I’ve been reading The Disordered Cosmos: A Journey Into Dark Matter, Spacetime, and Dreams Deferred, by Chanda Prescod-Weinstein. It’s the best science book I’ve read in a long while.
Part of it is a pop-science discussion of particle physics and cosmology, as they’re now understood. It may seem strange that the tiniest things and the biggest thing are such natural companion subjects. That is what seems to make sense, though. I’ve fallen out of touch with a lot of particle physics since my undergraduate days and it’s wonderful to have it discussed well. This sort of pop physics is for me a pleasant comfort read.
The other part of the book is more memoir, and discussion of the culture of science. This is all discomfort reading. It’s an important discomfort.
I discuss sometimes how mathematics is, pretensions aside, a culturally-determined thing. Usually this is in the context of how, for example, that we have questions about “perfect numbers” is plausibly an idiosyncrasy. I don’t talk much about the culture of working mathematicians. In large part this is because I’m not a working mathematician, and don’t have close contact with working mathematicians. And then even if I did — well, I’m a tall, skinny white guy. I could step into most any college’s mathematics or physics department, sit down in a seminar, and be accepted as belonging there. People will assume that if I say anything, it’s worth listening to.
Chanda Prescod-Weinstein, a Black Jewish agender woman, does not get similar consideration. This despite her much greater merit. And, like, I was aware that women have it harder than men. And Black people have it harder than white people. And that being open about any but heterosexual cisgender inclinations is making one’s own life harder. What I hadn’t paid attention to was how much harder, and how relentlessly harder. Most every chapter, including the comfortable easy ones talking about families of quarks and all, is several needed slaps to my complacent face.
Her focus is on science, particularly physics. It’s not as though mathematics is innocent of driving women out or ignoring them when it can’t. Or of treating Black people with similar hostility. Much of what’s wrong is passively accepting patterns of not thinking about whether mathematics is open to everyone who wants in. Prescod-Weinstein offers many thoughts and many difficult thoughts. They are worth listening to.
Nobody had particular suggestions for the letter ‘Y’ this time around. It’s a tough letter to find mathematical terms for. It doesn’t even lend itself to typography or wordplay the way ‘X’ does. So I chose to do one more biographical piece before the series concludes. There were twists along the way in writing.
Several problems beset me in writing about this significant 13th-century Chinese mathematician. One is my ignorance of the Chinese mathematical tradition. I have little to guide me in choosing what tertiary sources to trust. Another is that the tertiary sources know little about him. The Complete Dictionary of Scientific Biography gives a dire verdict. “Nothing is known about the life of Yang Hui, except that he produced mathematical writings”. MacTutor’s biography gives his lifespan as from circa 1238 to circa 1298, on what basis I do not know. He seems to have been born in what’s now Hangzhou, near Shanghai. He seems to have worked as a civil servant. This is what I would have imagined; most scholars then were. It’s the sort of job that gives one time to write mathematics. Also he seems not to have been a prominent civil servant; he’s apparently not listed in any dynastic records. After that, we need to speculate.
E F Robertson, writing the MacTutor biography, speculates that Yang Hui was a teacher. That he was writing to explain mathematics in interesting and helpful ways. I’m not qualified to judge Robertson’s conclusions. And Robertson notes that’s not inconsistent with Yang being a civil servant. Robertson’s argument is based on Yang’s surviving writings, and what they say about the demonstrated problems. There is, for example, 1274’s Cheng Chu Tong Bian Ben Mo. Robertson translates that title as Alpha and omega of variations on multiplication and division. I try to work out my unease at having something translated from Chinese as “Alpha and Omega”. That is my issue. Relevant here is that a syllabus prefaces the first chapter. It provides a schedule and series of topics, as well as a rationale for why this plan.
Was Yang Hui a discoverer of significant new mathematics? Or did he “merely” present what was already known in a useful way? This is not to dismiss him; we have the same questions about Euclid. He is held up as among the great Chinese mathematicians of the 13th century, a particularly fruitful time and place for mathematics. How much greatness to assign to original work and how much to good exposition is unanswerable with what we know now.
Consider for example the thing I’ve featured before, Yang Hui’s Triangle. It’s the arrangement of numbers known in the west as Pascal’s Triangle. Yang provides the earliest extant description of the triangle and how to form it and use it. This in the 1261 Xiangjie jiuzhang suanfa (Detailed analysis of the mathematical rules in the Nine Chapters and their reclassifications). But in it, Yang Hui says he learned the triangle from a treatise by Jia Xian, Huangdi Jiuzhang Suanjing Xicao (The Yellow Emperor’s detailed solutions to the Nine Chapters on the Mathematical Art). Jia Xian lived in the 11th century; he’s known to have written two books, both lost. Yang Hui’s commentary gives us a fair idea what Jia Xian wrote about. But we’re limited in judging what was Jia Xian’s idea and what was Yang Hui’s inference or what.
The Nine Chapters referred to is Jiuzhang suanshu. An English title is Nine Chapters on the Mathematical Art. The book is a 246-problem handbook of mathematics that dates back to antiquity. It’s impossible to say when the Nine Chapters was first written. Liu Hui, who wrote a commentary on the Nine Chapters in 263 CE, thought it predated the Qin ruler Shih Huant Ti’s 213 BCE destruction of all books. But the book — and the many commentaries on the book — served as a centerpiece for Chinese mathematics for a long while. Jia Xian’s and Yang Hui’s work was part of this tradition.
Yang Hui’s Detailed Analysis covers the Nine Chapters. It goes on for three chapters, more about geometry and fundamentals of mathematics. Even how to classify the problems. He had further works. In 1275 Yang published Practical mathematical rules for surveying and Continuation of ancient mathematical methods for elucidating strange properties of numbers. (I’m not confident in my ability to give the Chinese titles for these.) The first title particularly echoes how in the Western tradition geometry was born of practical concerns.
The breadth of topics covers, it seems to me, a decent modern (American) high school mathematics education. The triangle, and the binomial expansions it gives us, fit that. Yang writes about more efficient ways to multiply on the abacus. He writes about finding simultaneous solutions to sets of equations. And through a technique that amounts to finding the matrix of coefficients for the equations, and its determinant. He writes about finding the roots for cubic and quartic equations. The technique is commonly known in the west as Horner’s Method, a technique of calculating divided differences. We see the calculating of areas and volumes for regular shapes.
And sequences. He found the sum of the squares of natural numbers followed a rule:
And then there’s magic squares, and magic circles. He seems to have found them, as professional mathematicians today would, good ways to interest people in calculation. Not magic; he called them something like number diagrams. But he gives magic squares from three-by-three all the way to ten-by-ten. We don’t know of earlier examples of Chinese mathematicians writing about the larger magic squares. But Yang Hui doesn’t claim to be presenting new work. He also gives magic circles. The simplest is a web of seven intersecting circles, each with four numbers along the circle and one at its center. The sum of the center and the circumference numbers are 65 for all seven circles. Is this significant? No; merely fun.
Grant this breadth of work. Is he significant? I learned this year that familiar names might have been obscure until quite recently. The record is once again ambiguous. Other mathematicians wrote about Yang Hui’s work in the early 1300s. Yang Hui’s works were printed in China in 1378, says the Complete Dictionary of Scientific Biography, and reprinted in Korea in 1433. They’re listed in a 1441 catalogue of the Ming Imperial Library. Seki Takakazu, a towering figure in 17th century Japanese mathematics, copied the Korean text by hand. Yet Yang Hui’s work seems to have been lost by the 18th century. Reconstructions, from commentaries and encyclopedias, started in the 19th century. But we don’t have everything we know he wrote. We don’t even have a complete text of Detailed Analysis. This is not to say he wasn’t influential. All I could say is there seems to have been a time his influence was indirect.
So my attempt at keeping the Reading the Comics posts to Sunday has crashed and burned again. This time for a good reason. As you might have read between the lines on my humor blog, I spent the past week on holiday and just didn’t have time to write stuff. I barely had time to read my comics. I’ll get around to it this week.
In the meanwhile then I’d like to point people to the MathsByAGirl blog. The blog recently had an essay on Nicolas Bourbaki, who’s among the most famous mathematicians of the 20th century. Bourbaki is also someone with a tremendous and controversial legacy, one that I expect to touch on as I catch up on last week’s comics. If you don’t know the secret of Bourbaki then do go over and learn it. If you do, well, go over and read anyway. The author’s wondering whether to write more about Bourbaki’s mathematics and while I’m all in favor of that more people should say.
And as I promised a trifle, let me point to something from my own humor blog. How To Write Out Numbers is an older trifle based on everyone’s love for copy-editing standards. I had forgotten I wrote it before digging it up for a week of self-glorifying posts last week. I hope folks around here like it too.
When I saw the Maths History tweet about Edmond Halley’s birthday I wondered if the November 8th date given was the relevant one since, after all, in 1656 England was still on the Julian calendar. The MacTutor biography of him makes clear that the 8th of November is his Gregorian-date birthday, and he was born on the 29th of October by the calendar his parents were using, although it’s apparently not clear he was actually born in 1656. Halley claimed it was 1656, at least, and he probably heard from people who knew.
Halley is famous for working out the orbit of the comet that’s gotten his name attached, and correctly so: working out the orbits of comets was one of the first great accomplishments of Newtonian mechanics, and Halley’s work took into account how Jupiter’s gravitation distorts the orbit of a comet. It’s great work. And he’s also famous within mathematical and physics circles because it’s fair to wonder whether, without his nagging and his financial support, Isaac Newton would have published his Principia Mathematica. Astronomers note him as the first Western European astronomer to set up shop in the southern hemisphere and produce a map of that part of the sky, as well.
That hardly exhausts what’s interesting about him: for example, he joined in the late-17th-century fad for diving bell companies (for a while, you couldn’t lose money excavating wrecked ships, until finally everyone did) and even explored the bed of the English Channel in a diving bell of his own design. This is to me the most terrifying thing he did, and that’s even with my awareness he led two scientific sailing expeditions, one of which was cut short after among other things irreconcilable differences with the ship’s other commissioned officer, Lieutenant Edward Harrison (who blamed Halley for the oblivion which Harrison’s book on longitude received), and the second of which included a pause in Recife when Halley was put under guard by a man claiming to be the English consul, and who was actually an agent of the Royal African Company considering whether to seize Halley’s ship as a prize.
After his second expedition Halley published charts showing the magnetic declination, how far a magnetic compass’s “north” is from true north, and introduced one of those great conceptual breakthroughs that charts can give us: he connected the lines showing the points where the declination was equal. These isolines are a magnificent way to diagram three-dimensional information on a two-dimensional chart; we see them in topographic maps, as the contour curves showing where a hill rises or a valley sinks. We see them in weather maps, the lines where the temperature is 70 or 80 Fahrenheit (or 20 or 25 Celsius, if you rather) or where the wind speed is some sufficiently alarming figure. We see them (in three-dimensional form) in medical imaging, where a region of constant density gets the same color and this is used to understand a complicated shape within. Not all these uses derive directly from Halley; as with all really good, widely usable concepts many people discovered the concept, but Halley was among the first to put them to obvious, prominent use.
And something that might serve as comfort to anyone who’s taking a birthday hard: at age 65, Halley began a study of the moon’s saros, the cycle patterns of different relative positions the Sun and Moon have in the sky which describe when eclipses happen. One cycle takes a bit over eighteen years to complete. Halley lived long enough to complete this work.
 The Paramore, which — I note because this is just the kind of world it was back then — was constructed in 1694 at the Royal Dockyard at Deptford on the River Thames for a scientific circumnavigation of the globe, and first sailed in April 1698 under Tsar Peter the Great, then busy travelling western Europe under ineffective cover to learn things which might modernize Russia. Halley had hoped to sail in 1696, but he was waylaid by his appointment to the Mint at Chester, courtesy of Newton.
A couple weeks back I offered a challenge taken from Graham Farmelo’s biography (The Strangest Man) of the physicist Paul Dirac. The physicist had been invited into a game to create whole numbers by using exactly four 2’s and the normal arithmetic operations, for example:
While four 2’s have to be used, and not any other numerals, it’s permitted to use the 2’s stupidly, as every one of my examples here does. Dirac went off and worked out a scheme for producing any positive integer from them. Now, if all goes well, Dirac’s answer should be behind this cut and it hasn’t been spoiled in the reader or the mails sent out to people reading it.
I’ve been reading Graham Farmelo’s The Strangest Man: The Hidden Life of Paul Dirac, which is a quite good biography about a really interestingly odd man and important physicist. Among the things mentioned is that at one point Dirac was invited in to one of those number-challenge puzzles that even today sometimes make the rounds of the Internet. This one is to construct whole numbers using exactly four 2’s and the normal, non-exotic operations — addition, subtraction, exponentials, roots, the sort of thing you can learn without having to study calculus. For example:
Now these aren’t unique; for example, you could also form 2 by writing , or as . But the game is to form as many whole numbers as you can, and to find the highest number you can.
Dirac went to work and, complained his friends, broke the game because he found a formula that can any positive whole number, using exactly four 2’s.
I couldn’t think of it, and had to look to the endnotes to find what it was, but you might be smarter than me, and might have fun playing around with it before giving up and looking in the endnotes yourself. The important things are, it has to produce any positive integer, it has to use exactly four 2’s (although they may be used stupidly, as in the examples I gave above), and it has to use only common arithmetic operators (an ambiguous term, I admit, but, if you can find it on a non-scientific calculator or in a high school algebra textbook outside the chapter warming you up to calculus you’re probably fine). Good luck.
What interests me is that Lemoine looked into the problem of how complicated a proof is, and in just the sort of thing designed to endear him to my heart, he tried to give a concrete measurement of, at least, how involved a geometric construction was. He identified the classic steps in compass-and-straightedge constructions and classified proofs by how many steps these took. MacTutor cites him as showing that the usual solution to the Apollonius problem — construct a circle tangent to three given circles — required over four hundred operations, but that he was able to squeeze that down to 199.
However, well, nobody seems to have been very interested in this classification. That’s probably because the length doesn’t really measure how complicated a proof (or a construction) is: proofs can have a narrative flow, and a proof that involves many steps each of which seems to flow obviously (or which look like the steps in another, already-familiar proof) is going to be easier to read and to understand than one that involves fewer but more obscure steps. This is the sort of thing that challenges attempts to measure how difficult a proof is, even though it’d be interesting to know.
Here’s one of the things that would be served by being able to measure just how long a proof is: a lot of numerical mathematics depends on having sequences of randomly generated numbers, but, showing that you actually have a random sequence of numbers is a deeply hard problem. If you can specify how you get a particular digit … well, they’re not random, then, are they? Unless it’s “use this digit from this randomly generated sequence”. If you could show there’s no way to produce a particular sequence of numbers in any way more efficiently than just reading them off this list of numbers you’d at least have a fair chance at saying this is a truly unpredictable sequence. But, showing that you have found the most efficient algorithm for producing something is … well, it’s difficult to even start measuring that sort of thing, and while Lemoine didn’t produce a very good measure of algorithmic complexity, he did have an idea, and it’s difficult to see how one could get a good measure if one didn’t start with trying not-very-good ones.
The slightly dirty secret, though, is that it isn’t. It’s built around logical arguments, certainly, and the more rigorous the argument the better-proven a thing is usually considered to be. But you don’t get results proven with perfectly rigorously airtight deductive reasoning, at least not in the journals and monographs that report interesting new results, because it turns out this requires so much work that it takes forever. What you typically see is enough of an argument to be convincing that anything elided over could be filled in, if required. This is part of why huge results professing major new accomplishments, like a proof of Goldbach’s Conjecture, take time to verify: not only is there a lot that’s there, but suddenly the question of whether the elided steps really are secure has to be filled in.
Most of the big gaps-to-be-filled in basic mathematics were filled in a century ago. Pasch was among the people who found some points in Euclidean geometry where physical intuition about real-world things was assumed into mathematical arguments without it being explicitly stated. This didn’t mean any geometric results were wrong or counterintuitive or anything; just that there were assumptions in the system that Euclid — and everybody else — had made without saying they were making them, which is pretty impressive considering that Euclid thought to mention that he was assuming all right angles were congruent.
One of those discovered spots gets called now Pasch’s Axion, and it gives a good example of the kind of thing which can go centuries being assumed without drawing attention to itself: suppose you have a triangle connecting the points we label A, B, and C. And suppose you have a line which enters the triangle through the leg connecting points A and B, and which doesn’t pass through the point C. Then the line exits the triangle either through the leg between points B and C or through the leg between points C and A.
Obvious? Perhaps, but not more obvious than the axiom that a line segment can be drawn between any two different points, and it’s a special insight to notice these things are assumptions.
As promised I’m keeping and publicizing my statistics, as WordPress makes them out, the better I hope to understand what I do well and what the rest is. I’ve had a modest uptick in views from July — 341 to 367 — as well as in unique visitors — 156 to 175 — although this means my views-per-visitor count has dropped from 2.19 to 2.10. That’s still my third-highest views-per-visitor count since WordPress started revealing that data to us.
The countries sending me the most readers were the United States (202), Canada (30), and Denmark (19). Sending me just one each were Argentina, Bangladesh, Estonia, Finland, the Netherlands, Portugal, Singapore, Sri Lanka, Taiwan, and Viet Nam. Argentina, Estonia, and the Netherlands did the same last month; clearly I’m holding steady. And my readership in Slovenia doubled from last month’s lone reader.
The Maths History feed on Twitter mentioned that the 21st of August was the birthday of Augustin-Louis Cauchy, who lived from 1789 to 1857. His is one of those names you get to know very well when you’re a mathematics major, since he published 789 papers in his life, and did very well at publishing important papers, ones that established concepts people would actually use.
He’s got an intriguing biography, as he lived (mostly) in France during the time of the Revolution, the Directorate, Napoleon, the Bourbon Restoration, the July Monarchy, the Revolutions of 1848, the Second Republic, and the Second Empire, and had a career which got inextricably tangled with the political upheavals of the era. I note that, according to the MacTutor biography linked to earlier this paragraph, he followed the deposed King Charles X to Prague in order to tutor his grandson, but might not have had the right temperament for it: at least once he got annoyed at the grandson’s confusion and screamed and yelled, with the Queen, Marie Thérèse, sometimes telling him, “too loud, not so loud”. But we’ve all had students that frustrate us.
Amongst the Twitter feeds I follow and which aren’t based on fictional squirrels is the @mathshistory one, reporting just what it sounds like. It noted the 10th of July was the birthday of Roger Cotes (1682 – 1716) and I knew there was something naggingly familiar about his name. His biography at the MacTutor History of Mathematics Archive features what surely kept him in my mind: that on Cotes’s death at age 34 Isaac Newton said, “… if he had lived we might have known something”. Given Newton’s standing, that’s a eulogy almost good enough to get Cotes tenure, even today.
MacTutor credits Cotes with, among other things, inventing the radian measure of angles; I’m wise enough, I hope, to view skeptically all claims of anyone uniquely inventing anything mathematical, although it’s certainly so that radian measure — in which you give an angle of arc, not by how many degrees it reaches, but by how long the arc is, in units of the radius — is extraordinarily convenient analytically and it’s hard to see how mathematicians did without it. People who advanced the idea and its use deserve their praise. (Normal people can carry on with degrees of arc, for which the numbers are just more pleasant.) As a bonus it serves as one of the points on which people coming into trigonometry classes can feel their heads exploding.
Cotes’s name also gets a decent and, if I have it right, appropriate amount of fame for what are called Newton-Cotes formulas. These are methods for “numerical quadrature”, the slightly old-fashioned name we use to talk about numerical approximations of integrals. In an introductory calculus class one’s likely to run across a couple of rules for numerical quadrature — given names like the Trapezoid Rule, Simpson’s Rule, Simpson’s 3/8ths rule, or the Midpoint Rule — and these are all examples of the Newton-Cotes formulas. Teaching the routine for getting all these Newton-Cotes formulas was, for whatever reason, one of the things I found particularly delightful when I taught numerical mathematics; some subjects are just fun to explain.
MacTutor also notes that from 1709 through 1713, Cotes edited the second edition of Newton’s Principia, and apparently did a most thorough job of it. It claims he studied the Principia and arguing its points with Newton in enough detail that Newton finally removed the thanks he gave to Cotes in the first draft of his preface. A difficult but correct editor is probably more pleasant to have when the project is finished.
While following my own lightly compulsive tracking of the blog’s viewer statistics and wondering why I don’t have more followers or even people getting e-mail notifications (at least I’ve broken 2,222 hits!) I ran across something curious. I can’t swear that it’s still true so I’m not going to link to it, and I don’t want to know if it’s not true. However.
I assume it to be some sort of fluke. Possibly it reflects how the link I actually find useful is never the first one in the list of what’s returned, so perhaps they’re padding the results with some technically correct but nonsense filler, and I had the luck of the draw this time. Perhaps not. (I’m only third for “drabble math comic”, and that would at least be plausible.) But I’m amused by it anyway. And I’d like to again say that the MacTutor biographies at the University of Saint Andrews are quite good overall and worth using as reference, and are also the source of my discovery that Wednesday, March 21, is the anniversary of the births of both Jean Baptiste Joseph Fourier (for whom the Fourier Series, Fourier Transform, and Fourier Analysis, all ways of turning complicated problems into easier ones, are named) and of George David Birkhoff (whose ergodic theorem is far too much to explain in a paragraph, but without which almost none of my original mathematics work would have what basis it has). I should give both subjects some discussion. I might yet make Wikipedia.