*Warriors of the Mind: A Quest for the Supreme Genius of the Chess Board*(1989) by Nathan Divinsky and Raymond Keene, Divinsky uses mathematical techniques in an attempt to answer the question: "Who is the best chess player of all time?" Of course one has to realise that this book was completed in 1988 so its conclusions do not take into account nearly 30 years of later players and their results. We give some extracts from the Preface to this book below:

People interested in any sport often wonder who is the greatest artists or competitors in their beloved activity, and in particular, who was the very greatest. Chess enthusiasts are no different. Even though it is essentially impossible to compare Paul Murphy of 1858 with Gary Kasparov of 1988, chess authors and historians have filled the literature with long learned articles packed with charts and numbers, dealing with the chess champions of their era and comparing their champions with those of the past.

Sensible people may well throw their hands up and say that no real conclusion can be reached. It is, however, difficult to resist the fascination of meditating on Lasker and Capablanca versus Fischer, Karpov and Kasparov, not to mention Steinitz, Rubenstein, Keres and Botvinnik. The whole notion is simply too delicious to pass up.

We are going to embark on a careful analysis of such comparisons, using rather complicated mathematical ideas, messy equations and computer generated solutions. Such a plan was basically impossible before our computer age. we are in fact rather fortunate to live in a time when computers are available and when detailed data can be readily obtained. To this end we have gathered over 10,000 results and we have discovered several profound and intriguing insights. Of course the final conclusions are tentative, but do come along with us on our mathematical adventure.

...

Mathematics ignores all human prejudices and predilections. It is only interested in results and it analyses then in a cold, clinical fashion. Reputation means nothing to a formula. As Tarrasch once said: "It is not enough to be a good player, one must also play well!" Tal, for all his brilliance, has had some bad patches. These have certainly been caused by illness, but mathematics does not distinguish between illness and lack of talent. Mathematics is only interested in results, results and more results.

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Let us then take off all of our used clothing, divest ourselves of all conclusions and strip our souls bare. We can then put on white, sterilized jackets, rubber gloves and oxygen masks and walk through the magic door into the world of Pythagoras, Newton, Fermat and Gauss.

You must promise not to be surprised by what we discover, nor to beat your chest if one of your special favourites turns out to be a *patzer* [someone who plays chess badly]. Let us find out what is closest to the truth, with dignity and silence. When we return from mathematical wonderland, we can have a good yell and scream.