Thomas Hobbes and the Doing of Important Mathematics


One of my mathematics-trivia-of-the-day Twitter feeds mentioned that Saturday was the birthday of Thomas Hobbes (5 April 1588 to 4 December 1679), and yes, that Hobbes. I was surprised; I knew Hobbes had written Leviathan and was famous for philosophical works that I hadn’t read either. I had no idea that he’d done anything important mathematically, but then, the generic biography for a mathematician of the 16th or 17th century is “philosopher/theologian who advanced mathematics in order to further his astronomical research”, so, it wouldn’t be strange.

The MacTutor History of Mathematics archive’s biography explains that he actually came to discover mathematics relatively late in life. They quote John Aubrey’s A Brief Life Of Thomas Hobbes for a scene that’s just beautiful:

He was forty years old before he looked on geometry; which happened accidentally. Being in a gentleman’s library Euclid’s Elements lay open, and ’twas the forty-seventh proposition in the first book. He read the proposition. “By God,” said he, “this is impossible!” So he reads the demonstration of it, which referred him back to such a proof; which referred him back to another, which he also read. … at last he was demonstratively convinced of that truth. This made him in love with geometry.

And so began a love of mathematics that, if MacTutor is right, lasted the half-century he had left to his life. His mathematics work would not displace his place as a philosopher, but then, his accomplishments …

Well, that’s the less cheerful part of it. For example, says MacTutor, shortly before his 1655 publication of De Corpore (On The Body) Hobbes worked out a method of squaring the circle, using straightedge and compass alone. It’s impossible to do this (though that it is impossible would take two more centuries to prove), and Hobbes realized his demonstration was wrong shortly before publication. Rather than remove the proof he added text to explain that it was a false proof.

False proofs can be solid teaching tools: just working out where a proof goes wrong is a good exercise in testing one’s knowledge of concepts and how they relate, and whether a concept is actually well-defined yet. And it’s not like attempting to square the circle is by itself ridiculous. I suspect most mathematicians even today give it a try, at least before they can study the proofs that it’s impossible and they can go on to trying to do Fermat’s Last Theorem.

([Edited 31 May 2017 to change from “you can go on to trying” because I finally noticed the shift in pronouns was weird.])

But Hobbes also included a second attempted proof which he again realized was at best “an approximate quadrature”, and tried a third which he realized was wrong while the book was being printed, so he added a note that it was meant as a problem for the reader. Hobbes was sure he was close, and would keep on trying to prove he had squared the circle to the end of his life. These circle-squarings set off a long-running feud with John Wallis, a pioneer in algebra and calculus (and the person who introduced the ∞ symbol to mathematics), who attacked Hobbes’s mistakes and faulty claims.

Hobbes also refused to have anything to do with the algebra and calculus and the symbolic operations which were revolutionizing mathematics at the time; he wanted geometry and nothing but. MacTutor quotes him as insulting — and here we’re reminded that the 17th century was a golden age of academic insulting — Wallis’s Algebra as “a scab of symbols [ which disfigured the page ] as if a hen had been scraping there”.

The best that MacTutor can say about Hobbes’s mathematics is that while he claimed to do a lot of truly impressive work, none of the things which would have been substantial advances in mathematics were correct. And there is something sad that a person of great intellectual power could be so in love with mathematics and find that love wasn’t reciprocated. He wrote near the end of his life a list of seven problems “sought in vain by the diligent scrutiny of the greatest geometers since the very beginnings of geometry” that he concluded he’d solved; and, to be kind, he’s not renowned as the person who found the center of gravity of the quadrant of a circle.

But that sadness is taking an unfair view of the value of doing mathematics. So Hobbes spent a half-century playing with plane figures without finding something true that future generations would regard as novel — how is that a failing? How many professional mathematicians will do something that’s of any general interest, and won’t even write a classic on social contract theory that people will think they probably should’ve read at some point? He found in geometry something which brought him a sense of wonder, and which was delightful enough to keep him going through long and bitter academic feuds (I grant it’s possible Hobbes enjoyed the feuds; some people do), and without apparently losing his enthusiasm. That’s wonderful, regardless of whether his work found anything original.


Postscript: Some better-informed thoughts about this are in the article A bit more about Thomas Hobbes.

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