Seems the usual suspects have treked over here to start with their adhominen attacks (not including Elroch in this pool), and one of them could not resist mentioning the climate. The nit picky math analogies aren't even necessary as solving chess via a table base would be numbercrunching and retrograde analysis (working backwards from checkmate positions although I don't understand how that helps calculates the best forward moves), it's not a mathematical proof. Chess has mathematical patterns but the solution isn't mathematical.
Firstly, a tablebase encapsulates a huge number of mathematical proofs. Each step of its construction uses logic to add to the set of facts you know about chess. For example, suppose you have a position where black is to move where you have previously deduced that black will get mated in 548 moves with optimal play. Retrogade analysis from this position provides a set of positions where white has a mate in 549 moves by playing a specific move. Each of these is a mathematical fact.
Secondly, it is surprising that you have never noticed that if you look at a position in a tablebase it tells you what each move will achieve against optimal defence. This is plenty to tell you the best move to play!
That doesn't quite answer the question of how exactly retrograde analysis helps determine the best moves. Working backward from every possible checkmate/stalemate/insufficient material position is like generating a massive game tree in reverse. But how does that somehow tell you what the best branch is going forward? If it gives you a massive gametree back from some 2 knights vs pawn checkmate position, how does that reveal which of those branches is the optimal set of moves the other way?
the 'determining which moves are optimal' problem and 'constructing a very long mate chain' problems, while connected, are separate.
for 'determining which moves are optimal' - the retrograde analysis is not done ON the moves to be analyzed, but has been done BEFOREHAND in order to create the tablebase that is used, and referred to, in determining which moves are optimal.
it's not that a reverse tree has been created by a singular end checkmate position and traced backto the move to be analyzed, it's that a reverse tree has been created for EVERY SINGLE POSSIBLE end position, and ALL possible trees have ALREADY been traced to positions on the tablebase.
thus from reverse game trees to all positions, the true game tree has been created for your specific position.
@Maets-Nori
from your post:
"So "spotting errors" depends mainly upon the infallibility of the engines involved."
I would put it differently.
Fallibility. Not infallibility. With that second term suggesting 'binary'.
Fallibility. Degree of fallibility.
Infallibility of the engines never established. Might never be.
'Spotting errors' depends on degree of fallibility.
A good catch, I agree. That would have been a better way to phrase it.
I appreciate the correction.