in the interviews I’ve read with Soviet mathematicians and scientists, the things that come up over and over again are “mathematical circles,” a practice that originated in the pre-revolutionary Russian Empire and then spread far and wide through the Soviet Union. A mathematical circle is an informal group of teenagers and adults who really enjoy math and want to spend a lot of time thinking and talking about it. They’re a little bit like sports teams, in that they develop their own high-intensity internal culture and camaraderie, and often have a “coach” who is especially talented or famous. But they’re also very unlike sports teams, because they don’t compete with each other or play in leagues or anything like that, and usually any given circle will contain members of widely varying skill levels. Maybe a better analogy is a neighborhood musical ensemble that gets together and jams on a regular basis, but for math.
The most important thing to understand about mathematical circles is that the math they jam on is completely unlike the math you study in school, and also completely unlike the “competition” math that bright kids in the United States sometimes do. Both school math and competition math are primarily comprised of exercises. An exercise is a question concocted by a human being for a didactic purpose. Any bright kid with any amount of genre-savviness can immediately make a few assumptions upon being assigned an exercise. He or she can guess that the exercise is solvable in fewer than five minutes with the appropriate techniques, and that it is related to the material in the current chapter of the book. A clever student can often use psychological techniques to reverse-engineer what the teacher or the designer of the standardized test was trying to get at with the exercise, and answer it through a process of elimination or savvy guessing or pattern matching.
Solving an exercise is like hunting a neutered zoo animal. It may be a low-stress environment for polishing particular aspects of your technique, but it will not help you to survive in the wilderness. For that, you need to see people solving problems. A problem is a question of interest that comes up when somebody is trying to do something real. A problem may not be solvable by you, or by your coach, or by any human being. Even if the problem is solvable, it may require weeks or months of dedicated, painful pursuit. It may not be obvious what techniques are required to solve a problem, they may not be techniques that you know, or it may require a surprising combination of techniques. The problem is mathematical nature red in tooth and claw. There are no guardrails. There are no hints or answers at the back of the book. There is no book. It may eat you.
The bread and butter of the mathematical circle is solving problems together, as a team. There is no time here for exercises; you can do that lame stuff at school. Sometimes the coach picks a problem for you, something just beyond your ability, just the thing you need to hone your edge. But sometimes the whole circle works together on a problem that nobody has the answer to and that challenges the very best members. These problems are the most important, because with them you see great minds, men older and more talented than you, stretched to the breaking point and occasionally beaten. You see them grind and grind and try every possible attack on a problem and sometimes lose anyway. And you see them not run from being defeated, but cheerfully charge in again, because losing is good for you, losing is how you know you’ve picked an opponent worthy of a man. You learn to love things that are hard. And occasionally you win, and when you win it feels like you all win, like humanity wins, because you’re all in it together, all doing something beautiful and dangerous and exemplary of the best qualities that human beings have.
There are also times when everybody is too tired to work on a problem, and in those moments of recuperation, it’s the coach’s job to tell stories of legendary problems of the past and of the mathematicians who slew them. These stories often contain lessons, inspiration, or perspective on how mathematics evolved and got to be the way it is. Human history would look very different, after all, without the brachistochrone problemor the roots of a quintic polynomial problemor the icosahedron problemor the precession of Mercury’s perihelion problem. But other times there’s no hidden lesson, no grand perspective on the human story. They’re just ripping good yarns, and hearing them is a process of initiation into mathematical folklore, because every culture (and mathematics is surely a culture) has shared stories and references and inside jokes, even when they’re purely for fun.
You can start math circles really really young:
This book is the story of one such mathematical circle. But it’s an unusual one because…it’s for preschoolers.
The “coach” of this circle is Alexander Zvonkin, a professional mathematician frustrated that his kids are having all the wonder and life and joy crushed out of them by the grey functionaries at their school. So he starts a circle for his son Dmitry and a few of the neighbors’ kids, most of whom are around three or four years old. That’s young enough that according to Piaget’s experiments there are cognitive modules related to number and volume that simply haven’t come online yet. Fortunately, Zvonkin is familiar with the latest research on developmental psychology, and turns lemons into lemonade by using the kids’ lack of numerical intuition to introduce them to some pretty deep ideas about when two sets have equal cardinality. (If you’re curious, he talks more about these experiments in this journal article.)
At this point I expect you are rolling your eyes, especially if you have experience with three-year-olds. It can be difficult enough to get them to sit still, never mind ponder deep questions about the cardinalities of sets. And what exactly does it look like to pit somebody against a problem who is barely potty-trained? This is where the genius of Zvonkin’s format kicks in — it’s not really a book, it’s a journal, and one that is barely edited. So it’s full of failure after failure, entries like, “today I had a cool idea for a puzzle but everybody just screamed instead and then one of the kids vomited.” And yet, slowly, wondrously, over the four years of the circle’s existence, his patience pays off and the kids start doing really incredible things.
(Sadly I only learned of the existence of math circles well after graduation, a few years ago when I used to spend more time on Quora and noticed that Alon Amit, the most respected writer on math topics and someone who’d done many interesting things in his life, described himself simply as a “mathcircler”.)
I feel like we perhaps need to reach some “escape velocity” to get something like that going, but for ~rationality / deliberately figuring out how to think and act better.
I really liked this extended passage on math circles from John Psmith’s REVIEW: Math from Three to Seven, by Alexander Zvonkin, it made me wish math circles existed in my home country when I was younger:
You can start math circles really really young:
(Sadly I only learned of the existence of math circles well after graduation, a few years ago when I used to spend more time on Quora and noticed that Alon Amit, the most respected writer on math topics and someone who’d done many interesting things in his life, described himself simply as a “mathcircler”.)
Really appreciate the random excerpts you post regarding math philosophy/reasoning/etc. They’re consistently interesting and insightful.
I feel like we perhaps need to reach some “escape velocity” to get something like that going, but for ~rationality / deliberately figuring out how to think and act better.