FOM: Why Hersh can't distinguish between math and chess

Shipman, Joe x2845 shipman at
Thu Jan 22 09:14:14 EST 1998

> > Are the following statements "mathematics" by your definition?  Why or why not?
> >  1) White wins in chess if you remove Black's Queen from the initial position.
Indeed!  But the proof is NOT a mathematical one!

An aside to state those of my credentials relevant to the following: 
I currently compete in correspondence chess at a high level, and made 
a plus score in the most recently completed United States 
Championship (Invitational).  My peak rating is 2405 (Senior Master) 
and I have also had a Master's rating in ordinary tournament chess.
>From 1987 to 1991 I was a regular chess writer for Chess Horizons 
Magazine on the theory of Chess Openings and won four awards from the 
Chess Journalists of America for this work (including two for "Best 
Regular Magazine Column" and one for "Best Analysis").

Every chessplayer with significant practical experience knows that 
the advantage of a Queen in the initial position is enough to win.  
This is truly a known fact, in the philosophical sense of justified 
true belief.  It is also clearly a mathematically formalizable 
statement.  But this fact is not (yet) mathematical knowledge because 
we have nothing even remotely approaching a mathematical proof.  

It seems, from his response, that Hersh mistakenly thinks we have 
such a proof.  But this isn't necessarily so.  Recall that his 
definition of mathematics is simply that area of abstract human 
thought which is uniquely objective,  characterized by the full 
reproducibility of and social consensus about its results.  (Other 
sciences don't qualify because they are not abstract, other 
humanities don't qualify because there is no reproducibility and 
consensus.)  Chess is abstract.  There is a full consensus among
everyone competent in the subject (let us say, everyone with a 
rating equivalent to 1600 Elo points or higher; there are hundreds of
thousands if not millions of such players worldwide) about this 
result.  The result is reproducible in a very strong sense (if you 
take two players who are roughly equal in strength and make them play
a "Queen-odds" game the player with the Queen will always win).

Even if the successor to Deep Blue could somehow prove by exhaustive 
search that White wins when you remove Black's Queen (which I don't
think will ever happen), Hersh's definition also applies to much 
more complex results, such as "The initial position is not a forced 
win for the second player".  No machine could ever establish this,
and I doubt it will ever be shown in a mathematically rigorous sense 
(I think it is more likely that it will be rigorously shown that any 
proof of that statement would be astronomical in size).  But you
again have consensus and reproducibility (ask any good player to 
choose a color in a money game against an unidentified opponent and 
he will choose White; I specify "unidentified" because against a 
specific opponent he might have a surprise Black opening prepared).

Hersh's definition fails because it is purely external; there is 
something about the nature of mathematical proofs that distinguishes
them from Hershian ones.  To fully "mathematize" proofs of statements 
like 1) above for games like Chess or Go is a fascinating research 
program, but so far progress has only been made in endgames with so
few pieces that an exhaustive search of all positions is possible.
Unless and until this hugely ambitious program is accomplished (I 
know it is on Harvey's list of things to do and he'll undoubtedly get
to it within a few centuries), I maintain that Hersh's definition of 
what mathematics "really" is invalid and challenge everyone on the 
FOM list who agrees with this to come up with a better definition 
mathematics which properly distinguishes mathematical proof from
the kind of proof we have of statement 1).

By the way, Reuben, I intend to argue in a later posting for the 
expansion of the notion of proof to include what computer scientists 
call "interactive proofs", which will in fact make my chess example 
somewhat closer to what is to be regarded as mathematically provable 
and partially vindicate your definition (extensionally in the sense 
that I can't think of any better counterexamples, not intensionally 
because your definition is external and has nothing to say about the 
internal nature of proofs).

-- Joe Shipman

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