Dropping In on Gottfried Leibniz—Stephen Wolfram Writings

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Dropping In on Gottfried Leibniz

May 14, 2013

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

Leafing through his yellowed (but still robust enough for me to touch) pages of notes, I felt a certain connection—as I tried to imagine what he was thinking when he wrote them, and 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):

Or here’s Leibniz try to 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 and thinking?

I have always found Leibniz a somewhat confusing figure. He did many seemingly disparate and 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, and gotten a better feeling for Leibniz as a person, I’ve realized that underneath much of what he did was a 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, and 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, and began to read his way through its diverse collection of books. He went to 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 to have been interested in systematization and 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 to 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". And for his philosophy graduation thesis, Leibniz tried to pursue this idea. He used some basic combinatorial mathematics to 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 to 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 logic and combinatorics.

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

One such activity in 1672 took Leibniz to 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 to learn all the latest ideas and methods. And for example he sought out Christiaan Huygens, who agreed to teach Leibniz mathematics—after he succeeded in passing the test of finding the sum of the reciprocals...