# Logic and intuition in math and chess

I recently came across a post about the roles of logic and intuition in mathematics. It presented ideas originally expressed by mathematician Henri Poincaré, as found in his book The Value of Science. In one fragment, Poincaré suggests an analogy between math and chess playing:

If you are present at a game of chess, it will not suffice, for the understanding of the game, to know the rules for moving the pieces. That will only enable you to recognize that each move has been made conformably to these rules, and this knowledge will truly have very little value. Yet this is what the reader of a book on mathematics would do if he were a logician only. To understand the game is wholly another matter; it is to know why the player moves this piece rather than that other which he could have moved without breaking the rules of the game. It is to perceive the inward reason which makes of this series of successive moves a sort of organized whole. This faculty is still more necessary for the player himself, that is, for the inventor.

In this analogy, Poincaré suggests mapping logic to the rules of chess and intuition to “the inward reason which makes of this series of successive moves a sort of organized whole”.

I find this analogy flawed. Chess is an adversarial environment, not a benign one. What makes a chess playing deep is not the difficulty of finding a sequence of legal moves that achieve a certain outcome, but finding a sequence of legal moves that, despite the opponent’s responses, achieves said outcomes. In mathematics there is no adversary, any set of legal inferences suffices. In chess the source of difficulty is not conforming to rules, but exerting more optimization power over the board’s state than your opponent.

Mapping logic to the rules of chess misses the better analogy found in the mental process of finding those moves.  It is this process, this “inward reason”, that itself exhibits both components, varying from the mostly logical, to the mostly intuitive. This makes for a more natural analogy seeing that those concepts map very well to the already existing ideas in chess of tactics vs strategy.

Thus logic maps to tactics, and intuition maps to strategy. And we can recover the same properties Poincaré mentions about invention and proofs. A chess player may make strategic judgements about overall courses of action or positions, but victory itself must always materialize with tactical play.

Another matter is how logic and intuition correspond to brain processes. Intuition has the property that you cannot describe explicitly exactly how you arrived at a conclusion, whereas logic is always explicit. Add a bit of mind projection fallacy into the mix and you start getting fancy ideas about the Power of Intuition.