Are these the most beautiful? Are T h e s e the M o s t Beautiful? D a v i d W e l l s In the Fall 1988 Mathematical Intelligencer (vol. 10, no. 4) (11) readers were a s k e d to evaluate 24 theorems, o n a scale from 0 to 10, for b e a u t y . I received 76 c o m p l e t e d ques- (12) tionnaires, i n c l u d i n g 11 f r o m a p r e l i m i n a r y v e r s i o n (plus 10 extra, n o t e d below.) O n e p e r s o n a s s i g n e d each t h e o r e m a score of 0, (13) w i t h the comment, " M a t h s is a tool. Art h a s b e a u t y " ; that r e s p o n s e w a s excluded from the averages listed below, as w a s a n o t h e r that a w a r d e d v e r y m a n y zeros, (14) four w h o left m a n y blanks, a n d two w h o a w a r d e d nu- m e r o u s 10s. The 24 t h e o r e m s are listed below, o r d e r e d b y their average score from t h e remaining 68 responses�9 Rank (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Theorem Average d ~' = - 1 7.7 Euler's formula for a polyhedron: 7.5 V + F = E + 2 The n u m b e r of primes is infinite. 7.5 There are 5 regular polyhedra. 7.0 1 1 1 1 + ~ + ~ + ~ + . . . = "rr2/6. 7.0 A c o n t i n u o u s m a p p i n g of the 6.8 closed u n i t disk into itself has a fixed point. There is no rational n u m b e r w h o s e 6.7 square is 2. ~r is transcendental. 6.5 Every plane m a p can be coloured 6.2 with 4 colours. Every prime n u m b e r of the form 6.0 4n + 1 is the s u m of t w o integral squares in exactly o n e way. (15) The order of a s u b g r o u p divides 5.3 t h e order of the group. A n y square matrix satisfies its 5.2 characteristic equation. A regular icosahedron inscribed in 5.0 a regular o c t a h e d r o n divides the e d g e s in the G o l d e n Ratio. 1 1 4.8 2 x 3 x 4 4 x 5 x 6 1 + 6 x 7 x 8 , r r - 3 �9 " 4 If the points of the plane are each 4.7 coloured red, yellow, or blue, THE MATHEMATICAL INTELLIGENCER VOL. 12, N O . 3 �9 1990 Springer-Verlag New York 3 7 there is a pair of points of the s a m e colour of m u t u a l distance unity. (16) The n u m b e r of partitions of an 4.7 integer into o d d integers is equal to the n u m b e r of partitions into distinct integers. (17) Every n u m b e r greater than 77 is 4.7 the s u m of integers, the s u m of w h o s e reciprocals is 1. (18) The n u m b e r of representations of 4.7 an o d d n u m b e r as the s u m of 4 squares is 8 times the s u m of its divisors; of an e v e n number, 24 times the s u m of its o d d divisors. (19) There is n o equilateral triangle 4.7 w h o s e vertices are plane lattice points. (20) At a n y party, there is a pair of 4.7 p e o p l e w h o h a v e the same n u m b e r of friends present. Write d o w n the multiples of r o o t 2, ignoring fractional parts, a n d u n d e r n e a t h write the n u m b e r s missing from the first sequence. 1 2 4 5 7 8 9 1 1 1 2 3 6 10 13 17 20 23 27 30 The difference is 2n in the nth place. The w o r d p r o b l e m for g r o u p s is unsolvable. (21) 4.2 (22) 4.1 (23) The m a x i m u m area of a 3.9 quadrilateral with sides a , b , c , d is [(s - a ) ( s - b ) ( s - c ) ( s - d)] w, w h e r e s is half the perimeter. 5 [ ( 1 - - X 5 ) ( 1 - - x l O ) ( I - - X 1 3 . . �9 15 (24) [(1 - x)(1 - x2)(1 - x3)(1 - x4)... 16 3.9 = p(4) + p(9)x + p(14)x a + . . . . w h e r e p ( n ) is the n u m b e r of partitions of n. The following c o m m e n t s are divided into themes. Unattributed q u o t e s are from r e s p o n d e n t s . T h e m e 1: A r e T h e o r e m s B e a u t i f u l ? T o n y Gardiner a r g u e d that " T h e o r e m s aren't usually 'beautiful'. It's the ideas a n d p r o o f s that a p p e a l , " and r e m a r k e d of the t h e o r e m s he h a d n o t s c o r e d , " T h e r e s t are h a r d to s c o r e - - e i t h e r b e c a u s e t h e y a r e n ' t really beautiful, h o w e v e r important, or b e c a u s e the formulation given gets in the w a y . . . . " Several re- s p o n d e n t s disliked j u d g i n g t h e o r e m s . ( H o w m a n y r e a d e r s did n o t reply for such reasons?) Benno A r t m a n n w r o t e "for m e it is impossible to j u d g e a 'pure fact' "; this is consistent with his interest in Bourbaki and the axiomatic d e v e l o p m e n t of struc- tures. T h o m a s Drucker: " O n e d o e s n o t have to be a Rus- sellian to feel that m u c h of mathematics has to d o w i t h deriving consequences from a s s u m p t i o n s . As a result, a n y ' t h e o r e m ' cannot b e i s o l a t e d from the a s s u m p - tions u n d e r w h i c h it is d e r i v e d . " G e r h a r d Domanski: " S o m e t i m e s I find a p r o b l e m m o r e beautiful than its solution. I find also b e a u t y in m a t h e m a t i c a l i d e a s or c o n s t r u c t i o n s , s u c h as t h e Turing machine, fractals, twistors, and so on . . . . The ordering of a w h o l e field, like the w o r k of Bourbaki �9 . . is of great b e a u t y to m e . " R. P. Lewis writes, ' ( 1 ) . . . I a w a r d 10 points n o t so m u c h for the equation itself as for Complex Analysis as a w h o l e . ' To w h a t extent w a s the g o o d score for (4) a v o t e for the b e a u t y of the Platonic solids themselves? T h e m e 2: S o c i a l Factors Might s o m e votes have g o n e to (1), (3), (5), (7), a n d (8) b e c a u s e t h e y are ' k n o w n ' to b e beautiful? I am suspi- cious t h a t (1) received so m a n y scores in the 7 - 1 0 range. This w o u l d surprise me, because I suspect that m a t h e m a t i c i a n s are m o r e i n d e p e n d e n t t h a n m o s t p e o p l e [13] of others' opinions. (The ten extra forms referred to a b o v e came from Eliot Jacobson's s t u d e n t s in his n u m b e r theory course that emphasises the role of b e a u t y . I n o t e d that t h e y gave no zeros at all.) T h e m e 3: C h a n g e s i n A p p r e c i a t i o n o v e r T i m e There w a s a notable n u m b e r of l o w scores for the high rank theorems�9 Le Lionnais has one explanation [7]: "Euler's formula e i~' = - 1 establishes w h a t a p p e a r e d in its time to be a fantastic connection b e t w e e n the m o s t i m p o r t a n t n u m b e r s in mathematics . . . It w a s g e n e r a l l y c o n s i d e r e d 'the m o s t b e a u t i f u l formula of mathematics' . . . T o d a y the intrinsic reason for this c o m p a t i b i l i t y has b e c o m e so o b v i o u s that the s a m e formula n o w seems, if n o t insipid, at least entirely nat- u r a l . " Le Lionnais, u n f o r t u n a t e l y , d o e s n o t qualify " n o w s e e m s " b y asking, "'to w h o m ? " H o w d o e s j u d g m e n t c h a n g e w i t h time? B u r n s i d e [1], r e f e r r i n g to % g r o u p w h i c h is . . . a b s t r a c t l y e q u i v a l e n t to t h a t of t h e p e r m u t a t i o n s of f o u r s y m b o l s , " wrote, "in the latter form the problem pre- s e n t e d w o u l d to m a n y minds b e almost repulsive in its n a k e d f o r m a l i t y . . . " Earlier [2], perspective projection was, "'a process o c c a s i o n a l l y r e s o r t e d to b y g e o m e t e r s of o u r o w n country, b u t generally e s t e e m e d . . , to b e a species of 3 8 THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990 'geometrical trickery', b y which, 'our notions of ele- gance or geometrical purity m a y be violated . . . . ' " I a m sympathetic to Tito Tonietti: "'Beauty, even in m a t h e m a t i c s , d e p e n d s u p o n historical a n d cultural contexts, and therefore t e n d s to elude numerical inter- pretation." Compare the psychological concept of habituation. Can a n d do mathematicians deliberately u n d o such ef- fects b y placing themselves empathically in t h e posi- tion of the original discoverers? G e r h a r d Domanski w r o t e o u t the entire question- n a i r e b y h a n d , e x p l a i n i n g , " A s I w r o t e d o w n t h e t h e o r e m I tried to r e m e m b e r t h e feelings I h a d w h e n I first h e a r d of it. In this w a y I gave the scores." Theme 4: Simplicity and Brevity N o criteria are m o r e often associated with b e a u t y than simplicity a n d brevity. M. Gunzler w i s h e d (6) h a d a simpler proof. David Halprin wrote "'the b e a u t y that I find in mathematics �9 . . is m o r e to be f o u n d in the clever and/or succinct w a y it is p r o v e n . " D a v i d S i n g m a s t e r m a r k e d (10) d o w n somewhat, because it does n o t have a simple proof. I feel that this indicates its d e p t h a n d m a r k it u p accordingly. Are there no s y m p h o n i e s or epics in the world of beautiful proofs? Some chess players prefer t h e ele- g a n t simplicity of t h e e n d g a m e , others appreciate the complexity of t h e m i d d l e game. Either way, pleasure is derived from t h e reduction of complexity to sim- plicity, b u t the p r e f e r r e d level of complexity differs from player to player. Are mathematicians similarly varied? Roger P e n r o s e [10] a s k e d w h e t h e r a n u n a d o r n e d square grid was beautiful, or was it too simple? He c o n c l u d e d that h e p r e f e r r e d his non-periodic tessella- tions. But the question is a good one. H o w simple can a beautiful entity be? Are easy t h e o r e m s less beautiful? O n e r e s p o n d e n t m a r k e d d o w n (11) a n d (20) for being "too e a s y , " a n d (22) for being "'too difficult." David Gurarie m a r k e d d o w n (11) a n d (1) for being too simple, a n d a n o t h e r r e s p o n d e n t r e f e r r e d to t h e o r e m s t h a t a r e t r u e b y v i r t u e of t h e definition of their terms, w h i c h could have b e e n a dig at (1). T h e o r e m (20) is extraordinarily simple b u t m o r e t h a n a quarter of t h e r e s p o n d e n t s scored it 7 + . Theme 5: Surprise Y a n n i s H a r a l a m b o u s w r o t e : " a b e a u t i f u l t h e o r e m m u s t be surprising a n d deep. It m u s t provide y o u with a n e w vision o f . . . m a t h e m a t i c s , " a n d m e n t i o n e d the p r i m e n u m b e r t h e o r e m (which was by far t h e most popular suggestion for t h e o r e m s that o u g h t to h a v e b e e n i n c l u d e d in the quiz). R. P. Lewis: "(24) is top of m y list, because it is sur- prising, n o t r e a d i l y g e n e r a l i z a b l e , a n d difficult to prove. It is also i m p o r t a n t . " (12 + in the margin!) J o n a t h a n Watson criticised a lack of novelty, in this sense: "(24), (23), (17) . . . s e e m to tell u s little that is n e w about the concepts that a p p e a r in t h e m . " Penrose [11] qualifies Atiyah's suggestion "that ele- gance is m o r e or less s y n o n y m o u s with simplicity" b y d a i m i n g that " o n e should say that it has to do w i t h unexpected simplicity." Surprise a n d novelty are expected to provoke e m o - tion, o f t e n pleasant, b u t also o f t e n negative. N e w styles in p o p u l a r a n d h i g h c u l t u r e h a v e a n o v e l t y value, albeit temporary. As usual there is a psycholog- ical connection. H u m a n beings d o not r e s p o n d to just a n y stimulus: t h e y do t e n d to r e s p o n d to novelty, sur- p r i s i n g n e s s , incongruity, a n d complexity. But w h a t h a p p e n s w h e n t h e novelty wears off? Surprise is also associated with mystery. Einstein asserted, " T h e m o s t beautiful t h i n g w e can experience is t h e mysterious. It is the source of all true art a n d science." But w h a t h a p p e n s w h e n the m y s t e r y is re- solved? Is t h e b e a u t y t r a n s f o r m e d into a n o t h e r beauty, or m a y it evaporate? I i n c l u d e d (21) a n d (17) because t h e y initially mysti- fied a n d surprised me. At second sight, (17) remains so, a n d scores quite highly, but (21) is at most pretty. (How do mathematicians t e n d to distinguish b e t w e e n beautiful a n d pretty?) Theme 6: Depth Look at t h e o r e m (24). Oh, come o n now, Ladies a n d G e n t l e m e n ! Please! I s n ' t this difficult, d e e p , sur- prising, a n d s i m p l e relative to its s u b j e c t matter?! What m o r e d o y o u want? It is q u o t e d by Littlewood [8] in his r e v i e w of R a m a n u j a n ' s collected w o r k s as of "'supreme b e a u t y . " I w o n d e r e d w h a t readers w o u l d think of it: b u t I n e v e r s u p p o s e d that it w o u l d rank last, with (19), (20), a n d (21). R. P. L e w i s i l l u s t r a t e d t h e v a r i e t y of r e s p o n s e s w h e n h e suggested that a m o n g t h e o r e m s n o t included I could h a v e c h o s e n " M o s t of R a m a n u j a n ' s work,'" adding, "'(21) is pretty, but easy to prove, a n d not so d e e p . " D e p t h s e e m s n o t so i m p o r t a n t to r e s p o n d e n t s , w h i c h m a k e s m e feel that m y interpretation of d e p t h m a y be idiosyncratic. I was surprised that t h e o r e m (8), w h i c h is s u r e l y d e e p , ranks b e l o w (5), to w h i c h Le L i o n n a i s ' s a r g u m e n t m i g h t a p p l y , b u t (8) has n o simple proof�9 Is simplicity that important? (18) also scored poorly. Is it n o longer deep or diffi- cult? Alan Laverty a n d Alfredo Octavio suggested that it w o u l d be h a r d e r a n d more beautiful if it a n s w e r e d THE MATHEMATICAL INTELL1GENCER VOL. 12, N O . 3, 1990 ~ 9 the same problem for non-zero squares. Daniel S h a n k s once a s k e d w h e t h e r t h e quadratic reciprocity law is deep, a n d concluded that it is not, a n y longer. C a n loss of d e p t h h a v e d e s t r o y e d t h e b e a u t y of (24)? T h e m e 7: F i e l d s o f Interest Robert A n d e r s s e n a r g u e d that j u d g e m e n t s of mathe- matical b e a u t y "will n o t be universal, b u t will d e p e n d on the b a c k g r o u n d of the mathematician (algebraist, geometer, analyst, etc.)" S. Liu, writing from P h y s i c s R e v i e w (a h a n d f u l of re- s p o n d e n t s identified themselves as n o n - p u r e - m a t h e - maticians), a d m i t t e d "'my answers reflect a preference for the algebraic a n d number-theoretical over the geo- metrical, topological, a n d analytical t h e o r e m s , ' a n d c o n t i n u e d : "I love classical Euclidean g e o m e t r y - - a subject w h i c h originally attracted m e to mathematics. However, within t h e context of y o u r questionnaire, the p u r e l y geometrical t h e o r e m s pale by comparison." Should readers h a v e been asked to r e s p o n d only to those t h e o r e m s w i t h w h i c h t h e y w e r e extremely fa- miliar? (22) is t h e only item that should n o t have been included, because so m a n y left it blank. Was it outside the m a i n field of interest of most r e s p o n d e n t s , a n d rated d o w n for that reason? T h e m e 8: D i f f e r e n c e s i n Form T w o r e s p o n d e n t s s u g g e s t e d that e i " + 1 = 0 w a s (much) superior, combining " t h e five m o s t important constants." Can a small a n d "inessential" c h a n g e in a t h e o r e m c h a n g e its aesthetic value? H o w w o u l d i i = e -~'t2 have scored? Two n o t e d that (19) is equivalent to t h e irrationality of V 3 a n d o n e s u g g e s t e d that (7) a n d (19) are equiva- lent. Equivalent o r related? W h e n inversion is applied to a t h e o r e m in Euclidean g e o m e t r y are the n e w a n d original t h e o r e m s automati- cally perceived as equally beautiful? I feel not, a n d nat- urally n o t if surprise is an aesthetic variable. A r e a t h e o r e m a n d its d u a l e q u a l l y b e a u t i f u l ? D o u g l a s H o f s t a d t e r s u g g e s t e d t h a t D e s a r g u e s ' s t h e o r e m (its o w n dual) might have b e e n included, a n d w o u l d h a v e g i v e n a v e r y h i g h s c o r e to M o r l e y ' s t h e o r e m on t h e trisectors of the angles of a triangle. Now, Morley's t h e o r e m follows from t h e trigonomet- rical identity, 1/4 sin 30 = [sin 0] [sin (~/3 - 0)] [sin ('rr/3 + 0)]. H o w come o n e particular transformation of this iden- tity into triangle terms is t h o u g h t so beautiful? Is it partly a surprise factor, w h i c h the p e d e s t r i a n identity lacks? T h e m e 9: General versus S p e c i f i c H a r d l y t o u c h e d on by r e s p o n d e n t s , t h e question of general vs. specific seems important to me so I shall quote Paul Halmos [5]: "'Stein's (harmonic analysis) a n d Shelah's (set theory) . . . r e p r e s e n t what s e e m to be t w o diametrically opposite psychological attitudes to m a t h e m a t i c s . . . The contrast b e t w e e n t h e m can be described (inaccurately, b u t p e r h a p s suggestively) b y the w o r d s special a n d general . . . . Stein talked about singular integrals . . . [Shelah] said, early on: 'I love m a t h e m a t i c s because I love generality,' a n d he was off a n d r u n n i n g , classifying structures w h o s e elements w e r e structures of structures of structures." F r e e m a n D y s o n [4] has discussed w h a t he calls "ac- cidental b e a u t y " a n d associated it with unfashionable mathematics. Roger Sollie, a physicist, admitted, "I t e n d to favour 'formulas' involving ~r," a n d scored (14) almost as high as (5) a n d (8). Is "rr, a n d a n y t h i n g to do w i t h it, coloured b y the feeling that -a" is unique, that there is n o other n u m b e r like it? T h e m e 10: Idiosyncratic R e s p o n s e s Several readers illustrated t h e b r e a d t h of individual responses. Mood was relevant to Alan Laverty: " T h e scores I gave to [several] w o u l d fluctuate according to m o o d a n d c i r c u m s t a n c e . E x t r e m e example: at o n e point I w a s considering giving (13) a 10, but I finally d e c i d e d it just d i d n ' t thrill m e v e r y m u c h . " He gave i t a 2. Shirley Ulrich "'could n o t assign comparative scores to t h e . . . items considered as one g r o u p , " so split t h e m into geometric items a n d n u m e r i c items, a n d scored each group separately. R. S. D. Thomas wrote: "I feel that negativity [(7), (8), (19) a n d (22)] makes b e a u t y h a r d to achieve.'" Philosophical orientation came out in the response of J o n a t h a n W a t s o n (software designer, p h i l o s o p h y major, r e a d s M a t h e m a t i c a l I n t e l l i g e n c e r for foundational interest): "I a m a c o n s t r u c t i v i s t . . , a n d so l o w e r e d t h e score for (3), a l t h o u g h y o u c a n also e x p r e s s t h a t t h e o r e m constructively." He adds, " . . . the question- n a i r e i n d i r e c t l y r a i s e s f o u n d a t i o n a l i s s u e s - - o n e t h e o r e m is as true as another, b u t b e a u t y is a h u m a n criterion. A n d b e a u t y is tied to u s e f u l n e s s . " C o n c l u s i o n From a small survey, crude in construction, no posi- tive conclusion is safe. H o w e v e r , I will d r a w the nega- tive c o n c l u s i o n t h a t t h e i d e a t h a t m a t h e m a t i c i a n s largely agree in their aesthetic j u d g e m e n t s is at best grossly oversimplified. Sylvester described m a t h e - matics as t h e s t u d y of difference in similarity a n d simi- larity in difference. He w a s n o t characterising o n l y 4 0 THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990 mathematics. Aesthetics has the same complexity, a n d b o t h perspectives require investigation. I will c o m m e n t o n s o m e possibilities for further re- search. H a r d y a s s e r t e d that a beautiful piece of mathe- matics s h o u l d d i s p l a y generality, u n e x p e c t e d n e s s , depth, inevitability, a n d economy. "Inevitability" is p e r h a p s H a r d y ' s o w n idiosyncracy: it is n o t in other analyses I have come across. Should it be? Such lists, n o t linked to actual examples, perhaps r e p r e s e n t the m a x i m u m possible level of agreement, precisely because t h e y are so unspecific. At the level of this questionnaire, the variety of r e s p o n s e s suggests that individuals' interpretations of t h o s e generalities are quite varied. Are t h e y ? H o w ? W h y ? H a l m o s ' s generality-specificity d i m e n s i o n m a y b e c o m p a r e d to this c o m m e n t b y Saunders Mac Lane [9]: "I a d o p t e d a s t a n d a r d p o s i t i o n - - y o u m u s t specify the subject of interest, set u p the n e e d e d axioms, and de- fine the terms of reference. Atiyah m u c h preferred the style of the theoretical physicists. For t h e m , w h e n a n e w idea c o m e s u p , o n e d o e s not p a u s e to define it, b e c a u s e to do so w o u l d b e a damaging constraint. In- s t e a d t h e y talk a r o u n d a b o u t the idea, d e v e l o p its various connections, a n d finally come u p w i t h a m u c h m o r e supple a n d richer notion . . . . H o w e v e r I per- sisted in the position that as mathematicians w e m u s t k n o w w h e r e o f w e s p e a k . . . . This instance m a y serve to illustrate the p o i n t that there is n o w no a g r e e m e n t as to h o w to d o mathematics . . . . " A p a r t from a s k i n g - - W a s there ever?---such differ- ences in approach will almost certainly affect aesthetic j u d g e m e n t s ; m a n y o t h e r b r o a d d i f f e r e n c e s b e t w e e n mathematicians m a y h a v e the same effect. Changes over time s e e m to be central for the indi- vidual a n d explain h o w one criterion can contradict another. Surprise a n d m y s t e r y will b e s t r o n g e s t at the start. A n initial s o l u t i o n m a y i n t r o d u c e a d e g r e e of generality, depth, a n d simplicity, to be followed b y f u r t h e r q u e s t i o n s a n d f u r t h e r s o l u t i o n s , since t h e richest p r o b l e m s d o n o t reach a final state in their first incarnation. A n e w p o i n t of v i e w raises surprise anew, m u d d i e s the a p p a r e n t l y clear w a t e r s , a n d s u g g e s t s greater d e p t h or b r o a d e r generality. H o w d o aesthetic j u d g e m e n t s c h a n g e a n d d e v e l o p , in q u a n t i t y a n d quality, during this temporal roller coaster? Poincar~ and v o n N e u m a n n , a m o n g others, have e m p h a s i s e d the role of aesthetic j u d g e m e n t as a heu- ristic aid in the process of mathematics, t h o u g h liable to mislead on occasion, like all such assistance. H o w d o individuals' j u d g e m e n t s aid t h e m in their work, at e v e r y level from preference for g e o m e t r y o v e r anal- ysis, or whatever, to the m o s t microscopic levels of mathematical thinking? Mathematical aesthetics shares m u c h w i t h the aes- thetics of other subjects a n d n o t just the h a r d sciences. T h e r e is no space to d i s c u s s a variety of e x a m p l e s , t h o u g h I will m e n t i o n the related concepts of isomor- p h i s m a n d metaphor. H e r e is o n e v i e w of surprise [6]: "Fine writing, according to A d d i s o n , consists of senti- m e n t s w h i c h are natural, w i t h o u t being obvious . . . . O n the o t h e r hand, productions w h i c h are merely sur- prising, w i t h o u t b e i n g natural, can n e v e r give a n y lasting e n t e r t a i n m e n t to the m i n d . " H o w might "natural" b e interpreted in mathemat- ical terms? Le Lionnais u s e d the same w o r d . Is it t r u t h that is b o t h natural and beautiful? H o w a b o u t H a r d y ' s "inevitable?" Is n o t g r o u p t h e o r y an historically inevi- table d e v e l o p m e n t , and also natural, in the sense that g r o u p structures w e r e there to b e detected, sooner or later? Is n o t the naturalness a n d b e a u t y of such struc- t u r e s r e l a t e d to d e p t h a n d t h e role of a b s t r a c t i o n , w h i c h p r o v i d e s a ground, as it were, against w h i c h the individuality of other less general mathematical entities is highlighted? Mathematics, I a m sure, can only be m o s t d e e p l y u n d e r s t o o d in the context of all h u m a n life. In partic- ular, b e a u t y in mathematics m u s t be incorporated into a n y a d e q u a t e e p i s t e m o l o g y of mathematics. Philoso- phies of mathematics that ignore b e a u t y will be inher- e n t l y d e f e c t i v e a n d i n c a p a b l e of e f f e c t i v e l y inter- preting t h e activities of mathematicians [12]. R e f e r e n c e s 1. W. Burnside, Proceedings of the London Mathematical So- ciety (2), 7 (1980), 4. 2. Mr. Davies, Historical notices respecting an ancient problem, The Mathematician 3 (1849), 225. 3. T. Dreyfus and T. Eisenberg, On the aesthetics of mathe- matical thought, For the Learning of Mathematics 6 (1986). See also the letter in the next issue and the author's reply. 4. Freeman J. Dyson, Unfashionable pursuits, The Mathe- matical Intelligencer 5, no. 3 (1983), 47. 5. P. R. Halmos, Why is a congress? The Mathematical Intel- ligencer 9, no. 2 (1987), 20. 6. David Hume, On simplicity and refinement in writing, Selected English Essays, W. Peacock, (ed.) Oxford: Oxford University Press (1911), 152. 7. F. Le Lionnais, Beauty in mathematics, Great Currents of Mathematical Thought, (F. Le Lionnais, ed.), Pinter and Kline, trans. New York: Dover, n.d. 128. 8. J. E. Littlewood, A Mathematician's Miscellany, New York: Methuen (1963), 85. 9. Saunders Mac Lane, The health of mathematics, The Mathematical Intelligencer 5, no. 4 (1983), 53. 10. Roger Penrose, The role of aesthetics in pure and ap- plied mathematical research, Bulletin of the Institute of Mathematics and its Applications 10 (1974), 268. 11. Ibid., 267. 12. David Wells, Beauty, mathematics, and Philip Kitcher, Studies of Meaning, Language and Change 21 (1988). 13. David Wells, Mathematicians and dissidence, Studies of Meaning, Language and Change 17 (1986). 19 Menelik Road London NW2 3RJ England THE MATHEMATICAL INTELLIGENCER VOL. 12, NO. 3, 1990 41