Picture a universe made up of an infinity of points of view, each of them a minuscule soul that, without ever looking out of a window, mirrors the entire cosmos within itself — and which is nonetheless in perfect agreement with all the others, as though every clock in a vast hall had been set once and for all to strike the same hour for eternity. This is the world of Gottfried Wilhelm Leibniz (1646–1716): the last great universal scholar of the modern age — mathematician, logician, jurist, diplomat, historian and theologian — who attempted the most ambitious of syntheses, reconciling the mechanical science of his time with a metaphysics of substances and with Christian theology. His philosophy is at once one of the most coherent systems ever built and one of the bridges that connect Aristotle to the mathematical logic of the twentieth century.

1. Context: rationalism and the polymath

1.1 Life and work

Born in Leipzig in 1646, two years before the end of the Thirty Years’ War, Leibniz trained as a lawyer and spent most of his life in the service of the House of Hanover, as counsellor, librarian and historian. He was among the founders of the Berlin Academy of Sciences. In mathematics he developed the infinitesimal calculus independently of Isaac Newton — the notation we still use today (the d of the differential, the integral sign) is Leibniz’s. The priority dispute between the two men, stoked by their respective partisans and adjudicated partially by the Royal Society of London in 1712, poisoned his final years; the present historical consensus is that both reached the calculus independently.

Together with Descartes and Spinoza, Leibniz belongs to the tradition of continental rationalism: the conviction that reason, not sense experience, is the ultimate source of genuine knowledge, and that reality has an intelligible structure that thought can reconstruct a priori.

1.2 The genre of his works

Unlike Spinoza, who compressed his system into a single geometrical treatise, Leibniz scattered his across letters, short pieces and replies to interlocutors. The decisive texts are:

  • The Discourse on Metaphysics (1686), a concentrated exposition of the system, accompanied by a rich correspondence with the theologian Antoine Arnauld;
  • The New Essays on Human Understanding (written around 1704, published posthumously in 1765), a point-by-point refutation of the empiricism of Locke’s Essay;
  • The Essays of Theodicy (1710), the only major work published in his lifetime, devoted to the problem of evil;
  • The Monadology (1714), a synthesis of ninety numbered paragraphs that condenses the whole of his mature metaphysics.

2. The great principles

The entire Leibnizian philosophy rests on a handful of logical principles that he raises to the rank of laws of being.

2.1 Non-contradiction and sufficient reason

The principle of non-contradiction governs necessary truths: nothing can both be and not be at the same time and in the same respect. But Leibniz adds a second principle that becomes his signature: the principle of sufficient reason, according to which nothing happens without a reason that explains why it is so and not otherwise (Monadology, §§31–32). No fact is true, no statement true, without there being a sufficient reason for it — even if, in most cases, that reason lies beyond our reach.

2.2 The identity of indiscernibles and the principle of the best

Two celebrated corollaries follow from the principle of sufficient reason. The identity of indiscernibles holds that there cannot be two absolutely identical substances: if two supposed beings shared all their properties, there would be no sufficient reason for God to create them as two, and they would therefore be one. In nature, Leibniz said, there are no two perfectly identical leaves.

The principle of the best applies sufficient reason to creation: among the infinity of possible worlds, God, being supremely wise and good, had to have a reason for choosing one — and that reason can only be that the chosen world is the best possible.

3. Truths of reason and truths of fact

Leibniz distinguishes two kinds of truth. Truths of reason are necessary: their opposite is impossible because it implies a contradiction, and they reduce, through analysis, to identities (the whole is greater than the part; the theorems of geometry). Truths of fact are contingent: their opposite is possible, and they depend on the principle of sufficient reason (Caesar crossed the Rubicon; this world exists and not another).

The key to the system is the thesis that, in every true proposition, the predicate is contained in the subject (praedicatum inest subiecto). This leads to the notion of the complete concept of an individual substance: the concept of Caesar includes, from all eternity, everything that will truly be said of him — that he would cross the Rubicon, that he would be murdered in the Senate. Whoever possessed the complete concept of a substance would read in it the whole of its history. The difference from truths of reason is only that, in the case of contingencies, this analysis is infinite and only God can complete it.

4. The monads

4.1 Simple substances

The Monadology opens with a definition: the monad is a simple substance, that is, one without parts (§1). Since the composite is nothing but an aggregate of simples, there must be simple substances; and what has no parts can be neither extended nor divided nor destroyed by natural means — monads can only begin by creation and end by annihilation. They are therefore the true atoms of nature, but metaphysical and immaterial atoms, not physical corpuscles.

4.2 Windowless, yet alive

The most famous formula is that monads have no windows through which anything could enter or leave (§7). Nothing external modifies them; every change springs from an internal principle. This internal principle has two aspects: perception — the state by which the monad represents the manifold in the one, mirroring the universe from its point of view — and appetition — the tendency that carries it from one perception to another. Each monad is thus a living, perpetual mirror of the universe (§56): it contains the whole in some manner, but represents it distinctly only in a slice and confusedly in the rest.

4.3 The hierarchy of monads

Monads differ in their degree of perceptual clarity. The simplest have only confused perceptions, without consciousness (the “bare” monads). Souls add memory and more distinct perceptions (animals). Spirits — rational souls such as the human one — add apperception, reflective consciousness and the knowledge of necessary truths and of God. At the summit stands God, the Monad of monads. What we call a body is a well-founded aggregate of monads, dominated by a central monad that serves as its soul or entelechy.

5. Pre-established harmony

If monads have no windows, how can we explain that soul and body seem to act upon one another — that the will to raise an arm is followed by the arm rising? Leibniz rejects both of his predecessors’ solutions. Cartesian interactionism, which has the soul act on the body through the pineal gland, violates the physical laws of conservation. Malebranche’s occasionalism, which makes God the cause that moves the body on each occasion of an act of will, turns the divinity into a perpetual and unworthy intervention.

Leibniz’s solution is pre-established harmony: God created each substance such that, unfolding according to its own internal laws, it agrees exactly with all the others. Soul and body do not interact; they correspond. The image Leibniz himself uses to explain it — notably in his reply to Foucher and in examining the hypotheses about the union of soul and body — is that of two clocks set in perfect agreement: they are joined by no thread, nor adjusted at every instant by a clockmaker, but, having been built with such skill and regulated from the start, they always strike the same hour. Such is the universe: a concert with no conductor present, because the conductor composed it in advance.

6. The Theodicy and the best of all possible worlds

6.1 The argument

The term theodicy was coined by Leibniz from the Greek theós (God) and díkē (justice): it is a matter of “justifying God” in the face of the evil of the world. The problem is a classic one: if God is omnipotent, omniscient and supremely good, where does evil come from?

The answer marshals the principles already seen. In the divine mind there are infinite possible worlds — infinite compossible combinations of substances. God, through his understanding, knows them all; through his goodness, wills the best; through his power, creates what he has chosen. By the principle of the best, the existing world is the best of all possible worlds: not a world with no evil at all (which would be impossible, since it would amount to creating other gods), but the one in which the greatest quantity of perfection, order and variety is achieved with the minimum necessary imperfection.

6.2 The three kinds of evil

Leibniz distinguishes metaphysical evil — the finitude and imperfection inherent in every creature, simply by virtue of not being God — physical evil — suffering and pain — and moral evil — sin. Metaphysical evil is the root: being limited, the creature errs and suffers. God permits evil without directly willing it, because it is the condition of greater goods and of the harmony of the whole, as a shadow brings out a painting or a dissonance enriches a piece of music.

6.3 Voltaire and Pangloss: the satire and the thesis

No one shaped the popular image of Leibniz more than Voltaire in Candide, or Optimism (1759). The character Pangloss, a teacher of “metaphysico-theologo-cosmolo-nigology,” repeats in the face of every catastrophe — the Lisbon earthquake of 1755, war, disease — that “all is for the best in the best of all possible worlds.” The satire is biting, and deserved against a certain complacent optimism.

It must be said, however, in fairness to Leibniz: his thesis does not claim that everything is good, nor that there are no real evils to combat. It claims that, given that a finite world had to be created, this is the one with the best possible balance — a metaphysical claim about the divine choice, not an invitation to passivity. The Pangloss caricature and the Leibnizian thesis do not coincide, and conflating them is a common error worth undoing.

7. Logic, language and the dream of a calculus

Perhaps the most surprising legacy of Leibniz lies in his logical projects. He dreamed of a characteristica universalis: a language of symbols that would represent concepts as numbers represent quantities, decomposing complex ideas into simple ones. Coupled with it, a calculus ratiocinator would allow disputes to be settled by calculation — hence his famous remark that, faced with a disagreement, two philosophers could simply take up their pens and say: “Let us calculate.”

Although the project was not realized in his own day, it remarkably anticipates the symbolic logic that Gottlob Frege and Bertrand Russell would found in the late nineteenth and early twentieth centuries. Russell himself wrote a celebrated study, A Critical Exposition of the Philosophy of Leibniz (1900), recognizing him as a forerunner. The reduction of all reasoning to operations with symbols over a finite alphabet is, in turn, one of the intellectual roots of mathematical logic and, by extension, of computer science.

Critical analysis

The strength of the Leibnizian system lies in its rare coherence: few bodies of thought derive so much consequence from so few principles. And his logical pioneering is undeniable — in metaphysics, in mathematics and in the vision of a formal language of thought, Leibniz looks centuries ahead.

The objections are equally serious. The first concerns freedom: if the complete concept of a substance already contains everything it will do, in what sense was Caesar free in crossing the Rubicon? Leibniz answers that contingency is preserved because the opposite implies no contradiction (God could have created another Caesar), and that a free action is one that springs from the inclination of the substance itself without absolute necessity — but many critics find the way out insufficient to escape an underlying determinism.

The second objection is the theological leap of the “best”: that among infinite worlds there should be a single best one, and that this should be ours, are claims that depend entirely on divine goodness and wisdom, and that the experience of evil makes hard to assimilate — which was precisely the point of Voltaire’s satire. The third is the obscurity of windowless monads: immaterial substances, without interaction, whose agreement rests on a harmony guaranteed only by God, verge, for the empiricist critic, on an unverifiable postulate.

Legacy

Leibniz’s influence branches out in unexpected directions. In German idealism, Kant engages with him critically (the Critique of Pure Reason discusses the “amphiboly of the concepts of reflection” against the identity of indiscernibles), and Hegel sees in the monad a moment of consciousness developing out of itself. In modern logic, his characteristica prefigures Frege and Russell. In the philosophy of computation, the idea of reducing thought to symbolic calculation is one of the discipline’s conceptual matrices, and Leibniz is often remembered for his interest in binary arithmetic.

Finally, in contemporary modal metaphysics, the Leibnizian notion of possible worlds is reborn with extraordinary vigor: Saul Kripke’s possible-worlds semantics for modal logic, and David Lewis’s modal realism — for whom possible worlds are as real as our own — inherit, even as they radically transform it, the vocabulary that Leibniz coined three centuries earlier. Few philosophers of the past converse so directly with the analytic philosophy of the present.

  • Leibniz, Discourse on Metaphysics (1686).
  • Leibniz, The Monadology (1714).
  • Leibniz, Essays of Theodicy (1710).
  • Leibniz, New Essays on Human Understanding (c. 1704, posthumous publication).
  • Bertrand Russell, A Critical Exposition of the Philosophy of Leibniz (1900).

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