DETAILS of your problem, and SWITCH THE METHOD ON! Gottlob Frege, who was a mathematician as well as a logician and philosopher, wrote that,
The aim of proof is…not merely to place the truth of a proposition beyond all doubt, but also to afford us insight into the dependence of truths upon one another.
[Frege 1884/1953: 2 e ]
The explanations in mathematicians’ published papers may be logically convincing yet unconvincing in every other respect which makes them much harder to read and understand. As Hermann Weyl wrote:
We are not very pleased when we are forced to accept a mathematical truth by virtue of a complicated chain of formal conclusions and computations, which we traverse blindly , link by link, feeling our way by touch. We want first an overview of the aim and of the road; we want to understand the idea of the proof, the deeper context.
[Weyl 1995 : 453]
The italics are ours. The emphasis on perception could not be clearer. ‘Aha! Now I see it!’ is the hope and expectation of all mathematicians (and the despair of many school pupils.) To play around with a problem, to ‘suck it and see’, to turn it over and look at it from another perspective, is the very essence of mathematics. It is no coincidence, of course, that playing is just what game players do.
The three aspects cannot be separated. There are few examples of purely perceptual maths and none at all if you argue that perception in mathematics is always game-like because the ideas are already abstract. Nor can mathematics be purely scientific: it must in the final analysis be game-like. That is one of the differences between maths and physics, or chemistry. Experiment may convince a mathematician psychologically that a result must be correct but he or she still wants a proof, not just because proofs are convincing but because the effort of creating a proof for a challenging solution creates new ideas that illuminate the problem left behind and the path ahead as the sun rises on yet another region of the endless mathematical landscape that mathematical gold diggers rush to explore.
We shall start with game-like mathematics, then turn to the scientific aspect, and then finally to the foundation of all art and science – and mathematics – perception.
For the reasons already given, the distribution of topics between the following chapters is rather arbitrary. With a slight change in perspective or a different emphasis, an illustrative example in one chapter could readily be moved to another.
6 Game-like mathematics
Introduction
Game-like mathematics is a world of brilliant moves and cunning tactics, subtle and deep strategy, beautiful combinations and transformations, profound intuitions, and proof .
We first glance at the abacus and Cuisenaire rods, and examples of game-like tactics and strategy in ordinary arithmetic. Even small children can use tricks and short cuts to solve arithmetical problems more easily. These are their introduction to game-like tactics and stratagems and to the importance of not simply relying on routine algorithms – let alone on calculators!
Hidden connections in algebra is a fascinating theme: repeatedly, patterns and analogies turn up which seem to have a deeper significance. Usually they do, but how to find it? Geometrical figures always mean something but algebraic expressions or equations often seem to mean little – but then some neat move or an elegant transformation or a simplification is made, and pattern and structure and meaning appear, almost as if by magic.
Jean le Rond d’Alembert (1717–1783) said that, ‘Algebra is generous. She often gives more than is asked for’ [Boyer 1991 : 439] but only to those who learn to play the game of algebra fluently and with insight and imagination. Later we shall see how what appears to be a mere coincidence is connected to a surprising theorem by Liouville. This will be followed by a masterpiece by the incomparable Euler, that has
The aim of proof is…not merely to place the truth of a proposition beyond all doubt, but also to afford us insight into the dependence of truths upon one another.
[Frege 1884/1953: 2 e ]
The explanations in mathematicians’ published papers may be logically convincing yet unconvincing in every other respect which makes them much harder to read and understand. As Hermann Weyl wrote:
We are not very pleased when we are forced to accept a mathematical truth by virtue of a complicated chain of formal conclusions and computations, which we traverse blindly , link by link, feeling our way by touch. We want first an overview of the aim and of the road; we want to understand the idea of the proof, the deeper context.
[Weyl 1995 : 453]
The italics are ours. The emphasis on perception could not be clearer. ‘Aha! Now I see it!’ is the hope and expectation of all mathematicians (and the despair of many school pupils.) To play around with a problem, to ‘suck it and see’, to turn it over and look at it from another perspective, is the very essence of mathematics. It is no coincidence, of course, that playing is just what game players do.
The three aspects cannot be separated. There are few examples of purely perceptual maths and none at all if you argue that perception in mathematics is always game-like because the ideas are already abstract. Nor can mathematics be purely scientific: it must in the final analysis be game-like. That is one of the differences between maths and physics, or chemistry. Experiment may convince a mathematician psychologically that a result must be correct but he or she still wants a proof, not just because proofs are convincing but because the effort of creating a proof for a challenging solution creates new ideas that illuminate the problem left behind and the path ahead as the sun rises on yet another region of the endless mathematical landscape that mathematical gold diggers rush to explore.
We shall start with game-like mathematics, then turn to the scientific aspect, and then finally to the foundation of all art and science – and mathematics – perception.
For the reasons already given, the distribution of topics between the following chapters is rather arbitrary. With a slight change in perspective or a different emphasis, an illustrative example in one chapter could readily be moved to another.
6 Game-like mathematics
Introduction
Game-like mathematics is a world of brilliant moves and cunning tactics, subtle and deep strategy, beautiful combinations and transformations, profound intuitions, and proof .
We first glance at the abacus and Cuisenaire rods, and examples of game-like tactics and strategy in ordinary arithmetic. Even small children can use tricks and short cuts to solve arithmetical problems more easily. These are their introduction to game-like tactics and stratagems and to the importance of not simply relying on routine algorithms – let alone on calculators!
Hidden connections in algebra is a fascinating theme: repeatedly, patterns and analogies turn up which seem to have a deeper significance. Usually they do, but how to find it? Geometrical figures always mean something but algebraic expressions or equations often seem to mean little – but then some neat move or an elegant transformation or a simplification is made, and pattern and structure and meaning appear, almost as if by magic.
Jean le Rond d’Alembert (1717–1783) said that, ‘Algebra is generous. She often gives more than is asked for’ [Boyer 1991 : 439] but only to those who learn to play the game of algebra fluently and with insight and imagination. Later we shall see how what appears to be a mere coincidence is connected to a surprising theorem by Liouville. This will be followed by a masterpiece by the incomparable Euler, that has
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