by Wells, David
Psychologists have studied the Aha! experience: the penny drops and you finally ‘see it’, whatever ‘it’ may be. This is closely related to the ‘kick’ that Hardy talked about and it depends on making connections and spotting surprising relationships, so it depends on structure, and especially on analogy which is especially beautiful, we might say, because you get two patterns for the price of one.
Another side of the same coin is that mystery creating a challenge can be beautiful – and when it is clarified it remains beautiful because it has now been illuminated – so once again the mathematician wins both ways.
So does the chess player. Tournament spectators watching a master game on giant boards may be mystified by a certain move or sequence but in the adjoining analysis room another master comments on the play and sooner or later the mystery is elucidated: if it is indeed brilliant rather than an error (even masters make mistakes) then the puzzle and its solution share a linked beauty.
A sense of mystery, however, depends on the mode of presentation. If Pythagoras’ theorem is displayed as a dry statement about right-angled triangles to pupils who lack the experience to find it surprising, the presentation will fail, but if it is used to predict the distance from one corner of the classroom to the opposite corner, pupils start by being surprised but, all being well, conclude by understanding as well.
Jim Henle has referred to Romantic mathematics as maths that has the qualities of ‘remoteness and strangeness’ and François Le Lionnais distinguished between Classical and Romantic beauty. We might well conclude that the beauty of the Mandelbrot set is Romantic.
Figure 19.1 The Mandelbrot set, with an internal point marked
Figure 19.2 The Julia set of the marked point
Figure 19.1 shows the Mandelbrot set. Figure 19.2 shows the Julia set of an internal point, in one piece. When a point is chosen outside the Mandelbrot set then its Julia set breaks down into separated pieces: the more distant the point chosen, the more broken-up the Julia set becomes, creating a beautiful and unique Fatou dust cloud for every external point (Figure 19.3).
Figure 19.3 Fatou dust [Diagram by John Sharp]
We might feel that any feeling of ‘remoteness and strangeness’ that we experience would be dissipated as soon as these objects were better understood: dissipated, no, transformed, yes, at least for those with a sense of history [Henle 1996: 21] [Lionnais, Le 2004].
Beauty and individual differences in perception
Beauty is in the eye of the beholder.
Mathematicians cannot escape this everyday cliché. The vast landscape of modern mathematics guarantees that most mathematicians will only be at home in a small corner, and must lack the deeper understanding needed to make very delicate aesthetic judgements about regions of which they know little.
‘Which Is the Most Beautiful?’ was a questionnaire for readers of the Mathematical Intelligencer [Wells 1988: 30–31] designed to test the plausible proposition that mathematicians no more agree on their judgements of beautiful mathematics than art lovers agree on favourite paintings or music lovers agree over Mozart and Beethoven. The items were chosen to be elementary, so that all respondents would be more-or-less familiar with all of them, (but see the comment on Ramanujan.)
It worked well! The introduction quoted several classic comments such as John von Neumann's that, ‘I think it is correct to say that [the mathematician's] criteria of selection, and also those of success, are mainly aesthetical,’ [von Neumann 1947: 2053] and Poincaré's comment on this phenomenon: ‘It is true aesthetic feeling which all mathematicians recognise. The useful combinations are precisely the most beautiful’ [Poincaré 1914: 59]. In other words, the beauty of mathematics is not an add-on, it's not a bonus, and attention to beauty is not an option but an essential feature of mathematical creativity.
The introduction also referred to theorems, proofs, concepts and strategies, because all these can be labelled beautiful. Readers were then given 24 theorems and asked to mark each of them from 0 to 10, the most beautiful getting the highest mark, and to add any comments they thought appropriate.
Responses from readers were predictable – they often agreed and often disagreed but – surprisingly, some of the disagreements were extreme. The Euler relationship, eiπ = −1 came top with an average of 7.7/10, but several readers thought it was too obvious and well-understood to be ranked highly.
Euler won again with his polyhedron formula, V + F = E + 2, which tied second with ‘The number of primes is infinite’ on 7.5. The greatest surprise, however, was at the other end of the rankings. Equal last was a partitions identity from the great Ramanujan which J. E. Littlewood only 25 years previously had described as of ‘supreme beauty’: p(n) is the number of partitions of n:
[Littlewood 1963: 85]
Had aesthetic taste changed so much, so quickly? Or was Littlewood, perhaps, idiosyncratic even then? There is no doubt tastes do change – but that fits von Neumann's and Poincaré's claims: mathematicians ought to appreciate the ‘beauty’ that is most relevant to their own work, and partition theory is very special. I sympathise with Tito Tonietti who claimed that, ‘Beauty, even in mathematics, depends on historical and cultural contexts, and therefore tends to elude numerical interpretation.’
When they came to explain their choices there was somewhat more agreement. Several thought that you couldn't separate a theorem from its proof or its context and motivating ideas, but just about everyone agreed that simplicity and brevity were a plus though even here David Singmaster marked down Fermat's theorem that every prime of the form 4n+1 is uniquely the sum of two integer squares because it has no simple proof, while I am inclined to mark it up because it is that much more mysterious and, I suppose, deep. Are the most beautiful creations short lyrics or longer symphonies? Might apparent depth merely be a sign that the theorem is not yet clearly understood? Simplicity contrasts also with surprise. 13+17=30 is incredibly simple but totally non-surprising, while Fermat's theorem is both simple and surprising.
The general versus the specific and contingent
Respondents did not highlight another difference which undoubtedly exists, as Freeman Dyson, a brilliant young mathematician who long ago became a world-class physicist, explains:
Unfashionable mathematics is mainly concerned with things of accidental beauty, special functions, particular number fields, exceptional algebras, sporadic finite groups. It is among these unorganized and undisciplined parts of mathematics that I would advise you to look for the next revolution in physics. They have a quality of strangeness, of unexpectedness. They do not fit easily into the smooth logical structures of Bourbaki. Just for that reason we should cherish and cultivate them, remembering the words of Francis Bacon, ‘There is no beauty that hath not some strangeness in the proportion.’
[Dyson 1983: 47]
Dyson could be proved right: there are currently believed to be deep connections between quantum mechanics, chaotic quantum systems, Riemann's zeta function, and random matrices.
Mathematicians have often found that progress lies through making sense of weird and strange phenomena – which are also surprising! – as we have already seen in the case of the complex ‘imaginary’ numbers.
Freeman Dyson continued: ‘The only thing these various discoveries had in common was a concrete, empirical, accidental quality, directly antithetical to the spirit of Bourbaki.’
Bourbaki was the pseudonym of a group of French mathematicians who promoted an extremely formal and abstract view of mathematics and ignored anything which didn't fit their pattern and their taste. Their perspective was very popular for a while and greatly influenced – for the worse – the school Modern Mathematics Movement of the 1960s but has since faded.
Paul Halmos highlights a similar contrast:
Stein's (harmonic analysis) and Shelah's (set theory) represent what seem to be two diametrically opposed psychological attitudes to mathematics…The contrast between the two can be described (inaccurately, but perhap
s suggestively) by the words special and general…Stein talked about singular integrals…[Shelah] said, early on, ‘I love mathematics because I love generality,’ and he was off and running, classifying structures whose elements were structures of structures of structures.
[Halmos 1987: 20]
It's a safe bet that Saharon Shelah, today a prolifically successful mathematician, professor at the Hebrew University of Jerusalem and at Rutgers University, is more sympathetic to Bourbaki.
On the other hand, even the most ‘exceptional’ objects such as the sporadic finite groups are known to have some connections with the rest of mathematics [Stillwell 1998]. We might leave the last word, not for the first time, with David Hilbert: ‘He who seeks for methods without having a definite problem in mind seeks in the most part in vain.’
Beauty, form and understanding
Beauty, like some mythical Greek goddess, guides mathematicians to insight and understanding which, as it deepens, changes their perception of the form of the original problem and highlights the temporal nature of their original aesthetic judgement, but not to their loss, because with new perceptions come new insights and problems. Here is an example.
Euler's relationship eiπ = −1 is a special case of his formula,
whence,
and so
Hence cos2h + sin2h = 1 the well-known formula which is equivalent to Pythagoras theorem – and which is extremely beautiful (to most mathematicians!) however you look at it. How do its attractions change, for better or for worse, if we write down the series for sin x and cos x,
then start to square them (you can never finish) and realise that the coefficient of every odd power of x in the product will be zero, and that the even powers in the product, beyond x0=1, start,
…and so on and so forth? This is a brute force calculation which is not in itself elegant. On the other hand, writing down the coefficients of x2n in the total you get a sequence of identities such as, from the x6 term,
plus similar identities for the higher coefficients, which are not obvious, so even this brutal calculation has suggested something – but in fact it suggests a possible conjecture too, that if a function which is represented by a power series is identically zero, then all its coefficients must be zero, and this happens to be a true theorem and a deep one.
Beauty depends on form but form also depends on beauty. The link is provided by the human brain's talent for spotting relationships which ensures that every step forward in understanding is accompanied by at least a pleasurable ‘kick’ and at best an overwhelming experience of euphoria. Different forms, however, display quite different properties and there is no guarantee that even the subtlest sense of beauty and sensitivity to patterns will necessarily allow us to resolve all the problems that we encounter.
20 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.
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.
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].
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.
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?
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.
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.
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.
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 th
en 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 play
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.
(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.)
