Mathematics and Beauty Paul Ernest unravels the complexity of the notion e are often confronted with complex and the signs as perceived. Some mathematicians have fascinating mathematics-based images in claimed that there are beautiful equations, such as W the media including television, magazines, eiπ + 1 = 0. In my view it is not the sign itself that is the books, newspapers, posters, films, internet, and so on. string of signifiers presented on the page that is judged These are often strikingly beautiful, for example multi- to be beautiful. It is rather the surprising relationship coloured pictures of fractals and complex tessellations. signified by the sign string. Thus, part of the public perception of mathematics is “It contains five of the most important numbers in that it can give rise to very beautiful images, in short, maths: 0, 1, e, i, and π, along with the fundamental that mathematics can be both beautiful and intriguing. concepts of addition, multiplication, and Consequently, many agree that beauty is central exponentiation - if that’s not beautiful, what is?” to mathematics. Certainly the claim that aspects of (The Institute of Mathematics and its Applications n.d.) mathematics are beautiful is often heard both from Since mathematical beauty is appreciated indirectly, it members of the public, and from mathematicians must be experienced cognitively, through reason, the themselves. The following quotations illustrate this. intellect, intuition, and affect (feelings), rather than as Hardy (1941: 13) writes: something presented by the senses. But if they are not externally sensed things, what are the mathematical “A mathematician, like a painter or a poet, is a objects that we may call beautiful? In mathematics maker of patterns.”, and we have propositions, theorems, concepts, methods, The mathematician’s patterns, like the painter’s proofs, theories, applications and models, and any or the poet’s, must be beautiful; the ideas, like of these might be termed beautiful. So the question I the colours or the words, must fit together in a ask myself is: Is it possible to specify criteria for what harmonious way. Beauty is the first test: there is no is beautiful in mathematics? The term ‘pleasing to the permanent place in the world for ugly mathematics. eye’ cannot be applied in the same sense as it can to (Hardy 1941: p 14) paintings, scenery, etc., and ‘pleasing to the mind’s eye’ is a metaphor that does not take us far towards an Hardy wrote more about mathematical beauty understanding of mathematical beauty. So, what makes than almost any other, certainly at the time he was something mathematical beautiful? publishing his views, and I will look in more detail at what he says about it later on. The most obvious source of beauty in mathematics is pattern, structure, and symmetry, as in art. Betrand Russell writes, originally in 1919: But mathematics is abstract and so the patterns Mathematics, rightly viewed, possesses not only must be abstract, and some of the features of the truth, but supreme beauty — a beauty cold and abstractedness itself add to the beauty of mathematics. austere, like that of sculpture, without appeal to any Such features might be said to include the expression part of our weaker nature, without the gorgeous of abstraction and generality, and the simplicity and trappings of painting or music, yet sublimely pure, economy of expression used. Another pleasing aspect and capable of a stern perfection such as only the of mathematics is surprise and ingenuity in reasoning, greatest art can show. The true spirit of delight, the and interconnections between ideas in mathematics exaltation, the sense of being more than Man, which can appear beautiful. The use of mathematical is the touchstone of the highest excellence, is to be modelling to capture aspects of the world can be found in mathematics as surely as poetry. (Russell breathtaking, and also demonstrates its power. Lastly 1986: 60) the rigour of reasoning in proofs is noted, for example in the above quotation from Bertrand Russell, as a thing As this quotation implies, the beauty of mathematics of ’cold and austere’ beauty. is not a response to something perceived through our sense organs, as with paintings, music or even Developing these ideas more fully leads me to propose landscapes. In such cases the appreciation of beauty, seven dimensions of mathematical beauty. These are as well as a response to what is given by the senses, as follows. involves the cognitive discernment of features such 1. Economy, simplicity, brevity, succinctness, as structure, and of course our appreciations is elegance socially conditioned. But in mathematics nothing but The compression of a formula or a theorem of the symbols, figures or other representations can be wide generality or an argument (proof) into a sensed. Mathematical beauty is regarded as something few short signs in mathematics is valued and deeper in the domain of meaning and not just that of admired. September 2015 www.atm.org.uk 23 Mathematics and Beauty 2. Generality, abstraction, power Obviously there is nothing magic about the number The breadth and scope of a generality or a proof seven here, and another person may be able to point is one of the key characteristics of mathematics to another dimension of mathematical beauty that and evokes appreciation. I have overlooked, or to offer different components that make clearer distinctions. So this is a provisional These first two criteria overlap somewhat, but in analysis I offer for discussion. The immediate question my view are distinct enough to justify listing them is how does this analysis of beauty in mathematics fit separately. with others peoples’ ideas? The most comprehensive 3. Surprise, ingenuity, cleverness account of beauty in mathematics is given by the Unexpectedness, like wit, is appreciated and mathematician Hardy (1940) who proposed six features valued when it reveals a new knowledge of a beautiful mathematical proof. connection, method or short cut in solving a According to Hardy, such a proof should be: problem. General: the idea is used in proofs of different kinds 4. Pattern, structure, symmetry, regularity, visual (this relates to 2 above, generality) design The discernment of pattern in its various and Serious: connected to other mathematical ideas abstracted forms is the closest the values (this relates to 6, inter-connectedness) of mathematics come to those of art and Deep: ‘strata’ of mathematical ideas (this does not general aesthetics in the visual field, although correspond exactly to any of the above dimensions) in mathematics these properties are largely abstract. Nevertheless, mathematics is the Unexpected: the argument takes a surprising form science par excellence for elucidating the (this corresponds to 3, surprise) meaning of structure and pattern. Inevitable: there is no escape from the conclusion 5. Logicality, rigour, tight reasoning and deduction, (this corresponds to 5, logicality and rigour) pure thought Economical (simple): there are no complications The development of logical reasoning to its of detail (this corresponds to 1, economy and ultimate forms of rigour and purity of thought is simplicity) a valued part of mathematics and the steps in a well constructed mathematical proof evoke Hardy’s six features are attributes of a beautiful proof, admiration like a gold necklace with well forged as opposed my seven more general dimensions of links. mathematical beauty which are intended to apply to the full range of mathematical objects and constructions, 6. Interconnectedness, links, unification including formulas, theorems, proofs and theories. The evidence of connections between different Even so, there is a close correspondence with my concepts and theories within mathematics is seven dimensions which partially validates them, since intellectually exciting and attractive. It combines Hardy is one of the greatest pure mathematicians of economy, generality, ingenuity and structure and the twentieth century. He is one of the few to discuss so it could be argued that it is reducible to these the nature of mathematical beauty, so his opinions first four dimensions of beauty. Or it can be seen are significant. However, there are three mismatches. as sufficiently valuable in its own right so as Hardy’s feature C, depth, is not one of the dimensions. to deserve independent listing, as I have done It concerns a linking to deep, that is, to general ideas, here. so it seems to me to correspond to a combination of 2, 7. Applicability, modelling power, empirical generality, and 6, inter-connectedness. generality Two of my seven dimensions are missing: 7, empirical Like metaphors in poetry the capture of empirical applicability and 4, pattern and structure. Although situations in mathematical models and more dimension 4 seems the most obvious and foremost generally in applied theories and concepts is dimension of mathematical beauty, it might not be seen something appreciated both within and outside to be as applicable to mathematical proofs as it is to mathematics as a demonstration of its power results, theories and for the lay person, mathematically and ‘unreasonable effectiveness’ in the physical inspired designs. However I will show a proof that world (Wigner 1960), as opposed to the world of I believe is beautiful in this way. Hardy might well pure mathematics. discount this surface beauty, and perhaps regard Elegance is sometimes given as a dimension of pattern and structure as beautiful when they fulfil his mathematical beauty on its own, but I think it is other criteria, such as his first three. However, my reducible to several other simpler descriptors in the conclusion is speculative. above list such as economy, generality and power. This The omission of dimension 7, empirical applicability, fits with the views of Montano (2014: 182) who writes; is unsurprising for two reasons. First of all, Hardy is “elegance is sometimes defined as the quality of describing the beauty of mathematical proofs.