Feedback by Olivier Gerard on Opinion 57

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). 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 Friedmann 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 Friedmann) 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, Olivier.

Feedback by Dan Romik on Opinion 57


I think your Opinion merits some discussion on the semantic meaning of the
words "trivial", "deep" and "interesting". You criticize Hardy's statement
that chess is trivial, and offer your own opinion that chess is both much
deeper and more interesting than the Riemann Hypothesis.

The Oxford English Dictionary ( gives several
definitions for the word "trivial", of which the ones which seem pertinent
1. Of small acount, little esteemed, paltry, poor; trifling,
inconsiderable, unimportant, slight.
2. (math.) Of no consequence or interest, e.g. because equal to zero.

"Deep" is defined in various ways, among others as
1. Hard to fathom or get to the bottom of; penetrating far into a subject,
2. Said of actions, processes, etc. in which the mind is profoundly

As for "interesting", I won't bother defining it, except to recall the
Hebrew saying "al ta'am ve'al reach ein le'hitvakeach" (which I find more
evocative than the English equivalent "different strokes for different

What you explain about why you find chess deeper and more interesting than
the RH, is that the problem of determining if White has a winning move is
more difficult than RH (I agree), will probably never be solved (I agree),
and a solution, should one ever be found, will occupy more space than a
solution of the RH (I agree).

What I think Hardy meant when he used the word "trivial" with regards to
chess - and it is clear he meant to use it in a belittling sense, no
matter what dictionary he consulted - is that chess, being in some sense
an arbitrary problem dependent on human culture and history, is less
worthy of consideration as a mathematical problem. And this, regardless of
any issues of applicability - I'm sure Hardy, purist that he was, would
not find that the present-day uses of the Riemann Hypothesis in questions
of cryptography detract in any way from its beauty. Hardy, and I think
most present-day mathematicians, myself included, had a tendency to assign
a value to mathematical problems, based on criteria of elegance, beauty,
intrinsic interest, and most importantly, some elusive concept of
"naturality". You may find chess of more intrinsic interest than the RH -
it's really pointless arguing about interest. But I think even you would
be hard-pressed to argue that chess is a more natural problem. When I say
natural, I mean something such as: would intelligent beings on some remote
planet also come up with this problem? And even if they were living in a
different physical universe than ours, would they still come up with this
problem? I believe the answer is "yes" for the RH, and "most probably not" 
for chess (though they may come up with quite similar-looking problems to

To summarize, I claim that there is really no disagreement between you and
Hardy! You two simply enjoy different aspects of the practice of
mathematics - you have different utility functions, one might say. You may
be in it for the pure fun of it, the joy of taking a problem, no matter
where it came from, and attacking it with all you've got. Hardy, as you
correctly observed, had an almost religious sense concerning which
problems were the most important, and got an extra kick from the "added
value" of thinking about the problems which seemed to possess some kind of
transcendence or nobility. And to conclude on an optimistic note, I think
it's great that there are people like you out there, and also people like
Hardy! This shows that there are all sorts of ways of viewing and doing
mathematics. Why be so dogmatic about your own personal way of thinking?
There is room for everybody, don't you think?

Feedback by Apostolos Doxiadis on Opinion 57 (April 11, 2004)

Although I agree fully with all
you say there, I think there is a point where it may need
some revising, not to be unfair to that old Queen of Number Theory, as
Uncle Petros, in an angry moment, called G. H. Hardy.
You say that he says 'chess is trivial', which is indeed an amazing 

statement, if he made it. Although I have not a copy of the Apology 
here, I did some looking-up on the internet and I think that the only
connection he makes between chess and triviality is in the sentence "A
chess problem is genuine mathematics, but it is in some way 'trivial' 
mathematics..."  And of course a chess problem, of the type he means 
(mate in 2 or 3 moves) is rather trivial mathematically, but this is 
altogether a different kind of statement than saying 'chess is trivial'.
I just thought I'll tell you this, so you could check it -- maybe I'm 
wrong and he actually says 'chess is trivial' somewhere but, if not, the
statement about problems itself is. I'm sure you'll agree, not the ideal
target for your (justified) rage . But I think the main point of your 
Opinion can still be validly made, with a small amendation, because the 
amazing thing (or perhaps not so amazing in pre-Alan Turing times) is 
that Hardy does not think of chess as mathematics (too messy, too 
complex, too inelegant I'm sure) although he calls chess-problems the
'hymn-tunes' of mathematics, trivial ones of course! And his not 
considering chess to be mathematics, but only the 'trivial' chess 
problems, is perhaps as amazing, or more, than saying it is trivial

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