When Leibniz died in Hanover in November 1716, the man who had been called the last person to know everything worth knowing was buried with almost no one watching. He had been court librarian and historian to the dukes of Brunswick for forty years, and his own duke had just become King George I of Great Britain and moved to London without him. The learned societies of Europe were busy taking Newton's side against him in a fight over who had invented the calculus. So the funeral of one of the broadest minds the world has produced was, by the standard account, attended by his secretary and almost no other person from the court he had served for half his life.
What that lonely grave held was a polymath in the literal sense: a man who did first-rate original work in philosophy, in mathematics, in law, in physics, in geology, in the study of China, in the design of calculating machines, in the new science of running a library. He invented, independently of Newton, the differential and integral calculus, and the notation he chose for it is the one still taught in every school. He built a machine that could add, subtract, multiply, and divide. He founded the Berlin Academy of Sciences and corresponded with hundreds of people across Europe in several languages. And in his spare time, as it were, he produced a so original and so strange that three centuries later it is still not clear whether it is one of the deepest pictures of reality ever drawn or one of the most beautiful mistakes.
The strangeness can be put in one line. Leibniz held that nothing in the universe ever touches, pushes, or affects anything else, and that the world only looks like a place of causes and collisions because God arranged every separate thing, at the beginning, to keep perfect step with all the others forever. The eyes meeting these words are not moved by the page. The page is not lit by the lamp. Nothing reaches across to do anything to anything. And yet everything matches, down to the last detail, because it was all set running in advance to match.
This is the story of how a working mathematician and diplomat came to believe that, and of the machinery he built to make it hold together: the simple substances he called monads, the principle that nothing whatever is true without a reason, the proof that no two things in the world are exactly alike, and the claim he is most remembered and most ridiculed for, that this is the best of all possible worlds. The ridicule has a name, Dr Pangloss, and it has buried what Leibniz actually argued under a caricature for two hundred and fifty years. The argument underneath is harder, sadder, and far more interesting than the joke.
The break
The quarrel Leibniz walked into had two serious answers already on the table, and both deserve their strongest form before his break from them makes sense. Descartes had split the world into two kinds of substance: minds, which think and take up no space, and bodies, which take up space and do not think (the Descartes read). The two share no property, which is exactly why a thought has no obvious way to move an arm, the interaction problem Princess Elisabeth of Bohemia pressed on Descartes until it broke. That is a real difficulty, but the two-substance picture has a real virtue too: it keeps the mind genuinely distinct from mere matter, which is what most people, then and now, actually believe about themselves. It is not a foolish view. It is a strong view with one unpaid bill.
Spinoza paid the bill by tearing up the account. If two substances cannot interact, he reasoned, then stop saying there are two; say there is only one substance in all of reality, God-or-Nature, and that mind and matter are just two ways of describing the one thing (the Spinoza hub). Then the question of how they interact never comes up, because they were never two things to begin with. That is elegant and it is rigorous, and it has its own unpaid bill: it swallows the individual whole. On Spinoza's picture a particular person is not a thing in its own right but a passing ripple in the single substance, and the freedom and separateness each person feels turn out to be an illusion. One substance saves interaction by erasing the individual.
Leibniz refused both. He thought Descartes had too many kinds of substance and Spinoza had too few, and that both had made the same underlying error: treating the basic stuff of reality as something extended, something that takes up space and can in principle be pushed. Anything extended, Leibniz argued, can be divided, and anything divisible is made of parts, and parts are made of smaller parts, and so the search for what reality is ultimately made of never bottoms out in extended matter. It has to bottom out in something with no parts at all. The true atoms of nature are not little bits of matter. They are simple, partless, immaterial points, and there are infinitely many of them, and he called them monads.
The concrete move, the thing that makes this a genuine break and not just a third opinion, is what he does with interaction. Descartes could not explain how his two substances act on each other; Spinoza dissolved the problem by having only one substance. Leibniz does something neither considered: he keeps a plurality of real, distinct individuals, infinitely many of them, and then denies that they interact at all. No monad pushes, touches, or affects any other. The appearance of a world full of causes is real as an appearance and false as a description. What looks like one thing acting on another is two things independently unfolding programs that were written, at the first instant, to match. The problem of how substances interact is not solved. It is abolished, by denying that they ever do. Everything in the chapters ahead, the harmony, the sufficient reason, the best world, is the cost and the payoff of that one refusal.
Chapter 1
The last man who knew everything
"The Monad, of which we shall here speak, is nothing but a simple substance, which enters into compounds."
He was born Gottfried Wilhelm Leibniz on 1 July 1646, in Leipzig, in Saxony, in a Holy Roman Empire two years from the end of the Thirty Years' War that had wrecked the German lands. His father was a professor of moral philosophy at the University of Leipzig and died when the boy was six, leaving behind a large private library. Leibniz, by his own and the standard account, taught himself Latin as a child and was reading freely in that library by eight or nine, devouring whatever he found, ancient and modern, with no one directing him. It is the first sign of the thing that would define him: a mind that did not specialize because it could not see why it should.
He studied law and philosophy at Leipzig and then at Jena, and finished a doctorate in law in 1666–67, not at his home university but at Altdorf, near Nuremberg. The standard account is that Leipzig declined to grant the degree because he was thought too young, barely twenty, and that Altdorf, recognizing what it had, offered him a professorship on the spot. He turned it down. A university chair was a fixed and narrow life, and he wanted the world. He took service instead with the Elector of Mainz, one of the German princes, and became what he would be for the rest of his life: a man of affairs (a diplomat, a counselor, a fixer) who did the deepest philosophy and mathematics of his age in the margins of a working career, mostly in letters, mostly unpublished.
The decisive years were in Paris. From 1672 to 1676 he was there on a diplomatic errand (a scheme to distract Louis XIV from the German states by interesting him in an invasion of Egypt, which went nowhere), and Paris was then the center of the scientific world. He fell in with the Dutch physicist Christiaan Huygens, who took the gifted amateur in hand and showed him how far behind the frontier of mathematics he actually was. Leibniz caught up with frightening speed. Within a few years in and around Paris he had done the core of the work that founds the calculus (chapter 5), and he had built a machine, the Stepped Reckoner, that could carry out all four arithmetic operations by turning a crank, the first that could multiply and divide directly. He crossed to London in 1673 and was shown to the Royal Society, which made him a fellow. The machine impressed; the mathematics, in time, would start a war.
In 1676 he took the post he would hold for the next forty years: librarian, and later official historian and privy counselor, to the House of Brunswick-Lüneburg, the ruling dukes at Hanover. It was a strange harness for so wide a mind. He was set the task of writing a vast history of the Brunswick dynasty, traced back through the centuries to glorify the family, and he chased it across Europe through archives and monasteries, and never finished it; it was the great unfinished labor of his life. But the position gave him a base, an income, a magnificent library, and the freedom to correspond with everyone. From Hanover he wrote thousands of letters to hundreds of scholars, ran arguments across the continent by post, and in 1700 founded and became first president of the Berlin Academy of Sciences. He kept a foot in mathematics, law, theology, geology, engineering, and the study of China, where he saw in the ancient hexagrams of the I Ching a confirmation of his own discovery of binary arithmetic, counting in nothing but zeroes and ones, the system every computer now runs on.
The unifying thread under all of it (the law, the diplomacy, the math, the dynastic history, the letters to Jesuits in China) was a single conviction that sounds mad until it is set beside how he lived: that human knowledge and even human disagreement could in principle be made exact, calculated, settled the way a sum is settled. He spent his career trying to reconcile things other people thought irreconcilable: the Catholic and Protestant churches, the old Aristotelian physics and the new mechanical one, free will and a world fixed by God. He was constitutionally a reconciler, a man who assumed that behind every clash of views there was a deeper harmony if the right terms to state it in could only be found. The metaphysics of monads is that temperament turned into a picture of reality itself.
The end was bitter, and it matters because it is the shadow over everything he is remembered for. By 1716 he had fallen out of favor. His own employer, the Elector George Ludwig of Hanover, had in 1714 become King George I of Great Britain and gone to London, pointedly leaving his famous philosopher behind in provincial Hanover to finish the dynastic history, partly because Leibniz was by then a public embarrassment in England, locked in a vicious priority fight with Newton, the idol of British science (chapter 5). The learned world of Europe had largely sided with Newton. When Leibniz died in Hanover on 14 November 1716, of gout and other ailments, the standard account is stark: though he had served the court for forty years, his funeral drew almost no one, and only his secretary followed the coffin to the grave. The man who tried to harmonize all of Europe's quarrels died on the losing side of one of them, half-forgotten, in a town his king had left.
Chapter 2
The windowless points
"The Monads have no windows, through which anything could come in or go out."
The argument starts from a question so simple it sounds like a child's: what is the world ultimately made of? Take any ordinary thing, a stone, a table, a body. It is extended; it takes up space; and anything that takes up space has a left half and a right half, so it has parts. The parts are extended too, so they have parts, and those have parts, with no end in sight. So a stone is a compound, a heap, a thing made of other things. But a compound, Leibniz insists, is only real because the things it is made of are real. A pile is nothing over and above what it is a pile of. So if there are compounds at all, there must be simples: things that are not made of parts, that are not heaps of anything smaller. And whatever is not made of parts is not extended, because extension just is having parts laid out in space. The ultimate constituents of reality cannot be little bits of matter. They have to be unextended, partless units. These are the monads (from the Greek for "unit"), and §1 of the Monadology defines one as nothing but a simple substance that enters into compounds.
If a monad has no parts and no extension, it has no shape, no size, and no place in space in the ordinary sense; no one could ever see one or weigh one. So what is it? Leibniz's answer is the boldest turn in the system. A monad is not a speck of stuff at all. It is more like a mind, or the simplest possible relative of a mind: a single, indivisible center of perception. Each monad has an inner life, a sequence of representations, a point of view. It is not conscious in the way a person is (most monads, the ones making up a rock, perceive in a dim, confused, sleeping way, with no awareness at all), but the basic activity of a monad is perceiving, having states that represent things. Reality, at bottom, is not matter in motion. It is an infinity of partless, soul-like points, each one busy perceiving.
§7 is the line the whole system hangs from, the most quoted thing Leibniz ever wrote: "The Monads have no windows, through which anything could come in or go out." A monad cannot be entered. Nothing can be inserted into it from outside and nothing can leak out of it to act on another. There is no causal traffic between monads at all, no pushing, no signaling, no influence in either direction. Each one is a sealed world, generating its whole stream of perceptions from inside itself, out of its own nature, with no input from anything else in the universe. This is why Leibniz needs no theory of how substances interact. On his view they never do. The windows are not just shut. There are no windows.
That raises the obvious objection at once, and the answer is the heart of the system. If no monad ever affects any other, how is there a world? How do all these sealed, private, windowless points add up to one shared universe in which a hand grips a cup and the cup gets lifted? Leibniz's answer is that each monad, from its own sealed interior, perceives the entire universe (§56 calls each simple substance "a perpetual living mirror of the universe"), but it perceives the whole thing from its own particular point of view, and clearly only in its own small neighborhood while everything farther off shades into dimness. Every monad mirrors all the others; that is what its inner stream of perceptions is, a representation of the whole, angled from where it stands. There are not many little worlds. There is one universe, mirrored infinitely many times over, once in each monad, each mirror tilted differently.
He gives an image for this in §57 that does more work than any abstract statement could: the same town, seen from different sides, looks like wholly different towns, so that one town appears, as he puts it, "numerous in aspects" depending on where the viewer stands. There is one town and there are as many views of it as there are positions to view it from, and no two views are the same. A monad is a viewpoint on the one universe, and the universe is the single thing that all the infinitely many viewpoints are views of. The seeming paradox (every monad sealed off, yet all sharing one world) dissolves: they share a world not by exchanging anything but by each independently picturing the same single whole from its own angle.
Two consequences follow that show how far the picture reaches. The first: because each monad's present state grows out of its own preceding state by its own internal law, §22 says its "present is big with its future", every monad carries its entire history and its entire future folded up inside its current state, the way an entire plant is folded into a seed. Nothing comes to a monad from outside, so everything that will ever happen to it is already implicit in what it now is, waiting to unfold. The second: a human soul, on this account, is simply a monad of an unusually high grade, one that perceives clearly and remembers and reasons, the dominant monad governing the vast colony of dim, sleeping monads that make up its body. Mind and matter are not two kinds of stuff after all. They are the same kind of stuff, monads, at different grades of clarity, the clear ones we call souls and the confused crowds of them we call bodies.
Chapter 3
Two clocks that never touch
"The soul follows its own laws, and the body likewise follows its own laws; and they agree with each other in virtue of the pre-established harmony between all substances."
The windowless monad solves one problem and hands Leibniz a worse-looking one. If nothing acts on anything, the entire ordinary world is in trouble, not just the mind and the body but everything. A struck bell and the sound, a thrown stone and the broken window, a decision and the arm that carries it out: on Leibniz's account none of these is one thing causing another, because no monad can reach across to touch a second monad. So why does the world hold together so seamlessly? Why does the arm rise exactly when the will to raise it occurs, if the will does not move the arm? This is the question chapter 2 left open, and the answer is the single most distinctive doctrine Leibniz owns: pre-established harmony.
He explains it with an analogy he returned to again and again, the two clocks. Two clocks, or two good watches, hang side by side and keep perfectly identical time, second for second, day after day. How could two clocks agree that exactly? There are, Leibniz says, only three possible explanations. The first: one clock physically drives the other, through some linkage or shared rod, so that the motion of one is transmitted to its neighbor. That is the ordinary picture of causation, one thing pushing another, and it is the one Leibniz denies between monads, because monads have no windows for any such linkage to pass through. The second: a clockmaker stands by them constantly, watching, and resets each one by hand whenever they drift, keeping them in step by perpetual intervention. The third: the clocks were built so perfectly at the start, each so exactly true, that they keep identical time entirely on their own, forever, without ever being linked and without ever being touched again.
The three clocks are three theories of how mind and body, and how any two things in the world, manage to keep step. The first, mutual influence, is the everyday view and Descartes's hope, the one that founders on the interaction problem. The second, constant correction by a watchful maker, is the position of the occasionalists, the followers of Malebranche (the French priest who held that God Himself moves a person's arm on the occasion of that person willing it, because no created thing can really cause anything; the Descartes read). Leibniz rejects the second because it makes God a repairman who has to intervene at every instant of every life to keep the world's pieces aligned, which is, he says, to invoke a perpetual miracle and to insult God's craftsmanship, as if a watchmaker who had to keep adjusting his watches were a better watchmaker than one who made them so well they never needed it.
So Leibniz takes the third. God, in creating each monad, built into it from the first instant the entire script of its perceptions, all of them, in the exact order they will unfold, and God did this for every monad at once, fitting all the infinitely many private scripts together so that they correspond perfectly forever. A person's willing to raise an arm and that arm's rising are two separate clocks, the soul running its own program, the body running its own, neither touching the other, both wound at the creation to match to the instant. §78 states it for the soul and body directly: each "follows its own laws," and "they agree with each other in virtue of the pre-established harmony." The match is real and total; the contact is zero. It only looks like the mind moving the body, the way it would look if two perfectly synchronized clocks were wired together when in fact they are not.
Set against the rival answers, the appeal of this becomes clear, which is exactly why Leibniz thought it superior. Descartes could not say how an unextended mind pushes an extended body; pre-established harmony answers that it never does, so there is nothing to explain. The occasionalists had God intervening endlessly; harmony has God act once, at the beginning, with infinite foresight, and then never again. Spinoza had to make mind and body the same single thing; harmony keeps them genuinely distinct, two clocks and not one, while still guaranteeing they agree. Every other party had to either explain a contact that seems impossible or pay some heavy price to avoid it. Leibniz pays the price up front, in one lump, by loading the entire future of the universe into the act of creation, and buys a world that runs forever afterward with no further intervention and no interaction at all.
That purchase has a cost. If every monad's whole history is written into it at creation and unfolds from within with no outside input, then everything that will ever happen to anyone is, in a sense, already fixed in the script. Leibniz spent enormous effort arguing this is not the crushing fatalism it looks like, that the unfolding still runs through the soul's own perceptions and inclinations, so that a person's choices are genuinely theirs even though God foresaw and incorporated them in choosing this world to create. Whether that rescues real freedom is one of the oldest open questions about his system, and the fact-checkers of his own century pressed it hard. What is not in doubt is the boldness: to keep a universe of distinct individuals while denying that any of them ever acts on another, and to pay for it with a harmony fixed before the first morning.
Chapter 4
Nothing without a reason
"…there can be no fact real or existing, no statement true, unless there be a sufficient reason, why it should be so and not otherwise."
Underneath the monads and the harmony sits the engine that drives the whole system, two principles Leibniz treats as the bedrock of all reasoning. The first is the principle of contradiction: nothing can both be and not be the same thing at once; a square circle is impossible. Every philosopher grants that one. The second is the one that is distinctively his, the principle of sufficient reason, stated in §32: there can be no fact and no true statement "unless there be a sufficient reason, why it should be so and not otherwise." Nothing is ever just so, brute, for no reason. For anything that is the case, and for anything that is true, there is a reason why it is that way rather than some other way, even if no human being ever finds it.
It sounds almost too obvious to be a doctrine, the kind of thing everyone already assumes. But Leibniz wields it like a crowbar, and it pries open enormous conclusions. Consider the question that defeats most people the moment it is asked: why is there something rather than nothing? An ordinary causal story is no help, because it only ever explains one thing in the world by another thing in the world, this fire by that match, and never explains why the whole chain of things exists at all rather than a blank. The principle of sufficient reason says that even that question must have an answer, that the existence of the entire series of contingent things must itself have a sufficient reason, and that this reason cannot lie inside the series (every member of which is just one more thing needing explanation) but must lie in something outside it whose existence is not contingent but necessary. That something is God. The principle of sufficient reason, pushed all the way down, becomes an argument that God exists.
The same principle does sharper, smaller work too, and the sharpest is a consequence Leibniz drew that is genuinely surprising: the identity of indiscernibles. The reasoning runs like this. Suppose there were two things in the universe that were exactly alike in every single respect, identical in every property, distinguishable in no way whatever except that there are two of them. Then ask the Leibnizian question: what sufficient reason could God have had for putting this one here and that one there, rather than the other way around? By hypothesis there is no difference between them to ground any such reason; the arrangement would be a choice with no sufficient reason behind it, which the principle forbids God to make. So there cannot be two such things. Any two genuinely distinct things must differ in some property, however hidden. Perfect duplicates are impossible. He put it flatly in his late correspondence with the English theologian Samuel Clarke: "There is no such thing as Two Individuals indiscernible from each other."
He liked to make the point with a story from his own court. In the gardens at Herrenhausen, the palace grounds at Hanover, the Electress Sophie (his patron and friend, a sharp philosophical mind in her own right) was walking with her courtiers when the principle came up, and one gentleman of the court scoffed at the idea that no two things could be exactly alike, insisting that surely two leaves, at least, might be perfectly identical. Sophie challenged him to go and find a pair. He searched the gardens, holding leaf against leaf, and could not do it; every two leaves, examined closely enough, differed. The anecdote is the principle made visible: not that it would be hard to find identical leaves, but that, if Leibniz is right, it is impossible in principle, because to be two distinct leaves just is to differ somehow. Nature never repeats itself exactly, because there would be no sufficient reason for it to bother making the same thing twice.
The reach of the principle is what makes Leibniz Leibniz. From "nothing without a reason" he derives the existence of God, the impossibility of perfect duplicates, the rejection of empty space and atoms in the void (a vacuum would be a difference, here-empty rather than there-empty, with no sufficient reason for the emptiness to fall one way rather than another), and, in the next chapter, the most ambitious conclusion of all: that of all the worlds God could have made, there is a sufficient reason why this one is the one that exists. The principle is the thread that ties the whole metaphysics into a single argument. Grant Leibniz that nothing is true without a reason, and he claims the chain of reasoning runs all the way to the best of all possible worlds.
Chapter 5
The best of all possible worlds
"…as in the Ideas of God there is an infinite number of possible universes, and as only one of them can be actual, there must be a sufficient reason for the choice of God."
Now the principle of sufficient reason meets the oldest problem in religion, and produces the claim Leibniz is most famous for and least understood on. Start with God deciding whether and what to create. In the divine understanding, Leibniz says, there are infinitely many possible worlds: not just the world that exists, but every other complete, internally consistent total history that could have existed instead, every alternative universe that contains no contradiction. A world in which some person made a different choice this morning; a world with different laws of physics; any consistent total story is a possible world, and there are infinitely many. §53 lays it out: in the ideas of God there is an infinite number of possible universes, only one can be made actual, and so there must be a sufficient reason for which one God chooses.
What sufficient reason could a perfectly wise, perfectly good, perfectly powerful being have for choosing one world over all the others? Only that it is the best. A being with the wisdom to survey every possibility, the goodness to want the best, and the power to make it real, would have no sufficient reason to actualize any world except the finest of them all. §55 states the choice exactly: the best world exists because God's wisdom knows it, His goodness chooses it, and His power produces it. So the world that exists, this one, is the best of all the worlds that were possible. Not a good world, not an acceptable world: the best one, the single optimum that a perfect chooser, choosing among infinitely many candidates, would necessarily pick. That is Leibnizian optimism, and it is a conclusion driven by the machinery, not a sunny mood.
Everything depends on what "best" means, and the caricature gets this wrong. "Best" does not mean that every event in the world is good, or that nothing terrible happens, or that each individual thing is the finest it could be. It is a claim about the balance of the whole. The best world, for Leibniz, is the one that achieves the greatest richness of things, the most variety, the most reality, produced by the simplest and most general laws. §58 names the standard directly: the way to the best is "as great variety as possible, along with the greatest possible order," which is to get "as much perfection as possible." A world is graded as a total package, on how much it contains weighed against how simple the rules are that generate it. The best world is the most fruitful one runnable on the most elegant laws. It is an optimization over the whole, not a guarantee about any part.
That distinction is what lets Leibniz say something about evil that is far more serious than the joke it became. Why is there suffering, cruelty, disaster, in a world chosen by a perfect God? Because the best total package cannot be had without some local evil. A world with genuinely free creatures is richer than a world of puppets, but free creatures can choose wrongly, so freedom imports the possibility of moral evil. A world running on simple, constant natural laws is more perfect than one God has to keep interrupting, but constant laws mean fire that warms also burns and seas that bear ships also drown. The evils are real, and Leibniz does not pretend they are secretly pleasant. He argues that they are the unavoidable cost of features that make the whole better, and that a God optimizing the entire balance accepts the necessary local evil for the sake of the greater perfection of the whole. This is a theodicy, an attempt to justify the ways of God in the face of evil, and it gave its name to the book where he laid it out, the Theodicy of 1710, the one major work he published in his lifetime.
Then came the most famous demolition in the history of philosophy, and it was a novel. In 1759, decades after Leibniz's death and four years after the Lisbon earthquake of 1755 had killed tens of thousands on a church holiday and shaken Europe's faith that the world was benignly arranged, Voltaire published Candide. Its target was Leibnizian optimism, embodied in a ridiculous character, Dr Pangloss, a tutor who responds to every catastrophe (war, rape, plague, the hanging of his friends, the earthquake itself) by serenely repeating that all is for the best in this best of all possible worlds. Pangloss became, and remains, the popular image of Leibniz: the fool who looks at horror and calls it perfection. The caricature was devastating, and it stuck, and it is the reason "Panglossian" is now a synonym for idiot optimism.
It is also a misrepresentation, and the difference is exactly the distinction between the goodness of the whole and the goodness of the parts. Leibniz never claimed that each particular evil is itself good, or that the right response to suffering is to do nothing because all is already optimal. Pangloss draws that complacent conclusion; Leibniz's actual argument forbids it, because his claim is about the balance of the whole, not the goodness of the parts, and nothing in it says a person should accept a remediable evil rather than fix it. Voltaire fused Leibniz's hard metaphysical thesis (that even the best possible world contains some evil as a necessary cost) with a lazy moral one (that we therefore need not lift a finger), and the fusion is the joke. The joke is brilliant and it is unfair, and it has done what brilliant unfair jokes do: replaced the argument with the parody for two and a half centuries. The argument it buried is not comfortable. It says this world, with the Lisbon dead in it, is the best a perfect God could manage, and asks one to accept that the alternative worlds were all, on balance, worse.
Chapter 6
The calculus and the dream of a language
"…perception and that which depends upon it are inexplicable on mechanical grounds…"
The metaphysics was the work of his spare hours. The work the world could not ignore was the mathematics, and it produced both his most universal gift and his most disfiguring fight. In the mid-1670s, in and around Paris, Leibniz worked out the calculus: the mathematics of continuous change, of how to find the exact slope of a curve at a single point (the differential calculus) and the exact area under a curve (the integral calculus), and the deep fact that these two operations are inverses of each other. It is the tool that made modern physics and engineering possible, and Leibniz did not just discover it. He invented a notation for it of such clarity that it is the notation still used everywhere today: the d for a differential, as in dx and dy, and the long-S ∫ for an integral. Every calculus student alive is writing in Leibniz's hand.
The trouble was that Isaac Newton, in England, had developed his own version of the same mathematics earlier, in the 1660s, and had not published it. Leibniz developed his independently in the 1670s and published first, in the journal Acta Eruditorum in 1684. For a while the two great men were cordial. Then their followers, and eventually the men themselves, fell into one of the ugliest priority disputes in the history of science, each side accusing the other of theft. The English claimed Leibniz had seen Newton's unpublished work on a visit and stolen the ideas; the continentals claimed Newton had backdated his discovery to rob Leibniz of credit for what he had published openly.
The fight was settled, officially, by an act of rigged procedure that still stains the record. In 1711 Leibniz appealed to the Royal Society of London to adjudicate. The Society appointed a committee said to be impartial, which in 1712 issued a report, the Commercium Epistolicum, finding decisively for Newton and against Leibniz. What was not known at the time, and is now established, is that Newton was the president of the Royal Society, that he packed the committee, and that he anonymously wrote the supposedly impartial report himself, and then anonymously reviewed it favorably in the Society's journal. The verdict was Newton judging his own case under cover of an institution he controlled. The modern consensus is clean: Newton and Leibniz invented the calculus independently, neither stole from the other, and Leibniz's notation was the better one and is the one that survived. But the dispute embittered the last years of Leibniz's life, helped cost him his standing in England and his king's favor, and is part of why he died neglected (chapter 1).
Behind the calculus lay a dream larger than any single piece of mathematics, and it is the part of Leibniz that reads as the most startling glimpse of the future. He had wanted, since youth, to build a characteristica universalis: a universal symbolic language in which every basic concept would be assigned an exact character, an unambiguous sign, so that complex ideas could be written as combinations of simple ones the way numbers are built from primes. Paired with it he imagined a calculus ratiocinator, a method of computing with those characters, so that reasoning itself could be carried out as a kind of arithmetic. The vision was that truth could be calculated. He put it in an image that has become his most quoted hope: that one day, when two philosophers disagreed, they would not need to argue, but could take up their pens and say to each other, in effect, let us calculate, and resolve the dispute by reckoning, the way two accountants settle a sum.
He never built it; the project was centuries ahead of the tools it needed. But the aim, reducing reasoning to the mechanical manipulation of exact symbols, is precisely the aim that symbolic achieved in the nineteenth and twentieth centuries (in the work of George Boole and Gottlob Frege and those after them), and through it the aim at the root of the digital computer, a machine that does exactly what Leibniz wanted, computing with symbols according to fixed rules. Leibniz did not invent modern logic and did not foresee the computer. But he saw, earlier and more clearly than anyone, that thinking might be a form of calculation, and he reached for the tools to make it so, and the world he reached toward is, three centuries on, the world that built itself. The reconciler who wanted to settle every quarrel by computation was dreaming, it turns out, of the digital computer.
There is one last thread that ties the dream back to the metaphysics, and it is the deepest thing in him. The same Leibniz who wanted reasoning reduced to calculation also insisted, in §17 of the Monadology, that perception itself can never be explained mechanically. He offers a thought experiment: suppose a machine were built so that its workings produced thought and feeling, and suppose it blown up to the size of a mill, large enough to walk inside. A visitor walking through would see only parts pushing parts, gears and levers, and nowhere among them the perception, the experience, that the machine supposedly has. Mechanism can explain the body's motions; it cannot, he argued, explain the having of a point of view, which is why the ultimate things had to be perceiving monads and not mere matter. Three hundred years later, after everything his characteristica eventually became, the problem he posed in that imaginary mill, how the firing of physical parts could ever amount to the felt experience of being someone, is still open, still argued, still unsolved. The polymath who wanted to calculate everything drew, at the center of his system, the one thing he was sure no calculation could reach.