Games and Mathematics

by Wells, David

  The world ‘out there’ is full of objects, studied by scientists, which were created at some particular time or evolved over time and which may someday be destroyed – and we have conceptions of many of these objects in our heads – but we also have in those wonderful brains of ours, conceptions that were not created in the same way and which will never vanish in the same way either, strange objects that are simultaneously extremely private and also, potentially, extremely public and incredibly widely shared.

  The exceptions are people who have never learnt about them or find them difficult to learn, perhaps because it's an advanced topic in mathematics only understood by a few cognoscenti. That's why millions of players world wide share their understanding of the rules of chess and always play by those rules (occasional cheats excepted). It's also why mathematics can be so easily

  A deep philosophical error Philosophers have long distinguished between two basic types of objects (only), universals and particulars. Particulars, such as an apple or gate post, were no problem, and neither were the ideas of an apple or the idea of a gate post since everyone could agree that my idea of an apple might be subtly different from yours.

  Universals, such as the number 2 were a different matter. They seemed to exist independently of individual human minds, but how was this possible? The Greek philosopher Porphyry asked: ‘Are they beings with independent existence or do they exist only as human concepts?’

  The puzzle was: if the number 2 has an ‘independent existence’, then where is it? And if it does not have an existence independent of human minds, how can it be ‘the same’ in every mind? Surely the number 2 is the same for you as it is for me, and everyone else who is very basically numerate?

  According to the philosopher Bertrand Russell:

  The argument that 2 is mental requires that 2 should be essentially an existent. But in that case it would be particular, and it would be impossible for 2 to be in two minds, or in one mind at two times. Thus 2 must be in any case an entity, which will have being even if it is in no mind.

  [Russell 1903: 451]

  No wonder that, according to Russell,

  mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true,

  or that the number 2 is,

  a metaphysical entity about which we can never feel sure that it exists or that we have tracked it down.

  [Russell 1956: 542]

  Russell and other traditional philosophers have been mistaken because they have had no conception of the peculiar status of chess pieces and other game-like objects. They have never taken abstract games seriously as a subject for investigation and so have not realised that there are three fundamental kinds of objects, not two.

  [Wells 2010: Philosophy and Abstract Games]

  communicated. I once wrote the introduction to a book on aesthetics and mathematics by a Chinese author, Professor Wu Kai-Lang. Flipping the pages today, I grasp not a single Chinese character but I can read the mathematical statements because they use Arabic numerals and the usual symbols, a part of the universally shared language of mathematics.

  It seems then that there are three distinctly different kinds of objects, illustrated by these three examples: dreams

  mental and private

  apples

  physical and public

  the chess king

  mental and public

  The first type is difficult to communicate at all; the second we can talk about and describe pretty well but not perfectly; the third type we can communicate more-or-less perfectly, precisely because it was originally created in a mind.

  They are abstract

  No wonder, therefore, that chess is described as an abstract game and mathematics as an abstract activity. They both deal with situations in which all contingent features have been abstracted, stripped away.

  Chess is still an abstract game to players who cannot visualise the board and pieces mentally. (Individuals differ greatly in their ability to visualise.) You can tell this from the fact that the board and pieces can be many different sizes and shapes and made out of more-or-less any material. What's important for the play of the game is the structure of the board and the moves of the pieces: that chess once represented a battle between two armies no longer matters. The oriental game of Go (the Japanese name: it is called Wei-ch'i in Chinese) is also supposed to be a formalisation of conflict, but of guerilla warfare. Scott Boorman even wrote a book, The Protracted Game: a Wei-ch'i Interpretation of Maoist Revolutionary Strategy, but its original meaning matters not a jot when it is stripped down to a board divided by 19 lines in each direction plus two bowls full of black and white stones.

  They are difficult

  Many people find chess and maths difficult just because they are abstract and become more and more abstract as the player advances in understanding. There are also vast differences between the ability of the strongest players and the ordinary player. Even strong club players know that they are much weaker than International Masters who are much weaker than International Grandmasters who are much weaker than the small number of world-class players who are expected to win big international tournaments or even challenge for the world championship. Likewise, there are mathematical stars who shine brilliantly amongst the run of ordinary professional mathematicians.

  However, there is a crucial difference here between chess and maths. Although expert players of chess or Go see very deeply into positions and grasp subtle tactical and strategical ideas which are beyond the ordinary player, it is possible to help the ordinary reader appreciate their games by publishing them with annotations. The best annotations explain the strengths and weaknesses of each position, each player's strategical goals, and the tactical sequences which do or do not work.

  Even such annotations do not raise the ordinary player to grandmaster level – and world-class players will ‘see’ features of a position that even ordinary grandmasters will miss – but they do enable him or her to enjoy games played at even the highest levels.

  Unfortunately, this is not true of mathematics which is so abstract and ranges over so many different areas using so many technical terms and concepts that even professional mathematicians cannot hope to grasp more than a small portion of its vast landscape.

  There are, of course, many books of popular mathematics which entertain the reader and give some – faint – idea of the nature of advanced mathematics. We might say that they ‘annotate’ their themes – but no amount of annotation will enable the ordinary maths buff to really appreciate the proof by Andrew Wiles, for example, of Fermat's Last Theorem. The proof, first published in 1993, then finally completed in 1994, created a sensation yet it was only understood in depth by a few professional mathematicians familiar with particular areas of algebraic geometry and number theory, such as ‘the Taniyama–Shimura conjecture’ and ‘Iwasawa theory’.

  Also – unfortunately – tragically, even – the elementary mathematics that pupils learn in primary and secondary schools is not annotated either. It could be. It is perfectly possible to show pupils how and why each topic appears in their textbook and in their examination syllabus. It is possible, for example, to illustrate how so much of elementary mathematics is used in the sciences and indeed how it arose historically as scientists tried to answer questions about such everyday phenomena as floating bodies or soaring projectiles. It's possible but it almost never happens. To most pupils, the maths they learn in school, apart from arithmetic, could hardly be more abstract or more useless. No wonder that many of them are tempted to cry out, ‘What's the point of this? What's it FOR?’ without getting an answer [Wells 2008]. No wonder that maths in schools has such a bad reputation.

  Rules

  All you need for an abstract game are the rules. Choose the rules and everything else exists by virtue of the rules: so any board will do, if it fits the rules, and any shapes of pieces (in theory: in practice they are determined by convention and convenience.)r />
  The rules of a game can be changed provided everyone agrees – in fact children often change the rules of their playground games and families often play their own pet versions of MONOPOLY or CLUEDO, to the surprise of visitors invited to join in.

  The chosen rules have to be consistent with each other and with the goals of the game. For example, the game ought to come to an end. It is easy to invent a game which never ends. Indeed, Max Euwe, one-time world chess champion (1935–1937) and a mathematics teacher, proved that a chess game of infinite length was possible, given the rules of his time: the rule that a game is a draw if the same sequence of moves occurs three times in succession was not sufficient to prevent the possibility.

  The Nihon Kiin, the governing body of Japanese professional Go players, has more than once in the last century, several thousand years after the claimed origins of the game, made changes to the rules to allow for anomalous situations such as the triple ko (creating an endless repetition). Moreover, no one has proved that there are no more possible anomalous situations waiting to be discovered. The ancient rules of Go could be changed yet again in the twenty-first century.

  Hidden structures forced by the rules

  The complexity of an abstract game is created the moment the rules are laid down, instantly creating a rich miniature world, but their implications then have to be inferred which takes time and involves all three crucial aspects of mathematics: visual and mental perception, scientific exploration, and game-like calculation.

  Exploration leads – as it does in natural history and geography – to important structures and features being identified, named and classified, so that the game develops its own language. These structures make abstract games playable and mathematics manageable.

  Their existence also means that there are limitations on what you can do over the chess board. Every chess player is familiar with the scene in which a player gazing at a lost position exclaims plaintively, ‘There must be a move!’ but there isn't, the position is irretrievably kaput (with the qualifications that even master players do occasionally resign in salvable positions).

  It is this dialectic between the limitations forced by the rules and the imaginative creativity of the player that makes the best abstract games so extraordinarily rich and fascinating and attracts millions of players to games such as chess and Go.

  Argument and proof

  Proof can be a confusing idea. Scientists talk of proving this or that – that the universe started with a Big Bang, that the speed of light is constant, or that it is changing – but they actually prove nothing because it's always possible that a new chunk of evidence will be discovered next week that will undermine their theory. Scientists well know that the truth can be shocking and that they may get a personal shock tomorrow.

  In abstract games and puzzles we can prove many conclusions with total confidence and many more with a very high degree of confidence, and likewise in mathematics with the same qualifications. We'll meet examples later.

  Proof in a simple game Shirley Clarke wrote, describing two juniors playing a simple game, ‘They eventually “cracked it”, by convincing themselves that their strategy was a winning one, so that when they were asked, “Would you like to carry on playing?” they replied, “There's no point, it'll be boring.”’

  Figure 3.1 shows the very simple game: the players start bottom left, or any square on the left or bottom edges. The first player always moves to the right any number of squares and the second player moves towards the top edge, any number of squares. The player who reaches the top-right corner wins.

  What the young players realised after playing a number of experimental games was that the key squares all lie along the diagonal from bottom left to top right. In particular, they realised that any player who can move onto that diagonal, wins provided he or she does not later make an error. Once they had appreciated this fact, they also understood that the game had, as it were, been ‘spoiled’, it had lost its interest.

  Shirley Clarke concluded, ‘[this] theory about games seems to be true! The children started off by experimenting to test their theories but, in playing the game, it gradually dawned on them that there were mathematical reasons behind what was happening; they had moved on to the idea of proof’ [Wells & Clarke 1988: 4–5].

  Figure 3.1 Chess board and two dots marked

  Certainty, error and truth

  Mathematics has long been an example of perfection and absolute truth. What could be more certain than that 7 × 8 = 56 or that 13 + 17 = 30? Leaving aside smarty-pants responses on the lines of, ‘Ah! But I was using base 8! Tee hee!’ and focusing on the ordinary counting numbers, the answers are incomparably more certain than anything a scientist, let alone a lawyer, can claim.

  This certainty is rooted in the mathematics–abstract games analogy. It does not eliminate error but errors can usually be corrected, and vast swathes of mathematics are today, as far as we can tell, more-or-less error-free.

  Even when mistakes were found in the foundations of Euclid's geometry – his Elements has no axioms for betweenness, for example – Euclid's theorems did not collapse, in fact there was ‘no change’, partly because the ancient Greeks’ enthusiasm for drawing as many different ‘cases’ as possible was an excellent prophylactic against errors. Eighteenth-century mathematicians used divergent series carelessly (Chapter 9) and made mistakes – and some of their calculations were bizarre – but in the end divergent series were soundly understood and became a standard tool in the mathematician's toolbox.

  On the other hand, when mathematicians ‘step outside’ and pose questions about and use imprecise informal language, or when they create new mathematical games whose rules might be inconsistent, then they open the door to confusion and error. This is an especial danger when the concept of infinity appears. Here is one possible knot, first noticed by Galileo: 1

  2

  3

  4

  5

  6

  7

  8

  9

  10

  …

  1

  4

  9

  16

  25

  36

  49

  64

  82

  100

  …

  There are ‘obviously’ many more counting numbers than perfect squares because most of the counting numbers, starting with 2, 3, 5, 6, 7, 8,…are not perfect squares and yet, as the matching sequences illustrate, there is one perfect square for each counting number!

  It is actually quite easy to resolve Galileo's paradox by agreeing that, yes, if a set contains an infinite number of elements, then we can match a subset of its elements to itself. Here's another example: n

  1

  4

  9

  16

  25

  36

  49

  64

  81

  100

  …

  n3

  1

  8

  27

  64

  125

  216

  343

  512

  729

  1000

  …

  Clearly, ‘there are just as many cubes as there are squares’ though of course if we limit ourselves to numbers less than 1000000, or less than 101000000 for that matter, then that conclusion would be false.

  In contrast, argument in chess is either by analysis of possibilities (by calculation) which can become unmanageable after a few moves, or it is by human judgement which is fallible. Many facts about abstract games are true as soon as the rules are stated (assuming that they are consistent) but they can't be proved in a mathematical way, only checked by actually analysing the game. Mathematicians make judgements but usually expect to eventually prove or disprove them, even if there are many traps along the way and the longer the argument the more likely that it is flawed. Certainty, error and truth are tricky concepts in maths.

  Fortunately,
proofs, especially in new and poorly understood areas of mathematics, offer a great bonus. You not only convince yourself the theorem is true, which you probably believed already, but you are forced to use your imagination to create novel ideas which will be useful long after the proof has become familiar and taken for granted.

  Players make mistakes

  Chess players can all too easily make mistakes. They calculate wrongly or their beliefs about the resulting position are mistaken. Even the strongest players can show weakness of judgement. The leading British chess player after World War Two was C. H. O’D. Alexander. Playing the world title candidate David Bronstein in an Anglo-Russian radio match, Alexander made a mistake in a well-known opening by recapturing Bronstein's pawn with his knight rather than with his pawn. Bronstein's caustic comment was that ‘Every Russian schoolboy knows that you should recapture with the pawn.’

  Chess players hope, of course, that they will spot errors in their calculations or judgements during the game and before they have made their move, but this is often wishful thinking. Many mistakes in master games are only picked up when the games are analysed after the game in the analysis room – aided by the comments of spectators or maybe by computers – or when they are published with written annotations.

  Mathematicians also make mistakes although they hope (and expect) that errors will be corrected sooner rather than later. Ralph Boas, one-time editor of Mathematical Reviews, once claimed that most of the new results in the papers reviewed were true but about half the proofs offered were wrong [Hamming 1998: 649]!