matter and in actual realities the whole is a result of the parts; but in the ideas or in the possibles (which includes not only this universe, but also every other universe which can be conceived, and which the divine understanding effectively imagines), the indeterminate whole is anterior to the divisions... (p. 3)
[B]
To better conceive the actual division of matter to infinity, and the exclusion of all exact and indeterminate continuity, we ought to consider that God has already produced as much order and variety as it was possible to introduce in it up to now, and so no indeterminacy has remained in it; whereas indeterminacy is of the essence of continuity. (p. 3)
[C]
And it can be demonstrated that there is no line or shape in nature which gives exactly and keeps uniformly through the least space or time the properties of a straight or circular line, or of some other line of which a finite mind can grasp the definition.... [T]he divine wisdom does not will to trace exactly these shapes of limited essence, which presuppose something determined [!] and consequently imperfect in the works of God. (p. 4)
In framing the distinction between matter and space as a distinction between the "determinate" (déterminé) and the "indeterminate" (indéterminé), Leibniz cannot mean simply that bodies have determinate properties, whereas purely geometrical objects do not. Space, as such, does not have a determinate shape, but a triangle and a circle do, and these objects are every bit as ideal as space itself. Furthermore, it may be doubted whether bodies have a real determinate shape as opposed to a shape that can be assigned to them on the basis of our (more or less distinct) perception of them (more on this below).
What Leibniz seems to suggest most clearly in passage [A] is that any body, however small, is divided into actual parts. In geometrical objects, however, there is merely the possibility of division in an arbitrary manner. On this characterization of the distinction, it looks like the claim about determinateness is a consequence of the claim about a body's actual parts: if a body is actually divided into parts, then it has determinate parts (i.e. some definite parts or other); by contrast, if a geometrical object is not actually divided into parts, then it is an "indeterminate whole."
One question to ask about this account is whether having actual, and hence determinate, parts is necessary for a body to be real or actual (as opposed to ideal). I don't see that it is. It seems at least logically possible, even in Leibniz's metaphysics, for there to be a body that was actual but not actually divided into parts, e.g., an undivided material sphere. As I understand Leibniz's statements in [B] and [C], it is contrary to God's wisdom for there to be such a body, i.e. God wouldn't include such a thing in the best of all possible worlds; but that is different from saying that such a body is impossible. Indeed, Leibniz suggests that divine wisdom requires not just that bodies have actual parts but that they be actually infinitely divided. The force of this claim goes beyond what one might think to follow from the nature of matter: infinitely divisible, possibly; but actually infinitely divided? The support for this could only come from a separate commitment to the thesis that it is God's intention to maximize the variety of nature: consequently, wherever there could be a further division of a body into small bodies, there will be.
If there is a link between the determinateness of matter and its reality, it will have to come from the grounding principle discussed in my last post. Any body has a determinate existence, because its existence can be explained in terms of the prior existence of more basic things. Leibniz hints at this in passage [A] when he says: "In matter and in actual realities the whole is a result of the parts." As John suggested last time, however, it is not clear that Leibniz should have spoken of "parts" here, as opposed to monads. If any body is divided into actual parts ad infinitum, what sense does it make to think of the body as "resulting" from those parts? There are no smallest parts "in the limit" out of which the body could be constructed--that thought is the first step into the labyrinth. So if we are to explain the determinate existence of the body, it can only be in terms of the prior existence of non-spatial substances.
Leibniz's reasoning on this point is expressed more clearly in his letter to De Volder of 30 June 1704 (AG 178-9; passage [2] on John's handout):
From the fact that a mathematical body cannot be resolved into first constituents we can, at any rate, infer that it isn't real, but something mental, indicating only the possibility of parts, not anything actual.... But in real things, namely, in bodies, the parts are not indefinite... but are actually assigned in a certain way, in accordance with how nature has actually instituted divisions and subdivisions as a result of various motions; and although these divisions might proceed to infinity, nonetheless, everything results from certain first constituents, that is, real unities, though infinite in number.
Leibniz's first sentence implies that it is a necessary condition for something's being real that it be resolvable into "first constituents." He goes on to assert that bodies are "real things," and that they are actually infinitely divided. However, he comes back to state that it is not these infinite parts that determine the body's being real but the fact that it "results" from "first constituents" or "real unities." (Again, what this means is an open question, but we have to take things one step at a time.)
A final question concerns the link that John hypothesized between determinateness and a principle of individuation. I understood him to argue that Leibniz affirms such a link in general (in the case of substances), but that he fails to show how it can be upheld in the case of bodies. Let me say, briefly, how I think this should addressed. I'll leave it brief in the hope that others might jump in to take up the argument.
Philosophers mean different things by a principle of individuation. Here, I think, we are interested in a principle, according to which, for any two entities of a given type, there will be a basis for distinguishing them as two. In other words, we are looking for a principle which supports the general truth of the identity of indiscernibles. Leibniz says (Discourse on Metaphysics, secs. 8-9) that for any two substances, their complete concepts supply such a principle. But he doesn't commit himself to the view that bodies themselves have complete concepts, so the question is whether there is some other way of specifying the identity of a body, such that it can in principle be distinguished from any other body.
We can assume that Leibniz rules out relational properties as the basis for distinguishing them. (We need to explore in more detail his reasons for this, but let's just assume it for now.) So, it looks like any two bodies would have to be distinguished on the basis of their intrinsic geometrical properties: their shape and the arrangement of their parts (which in turn determines their shape).
Leibniz's standard line on this issue is that any two putatively indiscernible bodies (two leaves or two eggs) could be distinguished on the basis of some difference in the internal arrangement of their parts. Thus, while it might seem to the unaided eye that two bodies were perfectly similar, if we looked closely enough, we would find some microscopic difference in their parts. (see the letter to Sophie, p. 4). Assumed here is (i) that in any body, there is a determinate division into actual parts ad infinitum, and (ii) that for any two putatively indiscernible bodies, at some point in the process of division we will come upon an arrangement of parts that characterizes one and not the other.
Three quick remarks on this:
1. If my claim above about the contingency of the infinite division of matter is correct, then this principle of individuation for bodies is also only contingent.
2. The determinacy of the division of matter into parts is explained by Leibniz in terms of its composition from organic bodies (see Monadology, secs. 65-69). To the extent that each of these bodies has (if only phenomenally) the unity of a living body (e.g., my body, the cat's), we can infer that there is a fact of the matter about how any body is structured "all the way down."
3. Nevertheless, it is an open question whether any of these bodies can be characterized in terms of a set of precise geometrical properties. The exact shape of any macroscopic body will be determined by the shapes of its parts and how they are fitted together. Yet at each stage in its division we will confront exactly the same question as Leibniz raises about geometrical figures of "limited essence": the thing we thought was a circle, is "really" an extremely complex polygon. But which polygon? That depends on how far we push the analysis. In general, we know that no finite analysis will be sufficient to give us the "right" answer to this question. Similarly, in the case of living bodies, the parts that make up a three-dimensional solid don't have a shape that is precisely characterizable in finite terms. We get only progressively more complex characterizations of structure, each of which remains relative to the resolution of our senses. So, the question is this: if our first, "naked eye" estimate is "imaginary" because perceptually relative, why shouldn't we say the same about any characterization of the body's geometrical properties? But if this is so, then even if Leibniz were right to think that any two bodies could, in principle, be distinguished on the basis of their geometrical properties, this principle of individuation would remain mind-dependent (in the way, e.g., that color is). (For an opposing view, which argues for the reality of a body's shape, see Sam Levey's 1998 Phil. Review paper, and a newer piece "On Precise Shape and the Corporeal World," in Rutherford and Cover, Leibniz: Nature and Freedom (OUP 2005).
Comments?
[A]
[B]
To better conceive the actual division of matter to infinity, and the exclusion of all exact and indeterminate continuity, we ought to consider that God has already produced as much order and variety as it was possible to introduce in it up to now, and so no indeterminacy has remained in it; whereas indeterminacy is of the essence of continuity. (p. 3)
[C]
And it can be demonstrated that there is no line or shape in nature which gives exactly and keeps uniformly through the least space or time the properties of a straight or circular line, or of some other line of which a finite mind can grasp the definition.... [T]he divine wisdom does not will to trace exactly these shapes of limited essence, which presuppose something determined [!] and consequently imperfect in the works of God. (p. 4)
In framing the distinction between matter and space as a distinction between the "determinate" (déterminé) and the "indeterminate" (indéterminé), Leibniz cannot mean simply that bodies have determinate properties, whereas purely geometrical objects do not. Space, as such, does not have a determinate shape, but a triangle and a circle do, and these objects are every bit as ideal as space itself. Furthermore, it may be doubted whether bodies have a real determinate shape as opposed to a shape that can be assigned to them on the basis of our (more or less distinct) perception of them (more on this below).
What Leibniz seems to suggest most clearly in passage [A] is that any body, however small, is divided into actual parts. In geometrical objects, however, there is merely the possibility of division in an arbitrary manner. On this characterization of the distinction, it looks like the claim about determinateness is a consequence of the claim about a body's actual parts: if a body is actually divided into parts, then it has determinate parts (i.e. some definite parts or other); by contrast, if a geometrical object is not actually divided into parts, then it is an "indeterminate whole."
One question to ask about this account is whether having actual, and hence determinate, parts is necessary for a body to be real or actual (as opposed to ideal). I don't see that it is. It seems at least logically possible, even in Leibniz's metaphysics, for there to be a body that was actual but not actually divided into parts, e.g., an undivided material sphere. As I understand Leibniz's statements in [B] and [C], it is contrary to God's wisdom for there to be such a body, i.e. God wouldn't include such a thing in the best of all possible worlds; but that is different from saying that such a body is impossible. Indeed, Leibniz suggests that divine wisdom requires not just that bodies have actual parts but that they be actually infinitely divided. The force of this claim goes beyond what one might think to follow from the nature of matter: infinitely divisible, possibly; but actually infinitely divided? The support for this could only come from a separate commitment to the thesis that it is God's intention to maximize the variety of nature: consequently, wherever there could be a further division of a body into small bodies, there will be.
If there is a link between the determinateness of matter and its reality, it will have to come from the grounding principle discussed in my last post. Any body has a determinate existence, because its existence can be explained in terms of the prior existence of more basic things. Leibniz hints at this in passage [A] when he says: "In matter and in actual realities the whole is a result of the parts." As John suggested last time, however, it is not clear that Leibniz should have spoken of "parts" here, as opposed to monads. If any body is divided into actual parts ad infinitum, what sense does it make to think of the body as "resulting" from those parts? There are no smallest parts "in the limit" out of which the body could be constructed--that thought is the first step into the labyrinth. So if we are to explain the determinate existence of the body, it can only be in terms of the prior existence of non-spatial substances.
Leibniz's reasoning on this point is expressed more clearly in his letter to De Volder of 30 June 1704 (AG 178-9; passage [2] on John's handout):
From the fact that a mathematical body cannot be resolved into first constituents we can, at any rate, infer that it isn't real, but something mental, indicating only the possibility of parts, not anything actual.... But in real things, namely, in bodies, the parts are not indefinite... but are actually assigned in a certain way, in accordance with how nature has actually instituted divisions and subdivisions as a result of various motions; and although these divisions might proceed to infinity, nonetheless, everything results from certain first constituents, that is, real unities, though infinite in number.
Leibniz's first sentence implies that it is a necessary condition for something's being real that it be resolvable into "first constituents." He goes on to assert that bodies are "real things," and that they are actually infinitely divided. However, he comes back to state that it is not these infinite parts that determine the body's being real but the fact that it "results" from "first constituents" or "real unities." (Again, what this means is an open question, but we have to take things one step at a time.)
A final question concerns the link that John hypothesized between determinateness and a principle of individuation. I understood him to argue that Leibniz affirms such a link in general (in the case of substances), but that he fails to show how it can be upheld in the case of bodies. Let me say, briefly, how I think this should addressed. I'll leave it brief in the hope that others might jump in to take up the argument.
Philosophers mean different things by a principle of individuation. Here, I think, we are interested in a principle, according to which, for any two entities of a given type, there will be a basis for distinguishing them as two. In other words, we are looking for a principle which supports the general truth of the identity of indiscernibles. Leibniz says (Discourse on Metaphysics, secs. 8-9) that for any two substances, their complete concepts supply such a principle. But he doesn't commit himself to the view that bodies themselves have complete concepts, so the question is whether there is some other way of specifying the identity of a body, such that it can in principle be distinguished from any other body.
We can assume that Leibniz rules out relational properties as the basis for distinguishing them. (We need to explore in more detail his reasons for this, but let's just assume it for now.) So, it looks like any two bodies would have to be distinguished on the basis of their intrinsic geometrical properties: their shape and the arrangement of their parts (which in turn determines their shape).
Leibniz's standard line on this issue is that any two putatively indiscernible bodies (two leaves or two eggs) could be distinguished on the basis of some difference in the internal arrangement of their parts. Thus, while it might seem to the unaided eye that two bodies were perfectly similar, if we looked closely enough, we would find some microscopic difference in their parts. (see the letter to Sophie, p. 4). Assumed here is (i) that in any body, there is a determinate division into actual parts ad infinitum, and (ii) that for any two putatively indiscernible bodies, at some point in the process of division we will come upon an arrangement of parts that characterizes one and not the other.
Three quick remarks on this:
1. If my claim above about the contingency of the infinite division of matter is correct, then this principle of individuation for bodies is also only contingent.
2. The determinacy of the division of matter into parts is explained by Leibniz in terms of its composition from organic bodies (see Monadology, secs. 65-69). To the extent that each of these bodies has (if only phenomenally) the unity of a living body (e.g., my body, the cat's), we can infer that there is a fact of the matter about how any body is structured "all the way down."
3. Nevertheless, it is an open question whether any of these bodies can be characterized in terms of a set of precise geometrical properties. The exact shape of any macroscopic body will be determined by the shapes of its parts and how they are fitted together. Yet at each stage in its division we will confront exactly the same question as Leibniz raises about geometrical figures of "limited essence": the thing we thought was a circle, is "really" an extremely complex polygon. But which polygon? That depends on how far we push the analysis. In general, we know that no finite analysis will be sufficient to give us the "right" answer to this question. Similarly, in the case of living bodies, the parts that make up a three-dimensional solid don't have a shape that is precisely characterizable in finite terms. We get only progressively more complex characterizations of structure, each of which remains relative to the resolution of our senses. So, the question is this: if our first, "naked eye" estimate is "imaginary" because perceptually relative, why shouldn't we say the same about any characterization of the body's geometrical properties? But if this is so, then even if Leibniz were right to think that any two bodies could, in principle, be distinguished on the basis of their geometrical properties, this principle of individuation would remain mind-dependent (in the way, e.g., that color is). (For an opposing view, which argues for the reality of a body's shape, see Sam Levey's 1998 Phil. Review paper, and a newer piece "On Precise Shape and the Corporeal World," in Rutherford and Cover, Leibniz: Nature and Freedom (OUP 2005).
Comments?
posted by Don at 3:30 PM
2 Comments:
I have some concerns about this bit:
[Leibniz] doesn't commit himself to the view that bodies themselves have complete concepts, so the question is whether there is some other way of specifying the identity of a body, such that it can in principle be distinguished from any other body.
We can assume that Leibniz rules out relational properties as the basis for distinguishing them. (We need to explore in more detail his reasons for this, but let's just assume it for now.) So, it looks like any two bodies would have to be distinguished on the basis of their intrinsic geometrical properties: their shape and the arrangement of their parts (which in turn determines their shape).
I don't know what the alluded-to reasons for excluding relational properties are, but aren't the "intrinsic" properties mentioned themselves relational? As you say, the shape of a thing is determined by the arrangement of its parts, i.e. the relations between them. And the common ways of defining geometrical shapes are relational: a circle is the locus of all points equidistant from a center; a square is four sides of equal length joined at right angles. Perhaps this isn't the case for purely geometrical objects, but certainly for bodies, which do result from parts, shape's relational character is plain.
Furthermore, the shape of a thing is its arrangement in space [is it? I'm actually not clear on how ideal space and real bodies interact]. And since we know that for Leibniz space is relational, isn't shape by necessity also relational?
Which makes me wonder why Leibniz can't simply appeal to the more obvious relational properties to individuate bodies. The conditions on a world are, if I recall correctly, co-presence of elements, succession of elements, and mutual transactions among elements. These look to me to establish a Strawsonian spatiotemporal framework - any world will have one, and any elements not related by a spatiotemporal framework will belong to different worlds. Since the spatiotemporal framework provides a unique specification for any body in terms of its relations to all other parts of the framework, it follows that mundane relational properties like spatial ones are enough to individuate bodies for Leibniz. To distinguish any two objects a and b, simply find a third object or point in space cthat isn't equidistant from both, and presto - Rca but not Rcb.
So we have a principle of individuation that doesn't rely on complete concepts, and doesn't invoke any shifty parts. Nevertheless, Leibniz does make a fuss about being able in principle to find "internal" differences between bodies. What am I missing?
On the first point, I agree completely: geometrical properties of the sort I mentioned are relational. So a distinction needs to be drawn between properties that are determined by relations among a thing's parts and relations that it has to things that are not identical to it or its parts. The point was not to rule out relations in general, but to rule out the kind of account that you sketch in the last part of your post. While such an account might serve Leibniz's purposes in some instances, I'm assuming that he would reject it if there are models in which it would fail to deliver the desired principle of individuation. Such an account can be developed in different ways. If the objects in question already have intrinisic properties that allow us to distinguish them (e.g. red, blue, green), then this account might allow us to distinguish multiple individuals of the same kind. But it isn't guaranteed to do so. Isn't there always the possibility of symmetrical arrangements (e.g. an unending sequence of red, blue and green balls) in which we will have no basis for picking out any one of them in particular. The blue ball will always be between the red and the green, but I am interested in this one. Given Leibniz's relationalism about space, we can't appeal to primitive facts about location, so how are they to be individuated? The difficulties are compounded if we try to offer a purely relational account, e.g., the only kind of thing is a colorless, chargeless atom of fixed mass. How will (relative) spatial position pick out any one of these as distinct from any other?
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