Leibniz beyond the cartoon: the man who wanted to calculate thought

quando orientur controversiae, non magis disputatione opus erit inter duos philosophos, quam inter duos Computistas. Sufficiet enim calamos in manus sumere sedereque ad abacos, et sibi mutuo (accito si placet amico) dicere: calculemus.

When disputes arise, there will be no more need of argument between two philosophers than between two accountants. It will be enough to take up their pens, sit down at their reckoning-tables and say to one another, with a friend called in as witness if they like: let us calculate.

The remark is quoted constantly and usually badly. It comes from an unpublished manuscript, not from anything Leibniz sent to press, and he never built the system; he produced fragments of a logical calculus, some startlingly good, most abandoned. It is also worth resisting the claim, popular in excitable histories of computing, that Leibniz invented the computer. He did not. He built an arithmetical machine, an imperfect improvement on Pascal's, and the line from there to a stored-programme machine runs through two and a half centuries and several people who had never read him.

The calculus is the one item on the list that turned poisonous. Newton had the ideas first, in the mid-1660s; Leibniz worked his out from 1674 and published first, in 1684. A Royal Society committee, with Newton in the chair and holding the pen, found for Newton in 1713. Historians now accept independent discovery, and it is Leibniz's notation that every schoolchild learns.

Monads are an answer to a question about machines#

Here is where the caricature does most damage. Monads arrive in most surveys as spiritual atoms, baroque metaphysical furniture, and the student is told they have no windows, which sounds like a joke that failed.

Start instead from the question. The seventeenth century's great success was mechanical explanation: bodies are extended things, they push each other, and everything physical follows from shapes and motions. Leibniz was an enthusiastic mechanist about physics, and he also thought the programme had a hole in it. Extension is divisible without limit, so anything extended is a collection, and a collection borrows its reality from its members. If everything is extended, nothing is ultimately one thing, and there is nothing for the collections to be collections of. Unity cannot be got out of geometry.

So the ultimate constituents of reality must be unities that are not extended: simple substances, which he called monads. Having no parts, they have nothing to rearrange, which is why nothing can be pushed into or out of them. That is all "windowless" means. It follows from being simple, since there is no mechanism by which a partless thing could be mechanically affected. What monads have instead of parts is perception, defined as the representation in the simple of the compound, and appetition, the tendency carrying one perception into the next. Every monad represents the whole universe from its own position, more or less confusedly; a stone's monads very confusedly indeed.

The awkward consequence is that nothing acts on anything, and Leibniz paid for it with the pre-established harmony: each substance unfolds by its own internal law, and God has arranged the laws so that the unfoldings correspond. Your hand moves when you decide to move it, but the decision does not push it; the two series run in step because they were composed together. It is an extravagant solution and he knew how it looked. He preferred it to Spinoza's single substance, inspected at close range when the two met in Amsterdam in November 1676, and to Malebranche's God intervening at every moment, a perpetual miracle and so no explanation.

The argument's sharpest form is section 17 of the Monadology, written in Vienna between 1712 and 1714 and unpublished until 1840. Perception, Leibniz writes, is inexplicable in terms of mechanical reasons, that is, through shapes and motions:

If we imagine that there is a machine whose structure makes it think, sense, and have perceptions, we could conceive it enlarged, keeping the same proportions, so that we could enter into it, as one enters into a mill. Assuming that, when inspecting its interior, we will only find parts that push one another, and we will never find anything to explain a perception.

Anybody who has followed the last forty years of philosophy of mind will recognise the explanatory gap, stated in 1714 with a mill instead of a brain.

Two principles that are still being argued about#

The principle of sufficient reason says that for everything that is the case there is a reason why it is so and not otherwise. Leibniz called it one of the greatest and most fruitful principles of human knowledge, and conceded without embarrassment that in most cases we cannot know the reason. The identity of indiscernibles says that two things cannot share all their qualities and differ merely in number. In his fourth paper to Samuel Clarke — the correspondence of 1715 and 1716, conducted through Caroline, Princess of Wales, and cut short by his death — he reports that an ingenious gentleman, talking with him in the garden at Herrenhausen in the presence of the Electress Sophia, thought he could find two leaves perfectly alike. He looked. He could not.

Neither principle is a museum piece. Michael Della Rocca has spent two decades arguing that a serious commitment to it demolishes most of contemporary metaphysics; Shamik Dasgupta reformulated it in 2016 in terms of grounding, and Alexander Pruss has defended a version of it. The standard objection, that the principle cannot explain its own truth without circularity, remains.

The identity of indiscernibles has had a rougher time, and the trouble came from physics. Quantum statistics appear to describe particles of a kind that share every intrinsic and every relational property, precisely what Leibniz said could not happen; Steven French and others conclude that indiscernible physical particles exist. Simon Saunders and F. A. Muller reply that fermions are at least weakly discernible, each standing to the other in an irreflexive relation, at a distance from the other but not from itself. Bosons, which can be co-located, are the harder case and remain unresolved. Leibniz would have found the dispute intelligible, which is more than can be said of most seventeenth-century metaphysics after 1925.

Kripke and Lewis borrowed the phrase, not the theology#

Possible worlds are Leibniz's most conspicuous export. In his usage a possible world is a complete, consistent way everything could have been, one of infinitely many God contemplated before creating; God, being perfect, actualised the best. Twentieth-century modal logic uses the same words for something else. When Saul Kripke gave the subject its now-standard semantics in papers of 1959 and 1963, worlds became index points in a model related by an accessibility relation, there to make sense of the difference between necessity and possibility. Things are not found out about a counterfactual situation, Kripke wrote in Naming and Necessity, they are stipulated, and possible worlds need not be given purely qualitatively, as if we were looking at them through a telescope.

David Lewis went the other way and took the worlds to be as concrete as this one, the modal realism of On the Plurality of Worlds (1986). That looks superficially Leibnizian and is not. Lewis's worlds are causally and spatiotemporally isolated from one another, nothing selects among them, and "actual" is merely indexical. Leibniz's possibilia exist as ideas in the divine understanding and are ranked. Take away the ranking and the ranker and you have a different theory that has kept the name. Timothy Williamson's Modal Logic as Metaphysics (2013) proceeds in a register Leibniz would have needed a decade to learn. Metaphysics is one of the few fields where he is read as a contemporary rather than a source.

The Théodicée is a book about evil, not about contentment#

Essais de Théodicée sur la bonté de Dieu, la liberté de l'homme et l'origine du mal was published anonymously in Amsterdam in 1710, the most substantial philosophical book Leibniz put into print in his lifetime. It was written against Pierre Bayle, whose Dictionnaire historique et critique held that evil could not be reconciled with a good and omnipotent God by reason, only accepted on faith.

Its structure depends on a threefold distinction. Metaphysical evil is imperfection as such, the finitude belonging to any created thing because it is not God. Physical evil is suffering. Moral evil is sin. Leibniz argues that metaphysical evil is unavoidable in any created world whatever, and that the other two are permitted rather than willed, since removing them would require a worse world overall. The optimum is a global property, computed across the whole series, and not a claim that each item in the series is good. He is quite clear that this world contains horrors.

Voltaire read it, or read enough of it, and in Candide, ou l'Optimisme (1759) produced the most effective piece of philosophical demolition in European literature: twenty to thirty thousand copies in more than twenty editions within the year. The Lisbon earthquake of 1755 had killed tens of thousands on a Sunday morning in the churches, and Voltaire's objection is not stupid. A theory whose consolation consists in telling the survivors that the arrangement is optimal has said something obscene, whatever its logical merits, and Dr Pangloss is how such a theory sounds in the mouth of anyone less careful than its author.

That lands, and should be conceded rather than explained away. What it misses is the target. Pangloss holds that everything that happens is for the best, which Leibniz does not say, and Candide attacks the consolation while leaving the argument untouched. The real difficulty is whether a best possible world is a coherent notion at all, since for any world one names there seems to be a better. That objection was available in 1759. Voltaire, writing a satire, did not make it.

The question he would put to a language model#

The characteristica universalis is a real ancestor of the formal systems that run on machines. The line is not straight, and Leibniz was largely unread by the people who mattered most, but Frege — who wrote that Leibniz threw out such a profusion of seeds of ideas that in this respect he is virtually in a class of his own — meant the Begriffsschrift of 1879 as a partial execution of the programme, and from Frege the road to Turing is short.

What follows for artificial intelligence is less obvious than it is usually made to sound, and here I am speculating rather than reporting scholarship. A machine that manipulates symbols correctly satisfies half of what Leibniz wanted: the calculus ratiocinator, executed faultlessly and at speed. It does not touch the other half. Perception, in his sense, is the representation of a whole world in something genuinely one, from a particular point of view, with an appetition carrying it forward. The mill passage denies that any arrangement of interacting parts, however elaborate, could constitute that. A large language model is a very large arrangement of interacting parts.

So the question he would put to such a system is closer to John Searle's than to Turing's. Turing's 1950 test asks what a system can do, setting understanding aside if the behaviour is indistinguishable; Searle's Chinese Room argument of 1980 asks whether syntax could ever amount to semantics. Leibniz's mill is older than both and narrower than either. He is asking where the unity is: not whether the system is intelligent, nor quite whether it understands, but whether there is any one thing there with a point of view, or only parts pushing one another at a scale where we can no longer walk in and look. The framing is unfashionable, and it is where a good deal of current work on machine minds keeps ending up anyway.

He would have thought the machines wonderful and the question untouched, roughly what he thought of his own calculating engine.

The end was shabby. His employer had become George I of England in 1714 and left him in Hanover with the archives and the unfinished dynastic history. When Leibniz died in November 1716, no courtier attended the funeral apart from his secretary, and the grave in the Neustädter Kirche went unmarked for more than fifty years. The academy he had founded in Berlin said nothing. The papers went into boxes: between 150,000 and 200,000 pages, correspondence with more than 1,100 people, most of it never printed. Cataloguing began in 1901; the edition is still unfinished, the philosophical writings published only as far as 1690.

Three hundred years before anyone built a machine that could do arithmetic without a crank, he had asked the question the machines would force on everybody else: whether thought is a computation, and if it is, what is left over.

Confucius is a name most readers know and few can place. This profile introduces the philosopher behind it: his ideas of ren, ritual, the exemplary person and the rectification of names; his claim that good government begins in character; and the compiled, argued-over text we call the Analects.

Confucius