Games and Mathematics (8 page)
Read Games and Mathematics Online
Authors: David Wells
They are difficult
Rules
Hidden structures forced by the rules
Argument and proof
Certainty, error and truth
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.
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.
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.
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’.
. 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.
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.
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.)
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.)
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.
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.
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.
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).
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.
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.
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 gameShirley 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
Chess board and two dots marked
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.
'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
:
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!
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 | … |
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:
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 10
1000000
for that matter, then that conclusion would be false.
n | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | … |
n 3 | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1000 | … |
1000000
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.
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.
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