Re: The Future of Chess and Getting It "Unstuck"

In article
<cba1d838-f975-487c-b796-9e1e20786273@[EMAIL PROTECTED]
>,
Quadibloc  <jsavard@[EMAIL PROTECTED]
> wrote:
>>         Um.  Well, many infinite/unbounded games have "issues".
>> But standard chess, and every reasonably similar game, is finite,
>> and Goedel's Theorems really do not have anything to say about
>> such games.
>Godel's Incompleteness Theorem says that there are true statements in
>a mathematical system that cannot be proved from within that
>mathematical system.

	Yes;  but not for *every* mathematical system, only for
systems that include sufficiently powerful arithmetic.  I know
how to construct chess positions that depend on the truth of
[eg] 1 + 1 = 2, and an assortment of rather harder sums of
numbers-as-games [Noam Elkies has done some interesting work
in this area], but you will find it rather harder to construct
arbitrarily complicated arithmetic using only the finite set
of positions available in chess.  It is possible to encode an
arbitrarily complicated *statement* as a chess *game*, but that
is no more interesting than the fact that we can do the same
with sequences of letters -- the game itself, like the letters,
has no computational power of interest.

[...]
>Since physics uses mathematics - mathematics can predict the behavior
>of physical systems, and physical systems can calculate mathematical
>functions or otherwise do mathematics - statements about mathematics
>can impinge on the physical world.

	Yes, but a Goedel sentence is not a statement *about* maths
but a sentence *within* a mathematical system whose logical status
is being determined.  You have not explained why the statement that
*this* logical process will never produce *that* sentence as output
is any more interesting physically than the statement that doubling
an integer never produces an odd integer as a result.  [The interest
to pure maths and logic is a different matter.]

>Chess, too, can be played mathematically - in a sense, any computer
>chess program does exactly that.

	As David has pointed out, you are very close to a false
syllogism somewhere here.

>Godel's theorem might impinge on chess by making unavailable some
>infinite mathematical theorem that would be a simpler way to make a
>statement about chess than a calculation applying only to one finite
>form of chess.

	Goedel's theorem does not make any *particular* theorem
"unavailable".  Any theorem that was unprovable-but-true in some
axiom system could simply be added as a new axiom.

>		So even if it could not affect our _possible_ knowledge
>about a finite game, it could affect the knowledge that is
>_conveniently_ available about it.

	Not, however, because of Goedel.  There are indeed much
more general results that make [eg] standard chess easier, inc
alpha-beta pruning, but also "theorems" about endings such as
KQvK, KPvK, and others.  [The techniques that we all learned as
beginners about how to win KQvK, KQvKP, and so on, all work just
as well on a 1000x1234 board as on an 8x8 board, as long as the
50-move rule is extended appropriately.  Other endings, such as
KQvKR and KBNvK don't generalise so easily.]

>(sci.math pulled back in: it will take people from there to comment on
>this one!)

	Well, that's a little unkind.  There are regular contributors
to the chess groups [and no doubt to the other games groups] who are
professional mathematicians ....

-- 
Andy Walker
Nottingham