How My Mathematics Blog Was Read, For January 2015


And after reaching 20,000 views on the final day of December, 2014, could I reach 21,000 views by the end of January? Probably I could have, but in point of fact I did not. I am not complaining, though: I finished the month with 20,956 page views all told, after a record 944 pages got viewed by somebody, somewhere, for some reason. This is a record high for me, going well past the 831 that had been the January 2013 and December 2014 (tied) record. And likely I’ll reach 21,000 in the next couple days anyway.

According to WordPress, this was read by 438 distinct visitors, reading 2.16 views per visitor on average. That isn’t quite a record: January 2013 remains my high count for visitors, at 473, but it’s still, all told, some pretty nice numbers especially considering I don’t think I had my best month of blog-writing. I can’t wait to get some interesting new topics in here for February and see that they interest absolutely nobody.

The new WordPress statistics page is still awful, don’t get me wrong, but it has been getting a little bit better, and it does offer some new data I couldn’t gather easily before. Among them: that in January 205 I received 197 likes overall — a high for the past twelvemonth, which is as far as I can figure out how to get it, and up from 128 in December — and 51 comments, up from December 29, and also a high for the twelvemonth.

The three countries sending me viewers were, once again, the big three of the United States (594), Canada (56), and the United Kingdom (52), with Austria sending in 32 viewers, and Germany and Argentina ending 22 each. And India, for a wonder sent me a noticeable-to-me 18 readers, although on a per capita basis that still isn’t very many, I admit.

There was a bumper crop of single-reader countries, though, up from last month’s six: Belgium, Estonia, Finland, Greece, Hungary, Indonesia, Iraq, Japan, Kuwait, Libya, Mexico, Paraguay, Slovakia, and the United Arab Emirates each found only one person viewing anything around here. Greece and Mexico are repeats from December.

This month’s most popular articles were mostly comic strip posts, although they were a pretty popular set; none of these had fewer than 35 views per, which feels high to me. The top posts of the last 30 days, then, were:

  1. Reading the Comics, January 6, 2015: First of the Year Edition, in which I included drawing a sloppy `2′ as a snoring `Z’ as somehow connected to mathematics.
  2. Reading the Comics, January 24, 2015: Many, But Not Complicated Edition, which includes an explanation for why margins of errors on surveys are always like three or four percent.
  3. Reading the Comics, January 11, 2015: Standard Genres And Bloom County Edition, in which I reveal my best guess for Jon Bon Jovi’s shorts size in the late 80s.
  4. 20,000: My Math Blog’s Statistics, because my narcissism is apparently quite popular?
  5. Reading the Comics, January 17, 2015: Finding Your Place Edition, where, again, I can flog that thing about a watch as a compass.
  6. How Many Trapezoids I Can Draw, which also reveals how many trapeziums I think are different in interesting ways.
  7. A bit more about Thomas Hobbes, and his attempt to redefine the very nature of mathematics, which didn’t succeed in quite the way he wanted.

Among the interesting search terms that brought people here the past month have been ([sic] on all of them):

  • science fiction and trapazoids (Somebody should totally write the definitive SFnal treatment of trapezoids, I agree.)
  • food. stotagre nebus (I feel strangely threatened by this.)
  • a group of student offer at least one of mathematics,physics, and statistcs , 14 of them offer mathematics, 12 offer physics,and 16 offer statistics.7 offer statistics and maths 6 offer maths and physics, 4 offer physics and statistics only, while 5 offer all the three subject (Help?)
  • hiw to draw diffrent trameziums
  • soglow otto radio (Pretty sure I used to listen to that back on WRSU in my undergrad days.)
  • if a calendar has two consecutive months with friday the 13th which would they be (February and March, in a non-bissextile — that is, non-leap — year)
  • how to measure a christmas tree made of triangles and trapeziums (I would use a tape measure, myself)

So if I would summarize January 2015 in my readership here, I would say: tramezium?

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.

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.