Matt Parker on the Greatest Maths Mistakes

In Praise of 3Blue1Brown

I’ve been talking about bloggers I admire this past week, in one way or the other. But when it comes to consuming audio and video content, I’m at a comparative disadvantage. I much prefer reading to listening or viewing – partly because I read much faster. The other reason, of course, is that my preference for reading very quickly becomes a vicious cycle. Because I’m better at reading than at listening or viewing, I read more, and as a consequence, I get even better at reading. And so on.

But that’s just me, of course. Other folks might have (and do have) different preferences. In fact, I’ve often been told that I really should be creating videos in addition to writing this blog. And I don’t disagree, not even one little bit, and for the following reasons:

  1. If my purpose behind writing on this blog is to help people learn better (and that very much is the case), I’m not doing a very good job if I’m not optimizing for the medium that people prefer
  2. Creating YouTube videos really forces you to hone in on the exact message. Writing blogs allows me to be lazy while writing, and I needn’t worry about length and conciseness. I know this is a bad habit, and one of my to-do’s is to get better at writing bogposts.
  3. Creating videos allows you to be much, much more creative.

But that last point is precisely why I haven’t gotten around to creating videos just yet. Well, that last reason combined with my talent for procrastination, but let’s not go down that road. Let us, instead, go down that first road, the non-procrastination one.

It comes down to (surprise, surprise) an economic concept. Specifically, complements and substitutes. It would be the easiest thing in the world to set up a camera in a class in which I am already teaching something, and put up that video on YouTube. And folks who might watch this recording are simply substituting physically attending my class for virtually attending it. Which is great, of course, and I’ve have enjoyed watching videos that use this technique created by other folks.

But there are other videos on YouTube that don’t just substitute for the real world. The creator treats the format (video) as a complement to his content. The video is not a substitute for the physical classroom, it is a complement to what the creator is saying. And if you want to understand what that really means, try the following. Pick a math textbook of your choice, and try to understand linear algebra. Then watch Grant Sanderson weave his magic on the same subject.

And sure, maybe I was taught the subject badly in college. But even with a really good teacher and/or a really good math textbook in college, I cannot imagine not falling in love with the way Grant teaches us linear algebra. If you’ve watched even one of the videos in that series (and I really do hope you will watch all of them), I think you’ll agree that he comfortably ticks all of the boxes in my little list above.

And there’s so much to admire with all of his videos. The little “pi’s”, the music, his voice (an underrated part of what makes him such a good teacher. His pauses, his inflections and modulations, the pace at which he talks, all are always perfect), and the animations all end up making his videos so much better. And the content itself, and the insane amount of both coding and thinking that must go into each of them, is a whole other story.

My personal favorites from the channel are the Linear Algebra series, and the series on calculus. And as someone who teaches statistics, his video on Bayes Theorem is also fantastic. But let me be clear, these are simply the ones that resonated the most strongly with me. All of 3Blue1Brown’s videos that I have seen are fantastic. All. Every single one of them.

And if you left school/college with a slight dislike for mathematics (as I did), you couldn’t do yourself a better favor then spending a little bit of time everyday watching 3Blue1Brown’s videos. And if you are in love with mathematics (as I now am), you don’t need me to tell you to go watch his videos, now do you?

Grant, thank you very much for your work!

On Doing Different Things

One of the more interesting random questions that I was asked in class recently was about whether it makes sense to “do different things”.

By this, the student who asked the question meant the following: should one focus on doing one thing, and one thing only in terms of studies, in terms of a career and in terms of hobbies, or does it make sense to do a lot of different things in each case.

The obvious answer to give these days is that one should read Range, by David Epstein – and it certainly is a good way to start thinking about the subject. And I agree with the central idea in David’s book – variety is good! You can take a similar lesson from statistics, if you like, or you can focus on the English phrase “variety is the spice of life“… or you can read and reflect on a wonderful profile of one of the winners of the Fields medal this year.

One might say the same of his path into mathematics itself: that it was characterized by much wandering and a series of small miracles. When he was younger, Huh had no desire to be a mathematician. He was indifferent to the subject, and he dropped out of high school to become a poet. It would take a chance encounter during his university years — and many moments of feeling lost — for him to find that mathematics held what he’d been looking for all along.

The entire essay is, of course, well worth your time, and I would strongly recommend that you read it. Your eyes might glaze over some of the mathematics in the profile (mine certainly did), but there is a lot to be learnt by reading the essay carefully, especially if you want to think about whether “doing different things” is a good idea or not.

The profile is much about June Huh as it is about ruminations on productivity, daily routines, the connection between beauty and mathematics and the role that luck plays in our lives. Consider for example, the fact that Huh never works for more than three hours in a day. Folks who work in the corporate world, especially in India, take an almost perverse pride in the number of hours that they spend in office, but I would argue (from personal experience, more than anything else) that the actual work done doesn’t take more than four to five hours at best.

It gets even better, because there is mention later on in the article about an inversion of the Pomodoro technique:

Huh can still muster only enough energy to work for a few hours each day. “Other people work one hour and just take a five-minute rest,” Kim said. “He is like, one hour do something else, and just focus for five minutes, 10 minutes.”

Me, I’m very good at the doing-something-else-for-one-hour bit. It’s the five minutes of deep and focused work that I struggle with!

But the bit that I enjoyed the most was the part about somehow, just knowing:

For Huh, when he is working, there’s something almost subconscious going on. In fact, he usually can’t trace how or when his ideas come to him. He doesn’t have sudden flashes of insight. Instead, “at some point, you just realize, oh, I know this,” he said. Maybe last week, he didn’t understand something, but now, without any additional input, the pieces have clicked into place without his realizing it. He likens it to the way your mind can surprise you and create unexpected connections when you’re dreaming. “It’s just amazing what human minds are capable of,” he said. “And it’s nice to admit that we don’t know what’s going on.”

Two references that this reminded me of:

Poincaré then hypothesized that this selection is made by what he called the “subliminal self,” an entity that corresponds exactly with what Phaedrus called preintellectual awareness. The subliminal self, Poincaré said, looks at a large number of solutions to a problem, but only the interesting ones break into the domain of consciousness. Mathematical solutions are selected by the subliminal self on the basis of “mathematical beauty,” of the harmony of numbers and forms, of geometric elegance. “This is a true esthetic feeling which all mathematicians know,” Poincaré said, “but of which the profane are so ignorant as often to be tempted to smile.” But it is this harmony, this beauty, that is at the center of it all.

Pirsig, Robert M.. Zen and the Art of Motorcycle Maintenance (p. 240). HarperCollins e-books. Kindle Edition.

From further on in the same book:

Then Poincaré illustrated how a fact is discovered. He had described generally how scientists arrive at facts and theories but now he penetrated narrowly into his own personal experience with the mathematical functions that established his early fame. For fifteen days, he said, he strove to prove that there couldn’t be any such functions. Every day he seated himself at his work-table, stayed an hour or two, tried a great number of combinations and reached no results. Then one evening, contrary to his custom, he drank black coffee and couldn’t sleep. Ideas arose in crowds. He felt them collide until pairs interlocked, so to speak, making a stable combination. The next morning he had only to write out the results. A wave of crystallization had taken place. He described how a second wave of crystallization, guided by analogies to established mathematics, produced what he later named the “Theta-Fuchsian Series.” He left Caen, where he was living, to go on a geologic excursion. The changes of travel made him forget mathematics. He was about to enter a bus, and at the moment when he put his foot on the step, the idea came to him, without anything in his former thoughts having paved the way for it, that the transformations he had used to define the Fuchsian functions were identical with those of non-Euclidian geometry. He didn’t verify the idea, he said, he just went on with a conversation on the bus; but he felt a perfect certainty. Later he verified the result at his leisure.

Pirsig, Robert M.. Zen and the Art of Motorcycle Maintenance (pp. 239-240). HarperCollins e-books. Kindle Edition.

And finally, a fascinating paper, of which I learnt via a blogpost on MR, by Alex Tabbarok:

Many people have claimed that sleep has helped them solve a difficult problem, but empirical support for this assertion remains tentative. The current experiment tested whether manipulating information processing during sleep impacts problem incubation and solving. In memory studies, delivering learning-associated sound cues during sleep can reactivate memories. We therefore predicted that reactivating previously unsolved problems could help people solve them. In the evening, we presented 57 participants with puzzles, each arbitrarily associated with a different sound. While participants slept overnight, half of the sounds associated with the puzzles they had not solved were surreptitiously presented. The next morning, participants solved 31.7% of cued puzzles, compared with 20.5% of uncued puzzles (a 55% improvement). Moreover, cued-puzzle solving correlated with cued-puzzle memory. Overall, these results demonstrate that cuing puzzle information during sleep can facilitate solving, thus supporting sleep’s role in problem incubation and establishing a new technique to advance understanding of problem solving and sleep cognition.

And what better way to end then, than with this perfectly appropriate final extract:

The office is spare, practically empty. There’s a large desk, a couch for sleeping — Huh typically takes a nap later in the morning — and a yoga mat rolled out on the floor (just for lying down, he said; he doesn’t actually know how to do yoga).

On Mathematics, The Language

Josiah Willard Gibbs is a name likely to be unfamiliar to many people today. I must be honest and tell you that I myself wasn’t too familiar with it, and needed a couple of searches on Google to help me out.

I was familiar, though, with a quote that he is responsible for. It is a quote that appears in a book that I am not particularly fond of, and we will get into my reasons for my dislike in a bit. Here is the quote though, in all of its four word glory:

“Mathematics is a language” 

I was reminded of the quote when I read Dilip D’Souza’s lovely little rumination on just this question: whether mathematics ought to be considered a language or not. Dilip, in his essay, focuses on whether mathematics was invented or discovered, and says it is perhaps a bit of both. As an amateur student of mathematics, I cannot answer the question definitively one way or the other, and I’m happy to go along with his best guess.

But to go back to Dilip’s original question: is mathematics a language? Well, if it is one, it certainly isn’t one of the easier ones. Back in college, if you had given me the choice to study irregular verb conjugations in French or learn mathematics better, I would undoubtedly have chosen the former.

But over time, my attitude towards mathematics has changed, and I think for the better. The older I get, the more I am inclined to agree with Gibbs: it really is a language, and a beautiful one at that.

It is a language with beauty, as Euler’s identity makes abundantly clear. It is a language pregnant with mystery. And not just the kind of mystery that one associates with school-time tribulations! Try finding out what the sum of all natural numbers is, for example. And then wonder how Ramanujan thought the answer to be self-evident. If that doesn’t strike you as a mystery, nothing will.

And as with all things mysterious and beautiful, the more time you spend with mathematics, the more mysterious and beautiful it becomes. Dilip hints at the wonder that is Bolyai and Reimann geometry in his essay, and both are worth learning more about. But why stop there? Once you find yourself going down the rabbit-hole of discovering mathematics for its own sake, you find yourself in a wonderland that makes Alice’s look positively quotidian.

But as with all languages, so also with mathematics. Your love and affinity for it is very much a function of the way you were introduced to it. If your introduction to it was through dull and dreary classroom exercises, carefully designed to suck every single trace of fun out of the experience, then you are unlikely to have fallen in love with it.

And that, I suspect, is the case with most of us. Myself included, to be clear. But I was lucky. I was reintroduced to the subject by a professor of mine, who introduced to me the beauty that lies in wait beneath the seemingly impenetrable surface of the subject. And over the years, I have fallen in love with the language.

Noam Chomsky is famous for hypothesizing that we are all born with an innate ability to understand language – any language. Now, some languages may well be more difficult than others, but the more time I’ve spent with the subject, the more I have come to believe that Chomsky may well have a point. Mathematicians such as Steven Strogatz, Edward Frenkel and Grant Sanderson have helped me appreciate the language more.

And while I may never be able to compose even a limerick in this language, let alone author a magnum opus, the more I learn about it, the more I am able to appreciate the works of those who have gained some mastery over it.

A brief coda:

Gibbs’s quote, with which I started this essay, comes as an epigraph in a book that I don’t particularly like. As any economist of a certain age will tell you, that book is The Foundations of Economic Analysis, written by Paul Samuelson.

The reason I do not like it is because I think the book helped push the study of economics a little bit too far in the direction of mathematical analysis for its own sake. Mathematics helps make economic analysis more tractable, and more amenable to logical analysis, but it is the dose that makes the poison. The study of economics may have become more tractable because of mathematics, but in my opinion, the mathematical formalism has been taken too far.

As the wise Kenneth Boulding put it: “”Mathematics brought rigor to economics. Unfortunately, it also brought mortis.”

But make no mistake, this is at best a criticism of how economics as a field has developed over the years, if that. Samuelson himself admitted as much, and it is a sign of the greatness of the man that he did it in 1952(!). 1

For mathematics, today, I have nothing but gentle love and affection, and above all, a sense of awe and wonder. 

And I wish and hope for more of the same for you!

  1. Economic Theory and Mathematics–An Appraisal Paul A. Samuelson The American Economic Review Vol. 42, No. 2, Papers and Proceedings of the Sixty-fourth Annual Meeting of the American Economic Association (May, 1952), pp. 56-66 (11 pages). []