Somers-Hall, (1.10), Deleuze’s Difference and Repetition, ‘1.10 Leibniz (43–4/54, 46–52/56–63)’, summary

by Corry Shores
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[The following is summary. All boldface, underlining, and bracketed commentary are my own. Proofreading is incomplete, so please forgive my typos and other distracting mistakes. Somers-Hall is abbreviated SH.]

Summary of

Henry Somers-Hall

Deleuze’s Difference and Repetition:
An Edinburgh Philosophical Guide

Part 1
A Guide to the Text

 

Chapter 1. Difference in Itself

1.10 Leibniz (43–4/54, 46–52/56–63)

Brief summary:

Leibniz presents an infinite representational system. The world is composed of monads with predicates. This means the world takes on the subject-predicate structure of judgments, and is thus representational. Each monad is in some relation to every other monad, and these relations are expressed in a monad’s predicates. Therefore each monad expresses the world in its entirety. There are infinitely many monads, so there are infinitely many predicates for each one, and thus what each monad represents in its predication is infinite. Since we have a world of coexisting differences, we would think in Leibniz’ system that we might have non-oppositional difference. However, we have just one world with its own defined limits, since it stands opposed to other possible worlds that have inconsistencies and thus were not created. So while within this world there is non-oppositional difference, between this world and the others that God did not create there is in fact an oppositional difference.

 

Summary

[A while ago we examined Aristotle’s system of classification and definition. We examined the issue and problems of the highest genus or genera. Given that this is the most encompassing level of classification, we called it the Large, and problems regarding it we called the problems of the Large. In the prior section we saw Hegel’s infinite representation and dialectical movement. All categories of thought come about through this movement. The movement is somehow infinite. It is infinite either because it has no beginning or end (however, it would really seem to have a beginning and end, namely, indeterminate being and the absolute), or because at each stage there is a sort of unlimitedness. For example, at the beginning being and nothing seem to ceaselessly revolve one into the other in a movement that is genetic of a third concept, becoming (it is not clear to me if there are determinate stages and thus not endless interaction at each stage, or if it is one ceaseless interaction after another, all adding together and remaining in motion). At any rate, we need somehow to conclude that this dialectical movement is infinite. It is still representation, because it generates the categories of understanding that we use to make judgments, which have the form of representation. And also, we have a structure similar to the representational structure of genus-species. In Hegel we have something more like a process-product or pattern-instance structure. This is as clear as I can conceive matters at this point, but surely there is a better explanation. It would seem that since Hegel is dealing with the movement that generates all categories, then we are dealing with the Large, that is, with something roughly equivalent to the most basic category. SH will then speak of “the notion of contradiction as the largest difference” (48). I am not entirely sure what is meant by “largest difference”, but perhaps it can be understood in this way. What makes the dialectic movement the largest is that it is inherent to each stage and each instance of production. Also, at each stage, contradiction is like a motive force in the movement. Since contradiction is a motive force at each stage and in each production, it is also ‘largest’ in that sense of being all-encompassing. At any rate, Hegel introduces the infinite into the Large by saying that somehow it is ceaseless. Each stage is finite, but the entire process is infinite. In that way, Hegel introduces the infinite into the finite. Leibniz will also introduce the infinite into the finite, but this time on the level of the Small, that is, of the individual rather then of the highest grouping.] Deleuze has a problem with understanding the world by means of judgments, which normally take the subject-predicate form [exactly what is wrong with representation seems just to be the unresolved problems it creates. I am not sure if there are any other motivations to oppose representation. For example, I am not sure if there is an anti-totalitarian political motivation to be against systems that organize all their parts around some fixed center of power and that use fixed meanings in symbolic language to convince people of the legitimacy of that centralized power.] Leibniz, [like Hegel and Aristotle, is a representational thinker, because he] “holds to the view that all truths take the form of subject-predicate judgements” (48). We already saw for example: “man is a rational animal”. [So the first step we are now taking in our thinking here is to claim that such predicative formulations can be true. The next step is to say that if they are true, then they correspond to some reality in the world, Thus,] this means that in reality, there are things which can take predicates, since they are things with properties. [We should distinguish the language of predication we are using here from the way we were using it for Aristotle (and Porphyry). In that prior case, we said that the genus serves as a predicate to its species: “man (species) is an animal (genus) that reasons (specific difference).” Now instead in this current section, we are not thinking of the genus being a predicate. Rather, we are thinking of specific things having properties, and those properties being ‘predicated’ to the thing. We might for example say something like, ‘This apple is red’. Here the ‘subject’ of the sentence is the individual apple, and its predicate is something like ‘has redness’ or ‘is a red thing’ or ‘is something which is red’ or just of course ‘is red’.]

At the beginning of this chapter, we saw how Deleuze’s central claim is that we need to find an alternative way of conceptualising the world to that provided by judgement. Now Leibniz holds to the view that all truths take the form of subject-predicate judgements: ‘In every categorical proposition (for from them I can show elsewhere that other kinds of propositions can be dealt with by changing a few things in the calculus) there are two terms, the subject and the predicate’ (Leibniz 1989b: 11). It is certainly the case that some truths take this form, such as the claim that ‘man is a rational animal’, or that ‘seven is a prime number’. If we hold that our judgements are able to accord with the world, then it is going to be the case that the basic elements of existence are also going to be substances of some form possessing properties (what Leibniz calls monads).
(SH 48)

[Now we will address a problem that arises were we to suppose all this that we said above. Some predicates place the subject into a reciprocal relation with another thing. For example, Paul is taller than Peter. Our basic claim is that there are individuals that have predicates. The problem here is apparently that reciprocal relations like this make it confusing which is the subject and which is the predicate, because our example sentence means the same as John is smaller than Paul. I do not understand exactly what the problem is yet. It seems we have two individuals, Paul and John, both of which have properties, being smaller or larger than the other. Why is it that if something is found in the predicate of something else, then it cannot itself be a subject with reciprocal properties? But we need to figure this out in order to proceed to the next idea. One reason I can think of is that for some reason being in another’s predicate is to eliminate one’s own individual existence. So we have monads. We assume Paul and Peter are monads. If we put Paul into the predicate of Peter, we either have two Paul’s, which is  impossible, or we removed Paul from the world of monads and inserted it into Peter. The other reason I can think of is that we are claiming that all things fit a subject-predicate judgment form without any ambiguity, but for relational predicates there is no clear subject. Again, it is still not evident to me why even under this assumption we cannot have two clearly discernible individuals, Peter and Paul, both having a predicate that places each in relation to the other. Maybe the problem is that there is one relation but it is expressed with two different and opposite predicates. There is just one relation of largerness/smallerness between Peter and Paul, so there should be just one predicate for some reason, but in fact we have two. Still it is unclear why one relation cannot have two expressions depending on the perspective taken. There must be some better reason to explain the problem here. We have to assume it is a problem for all relational predicates, which include cause and effect. For, if x is the effect of y, then y is the cause of x. Which then is the subject (assuming there can only be one for each relation). We are going with this idea that this is so problematic as to call into question the viability of the judgment of cause and effect, and thus somehow no objects can be understood as causes of any other objects. This is very interesting, but I cannot grasp the reasoning for it yet.]

If we see the basic substances in existence as purely defined in terms of substances and properties, however, we encounter a problem when we deal with relations between substances, since these don’t seem to fit this structure. If we say, for instance, that ‘Paul is taller than John’, then it doesn’t seem clear what is the subject and what is the predicate (we might want to say that ‘Paul’ is the subject, and ‘is taller than John’ is the property, but what about if we rephrase the proposition as ‘John is shorter than Paul’?). Similarly, relations of cause and effect seem to involve two subjects and a relation between them. If all propositions can be reduced to judgements, therefore, we seem to be left with a world of non-causally interacting entities – ‘the monads have no windows through which something can enter or leave’ (Leibniz 1989a: §7).
(SH 48)

We now need to explain two problems. 1) [For the first problem, we are working with the notion that we cannot explain causality on the basis of relations. This still is not evident to me why, but it has something to do with the fact that our mode of subject-predicate judgment fails with reciprocal relations like cause-effect. X has the predicate ‘causes y.’ Y has the predicate, ‘is the effect of x. The claim is that we cannot answer, who is the subject? It cannot be X, since for the same relation, Y is also the subject when the relation is reformulated. For some reason, they both cannot be subjects even though they both are monads and even though there are no logical inconsistencies in these reciprocal formulations. At any rate, the first problem requires two ideas, a) we cannot explain causal interactions on the basis of relations between causing things and effected things, and b) the world is a system of causality that needs to be explained.] “First, how do we explain interactions without relations, given that we appear to live in a world of causally interacting substances” (48d). [SH will then tell us what the solution is, but I am unable to grasp it very well right now. The conclusion we will arrive at is that all things will contain relational properties to all other things, including causal relations. The way we get to this conclusion is by recognizing the above point that we have causality but we cannot understand causal interaction using the notion of relations. I do not grasp yet how this inference is made. If I wanted to come to this conclusion, I think I would first recognize that there are predicates expressing reciprocal relations, then next I would establish that for certain reasons all things in one way or another relate to all other things. One possible way are spatial relations of physical things. Each monad, if it lies in space, is to one side or another of some other monad, which is to one side or another of some other, and all are somehow spatially related. Another would be to see everything in the world as in a system of physical causality, where on the local level there are proximate causal interactions, but given that chains of effects spread that influence throughout the system, all things are either directly or indirectly related causally. I will quote.]

We now have to deal with two problems. First, how do we explain interactions without relations, given that we appear to live in a world of causally interacting substances; and second, how do we differentiate different monads? The solution to the first problem is to see each of these monads as somehow containing the relations between different substances as properties. This | means that ‘taller than John’ will be a property of Paul, and ‘shorter than Paul’ will be a property of John. If causal interactions are going to be understood purely as properties of each subject, then each monad will have to contain all of its causal interactions with the rest of the world. Leibniz therefore writes that [the following up to citation is Leibniz quotation]:

This interconnection or accommodation of all created things to each other, and each to all others, brings it about that each simple substance has relations that express all the others, and consequently, that each simple substance is a perpetual, living mirror of the universe. (Leibniz 1989a: §56)
(SH 48-49)

[So, by means of relational predicates, each smallest individual in the world express the entirety of the all other individuals in the world. [It seems we also assume that the world is infinite, thus] the infinite in a sense is contained in each monad. [Now, since each such relation is a relation of difference,] all the world’s difference in all its variety is contained in each smallest part.

Each monad is therefore made up of an infinite number of properties which together describe the totality of what would be its relations with the universe, and hence, in a sense, the universe itself. To this extent, the infinite, in the sense of even the smallest elements of the universe, is contained within each monad. The whole variety of difference is therefore brought into the notion of the essence of each particular monad. Deleuze writes that: ‘The inessential here refers not to that which lacks importance but, on the contrary, to the most profound, to the universal matter or continuum from which essences are finally made’ (DR 47/58).
(SH 49)

2) [The reasoning for the next problem is also a little hard for me to grasp. It seems like it goes like this. Suppose that each monad contains relational predicates that express its relation to every other part of the world, that is, to every other monad. Each one expresses the world in its entirety. Perhaps somehow then there is no way to distinguish each monad, since they all express exactly the same thing, the whole world. To continue our reasoning, recall the specific relational predicate Peter is taller than Paul. There is just one relation of size comparison between them, but two predicates, ‘is taller than’ and ‘is smaller than’. Both Peter and Paul can be the subject, so which is the subject? Well, it depends on which perspective you take. If we take Paul’s perspective, we make him the subject, and the appropriate predicates follow, and likewise for Peter. So return to the idea that each monad expresses every relation with every other monad. Each relation can put one or another monad as the predicate. Perhaps the idea is that if we take one monad and all its predicates, we can reexpress those predicates such that they become identical to the predicates of any of the other monads. Think of the simple world where there are just two monads, Peter and Paul.

(Peter > Paul) = (Paul < Peter).

So Peter can have this predicate: Peter is taller than Paul

And Paul can have this predicate: Paul is smaller than Peter.

Yet, the inverse is true for both.

So Peter can keep his predication: Peter is taller than Paul.

And Paul can invert his predication: Paul is smaller than Peter. Peter is taller than Paul.

Now, think of a world with infinitely many monads. The predicates for each can be made identical with the predicates for any, and thus, all others. All relational predicates of all monads can be made identical. So for monads, what distinguishes them is not what their predicates express, since they all express the sum of all differences constituting the world; rather, what distinguishes them is each one’s unique perspective, which might orient all the predications such that instead of taking the above third personal form, they instead substantiate their own selves in the formulations in a first personal way. So for example, Peter’s predication might be “I (Peter) am taller that Paul”. And Paul would say “I (Paul) am shorter than Peter”. But Paul’s predication would never be: Peter is taller than Paul (like in our substitution above), since we are taking his perspective and not Peter’s and so Paul will take the subject place of the sentence. Now, we need somehow to get to an idea of the distinct and confused expression of relations. This part I cannot follow well. But it seems that from one monad’s particular perspective, one can have clear knowledge of how it interacts with those other monads it is directly in relation to, but confused knowledge of its relation to all the other monads it is only indirectly related to, as mediated through the transitive chain of relations between continguously related monads. Thus perhaps we may say now that what distinguishes each one on the level of their predicates is that each will have some clear predicates and many other confusedly stateable predicates. But since each one takes a different relative position to the others and thus has different direct and different indirect relations, each monad will have as its ‘signature’ its own unique set of clearly stateable predicates. I am saying ‘clearly stateable’, because from the perspective of God, who created all the monads and their relations, I would think all is clear. The only way the predicates that a monad has, which were endowed by an omniscient being, would be unclear is if we think of the predicates being understood by means of a limited mind taking that particular limited perspective and stating the predicates as clearly as possible from that perspective. So for example, we can say that the earth’s gravity pulls on us and we on it to some much lesser extent. And we might also then say that at the furthest distance from us in the cosmos, we gravitationally influence those distant bodies but at some remarkably slight amount. We cannot however state what those bodies are, how we influence them, and so on.]

The second question was, how do we differentiate monads given that each expresses the whole of the universe? While each monad expresses the entire universe, each does so from a particular perspective, and so only that which is proximal to the monad is expressed distinctly. Events which are at some remove from the monad are only perceived confusedly [the following up to citation is Leibniz quotation]:

Monads are limited, not as to their objects, but with respect to the modifications of their knowledge of them. Monads all go confusedly to infinity, to the whole; but they are limited and differentiated by the degrees of their distinct perceptions. (Leibniz 1989a: §60)

[new paragraph] The difference between monads is therefore the difference between different perspectives on the world.
(SH 49)

[Note here that two monads relate without being opposed. It would seem then that we can have judgment without oppositional difference in Leibniz’ system. However, all the predicates must be consistent with one another for Leibniz. There is one consistent world to which all the monad’s perspectives are in harmony. We thus have a logic of identity, since we can only have one world and not many which are incompatible with that world. Somehow Deleuze sees there being as many worlds as there are perspectives, but the perspectives are all of one world. It is not clear to me how there are both many worlds and just one world at the same time, instead of there being one world with many perspectives.  Hopefully this idea becomes clearer as we proceed.]

Different perspectives are not opposed to each other, and so Leibniz appears to have succeeded in coming up with a form of non-oppositional difference which explains all of the accidents of entities. If he had done so, then he would have developed a conception of non-oppositional difference founded on judgement, thus providing an alternative to Deleuze’s philosophy. In the end, however, this project fails, because the concept of difference is still founded on an identity. If we ask what these different perspectives | are perspectives of, then we are given the answer that they are perspectives of the universe. The notion of the universe itself has to pre-exist the different perspectives of it, since it is through this notion that God determines which of the monads can exist and which cannot. Only those which are compossible, that is, can simultaneously co-exist within the same world, can exist. We cannot have a world in which Adam both sinned and did not sin, as this would be a contradiction, nor a world in which different monads see the world in such radically different ways, as then the impression of causality would break down. ‘There are, as it were, just as many different universes [as there are monads], which are, nevertheless, only perspectives on a single one’ (Leibniz 1989a: §57). Leibniz’s notion of difference therefore still relies on the convergence of these different perspectives on a single identity, the universe itself [the following is quotation of Deleuze]:

Leibniz’s only error was to have linked difference to the negative of limitation, because he maintained the dominance of the old principle, because he linked the series to a principle of convergence, without seeing that divergence itself was an object of affirmation. (DR 51/62)
(SH 50)

 

 

 

 

 

Citations from:

Somers-Hall, Henry. Deleuze’s Difference and Repetition. An Edinburgh Philosophical Guide. Edinburgh: Edinburgh University, 2013.

Or if otherwise noted:

DR:
[Deleuze] Difference and Repetition, trans. Paul Patton, New York: Columbia University Press, 1994/London: Continuum, 2004.

 

Leibniz, Gottfried Wilhelm (1989a), ‘The Principles of Philosophy, or, the Monadology’, in Roger Ariew and Daniel Garber (ed. and trans.), Philosophical Essays, Cambridge: Hackett Publishing, 213–24.

Leibniz, Gottfried Wilhelm (1989b), ‘Samples of the Numerical Characteristic’, in Roger Ariew and Daniel Garber (ed. and trans.), Philosophical Essays, Cambridge: Hackett Publishing, 10–19.