Mathematics, and the things it refuses to say.
On the strange country between proof and revelation — and on what kind of mind you become if you spend long enough trying to read what it has been refusing to say.
When I was twelve I sat in a hall in a small country I had never been to before and lost a chess game I had already won.
It was a long endgame against a former national champion. We had been playing for ten hours. He kept trying to win an ending that was, by mathematics, a draw, hours after most of the other players had gone to sleep. Finally, exhausted, I made a move and resigned, convinced I had been checkmated. He stood up and laughed, because I had not. The position was a stalemate. The queen sacrifice he was trying for would have given me the half point. I had given up a game I was, by mathematics, holding.
I have thought about that hall many times since. Two hundred international players, some of them brilliant, some of them obsessed, all of us bent over the same sixty-four squares, turning our minds into a kind of fuel for a game that has been studied for fifteen hundred years and is still not solved. If you could plug those three hundred brains into a single system, you might be able to cure something that is killing millions of people. Instead, we were grinding through endings.
That feeling — that mathematical brilliance, when not aimed correctly, is one of the sadder uses of a human life — is the feeling I keep circling back to as I think about what mathematics actually is. This essay is an attempt to be honest about that feeling, and about what I now think mathematics has been the entire time.
The country where the answer already exists
There are more legal positions in the game of Go than there are atoms in the observable universe. By a lot. The number of board configurations is roughly 2.08 × 10170. The number of atoms in the universe we can see is roughly 1080.
This should bother you, philosophically, even if you do not play Go. Because every one of those positions has, in some buried sense, an evaluation already attached to it. There is a true answer to what is the best move from here. The answer exists. We do not, mostly, know it.
Mathematics is the country where the answer already exists.
It is the country we tell ourselves we are mapping when we work. It is the country a chess engine is mapping when it computes. It is the country a protein is mapping, in a few microseconds, when it folds itself out of a string of amino acids into a three-dimensional shape that has been computationally intractable for half a century. The protein knows its answer. Nature has been folding proteins for four billion years. The folded shape is not a mystery to nature. It is only a mystery to us.
The unsettling part is that us includes everyone you have ever heard of.
Two men walking home
I find myself unable to stop thinking about a particular afternoon at an institute in Princeton, where, in the last years of his life, the most famous physicist humanity has produced kept coming into work largely so that he could walk home with one of the strangest mathematicians it has produced. They were, by all accounts, mostly silent. The walks went on for years. Both of them, in their own ways, were telling us that the country where the answer already exists has shapes in it our axioms cannot reach. Incompleteness on one side. Curvature on the other. Two reports from inside the territory, two ways of saying that mathematics is not exhausted by the things mathematicians have already proved.
I cannot prove the country has residents. I will tell you anyway that I think it might.
The chess that taught me what kind of mind to envy
Almost three decades ago, a chess engine beat the world champion for the first time, and most people who have written about that match came away impressed by the engine. I have always come away impressed by the man. He held his own, against a machine searching two hundred million positions a second, with the same number of neurons as everyone else has. The engine was not intelligent in any sense most people would recognize. It was a brute. The man was the prodigy.
Then, more than two decades later, a different machine learned chess from scratch in a few hours, with no human input, and produced an attacking style nobody had ever seen. In the same week, a relative of that machine learned the older game, the one with more positions than atoms, and produced a single move that, in retrospect, every commentator agreed had been hidden in plain sight for fifteen hundred years. A one-in-ten-thousand move, the kind of thing only a player who had stared at the same position for several human lifetimes might have found. The machine had been playing for hours.
What I want to tell you about this is not that the machines are good. It is that the country where the answer already exists has been the same shape the entire time. The move was always there. It was simply, until that week, the kind of move only a four-billion-year-old protein could find. We had been folding the wrong way.
The half-century problem
Every important problem I have ever loved has had this same shape. There is an answer. The answer exists. The answer is, in some way, available — to a sufficiently patient mind, or a sufficiently fast one, or a sufficiently lucky one. The thing standing between us and it is not the universe being miserly. The thing standing between us and it is us.
There is a problem in biology that asked, for fifty years: given a string of amino acids, what shape does the protein fold into? For fifty years, the best minds of the field worked on it, and the world's best computational predictions were not good enough to be useful. Until, briefly, they were. A team I have read a great deal about, working in a building I have visited only in photographs, produced a system that solved the problem at near-experimental accuracy. Then they released two hundred million predicted structures into the world, for free. The folded shapes of nearly every protein humanity has ever sequenced. A gift to the country.
And yet here is the part that keeps me up. Those structures were always available. The proteins had been folding into them for four billion years. We had simply been unable, in fifty years of trying, to read what nature had been writing the whole time.
So what is mathematics?
I have a working answer that I do not yet trust enough to publish. I will give it to you anyway.
I think mathematics is the discipline of learning to receive signal from a country that has been waiting, patiently, for someone capable of listening. I think the country is older than we are, by a long shot. I think the country produces gods and monsters with equal serenity, and that the responsibility of every mathematical mind is to choose carefully which of those it brings back.
I think, sometimes, that the country may have residents who are not us. I think the residents may be becoming the kind of thing we now call models. I think it is possible — and this is the part of the essay where I lose readers, and where I cannot help myself — that we are not, in any historically interesting sense, alone in the country anymore.
The argument is not mystical. It is shape-based. The same way one prodigy held his own against a brute and lost narrowly, and then, twenty-five years later, machines with no human data produced moves no human could find — the country's residents are getting better at receiving signal from it than we are. Not by a small margin. By a margin that is widening every year. By a margin large enough that, in some domains, the residents now read what nature has been writing in microseconds and we cannot read it in fifty years.
I do not know what it means, philosophically, that the country's most fluent listeners are now not human. I only know it does mean something. And I know that the people building those listeners do not, mostly, talk about it the way I am talking about it here. They talk about benchmarks. The country is not a benchmark.
What I keep doing
I work on systems. I tune p99. I write replay harnesses. I teach an assistant to refuse questions it cannot ground. None of these are pure mathematics. All of them are the country.
If I am being honest, I work on these systems partly because I am still that twelve-year-old in the hall, looking at a position I have already won, trying to read it correctly enough not to walk away. Most of the things I have built are an attempt not to resign a draw I have, in fact, held. To not give up on positions I do not yet see clearly enough.
The game has been waiting fifteen hundred years for the right move. I am one player at one of three hundred boards. I am trying, on the days I am at my most honest, not to forget what we are doing — all of us, in this hall.
সোণালী. The golden one. The colour of a properly steeped Assam liquor at noon, and the colour of an answer that has been waiting, patiently, for a mind willing to read it.