Thanks for taking the time to send me your thoughts on this issue. The
problem I have with your answer is, however, the same problem I started
with, namely, that to me, the idea of a solution to chess entails more than
an answer to the question "Is chess a draw or win, and if a win, for whom?"
For the time being, I am willing to leave aside the practical question of
whether a fast enough machine could ever do all the necessary calculations
before the end of time. (If however, you were to concede that it couldn't,
you could construct an argument that even theoretically, chess can never be
solved. Otherwise, to say it could be solved, you would have to x-out your
concession that it could never be solved within the bounds of time.) Anyways,
one could also fight against this line of reasoning, so I leave it aside for
the time being.
I am also not sold on the idea that a graph of all possible chess positions,
including all possible transpositions to one another, would answer the
solution question given above. If chess is a forced win, it might, but if
neither side can force a win, it isn't clear to me. One will eventually
cross the 50 move threshold or 3 time repetition, but, as someone else pointed
out, the problem with either of these thresholds is, there is no certainty
that there isn't some way one player or the other can't cross them and still
produce a win. It sometimes happens, for instance, that a draw via repetition
is agreed where one side could have won if they avioded the 3 repetition. Also,
some positions which are accepted as drawn after 50 moves without a pawn
move could be won il continued further. Why might someone on the losing
side of such a position continue play after he could invoke the draw, you
might ask? The answer might be simply to see if the opponent could win--
ie, curiosity (or possibly spite).
These considerations lead me to an earlier problem--how do you decide where
to cut off your analysis of these positions? If you have no 50-move rule
and no 3 time repetition limit (because you are concerned strictly with
whether or not a win could be achieved independent of these limits), then
you could be dealing with a potentially infinite number of ways of traversing
your graph of position (ie., an infinite number of games.) Of course, you
could try to evaluate each position in the graph independently, and in this
way immediately class a large number of positions as wins, but the problem
remains what to do with the left over positions where it wasn't so clear
whether they were wins or draws.
Lastly, and most importantly, I am troubled by the adequacy of the kind
to solution we are talking about here to really teach us much about chess.
Your view is that the solution would be to difficult for people to
comprehend--they could understand the "title" of the book (for instance,
'Chess is a Draw'), but they would be unable or unlikely to be able to
follow the huge set of moves that followed. To me, this is a hollow
solution. It answers the requirements of a game theorist's questions,
but not the require ents of someone who actually plays the game. That
person wants to know what is going on in a position and why. When my
chess students show me positions, they sometimes ask me, "Did I miss a
win here?". If I find one, and show it to them, their next question is
"how did I find that?" They want to be shown what was 'going on' in the
position--not merely in terms of sequences of moves, but in terms of
the strategical themes and nuances that will allow them to uncover these
sorts of possiblities when similiar positions arise for them in the
future. To me, a solution which no one can understand is 't much of a
solution. And to me, it seems doubtful that you will be able to arrive
at the kind of understanding I'm talking about from the kind of solution
you are talking about.
I agree with all of your reasoning and conclusions, up to *one* crucial point.
>[...] Of course, you
>could try to evaluate each position in the graph independently, and in this
>way immediately class a large number of positions as wins,
So far, so good (although the word "wins" should be replaced with the more
precise "forced wins for White with best play by White, or forced wins for
Black with best play by Black.")...
>but the problem
>remains what to do with the left over positions where it wasn't so clear
>whether they were wins or draws.
...and *there* is the flaw. These left over positions provably *cannot* be
wins for either White or Black (with best play by both sides). The only
possibilities (with best play) are (1) the position will lead to a draw or
(2) the game will never finish.
Most KQvsKQ positions are examples of this (e.g. Ka1Qa2 vs Kh8Qh7). Either
of the following constitute best play:
(1) exchanging queens and ending the game -or-
(2) shuffling queens for all eternity
Thus, (with best play) the game might end in a draw or it might never
end...but *clearly* neither side can force a win (against best play) from
such a position. In essence, such positions are drawn (with best play).
>Lastly, and most importantly, I am troubled by the adequacy of the kind
>to solution we are talking about here to really teach us much about chess.
You're certainly right about this. Such a solution teaches us next to
nothing about playing chess (it teaches us a great deal about math, however).