also been analysed by George Polya. Polya claimed that it was all about scientific induction but we think it is at least as much about game-like brilliance.
In Chapter 7 we turn to geometry and Euclid, for two thousand years the paradigm of game-like mathematics – which isn't to say that much of geometry hasn't first been discovered by experiment or that geometers haven't made mistakes. They have, but relatively few and geometry is far more notable forits wealth of surprises, the sheer elegance of its arguments and the depth of its insights.
Goodstein's analogies
R. L. Goodstein was a rare mathematician who took the analogies between mathematics and abstract games seriously. In his, Recursive Number Theory , he discussed ‘Arithmetic and the Game of Chess’:
To the numerals correspond the chess pieces, and to the operations of arithmetic, the moves of the game. But the parallel is even closer than this, for to the problem of defining number corresponds the problem of defining the entities of the game…
and a little later he added, as an aside, ‘Here at last we find the answer to the problem of the nature of numbers…’ [Goodstein 1957 ].
In his Essays on the philosophy of mathematics , he went further: ‘[mathematics] is the continual creation of new games and the comparative anatomy of games’ which shows much more insight. Even when he writes ‘If mathematics had no applications then it would indeed be “merely a game”’, he half-rescues himself by continuing, ‘Even so, not a meaningless game, for operating with symbols in a coherent structure is itself meaningful’ [Goodstein 1965 : 216, 111–112].
In Axiomatic Projective Geometry , written with E. J. F. Primrose, Chapter X is titled ‘Geometry as a Board Game’, and justifies its title literally: ‘The game is played by placing rows of letters…in columns on a “board”. Each column serves to express an attribute of the geometry…’ and so on [Goodstein 1953 ].
In an earlier paper, Geometry in Modern Dress , Goodstein had taken the axioms of projective geometry and presented them lightheartedly, first in terms of typists and the typewriters they used at the International Typewriting Agency, and then in terms of the Game of Letter-Board[Goodstein 1938 : 217].
Tactics and strategy
The abacus is one of the oldest calculating devices, whether using a board with pebbles or beads strung on wires or sliding in slots. A few fortunate Romans even used a hand-held abacus, made of a sheet with beads moving in parallel grooves. There is an example in the British Museum.
Mostly, the Romans used loose stones, or calculi (whence our words to calculate, and calculus) on a ruled board or cloth so it is no surprise that they also referred to ludus calculorum , meaning any game using the same equipment.
Addition on an abacus, or soroban , is like playing a simple board game, because you are physically moving pieces to achieve a desired result according to the rules. We might say that this is an empirical kind of arithmetic. Since an abacus expert hardly needs to think while using the device, experts can be extremely fast. In a contest in Tokyo in 1946, Kiyoshi Matsuzaki using a soroban easily defeated Private Thomas Wood using a hand-held electric calculator on addition, subtraction, multiplication and division problems [Tani 1964 : 7].
Small children who use Cuisenaire rods to do sums are also playing a game with simple rules – and whatever method you use, these games have tactics and strategies. What is 8 × 35? Well, 8 is 2 × 2 × 2, so to multiply by 8 we double three times: 35 – 70 – 140 – 280, the correct answer, easily calculated mentally.
What is 12 × 35? Since 12 = 8 + 4, the answer from the previous calculation is 280 + 140 = 420. However, you could also notice that 12 = 3 × 2 × 2 and 3 × 35 = 105, so doubling twice we get, 105 − 210 − 420.
As a last resort, you could use your 12 times table: 12 × 30 =
In Chapter 7 we turn to geometry and Euclid, for two thousand years the paradigm of game-like mathematics – which isn't to say that much of geometry hasn't first been discovered by experiment or that geometers haven't made mistakes. They have, but relatively few and geometry is far more notable forits wealth of surprises, the sheer elegance of its arguments and the depth of its insights.
Goodstein's analogies
R. L. Goodstein was a rare mathematician who took the analogies between mathematics and abstract games seriously. In his, Recursive Number Theory , he discussed ‘Arithmetic and the Game of Chess’:
To the numerals correspond the chess pieces, and to the operations of arithmetic, the moves of the game. But the parallel is even closer than this, for to the problem of defining number corresponds the problem of defining the entities of the game…
and a little later he added, as an aside, ‘Here at last we find the answer to the problem of the nature of numbers…’ [Goodstein 1957 ].
In his Essays on the philosophy of mathematics , he went further: ‘[mathematics] is the continual creation of new games and the comparative anatomy of games’ which shows much more insight. Even when he writes ‘If mathematics had no applications then it would indeed be “merely a game”’, he half-rescues himself by continuing, ‘Even so, not a meaningless game, for operating with symbols in a coherent structure is itself meaningful’ [Goodstein 1965 : 216, 111–112].
In Axiomatic Projective Geometry , written with E. J. F. Primrose, Chapter X is titled ‘Geometry as a Board Game’, and justifies its title literally: ‘The game is played by placing rows of letters…in columns on a “board”. Each column serves to express an attribute of the geometry…’ and so on [Goodstein 1953 ].
In an earlier paper, Geometry in Modern Dress , Goodstein had taken the axioms of projective geometry and presented them lightheartedly, first in terms of typists and the typewriters they used at the International Typewriting Agency, and then in terms of the Game of Letter-Board[Goodstein 1938 : 217].
Tactics and strategy
The abacus is one of the oldest calculating devices, whether using a board with pebbles or beads strung on wires or sliding in slots. A few fortunate Romans even used a hand-held abacus, made of a sheet with beads moving in parallel grooves. There is an example in the British Museum.
Mostly, the Romans used loose stones, or calculi (whence our words to calculate, and calculus) on a ruled board or cloth so it is no surprise that they also referred to ludus calculorum , meaning any game using the same equipment.
Addition on an abacus, or soroban , is like playing a simple board game, because you are physically moving pieces to achieve a desired result according to the rules. We might say that this is an empirical kind of arithmetic. Since an abacus expert hardly needs to think while using the device, experts can be extremely fast. In a contest in Tokyo in 1946, Kiyoshi Matsuzaki using a soroban easily defeated Private Thomas Wood using a hand-held electric calculator on addition, subtraction, multiplication and division problems [Tani 1964 : 7].
Small children who use Cuisenaire rods to do sums are also playing a game with simple rules – and whatever method you use, these games have tactics and strategies. What is 8 × 35? Well, 8 is 2 × 2 × 2, so to multiply by 8 we double three times: 35 – 70 – 140 – 280, the correct answer, easily calculated mentally.
What is 12 × 35? Since 12 = 8 + 4, the answer from the previous calculation is 280 + 140 = 420. However, you could also notice that 12 = 3 × 2 × 2 and 3 × 35 = 105, so doubling twice we get, 105 − 210 − 420.
As a last resort, you could use your 12 times table: 12 × 30 =
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