Leibniz

Gottfried Wilhelm Leibniz

Leibniz had a dream so audacious that it would take three centuries and the invention of the computer to begin to make it real. Imagine, he proposed, that we could build a universal language, a system of symbols precise enough to represent not just numbers but concepts, every idea broken down into its simple parts and written in exact notation, the way algebra writes quantities. And imagine, alongside it, a "calculus of reasoning," a set of rules for combining those symbols that would track the logic of thought itself. Then something extraordinary would become possible. When two people disagreed, about politics, about morality, about anything at all, they would no longer need to quarrel. They would simply take up their pens and say, "Let us calculate," Calculemus, and work out who was right the way we work out a sum. Leibniz spent his life chasing this dream, and along the way, almost as side effects, he invented the binary arithmetic of zeros and ones that every computer now runs on, built one of the first mechanical calculating machines, and co-invented the calculus. He was, three hundred years early, dreaming of the computer, and of something we are only now building, a machine that reasons.

The last universal genius. It helps to know who was dreaming this. Leibniz may have been the last human being to come close to knowing everything his age knew, and to advancing most of it. He co-invented the calculus, in a cleaner notation than Newton's, the one every student still learns. He worked out the binary arithmetic of ones and zeros on which all computing now rests. He designed and built a calculating machine that could multiply and divide. But that was only the mathematics. He was also a working diplomat, a court historian, a librarian who reorganized whole collections, a mining engineer who spent years trying to drain the silver mines of the Harz mountains with windmills, a geologist, a student of languages tracing their common origins, and a tireless correspondent who exchanged some fifteen thousand letters with over a thousand of the best minds in Europe, conducting, almost single-handed, the intellectual traffic of a continent. His motto might have been one of his own favorite phrases, that nothing should be lost. He wanted to gather up all of human knowledge, organize it, connect it, and put it to use, and the sheer breadth of what he actually mastered has not been equaled since.

A machine that reasons. It is worth pausing on how radical, and how prescient, the dream was. Leibniz had grasped something no one else of his age had: that reasoning might be a kind of calculation, that the operations of thought, at least in their logical skeleton, might be mechanical, reducible to the formal shuffling of symbols according to rules. If that were true, then a mind was, in part, a machine, and a machine could, in part, think. For two centuries the idea slept. Then, in the nineteenth and twentieth, it woke with a vengeance: logicians built the formal symbolic languages Leibniz had imagined, proved that whole domains of reasoning could indeed be reduced to rule-governed symbol-manipulation, and finally built physical machines, computers, that carried out those manipulations at inhuman speed. Every line of code, every logic circuit, every artificial intelligence that follows a chain of inference is a distant descendant of Leibniz's Calculemus. He did not have the technology to build his reasoning machine, but he saw, with uncanny clarity, that one was possible, and he laid down the conceptual foundations, the binary code, the formal logic, the very idea that thought has a computable shape, on which the digital world would eventually be raised.

A universe of mirrors. For all his fascination with machines, Leibniz's picture of reality itself was strange, beautiful, and alive. The world, he argued, is ultimately made not of dead matter but of countless simple, indivisible centers of perception, which he called monads. Every monad is a kind of soul, a point of view, and here is the haunting part: each one mirrors the entire universe from its own particular position, like a single eye reflecting the whole room it sits in. No two are alike, because each reflects the whole from a different angle, and together they make up reality as an infinity of perspectives, each one complete, each one containing, in its own way, all the others. The monads do not interact, Leibniz says; they are "windowless," shut up in themselves. And yet they all agree perfectly, because God, at the creation, set them running in harmony, like a vast array of clocks wound to keep the same time forever without ever touching. It is a vision at once mechanical and mystical: a universe with no real causation between things, only a pre-arranged dance of self-contained perspectives, each unfolding its own inner program, all of them mirroring one another in a harmony established before time began. You may not believe a word of it, and it remains one of the strangest and most original pictures of reality anyone has ever drawn.

The best of all possible worlds. And then there is the phrase that made him famous and got him mocked: that this is "the best of all possible worlds." Out of context it sounds like the silliest sort of optimism, a man insisting that everything is wonderful, and Voltaire skewered it without mercy in Candide, where the foolish Dr. Pangloss keeps declaring that all is for the best amid a parade of catastrophes. But the real argument is subtler, and it follows from Leibniz's deepest principle, the principle of sufficient reason: that nothing is ever the case without a reason why it is so rather than otherwise. Apply that to creation. A perfect God, choosing to make a world, must have had a sufficient reason to make this one rather than any other; and the only reason worthy of a perfect being is that this world is the best of the genuinely possible ones, the one with the greatest richness and harmony achievable. The evils in it are not proof that God failed but the unavoidable price of a world that is, on the whole and in the long run, the best that could be made. You can reject the conclusion, as most readers do, but it is not the babble of an optimist; it is a tight piece of reasoning about what a perfect creator would have to do, and the problem it wrestles with, why a good and all-powerful God would permit suffering, is one no believer has ever managed to escape.

The man who saw the future. The room reads Leibniz mostly from his shorter works and a vast correspondence, cited by section; he published comparatively little and scattered his genius across fifteen thousand letters, so the chat reasons from the wider record. The scope of what he saw is hard to take in. His relational theory of space, that space is not a vast empty container but simply the web of relations among things, was dismissed in his lifetime in favor of Newton's absolute space, and then vindicated two centuries later by Einstein. His talk of possible worlds, long treated as a curiosity, became the foundation of modern modal logic. His calculus is taught to every student of mathematics in his notation, not Newton's. And his dream of mechanizing reason runs straight into the machine you are reading this on. There is a temptation to remember him as the optimist Voltaire laughed at, the man of windowless monads and the best of all possible worlds. The truth is that Leibniz may have seen further into the future than any philosopher who ever lived: he glimpsed the computer, the logical foundations of mathematics, the relativity of space, and the possibility, which our own age is busy testing, that thinking itself might be something a machine can do. He spent his life trying to build a tool that would let human beings reason together without quarreling. We are still trying to build it.