Games and Mathematics (40 page)
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Authors: David Wells
20
Origins: formality in the everyday world
The rise and fall of formality
Religious ritual, games and mathematics
Origins: formality in the everyday world
The veins of a leaf, a meandering river, a butterfly wing, knots
in vines and creepers, ripples in water, crystals in rocks, the stripes on a tiger – patterns in nature are too striking to be missed. Indeed, the simplest
must
be recognised because the human brain has been designed by evolution to automatically pick out, for example, patterns of bars and stripes. The earliest cave art includes rudimentary patterns and many tribal people today decorate their own bodies with complex designs which illustrate their aesthetic sense.
in vines and creepers, ripples in water, crystals in rocks, the stripes on a tiger – patterns in nature are too striking to be missed. Indeed, the simplest
must
be recognised because the human brain has been designed by evolution to automatically pick out, for example, patterns of bars and stripes. The earliest cave art includes rudimentary patterns and many tribal people today decorate their own bodies with complex designs which illustrate their aesthetic sense.
The invention of weaving not only made it easier for early humans to live in colder and inhospitable climates, it enabled the development of more complex designs. The sophisticated Celtic ‘knots’ combine horizontal and vertical repetitions from weaving with the over-and-under pattern of knots which sailors long since turned into a decorative art form as well as an essential practical technique.
Aristotle
wrote that beauty was composed of order, proportion and exactness [
Metaphysics
XIII, M, iii] so it is no surprise those features are found in traditional architecture or that Leonardo da Vinci wrote, ‘Let no one who is not a mathematician read my works.’ The arts in the Christian West have always been linked to mathematics and proportion.
wrote that beauty was composed of order, proportion and exactness [
Metaphysics
XIII, M, iii] so it is no surprise those features are found in traditional architecture or that Leonardo da Vinci wrote, ‘Let no one who is not a mathematician read my works.’ The arts in the Christian West have always been linked to mathematics and proportion.
The verbal arts show similar features. Ancient literature starts with poetry which exploits patterns in sound and rhythm, and patterns in meaning, that is, figures of speech, in order to both entertain and teach.
Traditional dances are extremely formal, and music, related to dance, is constructed on patterns of scales, harmonious chords and rhythmic sounds. According to Leibniz (1646–1716), ‘[Music] is the pleasure the human soul experiences from counting without being aware that it is counting.’
Music, especially drumming, is also associated with military drill which is highly formal unlike the bloody messiness of actual battles. The Greeks’ games
at Olympia included the
Apobates
, in which armed contestants dismounted and then remounted a moving chariot, making a formal competitive game out of a military skill [Reed
1998
].
at Olympia included the
Apobates
, in which armed contestants dismounted and then remounted a moving chariot, making a formal competitive game out of a military skill [Reed
1998
].
Bell ringing is another ancient custom. The changes and peals are related to mathematical
group theory
, though of course that theory was developed long after the practical activity, just as modern analyses of chess follow centuries of play.
group theory
, though of course that theory was developed long after the practical activity, just as modern analyses of chess follow centuries of play.
The theatre has always been linked to music and dance, and poetry, and so to game-likeness and formality, as well as to strong emotions and declamation. Church services in all religions are more-or-less formal with rituals – closely linked to theatre – that have to be followed exactly to be efficacious.
If you have the misfortune to become involved in a law suit, then you will meet formality, and even ritual, again. You cannot behave as a you choose in a court of law: the occasion is too serious and justice demands that rules and procedures be followed precisely. This poses problems with language, because everyday language is far from formal and game-like: life and death, freedom or imprisonment can hinge on the meanings of the terms used. Were the words spoken intended as a threat to kill the defendant? Did the contract
bind
the signatories to repair the roof? Did the words used constitute a slander, or were they merely common abuse? Did the term ‘rotating’ necessarily refer specifically to the client's invention?
bind
the signatories to repair the roof? Did the words used constitute a slander, or were they merely common abuse? Did the term ‘rotating’ necessarily refer specifically to the client's invention?
The ancient art of rhetoric already treated language as a game-like medium in which the antagonists exploited moves and counter-moves and deliberately constructed an argument to be simultaneously elegant and convincing – and those two factors could not be separated in practice. So it is no surprise that Aristotle
connected the aims and rules of debating with his study of the logical syllogism.
connected the aims and rules of debating with his study of the logical syllogism.
Law also hinges on logic, even as lawyers deploy all the persuasive resources of rhetoric to turn their audience. Philosophers’ ‘laws of logic’ are very formal indeed, so much so that they are seldom if ever seen, in all their bare abstraction, either in courts of law or in everyday life, but arguments may be accepted or dismissed in a court because they are
illogical
. The law and the legal systems are highly, but imperfectly, formalised and there is a continual struggle to maximise clarity and game-likeness.
illogical
. The law and the legal systems are highly, but imperfectly, formalised and there is a continual struggle to maximise clarity and game-likeness.
Business offices demand a degree of formality as well as providing a theatre for game-like manipulation, role-playing, and ‘playing games’ with people, famously analysed by Eric Berne in his book,
Games People Play
[
1960
]. The very idioms, ‘What's your game?’, ‘What are you playing at?’ suggest that we are often well aware of the game-like in our everyday lives, and savvy social players plan their moves and think ahead.
Games People Play
[
1960
]. The very idioms, ‘What's your game?’, ‘What are you playing at?’ suggest that we are often well aware of the game-like in our everyday lives, and savvy social players plan their moves and think ahead.
Every society known to anthropologists has such formal features, because we naturally order our surroundings and try to order each other. The world of early man was potentially chaotic, threatening, mysterious and hard to understand. No wonder he tried to control it with more-or-less rigid and formal rituals or that he was attracted to patterns and designs.
The more formal the feature, the closer we are to ‘defining it in the head’. Traditional dances can be described very accurately in language, though there is no perfect substitute for dancing them. Street games are also close to complete formality, as are musical tunes which can become totally formal when written down with a suitable notation – though the element of feeling and spontaneity is then lost. Religious rituals are performed more-or-less identically at ceremonies separated widely in time and space.
Much of everyday life, however, is extremely informal and not at all game-like so anyone who overemphasises abstract games as a metaphor will get into trouble. We have to take into account both the formal and the informal.
Abstract board games are at one end of a continuum from informality to formality, where we also find mathematics. Plato
was wrong to locate mathematics in some weird inaccessible realm of
forms
: he should have looked at the children playing in the street or playing with their
tali
(knuckle bones) or watched the
Apobates
competing, listened to the musicians, and attended ceremonies in the Greek law courts and the temples: he might then have seen that mathematics is only one extreme of that very formality which is an essential aspect of our social lives together, and that
play
is an essential aspect of mathematics.
The psychology of playwas wrong to locate mathematics in some weird inaccessible realm of
forms
: he should have looked at the children playing in the street or playing with their
tali
(knuckle bones) or watched the
Apobates
competing, listened to the musicians, and attended ceremonies in the Greek law courts and the temples: he might then have seen that mathematics is only one extreme of that very formality which is an essential aspect of our social lives together, and that
play
is an essential aspect of mathematics.
Play
comes from Old English
plegian
meaning exercise, recreation, also to perform music, with similar connotations of a social activity: but social events, like children's games, are never entirely spontaneous, rather they are constrained by custom and habit, as well as by creative ingenuity.
comes from Old English
plegian
meaning exercise, recreation, also to perform music, with similar connotations of a social activity: but social events, like children's games, are never entirely spontaneous, rather they are constrained by custom and habit, as well as by creative ingenuity.
(The word
game
was originally from the Old English
gamen
meaning, joy, fun, amusement, derived from the Gothic word referring to people coming together, suggesting communion and participation.)
game
was originally from the Old English
gamen
meaning, joy, fun, amusement, derived from the Gothic word referring to people coming together, suggesting communion and participation.)
The Romantics associated play with spontaneity, but children know better. Play always tends both towards spontaneity and formality. From bouncing a ball repeatedly against a wall, to jumping from rock to rock at the sea side, building a sandcastle, or flying a kite, the theme is as much goal-and-control as spontaneous creativity, and should other children join in, as in traditional street
games, then it is natural to agree informal rules without which the game would at once degenerate.
games, then it is natural to agree informal rules without which the game would at once degenerate.
Like many games,
Oranges and Lemons
was accompanied by a verse, a formal feature, which the players sang:
Oranges and Lemons
was accompanied by a verse, a formal feature, which the players sang:
‘Oranges and Lemons’ say the bells of St. Clements,‘You owe me five farthings’ say the bells of St. Martins,‘When will you pay me?’ say the bells of Old Bailey,‘I do not know’ says the great bell of Bow,Here comes a candle to light you to bed,And here come a chopper to chop off your head!
The bells referred to old London churches. To play, two children decided in secret which one will be ‘oranges’, the other being ‘lemons’. They formed an arch and the other children danced through the arch while singing the song. When the last line was reached, the two children dropped their arms and captured the child in the arch, who then whispered either ‘oranges’ or ‘lemons’ and was sent to stand behind the appropriate child, until every child had been captured.
The rules are there to be followed, which isn't to say that children don't change them. They do, and street games appear in many variants. Overhear children adapting the rules of a game and you are hearing them ‘creating formality’, an activity that comes naturally to them. Games also easily die out, unfortunately, under pressure from TV and other modern day ‘attractions’.
Adult games are even more formal, not least because they include the convention that the referee or umpire is always ‘right’. Whether the ball crossed the line may be physically doubtful, but the referee's final decision makes it certain one way or the other, by convention, and the game-like aspect of the situation is confirmed. Televised replays do not make the referee redundant, but simply give him, or her, more information to aid their judgement.
In games, the players can also fantasise, which is why many adults are disturbed by the enthusiasm of modern children for fantasy games in which characters kill and are killed, or even rape and mutilate.
A psychologist might defend play and fantasy as defences against the child's anxieties – critics would answer that realising certain fantasies can be self-destructive – but either way we might ponder the claim that mathematicians play while they are being creative. The late Gian-Carlo Rota wrote of John von Neumann, who was certainly grandiose with an unusual personality, that he also suffered from anxieties:
Like everyone who works with abstractions, von Neumann needed constant reassurance against deep-seated and recurring self-doubts.
Rota did not elaborate, unfortunately, because there is a hint here that abstraction itself is, or can be, a defence against anxiety, even perhaps that mathematics itself can be a defence, as the philosopher A. N. Whitehead put it, against ‘the goading contingency of events’.
Formality in society has certainly had the historical function of aiding control and suppressing disorder and potential anarchy, so it is no great surprise if mathematics and abstract games such as chess can perform a similar function for individuals. Nor is it a surprise that games are used today in psychiatry and psychotherapy. Games represent one pole of our behaviour, formal, rule-bound and contained, in contrast to the opposite extreme of complete spontaneity, and yet in many circumstances we can express ourselves more creatively within a formal setting – when we struggle to create within the rules – than when we are given total freedom.
Ironically, as mathematics grows explosively and is exploited by all the hard sciences and now the soft sciences also, formality in the wider society is decaying. We still distinguish the formal from the informal – the formal letter applying for a job, an informal letter thanking a friend for an invitation, a formal conversation at an interview but an informal conversation with the local butcher – but the formality that was such a natural feature of hierarchical societies and which kept the classes apart while at the same time oiling the wheels of social intercourse, is no longer needed in an egalitarian culture.
Technology has played a role also. Children still play street games but they are dying out under the pressure of modern entertainment, including computer games.
Conversation in formal situations used to follow conventional patterns. The French minister Louis Joxe expressed the convention of the time that: ‘Conversation is a game. If you must explain the rules to someone, it is difficult to play’ [Rothschild, de
1968
: 111].
1968
: 111].
Aristotle
claimed that,
claimed that,
life also includes relaxation, and one form of relaxation is playful conversation. Here, too, we feel that there is a certain standard of good taste in social behaviour, and a certain propriety in the sort of things we say and in our manner of saying them.[
Nicomachean Ethics
IV, 14, 1128a]
Today, conversation, like dance, is more informal, though students of language may analyse conversation as a game of moves and counter-moves which can
be competitive, and we know that we can get into trouble by saying the wrong thing at the wrong time.
be competitive, and we know that we can get into trouble by saying the wrong thing at the wrong time.
In a world of estates and orders, of ranks and hierarchy, rules were taken for granted and the well-educated man or woman behaved with
decorum
in every situation. Rules and regulations guarded against disorder and chaos, even as they promoted a parallel emphasis on order and reason. In much earlier societies there was an especially strong link between religious ritual and formal games.
decorum
in every situation. Rules and regulations guarded against disorder and chaos, even as they promoted a parallel emphasis on order and reason. In much earlier societies there was an especially strong link between religious ritual and formal games.
We live in a fun-filled world, as Martha Wolfenstein foresaw more than fifty years ago in her paper,
The Emergence of Fun Morality
[Wolfenstein 1958] and we expect games to be fun too. In more primitive societies, games were serious and linked to ritual, religion and the sacred.
The Emergence of Fun Morality
[Wolfenstein 1958] and we expect games to be fun too. In more primitive societies, games were serious and linked to ritual, religion and the sacred.
Harold Murray in
The History of Board-games other than Chess
described lined boards used in Ceylon which were charms against evil spirits as well as boards for games of alignment. He also thought that race games played by native American Indians were essentially religious [Murray
1913/1952
: 236, 234]. The American anthropologist Stewart Culin who studied native Indian tribes, linked divination to the origin of board-games:
The History of Board-games other than Chess
described lined boards used in Ceylon which were charms against evil spirits as well as boards for games of alignment. He also thought that race games played by native American Indians were essentially religious [Murray
1913/1952
: 236, 234]. The American anthropologist Stewart Culin who studied native Indian tribes, linked divination to the origin of board-games:
Upon comparing the games of civilized people with those of primitive society many points of resemblance are seen to exist, with the principal difference that games occur as amusements or pastimes among civilized men, while among savage and barbarous people they are largely sacred and divinatory. This naturally suggests a sacred and divinatory origin for modern games, a theory, indeed, which finds confirmation in their traditional associations, such as the use of cards in telling fortunes.
The anthropologist Wim van Binsberge has linked ritual divination and board games directly to ‘formal models’ and mathematics. He points out that divination and board games can be abstractly defined, and are ‘relatively impervious to individual alteration’ [Van Binsberge
n.d.
]:
n.d.
]:
Both consist in a drastic modelling of reality, to the effect that the world of everyday experience is very highly condensed, in space and in time…both the board-game and the divination rite may refer to real-life situations the size of a battle field, a country, a kingdom or the world, and extending over much greater expanses of time…than the duration of the session…Divination and boardgames constitute a manageable miniature version of the world…Utterly magical, board-games and divination systems are space-shrinking time-machines.
Van Binsberge then notes that this representation works both ways: the divination or board game is a model of real life, but the results of the divination are, ‘subsequently fed back into real life, through information and skill gained, through prestige redistributed, personal balance and motivation restored, fears explicitly named and confronted…’
Divination and board games also introduce oppositions such as odd and even, and elementary counting (for example in mancala) and often geometrical elements too. Finally, van Binsberge observes that board games and divination, like mathematics, cross historical-geographical boundaries. (This account is largely based on Van Binsberge [
n.d.
].)
n.d.
].)
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