Reflections on Meeting #2-II

Continuing on from my last post, I want to say a few things about Leibniz's final stance on the labyrinth of the continuum. The following are two characteristic statements of his view:

“It is the confusion of the ideal with the actual which has muddled everything and caused the labyrinth of the composition of the continuum” (1696; AG 146)

“In actuals, simples are prior to aggregates, in ideals, the whole is prior to the part. Neglect of this consideration leads to the labyrinth of the continuum” (GP II 379)

Why do we muddle everything if we confuse the ideal with the actual? The short answer is that this follows from the attempt to conceive of a spatial (or temporal) continuum as composed of simplest elements. Given the nature of the continuum, these elements could only be spatial indivisibilia (points, infinitesimals). But here, Leibniz believes, we run up against insurmountable conceptual problems: we can't conceive of the composition of the continuum from indivisiblia. Therefore, the continuum isn't composed of simplest elements. But if the continuum isn't composed of simplest elements, then neither is it real. So, the continuum can only be an ideal thing in which there are no actual divisions but only the possibility of division in any arbitrary way.

Laying this out a little more carefully, we can see Leibniz relying on the following argument:

1. Any actual (or real) thing is either simple or composite.


2. For any actual x, if x is composite, then there are simples in terms of which x's existence can be explained.


3. No continuum (infinite or finite) is simple, since it is divisible into smaller parts ad infinitum.


4. If any continuum is composed from simples, those simples must be indivisibilia. (from 3)
   


5. No continuum is composed from indivisibilia.


6. Therefore, no continuum is actual (or real).

In Meeting #3 we'll look more closely at premise 5. It is certainly not self-evident, and most modern understandings of the continuum reject it, conceiving of space as a construction of points.

As Leibniz saw it, his major breakthrough in resolving the continuum problem came in distinguishing a purely geometrical continuum from matter. In the case of the former, he came to believe, there is no composition from simples (hence no reality); any mathematical continuum is a whole in which parts can be taken arbitrarily.

But in matter actual parts are given. At the macroscopic level this is obvious: things are composed of parts. But if bodies in general are composites, then premise 2 above asserts that their existence must be understood in terms of their composition from simples. However, no body, insofar as it is an extended thing, is simple: it is always divisible into further parts. So, it looks like the composition of matter is impossible.

This leaves us with a basic metaphysical problem that can be addressed in a variety of ways, only one of which Leibniz accepts:

• We can reject the claim that no bodies are simple by affirming the existence of material atoms: bodies which are divisible in thought, but which physically cannot be divided. {Leibniz rejects this as contrary to reason, affirming against it that matter is actually infinitely divided. Question: what is the basis for this assertion?}


• We can reject the claim that no bodies are simple by affirming the existence of indivisible, substantial points, out of which extended things are composed. {This runs afoul of the same arguments against indivisibilia mentioned above.}


• We can accept that matter is actually infinitely divided and that it is not composed of any ultimate simples (it is what David Lewis calls gunk), but deny that this has the consequence that matter fails the test of actuality/reality. {This is a non-starter for Leibniz: reality is always explained from the ground up.}


• We can accept that matter is actually infinitely divided and at the same time maintain that it is composed from genuine simples, but in this case the simples are not themselves extended material things and hence not divisible even in thought.

The last is Leibniz's position: bodies are actually divided into parts ad infinitum, but they are nonetheless "composed" of simple substances (monads). This allows him to affirm the reality of bodies as aggregates/composites of simples, but it introduces a new and equally daunting problem: if bodies are not spatial aggregates of monads, in what sense are they aggregates at all? We touched briefly on his answer last Wednesday; we'll return to the question in subsequent meetings.