Consequences of Leibnizian Complete Notions


In section eight of his Discourse on Metaphysics, Leibniz states: “…the nature of an individual substance or of a complete being is to have a notion so complete that it is sufficient to contain and to allow us to deduce from it all the predicates of the subject to which this notion is attributed.” This paper will first delve into the meaning of this definition and many of its logical consequences. An analogy will be drawn between Leibniz’s conception of the universe and the branch of mathematics known as dynamical systems in order to give a more complete understanding of necessity, contingency, and the implications these have on free will.

The formal definition above of a complete notion means that a substance and its characteristics and actions are inseparable. The detail about actions is the most unique to Leibniz’s account of a complete notion for while few people would disagree with the fact that the notion of my ‘self’ includes all my physical characteristics (although Leibniz could argue that matter is nothing but a phenomenon), a Leibnizian complete notion (LCN) would also include everything that has happened, is happening, and will happen to me (me being a representative case for all substances, meaning this argument applies in general). (I will discuss this concept and its implications on free will later.) Every substance must have such a complete notion because a substance is nothing more than the common subject to which many predicates belong. Grammatically, I, the person and substance, am only what could follow in a sentence whose subject is Shane Steinert-Threlkeld (or others depending on who is communicating; for example, I am the real substance to which various subjects [“that kid in the library cubicle,” i.e.] refer). While “God alone could recognize them,” a complete notion of a person contains within it “vestiges of everything that has happened to him and marks of everything that will happen to him” because all the predicates of a substance are completely interrelated in a way that one is not separable from the rest (602). I would not be writing this paper right now if I had not enrolled in this course; my LCN must contain these predicates, which contain infinitely more predicates through a continued regress.

Section 9 of the Discourse details many of the consequences of an LCN. There exist no two substances that resemble each other completely while remaining distinct. Either the two notions that one thinks of as separate substances actually represent the same substance or the knowledge one has of the notions does not constitute an LCN. If one has an LCN for two distinct substances, the two LCNs will not be the same. Therefore, the complete resemblance of two separate substances constitutes an illusion brought about by lack of knowledge, by having incomplete notions. Furthermore, “a substance can begin only be creation and end only by annihilation” (602). This claim follows logically from the fact that a substance contains all predicates from its past, present and future; these predicates will include creation and annihilation which precludes either from happening while the substance truly exists.

Additionally, a substance cannot be divided into two. If such a division were possible, then some predicates would belong to one of the new substances and the rest to the other. Recall, however, that the reason each substance has an LCN is that all its predicates depend on each other. Predicates of a substance, therefore, cannot be separated from any of the others. This same argument works in reverse; one substance cannot be created from two. If possible, the new substance would have two sets of predicates entirely distinct from one another and therefore would not be a substance since its predicates would not completely interrelate. Because a substance can begin and end only through creation and annihilation and substances cannot be recombined, “the number of substances does not naturally increase and decrease, though they are often transformed” (602).

Perhaps the most remarkable consequence of the LCN is that “every substance is like a complete world and like a mirror of God or of the whole universe” (602). Leibniz explains this idea by analogizing each substance to a city seen from various vantage points. Another way of thinking about this result comes from the question that launched modern chaos theory: does the flap of a butterfly’s wings in Brazil set off a tornado in Texas? While this question refers to the sensitivity of a dynamical system to its initial conditions (an idea returned to later), its image reflects the nature of substances, for even though I may not think I feel the flap of a butterfly’s wings in Brazil and the technology does not exist to measure such a miniscule effect, the flap still has an effect on me and every substance in the world. Leibniz makes this point parenthetically in section 15: “in fact every change affects all of them [substances]” (606). Since every change affects every substance, each of its predicates contains implicitly the similar predicates of every other substance (which are interrelated within each substance), the LCN of a substance, being the total of all its predicates, does mirror the entire universe. My LCN, in containing the affect of a butterfly’s flap, contains enough information (location of the butterfly, atmospheric conditions, laws of physics, etc.) to determine the affect the flap had on every other substance in the whole universe. So it goes for every action and every substance (though the form of the implicit containment will differ).

The course that a dynamical system (essentially a set of interdependent functions over time) follows is extremely sensitive to its initial conditions. Returning to the original notion of the butterfly effect, the universe, according to Leibniz, resembles a dynamical system in which God chose the initial conditions (footnote 1). God’s omniscience allows him to see all possible courses for all possible universes and, Leibniz argues, he has chosen the most perfect one. This analogy of the universe to a dynamical system (which is hardly an analogy given that the theory of dynamical system began as an attempt to model “simple” systems in the universe, i.e. the path of a pendulum, the weather, etc.) can extend to include Leibniz’s account of necessary and contingent truths. Even though the universe contains only one set of initial conditions and will follow only one course, the facts that would exist if another course had been chosen are contingent. That I am writing this paper now is a contingent truth because its opposite, even though not physically possible (I cannot both write and not write at the same time), does not represent a logical impossibility. From this perspective, the necessary truths—a truth whose opposite leads to a logical impossibility—are the rules that govern the dynamical system and thus determine the possible courses of the universe. If these rules were not true, the universe could not possibly exist as it does.

This distinction between necessary and contingent truths has many implications for the nature of human freedom in Leibniz’s conception of the universe. While at first glance it may appear that there is room for free will since the facts of our everyday lives are contingent truths, a closer inspection reveals a different realization. Returning to the dynamical system analogy, we see that even though I could theoretically be doing something other than writing this paper right now, because the course of the dynamical system that is the universe has been determined already, it has been inevitable for as long as I have existed that I would be writing this paper in this cubicle at this moment in time. Because only God has complete knowledge of the governing laws of the dynamical system, of all of the necessary truths of the universe, I did not know three weeks ago that this is where I would be right now. Whence comes the illusion of free will. God, however, has always known this fact because my complete notion includes it; I would not be who I am if I were not exactly here right now.

Even though my LCN possesses in it everything that has happened, is happening, and will happen to me, this sort of determinism does not mean that everything that happens to me does so necessarily since the things that happen to me are contingently true. They are true in this universe (which according to Leibniz is both the only and the most perfect one) that I live in but not in all possible universes. It is not necessary that anything that happens does happen, for it is theoretically possible that something else could have happened. But everything that has happened, is happening, and will happen in the universe has been determined by God’s choice of this, the most perfect universe.

While this brand of determinism may seem depressing, to an extent it is very satisfying, especially to anyone with a rationalist bent. According to Leibniz, everything God does is orderly (this makes sense with the dynamical system argument in that he operates within the governing rules of the system) and so “what passes for extraordinary is extraordinary only with some particular order established among creatures; for everything is in conformity with respect to the universal order” (600). If one accepts the Leibnizian conception that God has chosen the most perfect universe, then one should be grateful for having incomplete knowledge of complete notions. The human experience would be extraordinarily less rich without the constant quest for what is not yet known. To want to discover the universal order by which extraordinary is ordinary is to be a human being. Even though accepting Leibniz’s metaphysics means accepting a certain degree of determinism, it also instills a great sense of wonder and the greatest richness possible to the human experience. While many people are satisfied to ignore the great metaphysical questions that Leibniz addresses, those who do entertain them will find the process of scientific and philosophic inquiry as something resembling a gift from God.

Works Cited

Cahn, Steven M, ed. Classics of Western Philosophy. Indianapolis, IN: Hackett Publishing Company, 2006.

Leibniz, Gottfried. “Discourse on Metaphysics.” Cahn 598 – 618.

–“Monadology.” Cahn 619 – 626.

1. Leibniz could not himself say this because the formal theory of dynamical systems did not yet exist. The theory does, as a matter of fact, depend heavily on calculus, a branch of mathematics which Leibniz is widely credited with discovering.

Shane Steinert-Threlkeld (’11) is a Philosophy & Mathematics major at Johns Hopkins University.

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