Vanishing Difference

Continuity and Expression, in the Leibnizian World

Leibniz formulates the view of the world as fundamentally expressed through a subject. Subject is understood here as a metaphysical point of view or a monad rather than psychologically. However, even if each subject expresses the totality of the world, most of this expression is confused and obscure, articulated through infinitely small perceptions that function as differentials of consciousness. These perceptions are not available to the subject’s consciousness, even if they are fully operational. Next to this large, obscure area, there is a smaller portion of the world that is clear and distinct. This is the portion of the world directly affecting the subject.

Leibniz’s formulation of expression in this context seems strikingly similar to our dominant understanding of perception. This, however, proves to be a deceptive first impression. The dominant model of perception is that of impression. Perception is based on imprints of the world on a subject. Expression, on the other hand, is the very opposite of that; it is an internal articulation of the world from a situated position within a larger field. In the model of expression, the subject, instead of being a recipient of impressions, is expressive, obscurely for the most part but clearly and distinctly as far as its body is concerned.

The body should not be understood here in a narrowly biological or corporeal sense but rather as a locus of expressive articulation of the individual. Clarity in this context arises through the engagement with the world. It is the portion of the world that one is interacting with that is expressed clearly, while the rest remains in relative obscurity. Consequently, the point of view, in this context, is not a cognitive standpoint. It is constituted through the body’s interaction with the portion of the world that it expresses clearly and distinctly. Two individuals can never occupy the same point of view. Even if each expresses the same world, they do so from the singular distribution of clarity and obscurity determined through their bodily engagement.

​Compossibility

For Leibniz, the world has no existence outside the point of view that expresses it. There is no pre-given world. An individual is inseparable from it. The world only exists through the expressions of individuals. Even if the world is common to all individuals, it has no in-itself that is separate from its expressions. There is nothing outside points of view. Yet somehow the same world is expressed across a multiplicity of singular points of view.

This is where an obvious dilemma emerges in reading Leibniz, nicely articulated by Daniel Smith. If the world only exists through expressions of individuals, yet this is the same world, how do we then reconcile with the fact that individuals tend to express their world in infinitely different ways? If we follow the logic of the principle of sufficient reason, all these differences must already be contained in the concept of each individual. On the one hand, the individual contains the entire range of possible expressions of the world, yet on the other hand, the world is only expressed in one determinate version among the range of infinite possibilities.

How does an individual express one particular version of the world among all the possibilities? What determines this path or this selection? This is where Leibniz introduces his famous concept of compossibility. Each individual only expresses the version of the world that is compossible or compatible with all other individuals. This means that each point of view has internal constraints and dependencies on other points of view. In this manner, what is expressed is never arbitrary; it is a result of complex algorithmic constraints that govern which expressions come to pass. This means that identity, difference, and world are no longer separable. They are intertwined in a single logical fabric where the individual is fundamentally understood as a determinate way in which the world is conceptually expressed.

​Differential Relation

This view is further reinforced by Leibniz’s understanding of continuity based on his calculus. For Leibniz, continuity is defined by a vanishing difference rather than the persistence of identity. Simply put, vanishing difference is a difference that tends to disappear without the difference ever being canceled. We can find a good illustration of this in color gradients. One end of the spectrum here is clearly of a distinct color front hat on the other end, yet if we follow the passage between the two poles, the difference between the two colors will slowly diminish until we reach the point where the two are indiscernible. This is an expression of continuity that includes difference within itself.

In Leibniz’s world, we cannot demonstrate identity between two clearly distinct sets, such as Caesar and Brutus, for example. Each of the two is a singular expression of the world. However, following the principle of sufficient reason, we can still demonstrate continuity. We are talking here about conceptual continuity rather than an empirical one.

Insofar as following Leibnizian logic, each individual concept must encompass all its predicates and ultimately express the entire world; continuity between any two distinct sets must be in principle demonstrable, provided that they do not express incompossible worlds. One cannot understand the concept of Caesar without including Brutus and vice versa. Insofar as their worlds are not contradictory, the two sets can be unfolded without any rupture, as a continuous variation, much like in the example of color gradients. This formulation of continuity, far from asserting sameness, makes difference intelligible. Difference is always situated within a field of compossible relations that allow it to vary without collapsing into identity.

The best way to understand this continuity is through the notion of differential relation. A differential relation is a relation that persists even when its terms disappear. The terms may become indscernible yet the relation itself remains intelligible and mathematically expressible.

Once again, the gradient can be a useful illustration here. Suppose the relation between two contrasting colors at the opposite ends of a gradient can be expressed as a ratio, x/y. If we move from both ends of the gradient at an equal rate, the difference between the two colors diminishes, yet even when the two become indiscernible, the relation of x/y will remain mathematically determinable and expressible in non-zero values.

The understanding of continuity here is grounded in the endurance of relations. Difference can infinitely tend toward zero without ever reaching it, because it is grounded in the relation that produces difference. In this manner, we can see the Leibnizian world as composed of differential relations that constrain the way expressions pass into one another while maintaining mutual compossibility within the shared world.