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Leibniz and the Leaves: Beyond Identity

Angels, humans, the leaves on a tree; is each one unique or just an example of its kind? Peter Pesic explains why Leibniz thought even leaves are individuals.

Identity is a central question in philosophy, for it touches the deepest issues of being and individuality. Our ordinary word ‘identity’ covers several crucial concepts that should be clarified. Individuality indicates what makes an individual be an individual, rather than simply a member of a species or an instance of some universal quality. Identity, as used strictly by philosophers, indicates how an individual remains the same in different times and places, its self-sameness. Distinguishability or discernibility means how an individual can be told apart from all others.

In the long debate on these issues, Gottfried Wilhelm Leibniz (1646-1716) articulated his famous Principle of the Indiscernibility of Identicals: “There is no such thing as two individuals indiscernible from each other… To suppose two things indiscernible is to suppose the same thing under two names.” That is, each individual is absolutely unique, distinguishable from every other individual. Leibniz had a logical proof of this principle that goes back to arguments given by the Stoic philosophers: If there were two truly identical objects A and B, there would be no reason for God to put A here and B there, and so God (who always does things for a sufficient reason) could never have created such beings. Moreover, Leibniz notes the intimate connection between his principle and the axiom that no two bodies can occupy the same place at the same time; if so, the identity of a body corresponds to its impenetrability. Also, Leibniz holds that if every being is unique, the world is continuous and unbroken; nature does not make jumps but always moves smoothly.

Leibniz often presented this principle in connection with a story, which took place in the garden of Princess Sophie at Herrenhausen about the year 1695. Conversing with Leibniz and another “ingenious gentleman” (gentilhomme d’esprit), the princess observed that she did not believe there were two identical leaves. Yet the “ingenious gentleman” thought he could find some; after an extensive search, he failed because “he was convinced by his eyes that he could always note the difference.”

Leibniz told the story of the leaves as if it exemplified the futility of opposing his Principle not only through argument but also through experience; it is a fixture in many discussions of individuality in metaphysics. Yet the extensive literature on Leibniz’s arguments has not examined why the gentleman was so skeptical. After all, nowadays even children learn that no two snowflakes are the same; why did the gentleman seem so convinced he could easily find identical leaves? I wish to explore his probable assumptions, emphasizing what is new and surprising in Leibniz’s Principle. In so doing, I hope to clarify both the historical background of the Principle and also the remarkable shift that Leibniz implied in the concept of individuality.

Aristotle’s Conception of Identity

The ingenious gentleman was hardly naïve; he was a distinguished diplomat, Carl August von Alvensleben (1661-1697), who corresponded with Leibniz on many occasions before his untimely death. Why, then, did von Alvensleben doubt Leibniz’s assertion to the degree that (as Leibniz put it) he “ran all over the garden a long time” trying to disprove it? Though there is no direct evidence of von Alvensleben’s reasoning in his correspondence, reconsidering earlier ideas about identity can lead to a plausible account because they were likely to have informed his thinking. Among these earlier ideas, the treatments of Aristotle and Plato are especially important.

Though Aristotle considered the individual to be the basis of all classification, he also thought that being a member of a species provided a higher kind of identity than isolated selfhood (which is, in fact, impossible for natural beings), and being a member of a genus higher still. No being in nature could have the kind of uniqueness that is the hallmark of the prime mover, the source and goal of all motion in the earthly and heavenly realms. Hence, Aristotle would not have agreed to the claim that each leaf has a radical uniqueness in the sense that the prime mover has. Here he draws close to Plato’s more extreme position, in which the being of individuals is so grounded in the primal One that the differences between individuals cannot be essential, but reside only in the shadowy realm of images.

Both Aristotle and Plato consider that there is a certain irreducible irrationality in the material world that defies the imposition of intelligible form. In Plato’s version, the creator or demiurge tried to imprint the Forms on primal matter, which only imperfectly received the impression. Aristotle recognizes the power of chance, which cannot be studied scientifically and gives rise to manifold differences between individual instances. In this view, the differences between the leaves of the same species are accidental and should not confuse one as to their essential sameness. The astute Aristotelian scientist classifies leaves without being distracted by their insignificant irregularities, which do not alter their species. Essentially, such leaves are identical. Guided by an Aristotelian education, the ingenious gentleman might well have thought that he could easily find two leaves whose accidental dissimiliarities were so insignificant that he could meet the princess’s challenge.

Aquinas and the Angels

Beyond these powerful teachings of ancient philosophy, von Alvensleben may also have been influenced by medieval treatments, particularly by Saint Thomas Aquinas, who introduced a crucial innovation. His God has a very different relation to the cosmos than the unmoved mover, which is not a creator, since Aristotle considered the order of nature to be eternal. As creator, God gave some creatures personhood, an essential quality of His own essence; as Aquinas puts it, “Person in God is ‘the incommunicable existence of the divine nature’,” quoting Richard of St Victor. Aquinas argues that personhood is a special kind of individuality that can be found in singular, rational beings. Furthermore, there are three persons in God, all equal, and yet each has an incommunicable quality that sets it off as a different genus from the others: the Son is different from the Father, as is the Holy Spirit.

This incommunicable quality marks all true persons, as Aquinas notes, and it raises the question: do persons differ from each other so deeply that each person is a separate species with respect to personhood, or do all persons form a species in the way that all ponderosa pines form a species? There is a disturbing issue here that tends to be hidden for us: in earlier times to be a person was a rare and high dignity, originally applied only to the most eminent human beings. The Greek word for ‘person’ denoted the mask (called in Latin persona) worn by the protagonist of a tragic drama, as if only the crucial figure – for instance, Oedipus – was the person in the play, while all the others have lesser dignity. The Greek theater even reserved a central door for the person, of the three doors shown on stage; none but the person could pass through it. This structure continued to be reflected in the architecture of the Orthodox Christian church: of the three doors in the iconostasis (the screen before the altar), the central (or ‘royal’) door is reserved to the priest alone.

In Roman law, only the paterfamilias was truly a person in the eyes of the law; all the members of his household, including his children and slaves, fell under his authority, which gave him absolute right over their lives, including the right to kill them at will. Not to speak of the subjection of his slaves or wife or daughters, even his sons did not become persons while their father lived, no matter how old they were, unless their father specifically gave them their personhood. In order to give public form to this ceremony in a period when fewer and fewer Roman citizens spoke Latin, Roman jurists evolved a kind of legal pantomime in which the father ceremonially slapped his son in the face and, in so doing, turned the son around to face the court. This legal mime of a violent encounter between father and son reminds one of the tragic drama that seemed to the Greeks the crucible of personhood. Following this confrontation, the son metaphorically puts on the mask by which the court recognizes him as a being of sufficiently independent power that he may be heard in legal deliberations.

From this perspective, personhood regains its original, weighty meaning. As Aquinas confronted the difficult questions concerning the mode of existence of angels, he decided that they must share in the superlative individuality of God, for they too are immaterial spirits. Thus, the kind of uniqueness and individuality that Aristotle reserved to his prime mover alone, Aquinas extended to the angels. Each angel is a person, whose essence is incommunicable and unique, and each is a species unto itself, not to be confused with any other angel. In contrast, all ponderosas share a single species, regardless of their different shapes or sizes; in that sense, they do not have as much individuality as angels. Human beings lie somewhere in between; as animals, they belong to a single species, Homo sapiens, but as persons each is unique.

Leibniz’s Innovations

Leibniz was steeped in scholastic thought, which influenced him even as he reached beyond it. In his Discourse on Metaphysics, Leibniz explicitly extends Aquinas’ claim about angels to all substances: “It is not true that two substances can resemble each other completely and differ only in number [sole numero], and that what Saint Thomas asserts on this point about angels or intelligences (quod ibi omne individuum sit species infima) [that here every individual is a lowest species] is true of all substances, provided that one takes the specific differences as the geometers do with respect to their figures.” This is a startling generalization, and could well have seemed so to von Alvensleben; how can each leaf be essentially different from every other, even though they came from the same species of plant? How could the dignity and uniqueness appropriate to an angel be appropriate to a leaf?

Leibniz takes the little differences between leaves to be allimportant, whereas Aquinas and Aristotle would consider them insignificant compared to their shared identity given by their parent trees. For Aquinas, angels and immortal souls were unique and distinguishable because of their divine provenance, while leaves lack the exalted dignity of these unique spirits. To be sure, Leibniz certainly did not mean that the leaf had personhood. Yet how can a leaf gain a degree of individuality seemingly appropriate only to persons, something denied to many human beings under Roman law? And why does the individuality of a leaf stem from physical, rather than invisible, distinctions?

For Leibniz, external differences are the expressions of each individual’s uniqueness. However, he recognizes that if I pick up a leaf and put it somewhere else, that does not changes its individuality, which is independent of location in space or time. Similarly, I might change the leaf’s shape or paint it a different color without giving it a new individuality, much less one that is the same as another leaf. Moved by these considerations, Leibniz argues that, though not a person, each leaf has an individuality that sets it apart from all other leaves, just as each angel is set apart from all others. In his view, this realization extends to all beings, for he would have been deeply troubled if there were one kind of individuality for some beings, and a quite different kind for others. He rejected a sharp break between rational creatures having unique individuality and the rest of the world lacking it. For Leibniz, all nature is of a piece, a smooth, seamless whole in which there are no breaks or discontinuities.

In his later works, he came to consider this whole to be composed of monads, as he called them (the name means a unit or singleton), which are “the true atoms of nature,” each one “nothing but a simple substance that enters into compounds.” In his vision, there is no essential differences between the microscopic world and the macroscopic, or between animate and inanimate. Each monad is unique, so that “only someone who is capable of grasping the infinite could know the principle of individuation of a given thing.” All monads have the complex individuality of living things and they all mirror each other, each in their own peculiar way, because they share the same dignity.

Here Leibniz decisively goes beyond even those earlier thinkers who seemed to anticipate his formulations. When Nicholas of Cusa wrote in On Learned Ignorance that “No two things in the universe can be simply identical in everything,” he meant that all beings participate unequally in the One God, not that they are indivisible, isolated monads. Leibniz requires the pre-established harmony of monads to replace the divine unity of substance that is Cusa’s version of the Platonic One. This is the price of the radical individualism of each monad, and Leibniz was daring enough to accept it. Though he differed profoundly with Isaac Newton on other philosophical issues, here Leibniz articulated a crucial axiom of Newtonian dynamics; as Ludwig Boltzmann noted, the distinguishability and continuity of motion of each material point is the “first fundamental assumption of mechanics.”

Yet Leibniz does not locate the uniqueness of each leaf in a ‘primitive thisness’ that is separate from all its observable properties. Instead, all those properties, taken together, are what he means by the individuality of the leaf. Moreover, these properties do not exist in isolation from those of other leaves and of the rest of the world. As was just noted, each leaf has a place in the seamless fabric of nature. The leaf’s individuality is its ‘complete notion’, meaning the total of all the possible things that can be said of it, including everything it ever is, was, or will be. Each leaf, in its unique way, thus reflects the course of the whole universe.

This exalted vision may distract us from an important shift. True, each leaf is unique, but not because of some incommunicable ‘thisness’ hidden beneath its observable properties. Leibniz quietly removes that deeper ground of individuality and replaces it with something fundamentally collective in character. Individuality rests not selfishly ‘within’ the leaf but in its interrelationships with other beings in the cosmic community. The leaf is unique in the way that all beings are, so that each owes its individuality to all the others, never really claiming anything for itself alone. Equality and individuality for all may require unique ‘thisness’ for none. If so, the door stands open to even more radical kinds of equality, for nothing grounds the individuality of each in itself, apart from others. Though Leibniz did not believe in atomic theory, removing primitive thisness will later open the possibility to a thoroughgoing equality of atoms in the twentieth century.

Leibniz’s own political views were conservative, but one can compare his view of individuality with modern notions of personhood extended to all, not restricted to a privileged few. Holding such views in a democratic age, we tend to grant each snowflake its uniqueness, as if that individuality were its birthright. We have ceased to wonder whether the equality of the snowflakes (or people) does not contradict their essential inequality and uniqueness. Despite our passion for equality, the notion of divine providence remains compelling. How can God watch over the fall of a single sparrow unless He can always pick it out from all others?

However, the Princess was right in thinking that the question of identity may not be purely logical but rather that it requires experimental testing. It was admirable that von Alvensleben searched so long, though perhaps he gave up too soon. Ironically, if he had been aware of electrons (or any other fundamental particle of modern physics), he could have presented the Princess not just with two identical electrons, but with an endless stream of identical beings, for electrons altogether lack the individuality that Leibniz argued was an essential attribute of a substance. Some philosophers have tried to rescue Leibniz’s Principle by having each particle retain an underlying individuality, but not allowing it any expression in observable processes. I do not think Leibniz would have been satisfied with this compromise, for the essence of his Principle is that the unique identity of each substance is ineluctably manifest in all its outward interactions. If some monads had a hidden individuality, they would be set apart from those that did not, violating his axiom that nature is consistent and seamless.

Though Leibniz offers purely logical arguments against the possibility that two indiscernible things could exist, he also adds that “it is a great objection against indiscernibles that no instance of them is to be found.” This may be a unique case in which he calls on observation, not just logic, to buttress a philosophical axiom. Furthermore, in On Nature Itself (1698), Leibniz even envisaged the consequences if portions of matter were in every way identical, as in two concentric spheres, one revolving, one at rest: “no observer, not even an omniscient one, would detect even the slightest indication of change.” Though he immediately rejected this as “alien to the nature and order of things,” Leibniz recognized the hypothetical possibility of such identicality: “When I deny that there are two drops of water perfectly alike, or any two other bodies indiscernible from each other, I don’t say it is absolutely impossible to suppose them, but that it is a thing contrary to the divine wisdom, and which consequently does not exist.”

Yet Leibniz knew that there were other possible understandings of individuality. He had studied closely Spinoza’s assertion that “the whole of nature is one individual, whose parts, that is, all bodies, vary in infinite ways, without any change of the whole individual.” Spinoza also argued that parts of that substance “cannot be really distinguished” and therefore “bodies are distinguished from one another by reason of motion and rest, speed and slowness, and not by reason of substance.” As I have argued elsewhere, this is very close to the transcendence of identity in quantum theory.

Though he was mistaken in thinking that he could easily find two identical leaves, von Alvensleben was right to doubt Leibniz’s assertion. In a similar spirit, children keep looking for identical snowflakes, despite what they have heard. Ironically, by assuming that perfect sameness was contrary to the divine wisdom, Leibniz (like Einstein, long after him) ignored the possibility that God had, indeed, chosen identicality. Though he paid homage to “the art of inquiry into nature itself and of putting it on the rack … – the art of experimenting which Lord Bacon began so ably,” Leibniz did not see how deeply experiment might upset his reasoning. Nevertheless, Leibniz’s greatness lay in realizing that the issue of identity is crucial both for philosophy and for physics.

Peter Pesic’s book Identity: Physics, Philosophy and Literature will be published by MIT Press during the coming year.