by Wells, David
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.
Like many games, 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 1993: 49]
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.
The rise and fall of formality
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].
Aristotle 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.
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.
Religious ritual, games and mathematics
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.
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:
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.
[Culin 1975]
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.]:
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 ba
lance 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.].)
Formality and mathematics
Traditional religion is in decline in much of the developed West, even as mathematics is spreading into every nook and corner. More and more we find mathematicians spotting more-or-less formal aspects of Real Life which on closer examination, and with the application of a little abstraction, can be seen as mathematical. However you think, whatever work you do, whatever problems you are trying to solve, there is a high chance that behind the scenes, underneath the surface, there will be formal, game-like, mathematical structures, by grasping which you will understand the situation better.
It is no accident that mathematics has developed over the last few hundred years in parallel with the realisation that more and more of the world can been seen through mathematical lenses, to be explored scientifically and studied as formal and game-like, and this process is continuing.
The twentieth century was the century of physics and cosmology and chemistry which all depend deeply on mathematics. It has been claimed that this century will be the century of biology – a remarkable turnaround because biology used to be strikingly non-mathematical in contrast to the genuine ‘hard’ sciences. No longer. In the new world of ‘advanced’ biology, mathematics will be present because as biology has progressed it has become more and more apparent that mathematical structures – such as the helical structure of the DNA molecule – underlie biological organisms also.
Figure 20.1 is a face-on view of a molecule of Buckminsterfullerene, also known as the Buckyball or the football molecule, because modern footballs are made with the same surface pattern. It is one of a class of fullerenes, named after Buckminster Fuller, the visionary architect, designer and inventor, whose geodesic domes the molecule resembles. Harold Kroto, Richard Smalley and Robert Curl won the 1996 Nobel Prize for chemistry for discovering this extraordinary object.
Figure 20.1 Buckminsterfullerence molecule
The Buckyball is also linked to those very early mathematical objects, the Platonic solids. It is not completely regular, because some faces are pentagons while the rest are hexagons. However, all the vertices are identical, composed of 2 regular hexagons and a regular pentagon, and, naturally, the 90 edges, 32 faces and 60 vertices fit the Euler relation.
Regular hexagons themselves only fit together to cover a plane surface. In order to make the shape ‘bend round’ to form a closed surface, pentagons have been added. As it happens, there are an infinite number of closed surfaces, equivalent to the sphere, whose surfaces are made of hexagons and pentagons only, with three faces meeting at every vertex, but in every case, no matter how large or small the total number of faces, the number of pentagons is the same. A puzzle: what is that number? It is no surprise that the answer depends on the Euler relation.
Hidden mathematics
Scientists strive to reveal the underlying mathematical patterns in the world, to make them explicit – often in order to exploit them. There are many other patterns that are created by human beings who have not, however, stopped to ask questions about them – unlike mathematicians – but just used them for practical purposes, so that the underlying mathematics has only recently been appreciated. One example is knitting. Ironically, this extremely mathematical activity is usually the province of women and girls and is learned informally at home.
Aunt Mary did not learn to knit in her arithmetic lessons at school. She learned from her mother or other female relatives as girls have done throughout history not as an irksome task but as a common social activity which has a play-like aspect: it fills in time pleasantly and so it is re-creational. As laid out in books of knitting patterns, the instructions look extremely algebraic and frankly frightening to a non-expert. This is how to turn a heel:
1st row – K.12 [14, 16], sl.1, K.1, psso. K.1, turn.
2nd row – P.6 [6, 8], P.2 tog, P.1, turn.
3rd row – K. to last 6 [8, 8] sts., sl.1, K.1, psso, K.1, turn.
4th row – P. to last 6 [8, 8] sts., P.2 tog., P.1, turn.
5th row – K. to last 4 [6, 6] sts., sl.1, K.1, psso, K.1, turn.
6th row – P. to last 4 [6, 6] sts., P.2 tog., P.1, turn.
[Harris 1997: 220]
Good heavens! Now I can imagine what a page of ‘elementary’ algebra looks like to a beginning student. Mysterious, weird, and frightening, are three adjectives that come to mind. Yet to those ‘in the know’ who have been initiated into the secret, these instructions are clear, simple and easy to follow. Would that elementary mathematics were so clear to more of its students!
Anthropologists have noted that there are many practices in traditional societies that clearly have a mathematical aspect, though they are ignorant of Western-style mathematics. The mathematics educator Ubiratan D’Ambrosio introduced the term ethnomathematics to refer to just such practices and teachers sometimes introduce examples into their classrooms, such as the well-known sand drawings or sona, created by the Tchokwe of Angola.
As so often happens, the term ethnomathematics was at once interpreted differently by different writers – who sometimes had axes to grind. We can say, however, that most of the practices described as ethnomathematics have one feature in common: they are more concerned with problema than theorema, and do not involve asking questions about the practice. That is, they are more practical than theoretical, and so, plausibly, represent a pre-theoretical stage in the development of mathematics, just as chess was more-or-less pre-theoretical prior to Philidor in the eighteenth century.
Style and culture, style in mathematics
Although formality is everywhere in societies old and new, it appears in many disguises for which people have different preferences. If John enjoys traditional folk dancing and bell-ringing we cannot infer that he will enjoy the related mathematics of group theory. If Mary is an engineer we might be tempted to think that she will appreciate Fourier series which have so many practical applications – but she may use mathematics purely as an effective instrumental aid, and reserve her real enthusiasm for bridges and roofs.
We cannot even safely infer that if Peter is a mathematician he will appreciate – let's say – number theory. He may find it quite uninteresting and enthuse instead over differential equations. Just as players prefer different games – Edgar Allan Poe thought draughts much superior to chess – so chess players have different preferences, often linked to different ways of thinking.
‘The style is the man.’ In music, painting, architecture, fashion, reading, politics – you name it – people have their own preferences and styles, and so do mathematicians. Some think visually and geometrically (Klein and Atiyah), some think more algebraically (MacLane), some emphasise experiment while playing games brilliantly (Sylvester), some are adept at both pure and applied maths (Euler and Newton and Gauss), while others are one but not the other (Hardy the purist).
There is a historical factor here: Fermat and Euler and their many brilliant contemporaries were much like naturalists let loose for the first time into a wonderful forest, packed with exotic species: their descendants find the commoner and more brilliantly coloured creatures catalogued and locked up in zoos, and so have to search harder – but successfully – and with brilliant results.
Individual personal differences are easy to accept, if difficult for psychologists to explain. Racial differences are another matter. Felix Klein thought he could detect differences between French and German mathematicians. He claimed in 1893 that ‘a strong native space intuition seems to be an attribute of the Teutonic race, while the critical and purely logical sense is more developed in the Latin and Hebrew races’. At least Klein did not claim tha
t one ‘race’ was superior to the other – merely different. Jacques Hadamard in his book, The Psychology of Invention in the Mathematical Field, quoted Klein and then matched Klein's claim with counter-examples from French authors [Hardy 1946: 114].
Ludwig Bieberbach was a different case. Lecturing in Berlin under the Nazis, he claimed that nationality, blood and race influence the style of mathematics and that there were two types of mathematicians, the J-type, basically Germans, whose mathematics was superior to that of the S-type, who were basically Frenchmen and Jews. Hardy wrote a powerful letter to Nature, damning Bieberbach and regretting that he actually seemed to believe such nonsense. [Hardy 1934: 134].
Cultural influences are something else again. Continental mathematicians who followed Leibniz's superior notation and vocabulary for the calculus plausibly had a lasting advantage over British followers of Newton.
The simplest explanation for such cultural contrasts is transmission – students are taught differently by different teachers – but there are more complex explanations. It has been noted, for example, often by Americans themselves, that they are inclined to think in a mechanical manner: the very metaphors of checks and balances used to describe the Constitution have been interpreted as mechanical and a natural development from the original Newtonian model of the universe. Americans are happy to note also that this mechanical ‘genius’ has contributed to their culture in so many ways from Edison's brilliant inventions to their creative enthusiasm for computers and recent successes in Artificial Intelligence. This is clearly a question of culture since Americans are a smorgasbrod of immigrants from all over the world [Foley 1990].
