I’ve been curious about Gottfried Leibniz for years, not least because he seems khổng lồ have wanted to lớn build something lượt thích Mathematica and Wolfram|Alpha, & perhaps A New Kind of Science as well—though three centuries too early. So when I took a trip recently lớn Germany, I was excited to lớn be able lớn visit his archive in Hanover.

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Leafing through his yellowed (but still robust enough for me khổng lồ touch) pages of notes, I felt a certain connection—as I tried to lớn imagine what he was thinking when he wrote them, & tried to relate what I saw in them to what we now know after three more centuries:

This essay is also in Idea Makers: Personal Perspectives on the Lives & Ideas of Some Notable People»

Some things, especially in mathematics, are quite timeless. Like here’s Leibniz writing down an infinite series for √2 (the text is in Latin):

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Or here’s Leibniz try lớn calculate a continued fraction—though he got the arithmetic wrong, even though he wrote it all out (the Π was his earlier version of an equal sign):

Or here’s a little summary of calculus, that could almost be in a modern textbook:

But what was everything else about? What was the larger story of his work & thinking?

I have always found Leibniz a somewhat confusing figure. He did many seemingly disparate & unrelated things—in philosophy, mathematics, theology, law, physics, history, and more. And he described what he was doing in what seem to us now as strange 17th century terms.

But as I’ve learned more, & gotten a better feeling for Leibniz as a person, I’ve realized that underneath much of what he did was a chip core intellectual direction that is curiously close to the modern computational one that I, for example, have followed.

Gottfried Leibniz was born in Leipzig in what’s now Germany in 1646 (four years after Galileo died, và four years after Newton was born). His father was a professor of philosophy; his mother’s family was in the book trade. Leibniz’s father died when Leibniz was 6—and after a 2-year deliberation on its suitability for one so young, Leibniz was allowed into his father’s library, và began khổng lồ read his way through its diverse collection of books. He went khổng lồ the local university at age 15, studying philosophy and law—and graduated in both of them at age 20.

Even as a teenager, Leibniz seems lớn have been interested in systematization và formalization of knowledge. There had been vague ideas for a long time—for example in the semi-mystical Ars Magna of Ramon Llull from the 1300s—that one might be able khổng lồ set up some kind of universal system in which all knowledge could be derived from combinations of signs drawn from a suitable (as Descartes called it) “alphabet of human thought”. & for his philosophy graduation thesis, Leibniz tried to lớn pursue this idea. He used some basic combinatorial mathematics lớn count possibilities. He talked about decomposing ideas into simple components on which a “logic of invention” could operate. And, for good measure, he put in an argument that purported to prove the existence of God.

As Leibniz himself said in later years, this thesis—written at age 20—was in many ways naive. But I think it began khổng lồ define Leibniz’s lifelong way of thinking about all sorts of things. And so, for example, Leibniz’s law graduation thesis about “perplexing legal cases” was all about how such cases could potentially be resolved by reducing them to súc tích and combinatorics.

Leibniz was on a track khổng lồ become a professor, but instead he decided lớn embark on a life working as an advisor for various courts và political rulers. Some of what he did for them was scholarship, tracking down abstruse—but politically important—genealogy và history. Some of it was organization và systematization—of legal codes, libraries and so on. Some of it was practical engineering—like trying khổng lồ work out better ways to lớn keep water out of silver mines. & some of it—particularly in earlier years—was “on the ground” intellectual tư vấn for political maneuvering.

One such activity in 1672 took Leibniz lớn Paris for four years—during which time he interacted with many leading intellectual lights. Before then, Leibniz’s knowledge of mathematics had been fairly basic. But in Paris he had the opportunity khổng lồ learn all the latest ideas và methods. And for example he sought out Christiaan Huygens, who agreed lớn teach Leibniz mathematics—after he succeeded in passing the kiểm tra of finding the sum of the reciprocals of the triangular numbers.

Over the years, Leibniz refined his ideas about the systematization and formalization of knowledge, imagining a whole architecture for how knowledge would—in modern terms—be made computational. He saw the first step as being the development of an ars characteristica—a methodology for assigning signs or symbolic representations to lớn things, & in effect creating a uniform “alphabet of thought”. And he then imagined—in remarkable resonance with what we now know about computation—that from this uniform representation it would be possible to find “truths of reason in any field… through a calculus, as in arithmetic or algebra”.

He talked about his ideas under a variety of rather ambitious names like scientia generalis (“general method of knowledge”), lingua philosophica (“philosophical language”), mathematique universelle (“universal mathematics”), characteristica universalis (“universal system”) & calculus ratiocinator (“calculus of thought”). He imagined applications ultimately in all areas—science, law, medicine, engineering, theology và more. But the one area in which he had clear success quite quickly was mathematics.

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To me it’s remarkable how rarely in the history of mathematics that notation has been viewed as a central issue. It happened at the beginning of modern mathematical xúc tích và ngắn gọn in the late 1800s with the work of people lượt thích Gottlob Frege và Giuseppe Peano. & in recent times it’s happened with me in my efforts to lớn create Mathematica & the Wolfram Language. But it also happened three centuries ago with Leibniz. And I suspect that Leibniz’s successes in mathematics were in no small part due khổng lồ the effort he put into notation, và the clarity of reasoning about mathematical structures và processes that it brought.

When one looks at Leibniz’s papers, it’s interesting to see his notation and its development. Many things look quite modern. Though there are charming dashes of the 17th century, lượt thích the occasional use of alchemical or planetary symbols for algebraic variables: