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.
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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. These being pleasingly simple yet effective”. are primarily pure mathematics productions so this dimension is not applicable. Second, Hardy is well It is included above with economy and its synonyms as known as a purist, and for regarding utility as ugly, so this seems to be its primary meaning.
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it is unlikely that he would regard empirical applicability How does this illustrate the proof? The figure can be as a feature of mathematical beauty unless it is a by mentally divided into 6 vertical ‘zones’, three on the product of depth, or generality. left of the central, vertical dividing line and three on its right. There is a correspondence between the individual Aesthetic appreciation is ultimately irrational in that it terms in the compound algebraic sum in Table 1 and cannot be reduced to or replaced by rational analysis elements in these ‘zones’ in Figure 1. In the latter, small and logical reasoning. Such approaches can, however, black squares represent units, black solid areas under provide a partial illumination of its components, as I the horizontal dividing line on left-hand side and above have tried to do here. It the final analysis aesthetic it on the right-hand side implicitly represent n black appreciation depends on the positive responses and squares, and small white squares represent negative the feelings of humans, which in their turn give rise numbers. The figure illustrates the beautiful symmetry to preferred choices and actions. These need not be between the matching first three and the last three purely subjective and totally idiosyncratic, as that which terms in the general series being summed, exhibiting a is regarded as beautiful is sometimes shared within and rotational symmetry of order 2. But there is also a near possibly across cultures, and will be learned to some reflective symmetry about the horizontal and vertical extent. Even so, there are major differences in what is axes, if the complementary colours and some other considered to be beautiful between workers in different minor details are discounted. The figure brings out parts of mathematics, such as pure mathematicians, these pleasing symmetric and structural features of the applied mathematicians and statisticians. Without proof step, as can be discerned in Table 1. even looking at sub-divisions, research by Inglis and Aberdein (2015) found very widespread differences It can be asked whether the use of the complementary in the aesthetic appraisal of proofs among a large colours black and white in Figure 1 is a purely artistic sample of mathematicians. This confirms that flourish or whether it serves the mathematical proof although all mathematicians agree that mathematics idea. Undoubtedly the artist preferred the use of can be beautiful, there are significant differences complementary colours for aesthetic artistic reasons, in mathematicians’ opinions of what is beautiful in because of the dramatic contrasts in the final work. mathematics. So, it is hard to deny that there is a strong However, the use of contrasting colours also serves subjective element in mathematicians’ judgements and the mathematics, since it emphasises the difference opinions on mathematical beauty. between the sequence of growing numbers (black on white) and the sequence of diminishing numbers (white An example on black) and without it, it would be hard to illustrate the In order to make this discussion more concrete here all-important difference between positive and negative is an example which I believe exhibits mathematical numbers in the proof. beauty. This example draws on the proof that the sum Another feature of the plane relief shown in figure 1, of the first n natural numbers, that is the sequence indeed the main feature of the proof, is the fact that 1, 2, 3, ..., n is n(n + 1)/ . 2 the algebraic sums in Table 1 are mirrored. The first The standard elementary proof involves the following column (working from left to right) shows n + 1 black key step, see Table 1, the summing of n pairs of squares, the next shows n + 1 + (1 - 1), the third shows algebraic terms, each totalling n + 1: n + 1 + (2 - 2), and so on, thus illustrating some of the algebraic details of the proof. Working right to Table 1: The key step in the elementary derivation of left, the same sequence is shown in reverse but in the formula n(n + 1)/ 2 complementary colours, and with reversed vertical 1 2 3 … n - 2 n - 1 n positioning, thus providing the rotational symmetry of the figure. n n - 1 n - 2 … 3 2 1 + n + 1 n + 1 n + 1 … n + 1 n + 1 n + 1 Thus it can be said that the artistic work brings out and emphasizes dimension 4, beauty of the pattern, Figure 1 shows a small relief by the artist John Ernest structure and symmetry of the proof step shown in illustrating the structure of this proof (Ernest 2008). Table 1. It would have been possible for the artist to make a simpler plane relief as in Figure 2, and still illustrated the proof.
Figure 1: The sum of the first n natural numbers
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Figure 4: Sum of the first 4 natural numbers This is now so simplified that it no longer illustrates the proof, but just shows the proof idea in its basic concrete form. That is, it shows that doubling a triangle or triangular number (in this case the 4th one, namely 10) makes it into a rectangle or rectangular number (4 x 5 = 20), whose area is more easily calculated (length x height) and double that of the triangular Figure 2: Simplified illustration of derivation of formula number. While Figure 4 still has some beauty, it has lost summing the first n natural numbers the generality and much of the complexity that make Figures 1 and 2 so appealing. However, it still manifests Figure 2 offers an alternative and simpler visualisation ingenuity and pattern although it sacrifices abstraction of the derivation, but in skipping some of the illustrated and generality. It is a figure that has been used in algebraic complexities of the proof it also leaves out elementary teaching to communicate the idea of how a some of the visual complexity of the construction which finite sequence of natural numbers can be summed. add to its appeal and beauty. Of course this may be a matter of personal judgement, but it also sacrifices Going back to the proof illustrated in Table 1 and some of the algebraic complexity and detail shown in Figure 1, there are further aesthetic aspects beyond the mathematical proof and Figure 1. Although the artist the structural features noted above, in particular the chose to make his work as in Figure 1, some viewers ingenuity and cleverness of the proof. By taking the might like the simplified Figure 2 which also illustrates sum 1 + ... + n and reversing it, and combining the the proof, equally well, or even better, demonstrating two rows, the n actual column additions involved are the subjectivity of judgements of beauty. sidestepped, since there is a constant sum, introducing brevity. This features in the well known story of the It is possible to simplify the ‘proof’ still further as is mathematician Gauss in elementary school. He is shown in Figure 3. claimed to have summed the numbers 1 to 100 in a few seconds using this logic. Irrespective of its authenticity, this story is widely told to stress the teacher’s surprise at Gauss’s ingenuity and cleverness, dimension 3, surprise and ingenuity, in discovering a short and elegant, dimension 1, brevity, solution method despite his youth (Boyer 1989). Another pleasing aspect of the proof is its generality and power, dimension 2, applying to the first n numbers for any n. Also, the derived formula itself exhibits economy and simplicity, dimension 1, brevity. Thus the elementary derivation of the formula for the first n numbers discussed here illustrates four of the Figure 3 Sum of the first n natural numbers proposed dimensions of beauty: pattern and symmetry, generality, brevity and ingenuity. In addition, the This further simplification removes the separation of the illustration of the proof idea in Figure 1 also displays ‘zones’ mentioned above. The outcome is an attractive interconnectedness, dimension 6, between visual abstract pattern that still has rotational symmetry of and algebraic aspects of mathematics, because of order 2, as well as near reflective symmetries through the structural analogy between the algebra, Table 1, near diagonals of the figure. However, the figure no and the diagram Figure 1. This algebraic and spatial longer serves to illustrate the proof, for the columns interconnectedness is already implicit in the proof, merge into an overall pattern that does not bring out the Table 1, because the spatial disposition of the symbols features of the proof step in Table 1. is a necessary part of the argument. It is possible to simplify the ‘proof’ still further as is Of course, the elementary derivation of the formula n(n + 1) shown in Figure 4. /2 shown is one used only at school level. A more rigorous derivation employs mathematical induction, such as in the following deduction.
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1 The basis of the induction is ∑ 1 i = 1 its artistic representation? Based on the above text, the answer must be both. Although analysing artistic n n(n + 1) The inductive hypothesis is ∑ i = 1 2 representations shifts the discussion away from the The induction step is aesthetics of mathematics, this excursion has enabled n me to isolate and explore the dimensions of beauty ∑ n + 1 =∑ i + (n + 1) = n(n + 1) + (n + 1) = (n + 1) (n + 2) 1 1 2 2 in the proof itself. Some dimensions of mathematical beauty such as dimension 4, pattern and structure, are … thus proving the formula. perhaps more evident in the artistic representations However this form of the proof using mathematical than in the proof itself. However, the key element of induction although more rigorous loses the arguably dimension 4 utilised here is that of symmetry, and more beautiful step shown in Table 2 which both this is evident in the proof itself, Table 1. All the other explains as well as validates the formula (Hersh 1993), dimensions remarked on in the example, 1 economy, albeit at a more elementary level. 2 generality, 3 ingenuity and 6 interconnectedness, stem from the proof itself and not from its artistic It can be argued that the proof by mathematical representation. So I conclude that consideration of induction has the appeal of rigour, dimension 5, as well the artistic representations has not led as the brevity, generality and surprise of the simpler me away or detracted from my main focus proof. However, many students are unconvinced by it of this paper, namely the analysis and because the result proved, namely exemplification of the dimensions of beauty n n(n + 1) in mathematics. ∑ 1 i = , 2 is assumed as an hypothesis within the proof. This is a deductive fallacy in all proofs except those Paul Ernest, the University of Exeter using mathematical induction. Feeling cheated, or bamboozled, is not a positive aesthetic response, although it should be a temporary feeling until the References principle of mathematical induction is fully appreciated and understood. Mathematical induction is a difficult Boyer, C. B. (1989) A History of Mathematics (second idea and method (Ernest 1984), and it took me edition, revised by U. C. Merzbach), New York: Wiley. personally several years as a teenager to fully ‘get it’. Ernest, P. (1984) ‘Mathematical Induction : A In claiming that there is wide agreement that some Pedagogical Discussion’, Educational Studies in mathematical knowledge and objects are beautiful, Mathematics Vol. 15 (1984): 173-189. I am not proposing that this appreciation is intrinsic or Ernest, P. (2008) ‘John Ernest, A Mathematical Artist’, necessary. We acquire many of our values, like our The Philosophy of Mathematics Education Journal 24. knowledge, from our participation and immersion in social groups and cultures. Even despite such shared Hardy, G. H. (1941) A Mathematician’s Apology, immersion and influences as Inglis and Aberdein Cambridge: Cambridge University Press. (2015), show, there remain significant divergences in Hersh, R. (1993) Proving is Convincing and Explaining, mathematicians’ views of beauty. Educational Studies in Mathematics, 24 (4) 389-399. The one dimension of mathematical beauty that has Institute of Mathematics and its Applications, The (n.d.) not been illustrated in the above example is that of, ‘A beautiful equation’, The Institute of Mathematics modelling and applicability. This can illustrated in and its Applications consulted on 22 August 2013 at many ways such as through what is probably the
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