You might remember a post from last April, Thomas Hobbes and the Doing of Important Mathematics, timed to the renowned philosopher’s birthday. I talked about him because a good bit of his intellectual life was spent trying to achieve mathematical greatness, which he never did.

Recently I’ve had the chance to read Douglas M Jesseph’s Squaring The Circle: The War Between Hobbes And Wallis, about Hobbes’s attempts to re-build mathematics on an intellectual foundation he found more satisfying, and the conflict this put him in with mainstream mathematicians, particularly John Wallis (algebra and calculus pioneer, and popularizer of the ∞ symbol). The situation of Hobbes’s mathematical ambitions is more complicated than I realized, although the one thing history teaches us is that the situation is *always* more complicated than we realized, and I wanted to at least make my writings about Hobbes a bit less incomplete. Jesseph’s book can’t be fairly reduced to a blog post, of course, and I’d recommend it to people who want to really understand what the fuss was all about. It’s a very good idea to have some background in philosophy and in 17th century English history going in, though, because it turns out a lot of the struggle — and particularly the bitterness with which Hobbes and Wallis fought, for decades — ties into the religious and political struggles of England of the 1600s.

Hobbes’s project, I better understand now, was not merely the squaring of the circle or the solving of other ancient geometric problems like the doubling of the cube or the trisecting of an arbitrary angle, although he did claim to have various proofs or approximate proofs of them. He seems to have been interested in building a geometry on more materialist grounds, more directly as models of the real world, instead of the pure abstractions that held sway then (and, for that matter, now). This is not by itself a ridiculous thing to do: we are almost always better off for having multiple independent ways to construct something, because the differences in those ways teaches us not just about the thing, but about the methods we use to discover things. And purely abstract constructions have problems also: for example, if a line can be decomposed into nothing but an enormous number of points, and absolutely none of those points has any length, then how can the line have length? You can answer that, but it’s going to require a pretty long running start.

Trying to re-build the logical foundations of mathematics is an enormously difficult thing to do, and it’s not surprising that someone might fail to do so perfectly. Whole schools of mathematicians might be needed just to achieve mixed success. And Hobbes wasn’t able to attract whole schools of mathematicians, in good part because of who he was.

Hobbes achieved immortality as an important philosopher with the publication of Leviathan. What I had not appreciated and Jesseph made clear was that in the context of England of the 1650s, Hobbes’s views on the natures of God, King, Society, Law, and Authority managed to offend — in the “I do not know how I can continue to speak with a person who holds views like that” — pretty much everybody in England who had any strong opinion about anything in politics, philosophy, or religion. I do not know for a fact that Hobbes then went around kicking the pet dogs of any English folk who didn’t have strong opinions about politics, philosophy, or religion, but I can’t rule it out. At least part of the relentlessness and bitterness with which Wallis (and his supporters) attacked Hobbes, and with which Hobbes (and his supporters) attacked back, can be viewed as a spinoff of the great struggle between the Crown and Parliament that produced the Civil War, the Commonwealth, and the Restoration, and in that context it’s easier to understand why all parties carried on, often quibbling about extremely minor points, well past the point that their friends were advising them that the quibbling was making themselves look bad. Hobbes was a difficult person to side with, even when he was right, and a lot of his mathematics just wasn’t right. Some of it I’m not sure ever could be made right, however many ingenious people you had working to avoid flaws.

An amusing little point that Jesseph quotes is a bit in which Hobbes, making an argument about the rights that authority has, asserts that if the King decreed that Euclid’s Fifth Postulate should be taught as false, then false it would be in the kingdom. The Fifth Postulate, also known as the Parallel Postulate, is one of the axioms on which classical Greek geometry was built and it was always the piece that people didn’t like. The other postulates are all nice, simple, uncontroversial, common-sense things like “all right angles are equal”, the kinds of things so obvious they just *have* to be axioms. The Fifth Postulate is this complicated-sounding thing about how, if a line is crossed by two non-parallel lines, you can determine on which side of the first line the non-parallel lines will meet.

It wouldn’t be really understood or accepted for another two centuries, but, you *can* suppose the Fifth Postulate to be false. This gives you things named “non-Euclidean geometries”, and the modern understanding of the universe’s geometry is non-Euclidean. In picking out an example of something a King might decree and the people would have to follow regardless of what was really true, Hobbes picked out an example of something that could be decreed false, and that people could follow profitably.

That’s not mere ironical luck, probably. A streak of mathematicians spent a long time trying to prove the Fifth Postulate was unnecessary, at least, by showing it followed from the remaining and non-controversial postulates, or at least that it could be replaced with something that felt more axiomatic. Of course, in principle you can use any set of axioms you like to work, but some sets produce more interesting results than others. I don’t know of any interesting geometry which results from supposing “not all right angles are equal”; supposing that the Fifth Postulate is untrue gives us general relativity, which is quite nice to have.

Again I have to warn that Jesseph’s book is not always easy reading. I had to struggle particularly over some of the philosophical points being made, because I’ve got only a lay understanding of the history of philosophy, and I was able to call on my love (a professional philosopher) for help at points. I imagine someone well-versed in philosophy but inexperienced with mathematics would have a similar problem (although — don’t let the secret out — you’re allowed to just skim over the diagrams and proofs and go on to the explanatory text afterwards). But for people who want to understand the scope and meaning of the fighting better, or who just want to read long excerpts of the wonderful academic insulting that was current in the era, I do recommend it. Check your local college or university library.

From what I understood about this post was – aside from the king is god’s annointed and thus is always right and whatever he says is the rule regardless of its veracity – was that all accepted mathematicians of the day stuck with the classical models of mathematics, which in truth, can be difficult to use much less completely comprehend. So Hobbes wanted, like so many people these days, for mathematics to have a much more reasonable, understandable basis? From what i gather from your post, he offered various proofs for his assertions but none were complete or at least none proved his assertions when provided for peer review? And from what else you have said, this Hobbes guy had the deck stacked against him from the beginning because of his background in philosophy?

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Largely, yes, although I’m hesitant to declare exactly what Hobbes wanted because I haven’t read anything more than excerpts of his work, and he did a lot of work over the course of decades, and it can be hard to tell what the point of original work is until after it’s been thought and reworked, completed, and refined. In many ways mathematical constructs are like inventions, with the first ideas of something a complex and barely functional kludge that requires a couple generations of work to make into an elegant and understandable whole, and Hobbes’s mathematics didn’t get those generations of work.

I think it’s fair to say he wanted a more materialistic mathematics, thinking of the lines you draw with straightedge and pencil and the circles you draw with compass and paper. There’s obvious need for that, especially if you want to do something like turn an abstract concept into an actual machine or building or canal or such. But it isn’t the same kind of work that mainstream mathematics was doing at the time, so in some ways he was working in a different field from other mathematicians.

Hobbes’s background in philosophy didn’t by itself hurt him; the fields of mathematics and philosophy blur together on many points, and were even more blurry then. Both fields have about equal claims to Descartes as a founder of their modern incarnations, after all, and Berkeley and Pascal and Leibniz as lesser but still noteworthy figures. Philosophers can be surprised to know mathematicians get to put in a claim on Kant as one of their member; I’m sure the reverse happens. It was a more fluid era.

However, Hobbes’s

particularphilosophy worked against him, because it was frightfully controversial (then and since) and thus made it harder for people to stand behind him. And it put him politically at odds with the Oxford and Cambridge establishments — there was a fierce battle about how the Universities should be reformed and how free they ought to be — and these were the people who would form the Royal Society and the mainstream of English mathematical thought.And, yes, Hobbes didn’t manage to prove the big impressive things he wanted to prove, including some results that carried implication like pi being (if I have it correctly, as I can’t find the page for this right now) equal to 3.2, which everyone by then knew could not be so.

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If you liked

Squaring the Circle, you’ll likely enjoyInfinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, by Amir Alexander. He also touches on the Hobbes/Wallis feud, but talks about a number of other related discussions from 17th Century mathematics on the philosophical nature of the field.LikeLike

I’ve been looking at that! I’ve enjoyed Alexander’s other books, and mostly been figuring out whether my reading list has gotten short enough to start adding new things to it.

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Excellent post Joseph … I can’t but congratulate you!.

“Homo homini lupus”, as Hobbes would say!.

All the best to you. Aquileana :D

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Oh, I’d forgot that Hobbes quote, which is silly since it’s one of his top ones.

Thanks kindly; I’m glad you enjoyed the post.

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