How long will it be before computers beat every human player? I'd
accept an answer to within a year with a mathematical proof.  Please
note the wording of the first sentence.

turtlesallthewaydown,

For your question, I will assume that by "human player" you mean one
human acting alone, and that by "computers" you mean the most powerful
chess playing computer available.  I will also assume that the World's
Champion of Chess is the greatest human player alive, and that no
other player can defeat him (and even if he does, that player would
then become the World's Champion of Chess) So, rephrasing the question
with these assumptions in mind, how long will it be before the best
chess program can defeat the World's Champion of Chess?

If you want the answer to 100% certainity of a computer beating a
human, the answer is never.  The number of possible chess games has
been estimated at 10 to the 10th power to the 50th power.  Even a
computer that could process a trillion trillion moves per second could
never process all possible moves in a long match within the time
limits of an official chess match.

If you want a realistic answer of computer dominance, the answer would
appear to be fairly soon. In fact, just recently the "Brains in
Bahrain" battle between chess World Champion Vladimir Kramnik and the
reigning most powerful chess program, Deep Fritz.  In an epic eight
match struggle, Deep Fritz fought back from an early deficit to tie
Kramnik.  To emphasize the importance of this draw, Kramnik himself
was quoted as stating, "It is now clear that the top program and the
world champion are now approximately equal."

Now, chess was invented over 1,300 years ago, and humans have been
devising strategies ever since. So much of chess strategy has been
fleshed out that there remains very little left that man can do. In
addition, his actual brain capacity increase since those early times
has been very little.

Compare this with computers. In a fraction of that time, computers
have evolved to the point where they can make billions of calculations
per second. Moore's Law states that computer processing time will
approximately double every 18 months. Because this kind of raw
processing power is something humans cannot do, it opens up all sorts
of new stratagies that can be devised.

So, while humans remain largely stagnant, computers are doubling in
processing power and increasing their strategic horizons by at least
double every 18 months.  And currently, in a match between the best
human and best computer, there is a 33 1/3% chance of computer
victory, 33 1/3% chance of human victory, and 33 1/3% chance of a
draw.

In 18 months, the computer will be four times stronger, while the
human will be roughly the same.  Thus, the computer will become four
times more likely than the human to win a match.  Over an eight match
series, the chances of a human player pulling off a draw, let alone
winning the series, become microscopic.  After another 18 months, the
best human playing prowess will be 16 times less than the best
computer.  Although it is estimated that there are 10 to the tenth
power to the 50th power possible games of chess that can be played,
and computers will never be able to defeat a human with 100%
certainity, for all intents and purposes victory for the human will
become almost impossible at this point.

Because we are almost at the end of 2002, I will say that, within a
year of 2006, no human will stand a realistic chance of defeating the
best computer chess program.

Sites cited:

  Brains in Bahrain
  http://www.brainsinbahrain.com/

  Chess from Math World
  http://mathworld.wolfram.com/Chess.html

  Silicon Showcase
  http://www.intel.com/research/silicon/mooreslaw.htm

  Moore's Law
  http://www.webopedia.com/TERM/M/Moores_Law.html

  History of Chess
  http://math.uww.edu/faculty/mcfarlat/177hist.htm

Google search terms used:
  Moore's Law
  History of Chess
  Brains in Bahrain
  Deep Fritz

This is unquantifiable, as not every chess player will play against a
computer. If you want to know when they "think" computers will always
be able to beat the current -best- human being, then that is something
there have been studies on. Maybe you'd like to reword your question?

willie-ga

Yes, willie-ga is right . First there should had been a mathematical
definition of what's a mathematical proof, then --perhaps-- the
penalty of one star could have comne. Money isn't everything, right
willie-ga?

I'm afraid I have to disagree with hailstorm.  The improvement seen in
a chess program due to an increase in processing power is not a linear
relationship, and we are already at the point of seeing decreasing
returns with faster processors.  To say, "In 18 months, computers will
be 4 times stronger." is frankly absurd.  All the top chess programs
use some more of an ever-deepening mini-max search to "see" so many
moves into the future.  The problem is, at each new level of depth,
the complexity of the problem increases exponentially.  After a
certain depth, you can have a computer go 100 times faster and not be
able to search any deeper -- there are simply too many combinations. 
If you don't believe me, simply take any chess program and run it on
Pentium II computer and rate it some how.  Now run it on a Pentium 4
(at least 4 times faster, probably 20) and see if your rating for it
quadruples.  I don't think so!

Now, there are all sorts of strategies to continue to improve chess
programs, but most of them won't come "free" just because the
computers get faster -- they will come as the result of a clever
programmer trying something new.  In fact, we will probably have to
make "shallower" searches to make the next big breakthrough in chess
programming to better use all the processing power.

Basically, you can't predict a date at which computers are going to be
able to beat every human player and prove it with any scientific
rigor.  You can only make a guess.  The original question is like
asking, "When will there be no more war on Earth?  Please provide a
mathematical proof."