by Wells, David
Mathematicians have other sources of error too: when trying to create a new concept or to formalise some rather vague and intuitive idea, they can easily make mistakes, as we shall see.
Reasoning, imagination and intuition
The experience of playing chess for many years cannot be summed up in a few well-chosen sentences or indeed expressed in language at all. Experience develops intuition which guides the player when calculation is insufficient and the deeper your intuitions the better you will play. Intuition turns out to be essential in mathematics too.
One of the greatest pleasures of chess or Go is imagination. Chess columns in the newspapers invite readers to ‘Spot the Next Move!’ and the games of masters such as Botwinnik or Kasparov were packed with imaginative conceptions. Mathematics is just as much a matter of imagination. As Voltaire wrote,
There is an astonishing imagination, even in the science of mathematics…There was far more imagination in the head of Archimedes than in that of Homer.
[Voltaire 1764: #3]
The great David Hilbert was not the fastest thinker in the world but he showed extraordinary depth and imagination. Appropriately, with his pupil Cohn-Vossen he wrote a best-selling book on Geometry and the Imagination. When Paul Gordan (1837–1912) read one of Hilbert's papers which proved that a certain object existed but without showing how you could find it, he exclaimed, ‘This is not mathematics, this is theology!’ Hilbert himself said of a former student, ‘You know, for a mathematician he did not have enough imagination. But he became a poet and now he is doing fine’ [Wells 1997: 140].
The power of analogy
As we have noted, game players need a subtle sense of analogy to apply their past experience to a present situation. Mathematicians have the same faculty, as do scientists. George Gamow claimed that the greatest scientists even see ‘analogies between analogies’. No wonder that Japanese Go players still refer to traditional Go sayings or proverbs to guide them or that there is a book, Go Proverbs Illustrated by Kensaku Segoe, to aid the beginner.
Past experience has to be exploited by generalisation because it will seldom be repeated exactly, though there are exceptions in the chess openings and when mathematicians use standard methods and techniques: more often, however, a new situation is similar to past situations and the players has to spot the resemblance and then specialise from past experience.
Fortunately, this is enjoyable: both chess players and mathematicians get a psychological kick out of spotting analogies.
Simplicity, elegance and beauty
Play around with the Tower of Hanoi for a while and you ‘get the hang of it’, you feel there's a pattern there before you can follow it on every move, until finally the penny drops and you feel pleased with yourself. The chaos of the original play is replaced by structure, and you get an aesthetic ‘kick’, as you do from the proof that it can be solved in 2n − 1 moves. The ‘kick’ from playing the Tower of Hanoi successfully is limited, however, to beginners and children. To serious game players it is just too simple.
So, plausibly, are Nine Men's Morris and Hex. As we have seen, no-one has yet published a book of some world champion Hex player's Greatest Games but all the best games of all the great chess and Go players have been printed. Indeed all the games played in all the great international chess tournaments and matches have been published, sometimes in several languages, and often with annotations which help the amateur to understand what is going on. Chess is so complex, so subtle, and so hard to understand that without expert assistance most players would simply not have more than a slight idea what was happening in a typical grandmaster game – but that is where the greatest examples of the beauty and elegance of chess appear.
Moves can be surprising, even astonishing; they can develop a theme in an apparently inexorable sequence, or they can be so bizarre as to look at first sight like a mistake; they can be overwhelmingly powerful, or subtle and delicate. All these features contribute to the feeling that leads chess players – and Go players also – to use the language of art to describe individual moves, or sequences of moves or even the ideas themselves.
We said ‘the greatest examples’ but even the patzers in the local chess club can enjoy making moves which, at their own level, have the same delightful features, just as amateur mathematicians can enjoy any number of mathematical ideas and theorems and illustrations without being experts on the theory of sheaves.
Science and games: let's go exploring
What happens when you play around with a puzzle or game? Playing around is not the same as playing seriously, but rather a sort of loose exploration or experiment which helps to develop a feeling for what is happening. As this feeling develops you think perhaps of more serious experiments or tests that will push your exploration ahead scientifically. This points to another link with science: it is easy to draw conclusions that are false and which further experiment will show to be false, by counter-example.
We shall look at science and mathematics in Chapters 11 and 14.
4 Why chess is not mathematics
Competition
For all their resemblances, abstract games and mathematics are far from identical. Games, for a start, are competitive but everyday arithmetic is not, though even this difference should not be exaggerated.
Leonardo of Pisa, known as Fibonacci (1170–1250) was challenged by John of Palermo at the court of the Holy Roman Emperor Frederick II to solve a set of mathematical problems. Fibonacci solved them and won. Tartaglia (1500–1557) was challenged by Fior, a student of the famous Scipione del Ferro, to a fight not with swords but with cubic equations, each proposing 30 cubics for solution. Tartaglia solved all of Fior's in less than two hours while Fior struggled. Tartaglia later accepted a challenge to debate Cardan's assistant, Ferrari, in Milan with cubics and quartics, but the quartics were too much for him. At the end of the first day he crept out of the city and lost by default.
Vieta (1540–1603) was challenged to uphold the glory of French mathematics at the court of Henry IV against the sneers of the Dutch ambassador by solving an equation of degree 45. He succeeded by recognising it as a disguised trigonometrical identity.
The list goes on. Johann Bernoulli (1667–1748) challenged the mathematicians of Europe to solve the brachistochrone problem: to find the curve along which a small marble would roll in the shortest time from A to B. Bernoulli took the side of Leibniz in his dispute with Newton over the invention of the calculus so he was deliberately provocative:
there are fewer who are likely to solve our excellent problems, aye, fewer even among the very mathematicians who boast that [they]…have wonderfully extended its bounds by means of the golden theorems which (they thought) were known to no one, but which in fact had long previously been published by others.
Newton received the challenge at 4 o'clock in the afternoon after a long day as Master of the Royal Mint but by four in the morning he had solved it. He sent his solution anonymously by an intermediary to Bernoulli who, recognising the author, famously exclaimed, ‘The lion is known by his claw’ [O’Connor & Robertson 2006].
Priority disputes were an inevitable source of friction before a modern system of academic publishing was established although, according to Sal Restivo, competition between mathematicians in early Europe was not matched among either the Chinese or Hindu mathematicians [Restivo 1992: 18–19].
Felix Klein (1849–1925) was at the peak of his career when in 1881 Henri Poincaré (1854–1912) began publishing his work on automorphic functions. They first corresponded in a friendly fashion but Klein was stretched to keep up with his younger competitor and this plausibly contributed to the collapse of his health in 1882 and his subsequent depression: Klein's career as a leading research mathematician was largely over.
More recently, Andrew Wiles, having decided to devote the best years of his mathematical life to Fermat's Last Theorem, famously took precautions to hide his attempt from his colleagues, even publishing occasional papers on other
topics to discourage speculation that ‘he must be up to something’. He did finally ‘prove’ it but a gap was found in the proof and he had only a short time to fill it before his proof was, in effect, declared a failure. He succeeded, with the assistance of his pupil, Richard Taylor, in 1994.
Mathematical knights are still entering the lists today. Hofstadter's sequence (p. 144) prompted a challenge by John Conway, mathematician extraordinaire and inventor of the Game of Life, for a value of n which made the absolute value of a(n)/n − n/2 < 1/20. Conway offered $10000. Ouch! Mistake! C. L. Mallows answered the challenge and earned himself the tactfully ‘adjusted’ prize of $1000 by showing that n was a low 1489 [Weisstein 2006] [Schroeder 1991: 57–59].
Asking questions about
The important currents of mathematical thought develop from deep investigations of fundamental questions arising from mathematically natural phenomena.
[Williams 1998: xiv]
Practical chess players ask questions such as, ‘What are the best first moves? What move is best in this position? When should you double your rooks?’ They are frankly uninterested in questions such as: • How long is the longest possible game?
• How many possible distinct games can be played?
• What proportion of games should be won by white?
• Why do the knights move the way they do?
• What if the board were smaller, or larger?
These are what philosophers call meta-questions, questions about the game whose answers will not make you play better. They step outside the frame which is just what mathematicians – but not game players – do all the time.
Anyone who wonders, ‘Why do some numbers have more factors than others?’ has practically invented the idea of prime numbers, and several other kinds of numbers too, and could spend the rest of their life trying to answer that one question. When we ask about we step back and take a more distant view and even step outside the original rules of the game.
Asking questions is a crucial activity, but what questions should we ask? Which are the most effective? The easiest are ‘What if…?’ such as, ‘What if we asked Euler's question about a different map? What if the Tower of Hanoi puzzle had four pegs? What if we added up not the counting numbers, but their squares?
Other questions can start How or How Many or When or Which or Where…How can you draw a circle that goes exactly through three given points? In how many parts can a circle be cut by 5 straight lines? How many prime numbers are there? When will a prime number be the sum of three squares? Which numbers have the most prime factors? Which infinite series have sums which are irrational?
These questions can often be answered provisionally by experiment. Draw a different map for the ‘Bridges of Königsberg’ or try the Tower of Hanoi with four pegs and see what happens. Draw a circle and cut it up. Make a calculation, investigate a sample case – a dozen sample cases. Check on primes that are the sum of three squares, collect some data, get some idea of what is going on. Mathematicians spend a lot of time on such activities – that's why mathematics has such a strong experimental and scientific side.
Finally, however, we come to the Why questions. ‘Why?’ is often – usually – almost always! – a deeper question and harder to answer, not least because we are looking for proof. Ideas of proof have notoriously changed over the centuries. Today, many proofs by eighteenth-century mathematicians seem flawed, and ideas of problem solving have changed too: we now expect more than just An Answer.
Metamathematics and game-like mathematics
David Hilbert wanted to formalise all of mathematics, as if mathematics could be entirely reduced to a logical game. He failed, but his program did promote metamathematics in which mathematics is turned in on itself, to examine its own foundations, the logic of its proofs, and the soundness of its assumptions – a perfect example of asking questions about mathematics.
(Since pictures and diagrams, however useful to many mathematicians, are used sparingly in most published professional papers, as illustrations, and have a reputation for being potentially misleading, it is no surprise that metamathematics is extremely verbal, or that it is has links to the theory of computer programming, another very game-like activity.)
There are students of chess and other abstract games who specialise in asking questions about – they can join the International Society for Board Games Studies and contribute to its journal Board Game Studies – but they do not have to be chess or Go players to do so.
Changing conceptions of problem solving
The American Mathematical Monthly commenced in 1894 as a journal ‘Devoted to the solution of problems in pure and applied mathematics’, and the problems and solutions published much resembled textbook exercises.
In 1932 the problems were split into Elementary and Advanced and the latter section ‘[sought] problems containing results believed to be new, or extensions of old results…’ There is a hint here that a “result” might not be the end of a problem, that it might not be the last word, though in practice solutions were simply submitted and published as before.
Towards the end of the 1960s, however, a new section started called ‘Research Problems’, later renamed simply, ‘Unsolved Problems’. It was edited by Richard Guy from 1970 and in the odd numbered years included updates describing the progress made on the problems published. Over a period of a century or so the idea of a problem had changed from being little more than a textbook exercise to being a public challenge, open to development [Wells 1993].
Creating new concepts and new objects
Chess players create new concepts as they understand the game better but these are a part of their analysis and don't change the rules and don't change the game. When mathematicians pick out, let's say, the prime numbers and label them, they are changing the nature of the game and the prime numbers become an entity in a modified game that suddenly includes new ‘pieces’.
Sometimes a new idea is more-or-less forced by the situation, such as the radian as a measure of angle (p. 99). Other new concepts appear when we try to formalise some rather vague intuition, such as infinity which in everyday language has many variant meanings. Galileo's paradox is just one of many that would not exist if infinity were not such a subtle concept. To use the idea in mathematics it must be tied down and stripped of most of its everyday associations to prevent mathematicians tying themselves in knots.
Mathematicians exploring mathematical landscapes often find weird objects which seem at first sight to be of no particular significance: on deeper examination, this is hardly ever the case. Fractals are an example. The first fractals were regarded as weird monstrosities and appeared in books of mathematical recreations like strange animals exhibited in a Renaissance zoo. Today they are well understood, are recognised all over the place and have entered the mainstream.
Often, ‘weird’ objects turn out to be specific examples of some general phenomena, just as an explorer first discovering a carnivorous flower might be initially astonished but realise later that there are entire families of such strange plants.
Increasing abstraction
Asking questions about has another effect that students of mathematics know only too well: it makes mathematics even more abstract. The graphical picture of the Tower of Hanoi makes the puzzle easier to solve – it makes it trivial – but the graph itself takes some understanding: why does it perfectly represent the puzzle? Most players of the simple game of the Tower of Hanoi would not think of the graphical representation in a million years. It takes a mathematician – and a modern mathematician at that – to ‘naturally’ think of representing the positions in a game on a graph. It is a tactic, or strategy, that is now very well known and even seems not very abstract, though it was once novel and surprising.
The famous French mathematician Jean Dieudonné exhorted students to develop an ‘intuition for the abstract’, but that takes time and effort. John von Neumann once told a student complaining that he had not understood von Neumann'
s lecture, ‘You don't understand mathematics, you just get used to it!’ Not the advice you were given at school, but it does contain an important grain of truth.
Yes, intuition is the key and it can be developed, in part, by following von Neumann's advice. Experience and then more experience leads to intuition. Just as chess players become more-or-less familiar with the miniature world of the chess board by playing many, many games, so mathematics students become familiar with the miniature worlds of group theory or differential equations experience, by playing and experimenting, calculating and speculating and checking. Students who do not ‘play around’ with such new ideas, but stick to the textbook explanation and do no more than answer a few exercises, will never develop deep intuition, and never become real mathematicians.
Finding common structures
A game player would be amazed if a tactical or strategical idea that worked in chess also worked in Go (apart from the basic idea that a move which creates two threats is powerful) but mathematicians are used to finding matching features in algebra and geometry, or analysis and topology – or – you name it! The analogy is often more-or-less perfect, as when a geometrical diagram is translated into coordinates (Figure 4.1).
