Paul Erest uravels the complexity of the otio W e are ofte cofroted with complex ad fasciatig mathematics-based images i the media icludig televisio, magazies, books, ewspapers, posters, films, iteret, ad so o. These are ofte strikigly beautiful, for example multicoloured pictures of fractals ad complex tessellatios. Thus, part of the public perceptio of mathematics is that it ca give rise to very beautiful images, i short, that mathematics ca be both beautiful ad itriguig. Cosequetly, may agree that beauty is cetral to mathematics. Certaily the claim that aspects of mathematics are beautiful is ofte heard both from members of the public, ad from mathematicias themselves. The followig quotatios illustrate this. Hardy (1941: 13) writes: A mathematicia, like a paiter or a poet, is a maker of patters., ad The mathematicia s patters, like the paiter s or the poet s, must be beautiful; the ideas, like the colours or the words, must fit together i a harmoious way. Beauty is the first test: there is o permaet place i the world for ugly mathematics. (Hardy 1941: p 14) Hardy wrote more about mathematical beauty tha almost ay other, certaily at the time he was publishig his views, ad I will look i more detail at what he says about it later o. Betrad Russell writes, origially i 1919: Mathematics, rightly viewed, possesses ot oly truth, but supreme beauty a beauty cold ad austere, like that of sculpture, without appeal to ay part of our weaker ature, without the gorgeous trappigs of paitig or music, yet sublimely pure, ad capable of a ster perfectio such as oly the greatest art ca show. The true spirit of delight, the exaltatio, the sese of beig more tha Ma, which is the touchstoe of the highest excellece, is to be foud i mathematics as surely as poetry. (Russell 1986: 60) As this quotatio implies, the beauty of mathematics is ot a respose to somethig perceived through our sese orgas, as with paitigs, music or eve ladscapes. I such cases the appreciatio of beauty, as well as a respose to what is give by the seses, ivolves the cogitive discermet of features such as structure, ad of course our appreciatios is socially coditioed. But i mathematics othig but the symbols, figures or other represetatios ca be sesed. Mathematical beauty is regarded as somethig deeper i the domai of meaig ad ot just that of the sigs as perceived. Some mathematicias have claimed that there are beautiful equatios, such as +1 = 0. I my view it is ot the sig itself that is the strig of sigifiers preseted o the page that is judged to be beautiful. It is rather the surprisig relatioship sigified by the sig strig. It cotais five of the most importat umbers i maths: 0, 1, e, i, ad π, alog with the fudametal cocepts of additio, multiplicatio, ad expoetiatio - if that s ot beautiful, what is? (The Istitute of Mathematics ad its Applicatios.d.) Sice mathematical beauty is appreciated idirectly, it must be experieced cogitively, through reaso, the itellect, ituitio, ad affect (feeligs), rather tha as somethig preseted by the seses. But if they are ot exterally sesed thigs, what are the mathematical objects that we may call beautiful? I mathematics we have propositios, theorems, cocepts, methods, proofs, theories, applicatios ad models, ad ay of these might be termed beautiful. So the questio I ask myself is: Is it possible to specify criteria for what is beautiful i mathematics? The term pleasig to the eye caot be applied i the same sese as it ca to paitigs, sceery, etc., ad pleasig to the mid s eye is a metaphor that does ot take us far towards a uderstadig of mathematical beauty. So, what makes somethig mathematical beautiful? The most obvious source of beauty i mathematics is patter, structure, ad symmetry, as i art. But mathematics is abstract ad so the patters must be abstract, ad some of the features of the abstractedess itself add to the beauty of mathematics. Such features might be said to iclude the expressio of abstractio ad geerality, ad the simplicity ad ecoomy of expressio used. Aother pleasig aspect of mathematics is surprise ad igeuity i reasoig, ad itercoectios betwee ideas i mathematics ca appear beautiful. The use of mathematical modellig to capture aspects of the world ca be breathtakig, ad also demostrates its power. Lastly the rigour of reasoig i proofs is oted, for example i the above quotatio from Bertrad Russell, as a thig of cold ad austere beauty. Developig these ideas more fully leads me to propose seve dimesios of mathematical beauty. These are as follows. 1. Ecoomy, simplicity, brevity, succictess, elegace The compressio of a formula or a theorem of wide geerality or a argumet (proof) ito a few short sigs i mathematics is valued ad admired. September 015 3
. Geerality, abstractio, power The breadth ad scope of a geerality or a proof is oe of the key characteristics of mathematics ad evokes appreciatio. These first two criteria overlap somewhat, but i my view are distict eough to justify listig them separately. 3. Surprise, igeuity, cleveress Uexpectedess, like wit, is appreciated ad valued whe it reveals a ew kowledge coectio, method or short cut i solvig a problem. 4. Patter, structure, symmetry, regularity, visual desig The discermet of patter i its various ad abstracted forms is the closest the values of mathematics come to those of art ad geeral aesthetics i the visual field, although i mathematics these properties are largely abstract. Nevertheless, mathematics is the sciece par excellece for elucidatig the meaig of structure ad patter. 5. Logicality, rigour, tight reasoig ad deductio, pure thought The developmet of logical reasoig to its ultimate forms of rigour ad purity of thought is a valued part of mathematics ad the steps i a well costructed mathematical proof evoke admiratio like a gold ecklace with well forged liks. 6. Itercoectedess, liks, uificatio The evidece of coectios betwee differet cocepts ad theories withi mathematics is itellectually excitig ad attractive. It combies ecoomy, geerality, igeuity ad structure ad so it could be argued that it is reducible to these first four dimesios of beauty. Or it ca be see as sufficietly valuable i its ow right so as to deserve idepedet listig, as I have doe here. 7. Applicability, modellig power, empirical geerality Like metaphors i poetry the capture of empirical situatios i mathematical models ad more geerally i applied theories ad cocepts is somethig appreciated both withi ad outside mathematics as a demostratio of its power ad ureasoable effectiveess i the physical world (Wiger 1960), as opposed to the world of pure mathematics. Elegace is sometimes give as a dimesio of mathematical beauty o its ow, but I thik it is reducible to several other simpler descriptors i the above list such as ecoomy, geerality ad power. This fits with the views of Motao (014: 18) who writes; elegace is sometimes defied as the quality of beig pleasigly simple yet effective. It is icluded above with ecoomy ad its syoyms as this seems to be its primary meaig. Obviously there is othig magic about the umber seve here, ad aother perso may be able to poit to aother dimesio of mathematical beauty that I have overlooked, or to offer differet compoets that make clearer distictios. So this is a provisioal aalysis I offer for discussio. The immediate questio is how does this aalysis of beauty i mathematics fit with others peoples ideas? The most comprehesive accout of beauty i mathematics is give by the mathematicia Hardy (1940) who proposed six features of a beautiful mathematical proof. Accordig to Hardy, such a proof should be: Geeral: the idea is used i proofs of differet kids (this relates to above, geerality) Serious: coected to other mathematical ideas (this relates to 6, iter-coectedess) Deep: strata of mathematical ideas (this does ot correspod exactly to ay of the above dimesios) Uexpected: the argumet takes a surprisig form (this correspods to 3, surprise) Ievitable: there is o escape from the coclusio (this correspods to 5, logicality ad rigour) Ecoomical (simple): there are o complicatios of detail (this correspods to 1, ecoomy ad simplicity) Hardy s six features are attributes of a beautiful proof, as opposed my seve more geeral dimesios of mathematical beauty which are iteded to apply to the full rage of mathematical objects ad costructios, icludig formulas, theorems, proofs ad theories. Eve so, there is a close correspodece with my seve dimesios which partially validates them, sice Hardy is oe of the greatest pure mathematicias of the twetieth cetury. He is oe of the few to discuss the ature of mathematical beauty, so his opiios are sigificat. However, there are three mismatches. Hardy s feature C, depth, is ot oe of the dimesios. It cocers a likig to deep, that is, to geeral ideas, so it seems to me to correspod to a combiatio of, geerality, ad 6, iter-coectedess. Two of my seve dimesios are missig: 7, empirical applicability ad 4, patter ad structure. Although dimesio 4 seems the most obvious ad foremost dimesio of mathematical beauty, it might ot be see to be as applicable to mathematical proofs as it is to results, theories ad for the lay perso, mathematically ispired desigs. However I will show a proof below that I believe is beautiful i this way. Hardy might well discout this surface beauty, ad perhaps regard patter ad structure as beautiful whe they fulfil his other criteria, such as A, B ad C. However, my coclusio is speculative. The omissio of dimesio 7, empirical applicability, is usurprisig for two reasos. First of all, Hardy is describig the beauty of mathematical proofs. These are primarily pure mathematics productios so this dimesio is ot applicable. Secod, Hardy is well kow as a purist, ad for regardig utility as ugly, so 4 September 015
it ulikely that he would regards empirical applicability as a feature of mathematical beauty uless it is a by product of depth (C), or geerality (A). Aesthetic appreciatio is ultimately irratioal i that it caot be reduced to or replaced by ratioal aalysis ad logical reasoig. Such approaches ca, however, provide a partial illumiatio of its compoets, as I have tried to do here. It the fial aalysis aesthetic appreciatio depeds o the positive resposes ad the feeligs of humas, which i their tur give rise to preferred choices ad actios. These eed ot be purely subjective ad totally idiosycratic, as that which is regarded as beautiful is sometimes shared withi ad possibly across cultures, ad will be leared to some extet. Eve so, there are major differeces i what is cosidered to be beautiful betwee workers i differet parts of mathematics, such as pure mathematicias, applied mathematicias ad statisticias. Without eve lookig at sub-divisios, research by Iglis ad Aberdei (015) foud very widespread differeces i the aesthetic appraisal of proofs amog a large sample of mathematicias. This cofirms that although all mathematicias agree that mathematics ca be beautiful, there are sigificat differeces i mathematicias opiios of what is beautiful i mathematics. So, it is hard to dey that there is a strog subjective elemet i mathematicias judgemets ad opiios o mathematical beauty. A example I order to make this discussio more cocrete here is a example which I believe exhibits mathematical beauty. This example draws o the proof that the sum of the first atural umbers, that is the sequece 1,, 3,..., is ( + 1) /. The stadard elemetary proof ivolves the followig key step, see Table 1, the summig of pairs of algebraic terms, each totallig + 1: Table 1: The key step i the elemetary derivatio of the formula ( + 1) / 1 3 - - 1-1 - 3 1 + + 1 + 1 + 1 + 1 + 1 + 1 Figure 1 shows a small relief by the artist Joh Erest illustratig the structure of this proof (Erest 008). How does this illustrate the proof? The figure ca be metally divided ito 6 vertical zoes, three o the left of the cetral, vertical dividig lie ad three o its right. There is a correspodece betwee the idividual terms i the compoud algebraic sum i Table 1 ad elemets i these zoes i Figure 1. I the latter, small black squares represet uits, black solid areas uder the horizotal dividig lie o left had side ad above it o the right had side implicitly represet black squares, ad small white squares represet egative umbers. The figure illustrates the beautiful symmetry betwee the matchig first three ad the last three terms i the geeral series beig summed, exhibitig a rotatioal symmetry of order. But there is also a ear reflective symmetry about the horizotal ad vertical axes, if the complemetary colours ad some other mior details are discouted. The figure brigs out these pleasig symmetric ad structural features of the proof step, as ca be discered i Table 1. It ca be asked whether the use of the complemetary colours black ad white i Figure 1 is a purely artistic flourish or whether it serves the mathematical proof idea. Udoubtedly the artist preferred the use of complemetary colours for aesthetic artistic reasos, because of the dramatic cotrasts i the fial work. However, the use of cotrastig colours also serves the mathematics, sice it emphasizes the differece betwee the sequece of growig umbers (black o white) ad the sequece of dimiishig umbers (white o black) ad without it, it would be hard to illustrate the all-importat differece betwee positive ad egative umbers i the proof. Aother feature of the plae relief show i fig. 1, ideed the mai feature of the proof, is the fact that the algebraic sums i Table 1 are mirrored. The first colum (workig from left to right) shows + 1 black squares, the ext shows + 1 + (1-1), the third shows + 1 + ( - ), ad so o, thus illustratig some of the algebraic details of the proof. Workig right to left, the same sequece is show i reverse but i complemetary colours, ad with reversed vertical positioig, thus providig the rotatioal symmetry of the figure. Thus it ca be said that the artistic work brigs out ad emphasizes dimesio 4, beauty of the patter, structure ad symmetry of the proof step show i Table 1. It would have bee possible for the artist to make a simpler plae relief as i Figure, ad still illustrated the proof. Figure 1: The sum of the first atural umbers September 015 5
Figure : Simplified illustratio of derivatio of formula summig the first atural umbers Figure offers a alterative ad simpler visualisatio of the derivatio, but i skippig some of the illustrated algebraic complexities of the proof it also leaves out some of the visual complexity of the costructio which add to its appeal ad beauty. Of course this may be a matter of persoal judgemet, but it also sacrifices some of the algebraic complexity ad detail show i the mathematical proof ad Figure 1. Although the artist chose to make his work as i Figure 1, some viewers might like the simplified Figure which also illustrates the proof, equally well, or eve better, demostratig the subjectivity of judgemets of beauty. It is possible to simplify the proof still further as is show i Figure 3. Figure 3 Sum of the first atural umbers This further simplificatio removes the separatio of the zoes metioed above. The outcome is a attractive abstract patter that still has rotatioal symmetry of order, as well as ear reflective symmetries through ear diagoals of the figure. However, the figure o loger serves to illustrate the proof, for the colums merge ito a overall patter that does ot brig out the features of the proof step i Table 1. It is possible to simplify the proof still further as is show i Figure 4. Figure 4: Sum of the first 4 atural umbers This is ow so simplified that it o loger illustrates the proof, but just shows the proof idea i its basic cocrete form. That is, it shows that doublig a triagle or triagular umber (i this case the 4 th oe, amely 10) makes it ito a rectagle or rectagular umber (4 x 5 = 0), whose area is more easily calculated (legth x height) ad double that of the triagular umber. While Figure 4 still has some beauty, it has lost the geerality ad much of the complexity that make Figures 1 ad so appealig. However, it still maifests igeuity ad patter although it sacrifices abstractio ad geerality. It is a figure that has bee used i elemetary teachig to commuicate the idea of how a fiite sequece of atural umbers ca be summed. Goig back to the proof illustrated i Table 1 ad Figure 1, there are further aesthetic aspects beyod the structural features oted above, i particular the igeuity ad cleveress of the proof. By takig the sum 1 +... + ad reversig it, ad combiig the two rows, the actual colum additios ivolved are sidestepped, sice there is a costat sum, itroducig brevity. This features i the well kow story of the mathematicia Gauss i elemetary school. He is claimed to have summed the umbers 1 to 100 i a few secods usig this logic. Irrespective of its autheticity, this story is widely told to stress the teacher s surprise at Gauss s igeuity ad cleveress, dimesio 3, surprise ad igeuity, i discoverig a short ad elegat,dimesio 1, brevity, solutio method despite his youth (Boyer 1989). Aother pleasig aspect of the proof is its geerality ad power, dimesio, applyig to the first umbers for ay. Also, the derived formula itself exhibits ecoomy ad simplicity, dimesio 1, brevity. Thus the elemetary derivatio of the formula for the first umbers discussed here illustrates four of the proposed dimesios of beauty: patter ad symmetry, geerality, brevity ad igeuity. I additio, the illustratio of the proof idea i Figure 1 also displays itercoectedess, dimesio 6, betwee visual ad algebraic aspects of mathematics, because of the structural aalogy betwee the algebra, Table 1, ad the diagram Figure 1. This algebraic ad spatial itercoectedess is already implicit i the proof, Table 1, because the spatial dispositio of the symbols is a ecessary part of the argumet. Of course, the elemetary derivatio of the formula ( + 1) / show is oe used oly at school level. A more rigorous derivatio employs mathematical iductio, such as i the followig deductio. 6 September 015
1 The basis of the iductio is 1 i = 1 The iductive hypothesis is ( + 1) 1 i = The iductio step is 1 + 1 = ( + 1) 1 i + ( + 1) = + ( + 1) = ( + 1) ( + ) thus provig the formula. However this form of the proof usig mathematical iductio although more rigorous loses the arguably more beautiful step show i Table which both explais as well as validates the formula (Hersh 1993), albeit at a more elemetary level. It ca be argued that the proof by mathematical iductio has the appeal of rigour, dimesio 5, as well as the brevity, geerality ad surprise of the simpler proof. However, may studets are ucoviced by it because the result proved, amely ( + 1) 1 i =, is assumed as a hypothesis withi the proof. This is a deductive fallacy i all proofs except those usig mathematical iductio. Feelig cheated, or bamboozled, is ot a positive aesthetic respose, although it should be a temporary feelig util the priciple of mathematical iductio is fully appreciated ad uderstood. Mathematical iductio is a difficult idea ad method (Erest 1984), ad it took me persoally several years as a teeager to fully get it. I claimig that there is wide agreemet that some mathematical kowledge ad objects are beautiful, I am ot proposig that this appreciatio is itrisic or ecessary. We acquire may of our values, like our kowledge, from our participatio ad immersio i social groups ad cultures. Eve despite such shared immersio ad iflueces as Iglis ad Aberdei (015), show, there remai sigificat divergeces i mathematicias views of beauty. The oe dimesio of mathematical beauty that has ot bee illustrated i the above example is that of, modellig ad applicability. This ca illustrated i may ways such as through what is probably the most famous equatio of all time, amely Eistei s E=mc. This law iterrelates measures of eergy ad mass, ad makes the startlig assertio that a uit of mass is equivalet to c uits of eergy, where c is the speed of light. The equatio has simplicity ad geerality, However, its overwhelmig power ad beauty comes from the fact that it quatifies the vast eergies that are released i uclear explosios or other uclear reactios, ad that these predictios have bee repeatedly demostrated i the physical world. A simple mathematical equatio ecapsulates cataclysmic forces i the real world. The mai example discussed i this paper is that of the sum of the first atural umbers, both the proof step show i Table 1, ad its illustratio i artistic represetatios, icludig Figures 1,, 3 ad 4. Followig this discussio it is legitimate to ask whether I have bee explorig the aesthetics of the proof, or its artistic represetatio? Based o the above text, the aswer must be both. Although aalysig artistic represetatios shifts the discussio away from the aesthetics of mathematics, this excursio has eabled me to isolate ad explore the dimesios of beauty i the proof itself. Some dimesios of mathematical beauty such as dimesio 4, patter ad structure, are perhaps more evidet i the artistic represetatios tha i the proof itself. However, the key elemet of dimesio 4 utilised here is that of symmetry, ad this is evidet i the proof itself, Table 1. All the other dimesios remarked o i the example, 1 ecoomy, geerality, 3 igeuity ad 6 itercoectedess, stem from the proof itself ad ot from its artistic represetatio. So I coclude that cosideratio of the artistic represetatios has ot led me away or detracted from my mai focus of this paper, amely the aalysis ad exemplificatio of the dimesios of beauty i mathematics. Paul Erest, the Uiversity of Exeter QR Code Refereces Boyer, C. B. (1989) A History of Mathematics (secod editio, revised by U. C. Merzbach), New York: Wiley. Erest, P. (1984) Mathematical Iductio : A Pedagogical Discussio, Educatioal Studies i Mathematics Vol. 15 (1984): 173-189. Erest, P. (008) Joh Erest, A Mathematical Artist, The Philosophy of Mathematics Educatio Joural 4. Hardy, G. H. (1941) A Mathematicia s Apology, Cambridge: Cambridge Uiversity Press. Hersh, R. (1993) Provig is Covicig ad Explaiig, Educatioal Studies i Mathematics, 4 (4) 389-399. Istitute of Mathematics ad its Applicatios, The (.d.) A beautiful equatio, The Istitute of Mathematics ad its Applicatios cosulted o August 013 at <http://www.mathscareers.org.uk/viewitem.cfm?cit_ id=38931>. Iglis, M. & Aberdei, A. (015) Diversity i Proof Appraisal, I Press. Paper preseted at Mathematical Cultures Coferece 3, Lodo Mathematical Society, Lodo, April 014. Motao, U. (014) Explaiig Beauty i Mathematics: A Aesthetic Theory of Mathematics, Switzerlad: Spriger. Russell, B. (1986) Mysticism ad logic. Lodo: Uwi Paperbacks Wiger, E. P. (1960) The ureasoable effectiveess of mathematics i the physical scieces, reprited i Saaty, T. L. ad Weyl, F. J. Eds. (1969) The Spirit ad Uses of the Mathematical Scieces, New York: McGraw-Hill, 13-140. September 015 7