A non-geometric area of mathematics which applies to chess is symbolic logic. I think all of chess could be encoded into some very complicated logical statements, starting with p: black's king is in checkmate. Then a very large number of different statements involving pieces being on squares could imply p. In brief, chess is basically logic. It would be nice if it could be simplified.
In an effort to understand whether we can actually "solve" chess, I read a succinct summary of why it might be possible: data compression. If we can take the highly complicated state space of chess positions and lines that lead to results, and then encode and compress it to something we might conceive of actually writing to some storage medium (smaller than the number particles in the known universe), that would be a big accomplishment.
Many mathematical problems that orginally appear in a non-abstract form (for example geometric ones) can be brought down to an abstract one and can be understood better this way.
Is this also possible in chess, in an extreme case, having one formula with one variable for each piece that has certain values for certain squares, and the mathematical result can be translated to the chess result (with optimal play).
In evaluating positions, determining the evaluation on material, as it is taught to new players, could be a very rough approximation that helps in a few cases, but there are exceptions and in most positions there are multiple moves to preserve the same material count.
Another example could be tablebases, but they rather work as generalized "if->then" and cover all cases. In mathematics they would be the equivalent to a brute force proof, but not to an elegant proof.
There are multiple generalizations that work for a very specified set of positions, let's take the square rule for pawn promotion or the Troitsky line in the rare NNvP endgame (although the Troitsky line already has some exceptions)
It is the case that chess is too complex to have a simple method for all scenarios, but to solve chess, it could be more realistic to have a general method for an "elegant proof" than a 32-piece-tablebase. There will also be cases that could simplify the general method, and while tablebases get simpler with decreasing piece amount, this wouldn't necessarily be the case for other methods, which would then be more efficient in certain scenarios with high piece amount but other factors that simplify the method.