Feb 092025
 

So we studied high school chemistry. Covalent bonds. We learned about nice, well-behaved molecules. Carbon, for instance, with a valence of 4. Hydrogen, 1.

Next, shalt thou combine the two. For each carbon, shalt thou count four hydrogen atoms, no more, no less. Four shall be the number thou shalt count, and the number of the counting shall be four. Five shalt thou not count, neither count thou three, excepting that thou then proceed to four. Six is right out. Once the number four, being the fourth number, be reached, then regardest thou the newly made Methane Atom.

After these magic incantations, you turn around, smugly satisfied with your knowledge of chemistry, content that all is well in the world, and then someone shoves this under your nose:

It’s called methanium. It’s really just an ion, an extra proton stuck to that methane atom. It really does not want to exist, so much so, it’s a superacid, which is to say its acidity is greater than that of sulfuric acid.

Okay, so maybe methanium is not quite as evil as dimethylmercury, which really should have no right to exist in a sensible universe, but I daresay, the very existence of methanium already should inform us that we do not live in a sensible universe.

 Posted by at 3:36 pm
Feb 092025
 

I was reading about Borwein integrals.

Here’s a nice result:

$$\int_0^\infty dx\,\frac{\sin x}{x}=\frac{\pi}{2}.$$

Neat, is it not. Here’s another:

$$\int_0^\infty dx\,\frac{\sin x}{x}\frac{\sin (x/3)}{x/3}=\frac{\pi}{2}.$$

Jumping a bit ahead, how about

$$\int_0^\infty dx\,\frac{\sin x}{x}\frac{\sin (x/3)}{x/3}…\frac{\sin (x/13)}{x/13}=\frac{\pi}{2}.$$

Shall we conclude, based on these examples, that

$$\int_0^\infty dx\,\prod\limits_{k=0}^\infty\frac{\sin (x/[2k+1])}{x/[2k+1]}=\frac{\pi}{2}?$$

Not so fast. First, consider that

$$\int_0^\infty dx\,\frac{\sin x}{x}\frac{\sin (x/3)}{x/3}…\frac{\sin (x/15)}{x/15}=\frac{935615849426881477393075728938}{935615849440640907310521750000}\frac{\pi}{2}\approx\frac{\pi}{2}-2.31\times 10^{-11}.$$

Or how about

\begin{align}
\int_0^\infty&dx\,\cos x\,\frac{\sin x}{x}=\frac{\pi}{4},\\
\int_0^\infty&dx\,\cos x\,\frac{\sin x}{x}\frac{\sin (x/3)}{x/3}=\frac{\pi}{4},\\
…\\
\int_0^\infty&dx\,\cos x\,\frac{\sin x}{x}…\frac{\sin (x/111)}{x/111}=\frac{\pi}{4},
\end{align}

but then,

$$\int_0^\infty dx\,\cos x\,\frac{\sin x}{x}…\frac{\sin (x/113)}{x/113}\approx\frac{\pi}{4}-1.1162\times 10^{-138}.$$

There is a lot more about Borwein integrals on Wikipedia, but I think even these few examples are sufficient to convince us that, never mind the actual, physical universe, even the Platonic universe of mathematical truths is fundamentally evil and unreasonable.

 Posted by at 2:49 pm
Feb 082025
 

Here is one of my cherished possessions. A book, with an inscription:

The inscription, written just over 50 years ago, explains that I received this book from my grade school, in recognition for my exceptional results in mathematics as a sixth grade student. (If memory serves me right, this was the year when I unofficially won the Pest county math championship… for eighth graders.)

The book is a Hungarian-language translation of a British volume from the series Mathematics: A New Approach, by D. E. Mansfield and others, published originally in the early 1960s. I passionately loved this book. It was from this book that I first became familiar with many concepts in trigonometry, matrix algebra, and other topics.

Why am I mentioning this volume? Because the other day, the mailman arrived with an Amazon box containing a set of books. A brand new set of books, published in 2024. A series of mathematics textbooks for middle school and high school students, starting with this volume for 6th and 7th graders:

My instant impression: As a young math geek 50 years ago, I would have fallen in love with these books.

The author, André Cabannes, is known, among other things, as Leonard Susskind’s co-author of General Relativity, the latest book in Susskind’s celebrated Theoretical Minimum series. Cabannes also published several books in his native French, along with numerous translations.

His Middle School Mathematics and High School Mathematics books are clearly the works of passion by a talented, knowledgeable, dedicated author. The moment I opened the first volume, I felt a sense of familiarity. I sensed the same clarity, same organization, and the same quality of writing that characterized those Mansfield books all those years ago.

Make no mistake about it, just like the Mansfield books, these books by Cabannes are ambitious. The subjects covered in these volumes go well beyond, I suspect, the mathematics curricula of most middle schools or high schools around the world. So what’s wrong with that, I ask? A talented young student would be delighted, not intimidated, by the wealth of subjects that are covered in the books. The style is sufficiently light-hearted, with relevant illustrations on nearly every page, with the occasional historical tidbit or anecdote, making it easier to absorb the material. And throughout, there is an understanding of the practical nature, utility of mathematics, that is best summarized by the words on the books’ back cover: “Mathematics is not a collection of puzzles or riddles designed to test your intelligence; it is a language for describing and interacting with the world.

Indeed it is. And these books are true to the author’s words. The subjects may range from the volume of milk cartons through the ratio of ingredients in a cake recipe all the way to the share of the popular vote in the 2024 US presidential election. In each of these examples, the practical utility of numbers and mathematical methods is emphasized. At the same time, the books feel decidedly “old school” but in a good sense: there is no sign of any of the recent fads in mathematics education. The books are “hard core”: ideas and methods are presented in a straightforward way, fulfilling the purpose of passing on the accumulated knowledge of generations to the young reader even as motivations and practical utility are often emphasized.

This is how my love affair with math began when I was a young student, all those years ago. The books that came into my possession, courtesy of both my parents and my teachers, were of a similar nature: they offered robust knowledge, practical utility, clear motivation. Had it existed already, this wonderful series by Cabannes would have made a perfect addition to my little library 50 years ago.

 Posted by at 3:57 am
Feb 032025
 

How did we get here, asks the CBC rhetorically, as they recount the events that led to Trump’s announcement of across-the-board tariffs on Canadian imports to the United States.

On Nov. 5, Americans chose Donald Trump to be their next president. Twenty days later, Trump announced, via a post to his own social-media platform, that he would apply a 25 per cent tariff to all products imported into the United States from Canada and Mexico — a response, he claimed, to the fact that people and illegal drugs were entering the United States from those two countries.

At least in the case of Canada, this was an irrational justification. Seizures of fentanyl at America’s northern border represented 0.08 per cent of all fentanyl seized by American officials in the last fiscal year. The number of people entering the United States through Canada has also been a fraction of the total number of people entering via Mexico.

They also wonder if this might be a shot in the arm for Canadian patriotism. Damn right it will be and for a damn good reason:

But if American democracy continues down a dark path, not being American might be more than an argument against annexation. In that case, as Rob Goodman, an author and professor of politics and public administration at Toronto Metropolitan University, has written, “Canadian distinctiveness” might be not a “vanity object,” but an “essential safeguard of Canadian democracy.”

Again and again, I am reminded of the television adaptation of The Handmaid Tale, depicting a diminished, yet independent Canada where life remains reasonably normal even as south of the border, a country that no longer calls itself the United States of America but is renamed The Republic of Gilead, chooses totalitarianism. No wonder that even our cats seem to be concerned…

Some economists worry that fighting back against Trump’s tariffs is a losing proposition. I don’t think so. Canada’s economy is small compared to that of the US but not that small, and we have something America does not: the resilience of a people determined to fight back against a former friend who so blatantly betrays us. Yes, we will pay more at the grocery counter. We know that. Yes, American goods will disappear from shelves: in fact we will help remove them. But if this is how Trump thinks he can coerce Canada to become the “51st state”, I think I speak for the overwhelming majority of my compatriots when I respond with a resounding (even if un-Canadian in its directness) fuck off. Va chier.

In short: This is not a joke anymore. What Trump is doing is how a country treats its worst enemies, not its friends. If Trump thinks Canada is a pushover, I think he’s in for an even nastier surprise than his best buddy in Moscow when he attacked Ukraine. Let’s hope we never find out just how tough and resilient Canadians will be when backstabbed.

Friends of mine used to think (perhaps not anymore) that I went stark raving mad when I suggested that Canada should rapidly initiate an independent weapons program and build a credible nuclear deterrent. We have the know-how, the materials, we have the technology and the means. As to why? Consider this is a great, rich, but underpopulated country, sandwiched between tyrannical, warmongering Putinistan across the North Pole to the north and the rabid personality cult of Trumpland to the south, and you have your answer.

 Posted by at 2:54 am