FOM: chess "axioms" and untraversed gaps

Don Fallis fallis at
Sat Mar 21 12:10:55 EST 1998

"A chess problem is genuine mathematics."
- G. H. Hardy (in "A Mathematician's Apology")

I am in complete agree with Hardy, Shipman, Cabillon, and Friedman that
certain propositions about chess (e.g., "The initial position is a draw.")
are mathematical propositions (and if proved, they are mathematical
theorems).  However, I object to the suggestion (from Shipman) that
something like, (P) "White has a winning strategy when Black's Queen is
removed" might be a chess "axiom."

Charlie Silver is no doubt correct that "an axiomatic theory could be set
up that would completely define the game.  And, then theorems could be
proved" (FOM 3/20/98).   In this mathematical theory, the axioms are going
to be propositions that specify the rules of the game (e.g., that it is
played on an 8x8 board, that the pieces move and attack in certain ways,
etc.)  A statement like P above, no matter how compelling our inductive
evidence for it is, is something that must be proved (or disproved) from
these axioms that specify the rules of the game.  It would never be
appropriate to just take P as an axiom (except in the extremely unlikely
event that P turned out to be independent of the other axioms).

So P is not an axiom and it is not (yet) a theorem.  This still leaves open
the question of status of propositions (like P) that might legitimately be
used to derive further facts about winning positions in chess.  As it
happens, there are many cases where mathematicians draw out the
consequences of a proposition that is not itself an axiom.  For instance, a
number of things have been proved to be consequences of the Riemann
Hypothesis.  Now, in this case, there is no doubt among mathematicians that
RH must be proved before we can conclude that these consequences are true.
However, in "The Nature of Mathematical Knowledge," Philip Kitcher claims
that there are also cases where a theorem is applied by mathematicians even
though "no one has taken the trouble to prove it."  (By the way, the
problem here is not simply that the argument has not been formalized in
some suitable formal theory (e.g., first-order ZFC).  Kitcher goes on to
say that such theorems may "never receive general proofs which are rigorous
even by the standards of informal rigor that mathematicians accept.")  It
appears to me that such "untraversed gaps" (i.e., cases where, even though
no error may have been overlooked, the details of a proof have nevertheless
not been checked) occur all the time throughout mathematics (even if the
gaps are not usually as wide as ones Kitcher refers to).

I opined that it was legitimate to appeal to P because, even though it has
not been proved, there is little doubt that a proof could be constructed
given enough time, money, and graduate students (assuming that the universe
lasts long enough).  Unfortunately, the evidence that we have that the
proof could be constructed is clearly inductive (specifically, it is based
on chess players' extensive experience playing chess).  We cannot have
purely deductive evidence because we have not actually constructed the
proof.  But for the same reasons, don't we then have to conclude that
mathematicians themselves are basing some of their conclusions on inductive
evidence when they leave "untraversed gaps"?  (This would seem to be a
question that the General Theory of Proof (that Robert Tragesser calls for,
FOM 3/17/98) should answer.)

(By the way, while the discussion of chess "theorems" came out of the Hersh
debate on the degree to which mathematics is socially constructed, it
strikes me that the problem of "untraversed gaps" is independent of what it
is that makes mathematical propositions true or mathematical reasonings
correct.  Whatever the correct reasonings are, the mathematician that
leaves an "untraversed gap" has not performed them.)

Don Fallis
University of Arizona

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