[FOM] Doron Zeilberger's Opinion 57

Olivier Gerard ogerard at ext.jussieu.fr
Mon Nov 10 18:33:05 EST 2003

Dear Doron, Dear FOMers

As usual Doron don't hesitate to glue together several strong
opinions and important issues in mathematics with peripheral
provocative remarks and pretend they are related,
embarrassing both people sharing his views and people wanting to
discuss them.

There has already been remarks about the interpretation of
Hardy comments on chess.  There could have been about the
notion of pure and applied mathematics (raw and done,
boiled or roasted, salted or spiced) and the fact that
you want us to choose between an almost trivial formula
in one variable and the first 1000 terms of a sequence
extracted from a difficult problem. 

I will only write about the potential interest of chess problems,
and not really from a foundational point of view.

I have no special sympathy for Hardy as a commentator of mathematics.
But not much can be expected from someone as ignorant
of cricket as I remain.

I am not particular about the origin of a particular mathematical
concept or method. So chess, astronomy, theology, gambling,
litterature, music even mathematics and many more are correct.

But I do not consider the answer to a chess problem (however 
general it would be regarding the chosen rules of chess)
of any importance in itself for mathematics (and of any general
intellectual interest to speak like Dr Friedman and Dr Simpson)
and this is not by a kind of infinitarian bigotry.

What could be highly non-trivial, would be 

A) the possibility of answering a class of similar chess problems
relatively easily with predictable physical and human resources,
whereas previous knowledge and methods would not allow that.

B) (a la Friedman) the discovery of an intuitive "chess" theorem whose 
proof would require without remedy the use of unusually strong axioms of
set theory, the use of inaccessible cardinals or a similar
non naive foundational concept.

Let's suppose that, motivated, inspired, by a chess problem,
a team or a single mathematician young, old or in silicium succeeds
in making a breakthrough of this kind.
I think I would find this interesting, worth commending.

Probably because I bet it would certainly have interesting and similar
consequences on many other problems, of a less
arbitrary nature than XXth century european chessboard rules.

The funny thing, is that at soon as new methods of proof, computation,
estimation, checking, etc. would be devised out of such a solution
to a (chess or backgammon, or whatever) problem, it would render
the original question moot, trivial, at least to the average
mathematician. Footnotes would recall the origin of the
strange-looking terminology in the theory, such as the
"Grand Roque corollary" and the "Queen promotion index".
In fact the trivialization of the original question would
be the necessary mark of the mathematical interest of the ensuing theory.

But this is a point you already made about multiplication
in several articles and opinions as well as about
more sophisticated results which can now be routinely checked 
thanks to the WZ method and its descendants.

I suspect a large part of Opinion #57 of having been
directly dictated by your companion Shalosh B. Ekhad.

So I wonder, is it more interested (more motivated ?)
by artificial, human-culture originated problems than by
number-theoretical or classical combinatorial conjectures,
has it even any preferences whatsoever ?
But it probably can answer itself as it did in the past.

What would really interest me is to know why it is
interested in mathematics at all.

with my best regards,


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