I realize this question is risky (as the title and the tags indicate), but hopefully I can make it acceptable. If not, and the question cannot be salvaged, I'm sorry and ready to delete it or accept closure.
Oftentimes, when one listens to chess grandmasters commentating on games, they will say things like, "The engines give a +1 advantage to White, but the postion is dead-drawn." And they proceed to provide high-level explanations of such statements. I'm a bad chess player so I cannot really attempt to verify those, but they often sound extremely convincing. Also, when they teach simple endgames, they seem to consider analyzing the crucial ideas to be enough for proof (at least in the sense of "a completely convincing explanation") and it's difficult not to agree with that. Even certain positions with a lot of pieces on the board can apparently be treated this way. "There is no way to make progress" is something that one can frequently hear.
Sometimes these just turn out to be wrong. These statements are generally based on intuitions ("axioms") such as "you cannot give away your queen for a pawn", which are believed to be true in almost all situations and on already analyzed positions (like the Lucena position). The grandmasters' intuitions are very fine, but it can happen that they will miss a counterintuitive material sacrifice, a counterintuitive retreat or a counterintuitive something else. However, they can be extremely convincing sometimes (and never proven wrong).
It's clear that chess can't be solved this way any time soon. A clear obstacle is the "wild" or "unclear" positions, where "anything can happen". But there are also those "tame" positions, "dead-drawn" positions and "winning advantages". (Surely, some — or maybe a lot — of these statements are not correct or correct with incorrect reasonings behind them.) Another indication is that the humans who make these statements get beaten by computers, which primarily use low-level tree searches.
My question is how much, if anything, of this high-level reasoning can be put to a form that would make it mathematics and mathematical proof. I think brute force is clearly not the only way of evaluating positions rigorously. In a K+Q v. K endgame, one doesn't have to analyze each possible move of the lone king to show that it's doomed. It's enough to show that a stalemate is impossible when the king is not on the side of the board, that the king can be forced to the side of the board and then that whenever the king is on the side, it can be checkmated. For a rigorous proof, one can use all kinds of symmetries and translations without having to go through every single node of the tree.
The question
How much of that high-level thinking grandmasters employ can be made into mathematics?
is what I'm interested in, but this is vague — it's not clear what "how much", "high-level thinking" or even "made into" mean. But I think people must have tried to pursue this and some clear mathematical statements must have been produced. If not, is it at least possible to say why it's difficult, infeasible or impossible?
Is there a chess position with a large (enough to be intractable to a human with just a pen and a piece of paper) tree that has been solved without brute-forcing the tree? Has a rigorous proof accessible to a human been produced?
Best Answer
Let me prove, for example, that the following 7-piece position is a draw. 7-piece positions are about the borderline of what's doable by brute force: they were tabulated around 2010.
Black draws as follows:
1) if white queen captures the rook or the pawn, recapture.
2) else, in the event of check, move the king to h7, h8, or g8 (cannot all be controlled by the queen simultaneously)
3) else, if white queen or pawn moves to 6-th rank, capture it with the rook,
4) else, move the rook to f6 or h6.
Clearly, unless white sacrifices the queen, she cannot cross the 6-th rank with the king and thus cannot break the above routine. The possible sacrifices are:
A) by capturing the g7 pawn. Recapture with the king and continue moving the rook, securing a draw.
B) by taking the rook. Any resulting pawn endgame is drawn by moving the king between h7, h8, and g8;
C) by Qa6, Qb6, Qc6, Qd6 or Qe6, followed by rook takes queen and king takes rook. The very same idea, only be sure to take g7:h6 when you can.
D) By Qg6 R:g6 h5:g6. This is also a theoretical draw. Move Kh8-h7-h8-... and be sure not to take g7:h6 at a wrong moment.
Some details are left off here, but I think the level of rigour is fairly close to how math is written.
P. S. In a game between Mamedyarov and Caruana (world N17 and N3, respectively), played last Friday, a draw was agreed in the following position:
Computers wrongly give a decisive advantage to white: the idea of cutting the king with the rook is too "high-level" for them to understand it. And I believe, if needed, one can devise a rigorous proof here similar to the one above.