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Games and Mathematics: Subtle Connections

Cambridge University Press
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David Wells is a well-published British mathematician and youth master chess player, so he is fluently conversant with the characteristics mathematics shares with games — imagination, insight, intuition, and the playful dynamic of approaching problems lightly and adventurously. All this occurs within the mathematician’s “miniature world,” something a game-player shares as well. Wells relishes games, from their visual and tactile aspects to the need for the player to strategize blindly and build mental stamina. The chess piece itself is an icon for him. In fact, in the section “Why Chess is Not Mathematics” he comes just short of making the case that it is.

The typical reader will probably agree that a considerable amount of modern mathematics has grown out of games, and games can serve heuristically to understand mathematics. So it seems natural to find the book divided into two parts. In Part I, which we might label “games are mathematics,” we see strong squares in chess, Nash tackling the simple game of Hex, and the problem of filling a square with dominoes burgeoning into a question in statistical mechanics. Part II –ostensibly “mathematics is games” — cheerfully sweeps over the historical breadth of the discipline with clever examples from calculus, number theory, geometry. Invented or discovered? Intuitive? Sheer luck or grim determination? Many examples are familiar; some are obscure personal favorites; and some are just fiendishly ingenious, such as Euler’s work with pentagonal numbers. Probability and game theory are surprisingly omitted. Eventually Well’s interest in the visual leads to a discussion of mathematics and beauty.

The book is a pleasure to read, although one could stumble on some of his catechisms. As with, “The mathematician as game player observes and conjectures; the mathematician as scientist makes moves and spots possibilities; the mathematician as observer studies objects like the pieces in an abstract game of chess.” (p. 8.) Wells seems to say that mathematicians have the DNA of games in the analogies and other mental processes that create proofs, but one might ask: why is this DNA not shared by empirical scientists? “Empirical scientists can look, explore, conjecture and test but not prove, so we might say that, yes, mathematics is also empirical but it is empirical-plus and it’s plus because it is game-like, the crucial point.” (p. 123.) As for Lakatos, "he made the mistake of seeing mathematics as far too scientific… and had no conception of mathematics as game-like.” (p.117.)

But the historical references and the quotes alone are quite wonderful. Plato likens the practice of mathematics to closely fitting a shoe to the foot while not mistaking one for the other. (A rather non-Platonic sentiment!) Heaviside claims that to reduce physics to mathematics is wrong because physics gives life to the mathematician’s observations. And Bertrand Russell is wrong in thinking that there are only two fundamental kinds of objects — the mental and the existent, because tactile chess pieces comprise a third type! (C. S. Peirce was here.) Meanwhile “Goodstein was a rare mathematician who took the analogies between mathematics and abstract games seriously.”

So it’s better not to demand too much from this friendly, enticing, game-spirited book. We have a persuasive case that to understand and thoroughly enjoy mathematics, we must understand how it is rooted instinctively in gamesmanship. And even further, that it is related to our native drive to grasp a tool, be it a chess piece, or string, or pebble. Perhaps all would be well in mathematics departments if students would pick up a pencil and play Hex.

Sandra Z. Keith is professor emerita of St. Cloud State University, MN.04/23/2013

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Monday, January 14, 2013
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Part I. Mathematical recreations and abstract games:
1. Recreations from Euler to Lucas
2. Four abstract games
3. Mathematics and games: mysterious connections
4. Why chess is not mathematics
5. Proving versus checking
Part II. Mathematics:
game-like, scientific and perceptual: 6. Game-like mathematics
7. Euclid and the rules of his geometrical game
8. New concepts and new objects
9. Convergent and divergent series
10. Mathematics becomes game-like
11. Maths as science
12. Numbers and sequences
13. Computers and mathematics
14. Mathematics and the sciences
15. Minimum paths from Heron to Feynmann
16. The foundations: perception, imagination and insight
17. Structure
18. Hidden structure, common structure
19. Mathematics and beauty
20. Origins: formality in the everyday world

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Monday, January 14, 2013