On Chess V: Chess and Mathematics III

"The game of chess has always fascinated mathematicians, and there is reason to suppose that the possession of great powers of playing that game is in many features very much like the possession of great mathematical ability. There are the different pieces to learn, the pawns, the knights, the bishops, the castles, and the queen and king. The board possesses certain possible combinations of squares, as in rows, diagonals, etc. The pieces are subject to certain rules by which their motions are governed, and there are other rules governing the players. [...] One has only to increase the number of pieces, to enlarge the field of the board, and to produce new rules which are to govern either the pieces or the player, to have a pretty good idea of what mathematics consists." (James B Shaw, "What is Mathematics?", Bulletin American Mathematical Society Vol. 18, 1912)

"Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game." (Godfrey H Hardy, "A Mathematician's Apology", 1940)

"Pure mathematics is the world's best game. It is more absorbing than chess, more of a gamble than poker, and lasts longer than Monopoly. It's free. It can be played anywhere - Archimedes did it in a bathtub." (Richard J Trudeau, "Dots and Lines", 1976)

"There is still a great deal of uncharted territory in the opening phase of the game. New ideas, new concepts, new plans in old and forgotten variations, there is still much to discover in the opening. The tactical patterns and strategic concepts of the middle game have been well mapped out by generations of Grandmasters, although there are occasional fresh twists. In the endgame, however, the plans and possibilities are open and known to all, an almost mathematical exercise. This isn’t to say that everything is predetermined. With flawless play from both sides, the endgame will advance toward a predictable conclusion. But since humans are flawed, damage can be inflicted or repaired. Even if one player is at a clear disadvantage, he may simply outplay his opponent." (Garry Kasparov, "How Life Imitates Chess", 2007)

"As art, chess speaks to us of the personal decisions that are made in the course of a game. Looking at this facet of the game, the essential protagonist is the aesthetic sense rather than the capacity for calculation, which thus moves us closer to the human dimension and farther from mathematical algorithms." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"For all their wealth of content, for all the sum of history and social institution invested in them, music, mathematics, and chess are resplendently useless (applied mathematics is a higher plumbing, a kind of music for the police band). They are metaphysically trivial, irresponsible. They refuse to relate outward, to take reality for arbiter. This is the source of their witchery." (George Steiner, George Steiner at The New Yorker, 2009)

"In emergent processes, the whole is greater than the sum of the parts. A mathematical phenomenon that appears in certain dynamic systems also occurs within biological systems, from molecular interactions within the cells to the cognitive processes that we use to move within society. [...] Emergent patterns of ideas, beauty, desires, or tragicomedy wait, ready to trap the next traveler in their complex domain of neatly patterned squares - the never-ending world of chess metaphors." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"The master of chess is deeply familiar with these patterns and knows very well the position that would be beneficial to reach. The rest is thinking in a logical way (calculating) about how each piece should be moved to reach the new pattern that has already taken shape in the chess player’s mind. This way of facing chess is closely related to the solving of theorems in mathematics. For example, a mathematician who wishes to prove an equation needs to imagine how the terms on each side of the equal sign can be manipulated so that one is reduced to the other. The enterprise is far from easy, to judge by the more than two hundred years that have been needed to solve theorems such as that of Fermat (z^n = x^n + y^n), using diverse tricks to prove the equation." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Chess is a perfect arena for just such an exerted exploration of the possible. Its chequered sea is very deep indeed. The mathematics behind the game’s complexity are staggering. […] For all its immensity, chess is a finite game. It is therefore at least conceivable that a machine might one day be programmed with the knowledge, deep down in its nodes, of every possible sequence of moves for every possible game. No combination, however ingenious, would ever surprise it; every board position would be as familiar as a face." (Daniel Tammet, "Thinking in Numbers" , 2012) 

"Chess, with its straightforward rules and tiny Cartesian playing field, is a game tailor-made for computers." (John Horgan, "The End of Science", 2015)