Can’t deny that I will sometimes stockpile links of mathematics stuff to talk about. Sometimes I even remember to post it. Sometimes it’s a tweet like this, which apparently I’ve been carrying around since April:

I admit I do not know whether the claim is true. It’s plausible enough. English has many variants in England alone, and any trade will pick up its own specialized jargon. The words are fun as it is.

From the American Mathematical Society there’s this:

I talk a good bit about knot theory. It captures the imagination and it’s good for people who like to doodle. And it has a lot of real-world applications. Tangled wires, protein strands, high-energy plasmas, they all have knots in them. Some work by Paul Sutcliffe and Fabian Maucher, both of Durham University, studies tangled vortices. These are vortices that are, er, tangled together, just like you imagine. Knot theory tells us much about this kind of vortex. And it turns out these tangled vortices can untangle themselves and smooth out again, even without something to break them up and rebuild them. It gives hope for power cords everywhere.

Nerds have a streak which compels them to make blueprints of things. It can be part of the healthier side of nerd culture, the one that celebrates everything. The side that tries to fill in the real-world things that the thing-celebrated would have if it existed. So here’s a bit of news about doing that:

I like the attempt to map Sir Thomas More’s Utopia. It’s a fun exercise in matching stuff to a thin set of data. But as mentioned in the article, nobody should take it too seriously. The exact arrangement of things in Utopia isn’t the point of the book. More probably didn’t have a map for it himself.

(Although maybe. I believe I got this from Simon Garfield’s On The Map: A Mind-Expanding Exploration Of The Way The World Looks and apologize generally if I’ve got it wrong. My understanding is Robert Louis Stevenson drew a map of Treasure Island and used it to make sure references in the book were consistent. Then the map was lost in the mail to his publishers. He had to read his text and re-create it as best he could. Which, if true, makes the map all the better. It makes it so good a lost-map story that I start instinctively to doubt it; it’s so colorfully perfect, after all.)

And finally there’s this gem from the Magic Realism Bot:

A bit more about Thomas Hobbes

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.

John Dee, the ‘Mathematicall Praeface’ and the English School of Mathematics.

The 13th of July was the birth date for John Dee, one of those historical figures who seems ready-made for dopey historical thrillers as he combined the religious disputes of his time — the reigns of Queen Mary I and of Queen Elizabeth I — with astrology and astronomy and mathematics and possibly espionage, with navigation and the early days of England’s expansion to a world power; he even gets into such fascinating-to-the-fans issues like calendar reform.

The Renaissance Mathematician does him some justice by here writing a biographical sketch that focuses on, first, what he can actually be shown to have done (as opposed to the many and really too-far-reaching conspiracies that can touch on him), particularly in turning England from a mathematics desert to a place where people like Isaac Newton, John Wallis (you know his work in the &infty; symbol), or William Oughtred (of slide rule fame, as well as a pioneer in using “x” to symbolize multiplication) could thrive.

I have written about John Dee several times in the past but always in reaction to someone making stupid statements about him so I thought that today on his birthday, he was born 13 June July 1527, I would write something positive without prior provocation.

The world into which Dee was born was one in which mathematics did not play a very significant role and this was particularly true of England, which in this sense lagged severely behind the continent. During the High Middle Ages the European universities virtually ignored mathematics although the introductory degree was theoretically based on the seven liberal arts including the quadrivium consisting of arithmetic, geometry, music (theory of proportions) and astronomy. These subjects were only treated in a very superficial manner and there were no dedicated chairs for the study of mathematics.

This situation began to change in the fifteenth century at the humanist universities…

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