Math Should Be as Popular as Chess#
Author: Steve Kieffer
Some of you may have read the title of this blog post and thought, “Well…maybe it already is?” I would disagree. If you tuned into the 2021 FIDE World Chess Championship on twitch.tv, it was normal to see anywhere from ten- to a hundred-thousand viewers at any given time. By contrast, how many people have ever read the proof of Fermat’s Last Theorem?
It’s important to understand that in this essay we are talking about reading proofs, not writing them. Generally speaking, it is easier to consume an artifact than produce one. (Compare listening to a symphony to composing one.) The thousands who enjoyed the Championship chess games weren’t necessarily capable of choosing the same moves the players chose, and yet they enjoyed watching the games nonetheless. I’m asking why more people can’t at least enjoy reading proofs, even if they were not prepared to write these proofs themselves.
Why aren’t people reading proofs?#
Lots of spectators are watching chess games, so why aren’t just as many people reading proofs? I’ll argue that, while the challenges – I enumerate four – of understanding proofs and chess games are in many ways similar, proofs present fundamentally more difficult problems, and less has been done so far to address them. In the final section, I’ll talk about some things we can do.
Seeing and Confirming#
In his book, Math with Bad Drawings, Ben Orlin compared math to, “something dainty and elegant astride the wrinkled gray mass of reality below.” [O2018] With this as inspiration, I ask you the following question:
If a cat stands on top of an elephant’s head, then the cat’s eyes will be higher than the elephant’s eyes, right?
If you answered yes, then congratulations, you are capable of understanding the proof of Fermat’s Last Theorem!
What you just demonstrated was the ability to see and confirm. I described a situation, you saw it, I made a statement about the situation, and you confirmed that the statement was indeed true.
This, it turns out, is what mathematical proofs are made up of. You are repeatedly asked to picture a certain situation – a novel situation you have never actually encountered in the real world – and then confirm that a certain thing about that situation is true. “If we did this, do you see that that would follow?” “If this, that, and the other were put together in this way, do you see that this statement would then be true?”
Put this way, proofs may sound a bit dry, but the truth is that every story asks you to see and confirm…or at least see and believe. In Sherlock Holmes, a descriptive paragraph will paint a scene for you, but then you’ll be told what Holmes did next, and you’ll be expected to believe it. (Otherwise the story would seem far-fetched.) That’s a kind of “seeing and confirming.”
So why then is it difficult to read mathematical proofs, if you have just demostrated the basic ability – seeing and confirming – that is required?
To answer this question, let’s raise the difficulty level a little, and try a second example:
Consider a chess opening. White opens with the Queen’s Gambit, and black responds with the Tarrasch Defence. Do you see that in the fourth move, white can capture a pawn on
This time, most of us will have to answer, “No, I don’t see that. I know you’re talking about chess openings, but I’m not an expert in those. If I knew more about those openings, then I could determine whether what you said is true.”
In other words, the problem this time is that you cannot see. If the board were set up in front of you, then you could see, and you would have no trouble confirming.
In the same way, mathematics uses a specialized language, and until you learn what the words mean, you have no hope of seeing and confirming. When chess experts hear “Tarrasch Defence,” they can picture the board, and they know what can happen next. When mathematicians hear “the Galois group of the fifth cyclotomic field,” they know what this object is, and they’re ready to think about it.
So, it’s fair to say that if most people aren’t reading proofs, the first reason is that proofs are written in a foreign language. And yet there’s hope, because, if you only knew that language, then all the stories would ask you to do is repeatedly see and confirm. Once you’re ready for the words to conjure a picture in your mind – to make you see – you should be ready to confirm, right?
But you don’t get a chess board#
Unfortunately, even once you know the langauge of mathematics, another major challenge remains, and this is that, unlike in chess, you don’t get a board and pieces to move around. This is problematic, because the situations that can arise in the middle of a good proof can be a lot more complicated than one cat standing on one elephant’s head. They can easily be as complex as a crowded chess middlegame, or worse.
Now, some chess masters are able to play the game without a board and pieces, just leaning back, staring at the ceiling, and remembering where all the pieces are. But many more thousands of people have been able to enjoy the game because of the availability of a few crude tools: a flat piece of wood divided into 64 squares, and a bunch of figurines to move around on top of it. If chess had only ever been played “in the mind’s eye”, then only a few people throughout history would have ever participated in it.
For the most part, mathematics is practiced as a “mind’s eye only” game. True, there have always been at least half-hearted attempts to illustrate it, using messy pictures scribbled on blackboards, or on napkins, or even in the sand. But due to the abstract, often infinite, nature of the ideas and patterns mathematicians work with, they have, understandably, largely given up on trying to illustrate these ideas.
In a powerful essay, On Proof and Progress in Mathematics [T1994], Fields medalist Bill Thurston had much to say about the way we communicate math to one another, noting among other things that, “mathematicians usually have fewer and poorer figures in their papers and books than in their heads.”
The need for better visualization tools for mathematics, for a counterpart to chess’s board and pieces, is surely another reason why more people aren’t reading and enjoying proofs.
And the stories have holes#
Mathematical proofs at a high level are generally written by and for experts. As we’ve discussed, this means they will be filled with technical jargon, but it also means something else: it means lots of steps will be skipped, and as the reader you’ll be expected to fill in the gaps for yourself.
This is not a great way to make stories fly off the shelves. If every page of Dostoevsky said, “Solve this riddle to find out what Alyosha did next,” most of us never would have finished The Brothers Karamazov. Even mystery novels eventually fill in all the holes, after letting you ponder things for a while. Not so with proofs. For many proofs, there’s not a single written record, anywhere, that tells you all the steps.
Before talking about why this is so, let me first be clear that I am not blaming the authors of proofs for this situation. They have limited time and space in which to write, and they have to address their most likely audience.
But there’s a deeper problem, which is that no one can agree on what “all the steps” means. To the experts for whom the proof was written, all the steps are already there. For a first-year PhD student in math, a couple more steps could help. For a bright undergraduate math major, still more steps would be nice.
Once again, the chess world is better off, because in chess there’s no question as to
what constitutes a move. “Knight to
c3.” “Bishop takes
These are the moves, and you’ve made a complete account of what happened in a
game when you’ve written down not every second or third move, but every
While it’s true that formal mathematical logic offers theories of mathematical reasoning in which the legal steps are no less atomic or well-defined than the moves of chess games, there are many competing theories, and none represents the informal reasoning steps taken by humans thinking about mathematics at various levels of sophistication.
It seems fair to say that more people would read proofs if they could order a “custom proof,” written with just the granularity of steps, and the amount of explanation, that was right for them.
And the strategy is more obscure#
But is the list of moves really an account of everything that happened in a game of chess? What about what happened in the players’ heads? As any new student of chess knows, a mere record of the moves played on the board will not teach you much. You may know what happened, but do you know why?
This is why chess commentary is so popular. Whether it is live commentary during a televised game, or written annotations on historical games, chess grandmasters can provide insight, explaining what the players were probably thinking about at each stage of the game, and which heuristic ideas they might have used to help them choose their next move.
To recast the problem on the mathematical side, suppose all the steps of a proof were broken down to whatever size was just right for you, so that you were able to confirm that each step was indeed valid. Even then, you might still feel that you really didn’t understand the proof, because perhaps you couldn’t imagine how its author ever came up with all these ideas. Where did they come from? How could I have thought of these steps myself, you might ask.
Fields medalist Tim Gowers commented on this subject, writing that, “Many of the best proofs, or at least the best-written proofs, come with some indication of how they were discovered.” He further elaborated that a good proof ought to come with a “demystification,” i.e., “a convincing demonstration of how a mere (mathematically trained) human being could search for a proof and find this one, without relying on excessive amounts of luck.” [G2000] Gowers’ point was not that many proofs had ever been written this way, but rather a call to action, that we might see more of this sort of thing in the future.
At the start of this essay I said we were talking about what it takes to read a proof, not write one, but the truth is that a major part of feeling that you understand a proof is being able to at least imagine that, under the right circumstances, you could have written it yourself. The steps aren’t just true, they’re motivated, they’re natural. This point has been famously observed by Poincaré [P1908], and revisited by authors like Avigad in Understanding Proofs [A2008].
It begins to seem little wonder that chess is more popular than math. In chess, not only do you have all the moves laid out in front of you, visualized clearly on a board, but you even have experts teaching you why these moves make sense! Not much of the same can be said for proofs today.
Let’s do as much for math as has been done for chess#
We want more people to enjoy the amazing stories told by great mathematical proofs, and we’ve identified four major obstacles that stand in the way:
Understanding the words
You need to know the language of mathematics. This means being familiar enough with the “elephants” and “cats” of the mathematical universe, that you can picture one standing on the other’s head when you’re asked to do so.
Keeping track of all the objects in play
While the situations you’ll be asked to envision can be at least as complex as a chess middlegame, you have no board and pieces to help you keep track of what’s going on.
Filling in the gaps
Proofs are written for experts, and you are expected to fill in many missing steps for yourself.
Explaining the ideas
Even with every step of a proof laid out plainly, you can still wonder how these steps could have emerged from a system of ideas, heuristics, and strategies.
What can we do to improve the situation?
Perhaps with the use of information technology, the mathematical community can work together to better address all four of the problems identified above.
One idea is to use Proofscape, to address the problems in the following ways:
Understanding the words#
The way to know what elephants and cats are is to see these animals, observe their behavior, and even interact with them. In the same way, seeing and interacting with many examples of the entities that populate the mathematical world is the way to learn what the words mean. Proofscape therefore offers facilities to make it smooth and easy to explore numerical examples of the types of objects in play at any given step in a proof, and do things like compare multiple examples side by side. The intro to example explorers explains more.
For a “board and pieces,” Proofscape offers proof charts. These are flowchart-like diagrams that represent all the steps of a mathematical proof. The software also features visualization tools to help you see both the close up view and the overview simultaneously.
Proofscape is also part “project planner.” Every step in every proof has a checkbox and a place to jot down notes, so you can keep track of your progress in the project of understanding a proof.
You can find an intro to both the visualization, and the project planning features of Proofscape, in the software tutorial.
Filling in the gaps#
In Proofscape, anyone can author an expansion on any step in any proof, and thereby fill in the gaps. When an expansion is indexed, it becomes available to others. Then, when someone is struggling with a difficult inference, they might just open an expansion and see how the inference follows. Those who don’t need the expansion needn’t open it. In this way, the granularity of the proof is tailored to each reader. The idea is demonstrated in the intro to expansions.
Explaining the ideas#
One of the reasons it has been hard to provide “proof annotations” with the same readiness with which chess annotations are written, is that we do not have the kind of database of past proofs we would need. However, the Toeplitz Project aims to provide such a database, by representing historical mathematics in Proofscape modules.
Orlin, B. Math with Bad Drawings, Illuminating the Ideas that Shape Our Reality. Black Dog & Leventhal Publishers. New York. 2018.
Thurston, W. On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161–177. 1994.
Gowers, W. Rough Structure and Classification. Geometric and Functional Analysis, Special Volume – GAFA2000, 1-39. 2000.
Poincaré, H. Science et méthode. 1908.
Avigad, J. Understanding Proofs. Paolo Mancosu, editor, The Philosophy of Mathematical Practice, Oxford University Press, 317-353. 2008.