Gottfried Wilhelm Leibniz (1646–1716) was born in Leipzig and spent most of his life as librarian and dynastic historian to the Duchy of Hanover. Law, diplomacy, mining reports, library catalogues, the differential and integral calculus he developed independently of Newton, writings on Chinese philosophy — all belonged to his extreme intellectual range. One of the longest and most bitter priority disputes in the history of mathematics — over who had invented the calculus — peaked in 1712 when the Royal Society ruled, with a stacked committee, in Newton's favour. Two years later, in the last phase of his life, Leibniz wrote the densest summary of his philosophical system.

The Monadology is 90 short paragraphs. Its opening sentence states the thesis: "The monad, of which we shall here speak, is nothing but a simple substance, without parts." Matter is divisible and therefore cannot be ultimate reality; what is ultimate are simple, indivisible units — monads. But monads are not physical atoms: they are mental rather than material, each a "mirror" of the whole universe from its own point of view, equipped with internal perception and appetition. Monads have no "windows" — that is, they do not causally interact with one another. So where does the visible order of the universe come from? Leibniz's answer is pre-established harmony (harmonia praestabilita): God has set the inner series of each monad in such a way that they synchronize with all the others, as if separate clocks had been wound to chime in unison.

Beneath the Monadology lie two great principles: non-contradiction and sufficient reason. By the principle of sufficient reason nothing is without a cause; for every fact there is some reason why it is so and not otherwise. From this principle Leibniz draws his famous claim: God has chosen the actual world from among infinitely many possible worlds, and because the choice must rest on sufficient reason, the world chosen must be the best of all possible worlds. After the 1755 Lisbon earthquake, Voltaire turned this thesis into Pangloss's sneering slogan in Candide. Leibniz's original argument, however, is subtler: "best" does not mean the maximum of happiness but, roughly, the maximum of variety from the minimum of principles.

Leibniz is the third great figure of seventeenth-century continental rationalism, alongside Descartes and Spinoza; but unlike them he was not only a metaphysician. His characteristica universalis — a universal logical language in which every concept would be symbolised and reasoning turned into calculation — combined with his work on binary numbers and his mechanical Stepped Reckoner, makes him the philosophical ancestor of the modern logic and computation that runs through Frege, Boole, Russell, and Turing. "Calculemus!" he wrote: instead of arguing, let us take up our calculating machines, sit down together, and reckon which of us is right. Three centuries later this dream took shape as the computer.