What was Leibniz's dream of a calculus of reasoning?

Imagine two people arguing about a hard question, and instead of raising their voices, they pick up pencils and say, "Let's work it out." That scene was Gottfried Wilhelm Leibniz's lifelong dream. He wanted a language so exact that reasoning in it would be as mechanical as long division, so that a disagreement could be settled the way an accountant settles a sum.

The dream began with a short, ambitious booklet he published in 1666 in the university town of Leipzig. He later called it immature. It nevertheless contains, in seed form, the idea that thinking can be broken into symbols and rules, which is the starting point of logic as it is used in computers today.

What happened in 1666

The booklet's Latin title is Dissertatio de arte combinatoria, usually translated as "Dissertation on the Art of Combinations." It is an expanded version of Leibniz's first doctoral dissertation, written before he had seriously studied mathematics. It was printed in Leipzig in 1666 and gave him his first taste of fame among scholars.

The work had an odd afterlife. In 1690 it was reprinted without his permission, and Leibniz published a short notice in a learned journal distancing himself from it. He said several times that he wished it had not circulated. Even so, historians treat it as the opening move in a project he kept returning to for decades.

The idea: an alphabet of human thought

The core idea is simple to state. Every concept, Leibniz proposed, is a combination of a fairly small number of basic concepts, in the same way that every word is a combination of letters. If you could list those basic concepts and give each one a symbol, then:

• every complex idea would have a precise "spelling" made of basic symbols;
• every true statement would be a correct combination of those symbols;
• finding new truths would become a matter of trying combinations systematically.

Leibniz borrowed part of this from Ramon Llull, a 13th-century thinker from Majorca who had designed devices for combining basic truths. Leibniz admired the ambition but criticized Llull's categories as arbitrary. He also cited the English philosopher Thomas Hobbes, who had argued in 1651 that all reasoning is a kind of computation.

The booklet also contains real mathematics. Leibniz studied how many ways a set of items can be arranged or grouped, the branch of mathematics now called combinatorics. He applied it to law, to organ stops and to old theories about the elements.

A worked example from the booklet

His most careful example comes from geometry. He gives numbers to simple concepts and builds new concepts as combinations of them. Here is a simplified version:

Notice what happens. Once the simple concepts have codes, a definition becomes a formula. You could, in principle, check whether two definitions mean the same thing by comparing their formulas, without arguing about words.

From combinations to calculation

In later writings Leibniz went further. He described a characteristica universalis, a universal symbolic language for all ideas, and a calculus ratiocinator, a "calculus of reasoning" that would operate on those symbols by fixed rules. With both in hand, he wrote, philosophers would no longer need to dispute any more than accountants do; they could simply sit down with their slates and say:

"Let us calculate."

He also built mechanical calculators, improving on earlier designs, so he knew that arithmetic, at least, could be handed to a machine.

Why it mattered

Much of Leibniz's work on logic sat unpublished in his papers at Hanover for close to two centuries, so its direct influence is debated. Logicians of the 1800s found themselves walking his path anyway. In the middle of the century George Boole built an algebra of logic; his widow later wrote that when he learned how much Leibniz had anticipated, he felt as if Leibniz had reached across the centuries to shake his hand. In 1879 Gottlob Frege presented his own formal notation as a first step toward the universal language Leibniz had described, though he judged the full project too large for one person. Around 1900 Louis Couturat studied the unpublished papers and argued that Leibniz already held the key principles of the new logic.

That modern logic is what computer circuits and programming languages are built on. In 2019 Leibniz's dream shows up in programs that check mathematical proofs and in rule-based systems that draw conclusions from stated facts. Today's language programs, such as OpenAI's GPT-2, take a different route: they represent each word as a long list of numbers learned from text, not as a hand-built formula.

Did anyone ever build Leibniz's reasoning machine?

Not in the form he imagined. No one has found a small list of basic concepts from which all ideas can be spelled, and most researchers now doubt such a list exists. Everyday words are too flexible, and their meanings shift with context.

Parts of the dream, though, became real. Formal logic gave us exact symbols and rules for valid reasoning. Electronic computers, which Leibniz could only foreshadow with his calculators, apply those rules at enormous speed. Proof checkers can confirm that a long mathematical argument has no gaps, which is close to "let us calculate" for one carefully defined domain.

What remains out of reach is the step Leibniz took for granted: turning ordinary questions about law, ethics or politics into clean symbols that everyone agrees on. That translation is still a human job, and arguably the hardest part of reasoning.