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Conversation

Peter Sarnak on “The AlphaZero Test” for Mathematics<br>Peter Sarnak<br>Communicated by Notices Executive Editor Siobhan Roberts

I’m an old-fashioned number theorist. I don’t use artificial intelligence devices myself. But I’m spoiled because I’ve had a lot of students who are very capable — both these days with AI, and in the past with programming and running any technology that’s useful. I learned to program in Fortran when I was young, but that’s not what anybody’s using anymore. So, I appeal to my students and work with them.

When Jacob Tsimerman came along as my PhD student, it was clear from the very start that he was a superb mathematician, a superb talent. It was more a matter of guiding him. I was involved in training him in the old-fashioned way.

I know Jacob is very excited about AI. I ignored it for a while. But then I asked him, “Why are you so excited?” He said, “It’s solving these math competitions, the International Mathematical Olympiad competitions.” I suggested, “Well, maybe that says more about the math competitions than it does about AI.” His response: “What level does the AI need to perform at before you take it seriously? What would it need to do? When would you change your mind?” And since I take Jacob very seriously, I took note. It’s certainly a relevant question. Many of us mathematicians have our heads in the sand, ignoring AI, hoping it goes away.

Figure 1. Peter Sarnak and Jacob Tsimerman.

After that discussion with Jacob, I had a discussion with Akshay Venkatesh at the Institute. He is thinking similarly about AI. With those two influences, I really started thinking about it. I go to seminars. I invite people to talk about it. When I have the opportunity, I talk about it myself.

Last year in India I had the pleasure of being part of the awards ceremony for the winners of the Infosys Prize, and they asked me to give a speech. The title of my talk was “Number Theory Pure and Applied,” and I also philosophized about AI and its impact on mathematics. Drawing from my remarks on that occasion, here’s my position:

Calculations and computation have always been central to the development of number theory. The question of whether and how the recent explosion in statistical machine learning will impact number theory, or more generally pure mathematics in say the next 50 years, is one that working mathematicians are beginning to and should be addressing.

My perspective is that of a mathematician and chess player — I was a semi-professional chess player before I became a mathematician. I played from the age of 10 to 16, non-stop every day for six years, and I continued to play until I was about 24. At the beginning, I learned the theories of the famous old-world chess champions — Lasker, Alekhine, Botvinnik. They invented theories about what you do in different positions that blew me away. I was in awe.

When I encountered abstract math, I found it to be a different level of intellectual achievement, a different level of awe. And I still feel that that’s true of the great discoveries in math.

AlphaZero and AlphaGo, the AI game-playing machines from DeepMind, arrived on the scene about ten years ago. I was extremely impressed. They permanently changed chess and Go. With AlphaZero, the word “zero” is very important. It starts from zero. It doesn’t go to a database. It’s self-teaching. If that starts happening in mathematics, it will be a major change.

Taking a closer look at the chess playing machines may offer some insight as to how the trajectory of machines doing mathematics will evolve. Notably, while chess is finite and has artificial rules, it is too large to be analyzed by brute force. The strongest chess programs are Stockfish and AlphaZero and their derivatives. AlphaZero is stronger than Stockfish; both programs beat humans handily.

Stockfish is programmed according to chess theory, and it is an extension of chess master play. It calculates faster and deeper, never makes shallow tactical mistakes, and it accesses large databases of recorded games. Critically, because Stockfish is based upon and explained by chess theory, its play can be understood by strong players.

AlphaZero, on the other hand, makes no use of chess theory or databases. Instead, it generates a database by playing against itself many times and uses statistical machine learning to improve its guess for the best move in any position. It is better at guessing than any human or any other machine, and as such, it has passed some statistical complexity threshold special to the game of chess and its size. AlphaZero offers no understandable — to a human — explanation for its guesses. Nonetheless, some masters say that they can get...