Gottfried Wilhelm Leibniz

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Gottfried Wilhelm Leibniz was a German philosopher, mathematician, and universal scholar, one of the last figures who could plausibly claim command of the whole of learning in his day. Born in Leipzig in 1646, he worked as a diplomat, librarian, and court councillor, corresponded with hundreds across Europe, co-invented the calculus independently of Isaac Newton, and built a metaphysical system whose strangeness has kept it alive. He died in Hanover in 1716, in some disfavor, and was buried with little ceremony.

His life was lived in the antechambers of power and the back rooms of libraries rather than in a university chair, which he never held for long. The son of a Leipzig professor, he read his way through his father’s library as a boy, took his degrees in law and philosophy, and entered the service of the Elector of Mainz before he was twenty-five. A diplomatic mission carried him to Paris in 1672, where he spent four formative years among the leading mathematicians and natural philosophers of the age and worked out the differential and integral calculus — the notation he devised, with its d and its elongated S for the integral, is the one still in use. From 1676 he served the House of Brunswick-Lüneburg at Hanover as librarian and councillor, a post he held for the rest of his life: he reorganized the ducal library, chased the genealogy of his patrons through the archives of Europe, designed mining machinery in the Harz mountains, drafted schemes for the reunion of the churches, and founded the Berlin academy of sciences. The breadth was not a hobby. For Leibniz the unity of knowledge was a metaphysical conviction before it was a research program, and the same mind that pursued a binary notation pursued a universal symbolic language in which disputes might be settled by calculation. When the priority quarrel with Newton over the calculus turned bitter, and his Hanoverian masters ascended the British throne as the house of George I without taking their aging librarian to London, he ended his days largely alone with his papers. Only his secretary, by the surviving account, attended the burial.

The monad

His mature metaphysics turns on the monad — from the Greek for “unit.” In the Monadology, a short late summary written in French in 1714 and circulated only after his death, Leibniz held that the ultimate constituents of reality are not bits of matter but simple, indivisible, immaterial centers of perception, “windowless,” each mirroring the entire universe from its own point of view. The word simple is exact: a monad has no parts, and what has no parts cannot be broken, cannot begin or end by natural means, and cannot be entered. Extended matter, being endlessly divisible, can never reach a true unit; so the real, for Leibniz, must lie beneath the divisible world in points that are not points of space but points of view. Each is charged with perception — not necessarily conscious, but a representing — and with an inner appetite that drives it from one perception to the next. Nothing passes between them; there are no windows through which an influence might come or go. And yet each monad expresses the whole, as a town looks different from every quarter while remaining one town, so that the universe is multiplied without being divided, reflected entire in every one of its living mirrors.

The monads are not all of one rank. The bare ones perceive without remembering; souls add memory and feeling; rational spirits add reflection and the knowledge of necessary truths, and stand in a higher relation to God as minds capable of entering, in their small way, into the architecture of the whole. What we call a living body is a colony of monads gathered under a dominant one, the soul, so that an animal is not a machine with a ghost inside it but an organism alive all the way down, every part of it teeming with further organisms without end. No two monads are ever exactly alike, for if they were they would be one — Leibniz’s principle that there are no two indiscernible things forbids the bare repetition that an empty space or an inert atom would require. The world is variety without remainder, plenitude pressed as full as logic allows.

If the monads are sealed against one another, the order among them has to come from elsewhere. Leibniz located it in a pre-established harmony: God, surveying at creation all the possible monads and all their possible histories, chose and coordinated them so that each unfolds its own series from within while the series agree perfectly, the way two clocks set going together keep the same time without one driving the other. What looks like causation between bodies, or between body and soul, is the synchronized running of independent timepieces that God wound once. From this followed his most quoted claim, set out in the Theodicy of 1710 — the book that gave the word theodicy to the language, its full subject being the goodness of God, the freedom of man, and the origin of evil. A perfect God, choosing among all the worlds he might have made, must have chosen the best; so that ours is the best of all possible worlds, and the evil in it is the unavoidable shadow that a greater good casts, a dissonance resolved in a harmony too large for any creature to hear entire. The phrase was later turned against him as a byword for naive optimism — Voltaire built the whole of Candide around the chatter of Dr. Pangloss, who insists that all is for the best amid one catastrophe after another — but in Leibniz’s own argument it was a tight piece of reasoning about why a good God permits evil at all, anchored in his principle that nothing is so without a sufficient reason.

A word with a long charge

The word monad was not new. It carried a long Pythagorean and Neoplatonic charge. For the Pythagoreans, number was the substance of things, and the monad — the unit, the One — was the principle from which the whole series of numbers, and so the whole order of being, descended; Pythagoras himself stands at the head of that lineage of thought. The Neoplatonists raised the same intuition to its height: the One is the source beyond being from which all multiplicity flows by emanation, each lower level a diminished image of the level above, the cosmos a graded descent and return that the formula as above, so below compresses to a phrase. By the Renaissance the term had passed through the occult philosophers who fused that Platonism with magic and number. Cornelius Agrippa gave the monad a place in his arithmetic of powers; Giordano Bruno wrote a treatise De monade on the unit as the seed of all plurality; the language of the One as a coincidence of opposites came down from Nicholas of Cusa. When Leibniz reached for monad he was lifting a word saturated with this Neoplatonic inheritance and with the older science of number as the key to the real.

Scholarship has shown that the young Leibniz read deeply in this material. Christia Mercer’s study of the origins of his metaphysics traces the Platonist strain back to his Leipzig teachers, Jacob Thomasius and Johann Adam Scherzer, who handed him a metaphysics of God’s relation to the created world that he never abandoned even as he fitted it out with the rigor of the new mathematics. His contacts ran into the living currents of the occult and the heterodox-pious seventeenth century: he read and corresponded with the Cambridge Platonists, and his exchange with Anne Conway — whose own treatise argued for a world of nested living monads, the term itself appearing in her circle before his — left a mark he acknowledged, crediting her vocabulary as kin to his own. He kept the language of emanation and living force while recasting it in the severe, almost mathematical idiom that is unmistakably his.

The Nuremberg circle

His early years brushed the hermetic world directly. Around the winter of 1666–67, freshly graduated and not yet twenty-one, Leibniz served as secretary to an alchemical society in Nuremberg — his first paid employment. The fullest account is a famous anecdote from his early biographer Johann Georg Eckhart: that the young man won the post by composing a letter packed with imposing alchemical terms he did not himself understand, and so impressed the adepts that they took him for an initiate and made him their scribe. The story is repeated everywhere and corroborated nowhere else, and its reliability is open to doubt; what is firmer is that Leibniz moved for a season among men devoted to the work of transmutation, and that the circle has since been linked — loosely, and on later evidence — with the Rosicrucian enthusiasms then circulating through the German lands. The historian George MacDonald Ross, who weighed the episode most carefully, found the Rosicrucian label asserted more often than established, and the secretarial duties harder to pin down than the legend suggests. How much of Leibniz’s philosophy grew from that soil remains debated. He never renounced alchemy outright, kept a working interest in chymical matters into middle age, and carried forward a picture of nature as everywhere alive and active — but the line from the Nuremberg laboratory to the windowless monad is one of atmosphere and suggestion rather than of any document he left.

The binary and the hexagrams

One thread has drawn particular attention. Leibniz worked out the binary arithmetic that underlies modern computing, writing every number with only 0 and 1, and he read into it a theological figure — the creation of all things by God out of nothing. As early as 1697 he sketched a commemorative medal whose design would carry the idea as an emblem: from the unit, standing for God, and the zero, standing for the void, every number whatever can be generated, an image of creation ex nihilo struck in arithmetic. The notion fused his mathematics with the deepest of his theological convictions and the oldest of the Neoplatonic intuitions — that all multiplicity issues from a single source that is itself beyond multiplicity.

Then the figure seemed to be confirmed from an unexpected quarter. A Jesuit missionary in China, Joachim Bouvet, sent Leibniz a printed diagram he had encountered at the Beijing court: the arrangement of the sixty-four hexagrams of the I Ching attributed to the ancient sage Fu Xi, the figures built from broken and unbroken lines. When it reached Hanover in the spring of 1703, Leibniz saw in those lines an image of his own binary notation — the unbroken line a 1, the broken line a 0 — and took the correspondence as evidence that an archaic Chinese wisdom had grasped the same order he had reconstructed by reason. He published his binary system that year with the Paris academy, the title itself pointing to the light it threw, he believed, on the old Chinese figures. The reading let him fold a foreign antiquity into his own scheme of a prisca knowledge once held and since scattered, the same hope of a single recoverable order that ran beneath the whole hermetic enterprise. Historians treat the resemblance as real but the inference as his own enthusiasm rather than a recovered secret: the Fu Xi sequence can be mapped onto binary because any complete ordering of two-valued symbols can be, not because the diviners of antiquity were counting in twos.

What makes Leibniz a borderline figure for the esoteric tradition is exactly this doubleness. He is a founder of modern logic and mathematics who never quite left behind the world of correspondences, hidden harmonies, and a cosmos in which every smallest unit reflects the whole. In him the monad of the Pythagoreans becomes a substance to be reasoned about with the precision of a proof, and the creation of number from nothing becomes at once an arithmetic and a confession of faith. The system he left is severe and luminous at once, and it has been read in both directions ever since.

Texts and scholarship

The Monadology survives in many hands; a clear modern English version is the freely available edition by Jonathan Bennett of the Principles of Philosophy known as Monadology (1714), which sets out the windowless monad and the pre-established harmony in plain terms. The metaphysics of perception, expression, and harmony is treated with care in the Stanford Encyclopedia of Philosophy’s survey of Leibniz’s philosophy of mind, and the argument of the Theodicy — the best of all possible worlds and the place of evil within it — in its companion article on Leibniz on the problem of evil. The standard intellectual biography is Maria Rosa Antognazza’s Leibniz: An Intellectual Biography (Cambridge University Press, 2009), which won the History of Science Society’s Pfizer Award and remains the fullest single portrait of the man and his world. The Platonist roots of his system are the subject of Christia Mercer’s Leibniz’s Metaphysics: Its Origins and Development (Cambridge University Press, 2001), which dates the harmony far earlier in his development than once supposed and traces it to his Leipzig schooling. The Nuremberg episode and the larger question of his debt to alchemy are examined most soberly in George MacDonald Ross, Leibniz and the Nuremberg Alchemical Society, Studia Leibnitiana 6 (1974), 222–248. Behind all of these stands the Neoplatonic source that gave monad its weight, read most directly in Plotinus on the procession of all things from the One.

→ In the library: The Enneads (MacKenna) — V. 1, The Three Initial Hypostases

→ Related: Neoplatonism · The One · Emanation · Cornelius Agrippa · Pythagoras · Pythagoreanism · Rosicrucianism · Alchemy · I Ching · China · Numerology · Anne Conway · Giordano Bruno · Nicholas Of Cusa · Isaac Newton · Cambridge Platonism · As Above So Below