Is Chess Like Math?

Is Chess Like Math?


Hi all,
I am back today with a small opinion piece inspired by a comment on one of my other posts. We were discussing tactics and @little_ernie mentioned his past love of mathematics, which got me thinking about the link between math and chess.

It's a fairly accepted idea: chess is somewhat like math. And why not? If you look at the rules, you will see no endogenous source of randomness, no dice, no bluff, no risk preferences. It does look like a giant math problem. So it makes sense to equate chess and math.

But I disagree. And since I love to be contrarian, I wrote this little piece about it.

Chess is not the sum of its rules.

Yes, if you could solve chess, it would be a giant math problem: a giant, solved math problem. But so far, we haven't. Even AIs have not solved chess. The simple fact that they can beat each other should be enough to prove it.

But moving aside the AI question for a second, when I play against another human, I don't feel like I am doing math, precisely because the game is not solved, but also for a few other reasons:

Humans bring unpredictability:

The rules don't contain any randomness; the opponent does. Whether or not you will see my trap soon enough, whether or not you remember this opening line or this endgame trick, this is an unknown to me.

The perfect illustration is the resignation decision. When to resign, when to keep playing? If you treat the game purely as math, you should resign in any lost position. But you don't. You keep playing when:

  • the position is complex, and you evaluate the probability of a slip-up as high or,
  • the time constraints of your opponents also increase the likelihood of a slip-up or,
  • you don't believe your opponents know how to mate you with two bishops,
  • etc.

In other words, you make a probabilistic estimation of a god-sent mistake and balance this against how much you value your time and how much you hate playing a lost position.

Decisions are based on tactics (math) and strategy (something else):

If I put my knight on that outpost, it's not because I have calculated it to the end. I do it because I calculated no immediate cost (tactics) and I believe that my knight will end up being useful here in the long run (strategy).

Mathematically inclined people will recoil at the idea of using such a crass heuristic. They will want something clean, based on complete and correct reasoning that allows you to know with certainty. Choosing the best heuristic to make an educated guess is not math: at best, it is physics.

Habits and preferences do have an impact.

Anyone who has a friend who only loves to attack and play sharp positions knows that forcing him to play against a London System will crush his spirit.

Almost all players are better in some positions and worst in others. And it's almost always linked to personal preference. Even if the reason you are bad in a particular variation of your favorite opening is that you didn't study it enough, there is a reason why you studied this variation less; and it's probably your dislike for it.

Whether it's choosing the right opening, making the position sharper or more positional, dragging out the game for hours, or just trying to get your opponent to overthink, a bit of Machiavellian psychology will help you win if only marginally.

This is never the case when you attack some math problem.

In a game, we do not (always) look for the best move

In a game, you don't look for perfect. You don't even look for "good." You look for good enough.

99.9% of players will be completely indifferent between a mate in 5 and a mate in 3, or any two winning positions.  And when deciding our next move, we stop our search when we find one that should help us clinch victory or just maintain the current value of the position.

It is a very different story when doing tactics, theory, or just analyzing a game afterward, but during the game, our approach is very "non-math." Again, thinking "this is not correct, but it seems to do the job" looks more like physics or computer science.

If not math, then what?

When playing chess, we do not assume everything is predictable. We admit there is "noise" in the system. We rely on heuristics and tricks, taking into account psychology in order to find, not the best move, but a good enough one.

No mathematician I ever worked with would recognize this as an offshoot of his beloved language of logic.

I hinted at a comparison with physics throughout this piece, but this was only to keep my final comparison for the conclusion. Indeed, while physics is far less formal than math and can use heuristics without feeling dirty, physicists don't have to deal with humans. They have randomness in measurements and the weirdest part of modern quantum physics, but otherwise, they do approximate an objective reality.

No, only one science uses math, logic, statistics, heuristics, and common sense to find a good enough answer to some interesting question. Actually, make that one and a half sciences: Economics and Finance.

Full disclosure, I am finishing a Ph.D. in finance right now, so I might be a little bit biased. Nevertheless, I believe the comparison hold:

  • There are hard rules (although unknown as in all sciences) that we try to approximate.
  • We have very noisy measures of reality and are forced to go for approximations.
  • Economic realities are so complex that "good enough" is all we can hope for.
  • The single largest source of randomness in the economic system is that... humans are the ones playing.

As opinion pieces go, this is a fairly useless one, but it was fun to write, and I hope entertaining to read. If you have a better science or human activity you want to compare chess to, leave it in the comment.

That's it for today. Until next time, happy learning!