"The Fragility of Beauty in Mathematics and in Art"
Sarah Jones Nelson and Enrico Bombieri
A collaboration under the auspices of the School of Mathematics, Institute for Advanced Study, Princeton, New Jersey
Published in Art in the Life of Mathematicians
Anna Kepes Szemerédi, Editor, American Mathematical Society: Providence, Rhode Island, 2015
Does beauty exist in mathematics? The question concerns mathematical objects and their relations, the real subject of verifiable proofs. Mathematicians generally agree that beauty does exist in the structural beauty of theorems and proofs, largely visible only to mathematicians themselves, and in mathematical beauty everyone can see in art and nature. Irrefutably beautiful patterns emerge universally from the relation of elements and objects in mosaic tilings, for instance, in landscape painting, flowing rivers, and in the spiraling symmetries of pine cones and seashells. Mathematical patterns in physical formations give you a sense of beauty unchanged by the variation of its elements: universality, symmetry, simplicity, elegance, and power.
The critique of beauty in art is fragile because it depends upon relative standards of judgment that vary in time across cultures. To Leibniz, this explained the difference between truths of fact and truths of reasoning that reflect critical intention. Until the late Enlightenment, a formal distinction by Kant was unintelligible between perception of natural beauty as a beautiful object, and of artistic beauty as a beautiful representation of an object. In the tradition of his day, Kant used classical Greek theory of proportions in nature and in art to verify his beliefs in the factual truth of beauty. However, Hume’s analysis of facts, values, and taste, combined with Spinoza’s theory of ethics and emotions such as envy and love, had already made the inner life of strong emotions a permanent norm of critical judgment. This reversed the Renaissance idea that art is the mirror of nature unifying optical truth and factual beauty, first shown in Alberti’s new mathematics of perspective. The classical Greek unification of truth, beauty, and moral goodness – a platonic trope for like-minded realists such as Brunelleschi, Giotto, Leonardo, and Michelangelo – wildly contradicts any amoral judgment or critique of beauty in abstract and late modern art.
Leon Battista Alberti was a philosopher and abbreviator or secretary to the Papal Curia. He first formalized one-point perspective in De pictura (1435), and in the vernacular Della pittura (1436). Alberti collaborated in Florence with the architect Filippo Brunelleschi. In Rome he worked with Luca Pacioli, a mathematician and collaborator with Leonardo da Vinci in Milan on De divina proportione (1509). Brunelleschi had earned his brevetto in mathematics and had studied under Paolo dal Pozzo Toscanelli, the Florentine mathematician, astronomer, cosmographer, and adviser to Columbus. Pacioli probably was a student of Piero della Francesca, a painter, mathematician, and theorist of De prospectiva pingendi (1474). This treatise formalized his exquisitely symmetrical Brera Madonna (1472-1474) much in the same way as his pala d’altare of Saint Anthony (c.1470). Pictorial perspective, the two-dimensional representation of three-dimensional space, thus became a philosophy and rule of art to be preferred, Leonardo said, to all systems of learning because of its foundation in the certainties of physics and mathematics – and of commissioning patrons with taste for trompe-l’oeil.
The remarkable continuity of platonic realism in art and mathematics extends to a widely-held belief that numbers, geometry, and all of mathematics are revealed from an absolute realm of objects or pure forms paradoxically independent of the senses and of physical reality. In the platonist view, a mathematician discovers pre-existing objects such as the golden ratio; whereas the formalist invents and constructs proofs like an architect or a builder of mathematical objects made from the materials of a given culture. Most mathematicians are platonists on Sundays working in a formalistic way on weekdays. Many play upon a multitude of objectively distinct and valid mathematical systems – once universal to Plato – in a world of plentiful platonism.
The golden ratio φ is a simple object expressing a fundamental hidden structure of Renaissance art. In Euclidean geometry, φ is the ratio of the side of the regular star-pentagon to the side of the regular pentagon. In contemporary mathematics, φ equals 1 plus the square root of 5 divided by 2:
Properties of the golden ratio relate deeply to the number 5, a topic of fascination ever since Plato asserted that numbers emanate from pre-existing objects revealed and discovered from the realm of pure forms. The historian Annemarie Schimmel, in Das Mysterium der Zahl, the English edition The Mystery of Numbers (1983), documented the association in antiquity of the number 5 with the mystical pentagram of kabbalistic knowledge. Apocryphal variants of Genesis 1:27 described the Edenic beginning of the universe in which the number 5 originated from the union of numbers 2 and 3, symbolizing the first forms of woman and man, with 1 symbolizing God and the unification of physical reality.
Structures in fives emerge everywhere in nature and in art. For example, the proportion of the golden ratio is plainly visible in the five-fold symmetries in certain plants and flowers. Leonardo’s Vitruvian Man (1490) suggests that the human body itself is a star-pentagon expressing the golden ratio as a metaphor of nature and a model of symmetry and proportion in design. Piero’s works express five-fold structures in the linear geometry of commensurazione, his standard for judging contour and proportion. The pala d’altare of Saint Anthony features a central subdivision in five vertical spaces with a vertical subdivision in five sections. The Brera Madonna presents three saints and two angels to the left with and two angels and three saints to the right, completing the symmetry. A partly-armored patron on the right, Federigo da Montefeltro, Duke of Urbino, lends a strong element of tension that breaks the symmetry. In this way, Piero restores a harmonious equilibrium where the Duke kneels, hands in prayer, directing your gaze diagonally towards the Virgin and her infant Son. Another well-known painting by Piero, the Nativity (c.1470), sees the fragile Child surrounded at the left by five angels singing and playing the lute, with two shepherds, Saint Joseph, and the ass and ox at the right, backgrounding this masterpiece of asymmetry; the Virgin in adoration and the mysterious dove of the Holy Spirit at the top complete this tender image of humility. Even more elegant mathematically is his Madonna del Parto (c.1460), featuring the paradox of a pregnant Virgin contoured by a tent in the form of a pentagon. Piero intended his works to represent the elements of biblical narrative in a symbolic language that unified formalism with mathematical analysis. A pragmatic master of technique, he never stopped at mere prescience and left nothing to chance.
The golden ratio not only is a simple ancient object; φ also plays a deep role in the formation of modern mathematics. As the irrational number between 1 and 2 that is most distant from rational numbers, the golden ratio has the property of being the unique number between 1 and 2 that requires the largest value of the denominator q for reaching a given approximation by the rational number p divided by q. We cite two strikingly elegant formulas for the golden ratio. In the first, φ equals the square root of 1 plus the square root of 1 plus the square root of 1 in a nested construction to infinity; in the second, φ equals 1 divided by 1 plus 1 divided by 1 plus 1 divided by 1 in a nested construction to infinity. The second formula is the more interesting of the two as the initial point for the proof of the extreme irrationality of the golden ratio.
The Fibonacci sequence, named after Leonardo of Pisa (called Fibonacci), expresses the golden ratio with beautiful precision. Fibonacci first published it in Liber Abaci (1201). He had worked for his father in a custom house near Algiers and studied under Muslim mathematicians along Mediterranean trade routes to become the most renowned mathematician in medieval Europe. The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . . to infinity, where each number after the first two is the sum of the two preceding numbers. It gives the best rational approximations to the golden ratio; the fractions Fn+1/Fn get closer and closer to φ. The Fibonacci sequence is expressed also in terms of the golden ratio:
Euclidean geometry and the golden ratio form the earliest foundation of beauty in the history of mathematics. Imagine that in 1899, more than two millennia after Euclid wrote Elements, the mathematician Frank Morley discovered the last genuinely new theorem in euclidean geometry. The Morley theorem states that the three points where the trisectors of the angles of an arbitrary triangle meet are the vertices of an equilateral triangle. Beautiful! You may ask why a result such as Morley’s theorem never occurred in classical euclidean geometry: probably because the Greeks of classical antiquity were unable to obtain the trisection of an angle using euclidean constructions. Today mathematicians know that trisecting an angle cannot be done within the framework of plane euclidean geometry. They consider this ‘negative’ result a beautiful consequence of Galois’ theory (1832) on the resolubility of algebraic equations, obtained by means of profound results on permutation groups. So, what may once have been a blemish on euclidean geometry – the impossibility of proving Morley’s theorem in its context – now is a beautiful discovery augmenting our knowledge of geometry, logic, and the corresponding formations of symmetry.
You might infer from our history of the golden ratio that it is a fundamental pillar of all mathematics. But the Laplace equation Uxx + Uyy + Uzz = 0 is far more significant. Again and again it appears in analysis, in probability, mathematical physics, astrophysics, chemistry, even in financial engineering – indeed in all cases involving the equilibrium state of a system. It beautifully expresses the powerful relevance of mathematics to deep open questions, for example, in philosophy concerning causation. Laplace’s Mécanique céleste (1829) develops Newtonian mechanics and differential calculus to the point that determinism seems an inescapable state of physical reality: knowing the forces and knowing the position and velocity of each particle in the universe at a given time, the state of the universe at any later time is uniquely determined; thus with Laplace, chance, free action or agency are causal fictions about the true laws of nature. But his equation is just a simple mathematical model. Is it correct or elegant to draw such sweeping philo- sophical consequences from it, or to suggest that the laws of nature have never been revised by new discoveries?
Now, with the advent of quantum theory we know that Laplace’s system can- not describe all mechanisms of the observable universe. In quantum mechanics the state of the universe is given by a wave function ψ satisfying the Schrödinger equation, a close relative of the Laplace equation. For Schrödinger also the evolution of ψ is deterministic: knowing the wave function at one time it is uniquely determined at later times. However, the wave function only describes the probability of results from an observation. Nearly a century after the formulation of quantum theory there is still no consensus about its domain of validity, perhaps because it contra- dicts classical mechanics and the strong view – at the foundation of science and society – that natural processes and human actions are determined by the past and simple mechanisms of cause and effect. But how can actions producing anything as startlingly complex and undetermined as a beautiful new proof, or the exuberant realism of the Renaissance, emerge in a universe predetermined by fixed laws?
The view that mathematics is intrinsic to laws of nature, to human emotions and the arts shows marvelously in Albrecht Dürer’s most celebrated engraving, Melencolia I (1514). Its central image of a winged woman gazing darkly inward – like an oracle contemplating the shadowy forms of the cosmos – symbolizes melencholia imaginativa, an early modern trope from classical Greek medicine and analysis of the four humors or temperaments. She personifies the genius of art and deep mathematical reality: a sphere foregrounds the perfect finite universe; a polyhedron, the truncated rhombohedron, represents the descriptive geometry of Archimedean solids; astrolabes and quadrants suggest the measure of space and time. A subtle 4 x 4 magic square of integers at the tip of her wing refers to the Fibonacci number and is a sign of nature’s hidden mathematical order turning evil to good and vanquishing anxiety, said the Renaissance philosopher Marsilio Ficino. The date 1514, in the center of the last row of the magic square, celebrates the completion of this masterpiece. Dürer, a German artist and mathematician of treatises on perspective and proportion, shared with his theorizing Italian contemporaries a philosophy of art and a theology of the inner life that revolutionized Reformation Europe.
Dürer’s image of the inner life predicts the paradoxical joys of mathematical truths that inform even music. Consider your experience of how a Bach concerto or a Mozart sonata deepens your perception of reality. The structure of music is mathematical, as Pythagoras first showed. So it is with the durable objects of mathematics ranging from music to art and beyond to cultural and natural selection and onward to cosmology describing the symmetrical formation of matter at the beginning of the universe.
Symmetrical formations of matter express beauty in mathematics. The concept of a group expresses symmetry in mathematics. What is a group? Consider any object, concrete or abstract. A symmetry of the object – mathematically, an automorphism – is a mapping of the object onto itself that preserves all of its properties. The product of two symmetries, one followed by the other, also is a sym- metry, and every symmetry has an inverse that undoes it. Symmetries of a square can be obtained by rotating it 90 degrees or by reflecting it in the vertical axis. Mathematicians consider Lie groups (pronounced lee) to be a beautiful continuous foundation for a great portion of mathematics, and for physics as well. Besides continuous Lie groups there are noncontinuous finite and discrete groups; some are obtainable from Lie groups by reduction to a finite or discrete setting.
Anyone braving the labyrinth of interior design knows the fearful symmetries of wallpapers in the lattice types of mathematical tilings for the parallelogram, the hexagon, triangle, rectangle, square, and the rhombus, at various rotations and reflection axes. Unlike lattice tilings, which are periodic, there are also aperiodic self-similar quasicrystalline tilings called Penrose tilings, named after their discoverer, the mathematician and physicist Roger Penrose. While the number of distinct lattice types is finite – exactly 17 of them exist – there is a continuum of distinct Penrose tilings. But by self-similarity, every piece of a given Penrose tiling appears infinitely many times in every other Penrose tiling! Penrose tilings came as a big surprise to mathematicians, since their spectral properties are point-like, resembling the point-like X-ray diffraction patterns of natural crystals.
Even more surprising was the Nobel prize-winning discovery by Daniel Schechtman that quasicrystals exist in nature as five-fold symmetries of aperiodic alloys of certain metals. The physicist Paul Steinhardt, one of the discoverers of quasicrystals, has beautifully shown their connection to the stunning similarity between Penrose tilings and early medieval Islamic mosaics, or Girih tilings. Imagine that six centuries before Penrose, Islamic artists and architects introduced pentagon and decagon motifs, tilings of partial pentagonal symmetries that express the fragile beauty of timeless art. These abstract subtleties in mathematics are tools for describing nature and works of art such as Girih and Penrose tilings, simply done with two basic rhombic tiles, one narrow and one wide, in precise shapes linked to the golden ratio, the pentagon, and star pentagons. Mathematical tools of description help us see the otherwise hidden universal structures in nature, the golden ratio, the star pentagon, each giving rise to formal norms of precision in social abstraction from pure mathematics to abstract art.
Finite groups of symmetries, like the symmetries of a square or a cube, defied classification for a long time until mathematicians successfully classified all finite simple groups. The classification theorem is a proof that today runs more than 3,000 pages and took over 40 years of the collective efforts of more than 100 math- ematicians. This theorem brings order to the theory of finite groups. Simple groups are important because they are a kind of foundation stone from which every finite group is built. For example, a group of rotations of a polygon with 15 sides can be obtained by combined rotations of 120 degrees and rotations of 72 degrees, with the latter generating simple groups. Alternating groups, starting with the icosahedral group, and finite groups of Lie type, form finitely many families of simple groups, but with infinitely many members in each family. As the classical Lie groups, they are closely associated to symmetries of underlying geometries. There exist also 26 exceptional groups, quite unlike the groups of Lie type, known as sporadic groups. Two sporadic groups are relevant here: the Conway symmetry group, named after John Conway, the symmetries (up to a reflection) of a very remarkable lattice in 24 dimensions, the Leech lattice; and the Fischer–Griess group F1, proved to exist by R.L. Griess while visiting the Institute for Advanced Study. Also dubbed the Monster by mathematicians, the Fischer-Griess group is the largest, gigantic sporadic group containing
elements. It contains inside itself 21 of the 26 sporadic groups, and the Conway group is one of them! By a totally unexpected development from the curious numerology 196883 + 1 = 196884, where 196883 is a critical number needed to describe the Monster and 196884 is another number – from the 150-year-old study of elliptic and automorphic functions – the Monster has now been tamed by its clear connection to many distinct fields in mathematics and in mathematical physics. Isn’t it beautiful that the persistent cooperation of mathematicians would tame the Monster F1?
More could be said of beauty in mathematics, from the fragile process of robust peer review to any consensus of what is verifiably true and beautiful. Even the simplest thing in mathematics, namely, the number sequence 1, 2, 3, . . . from which all mathematics took life, contains within itself a deep mystery, the sequence 2, 3, 5, 7, . . . of prime numbers that form the building blocks of multiplication. Mathematicians already have uncovered beautiful relations among the properties of prime numbers, some firmly established. But the most important relations are still conjectural, raising open questions in analysis, geometry, and even physics. Dwelling further here would exceed our present scope, so we finish our list of beautiful examples with a famous construction connecting mathematics to logic and philosophy.
Mathematicians are always looking for a beautiful proof, never satisfied just knowing that something is true. They want to know why it is true. Take the continuum, an ancient source of contention among Greek philosophers such as Zeno, his paradox arising from infinite divisibility. George Cantor gave a precise mathematical definition of the continuum reflecting our naive view of its being the totality of all numbers, written in decimal notation as an integer followed by an infinite sequence of decimal digits, not all equal to 9 from some point onward. (Cantor’s definition is uncannily near Eudoxus’s conception of a number.)
Cantor produced a famous diagonal argument known as Cantor’s theorem of the uncountability of the continuum. This powerful simple proof shows that the continuum, namely, all real numbers between 0 and 1, cannot be listed in a list as first, second, third, and so on. Therefore the continuum is uncountable. Suppose by counterexample that it is countable in an infinite list:
0.643546675432534645600112 . . .
0.100053453647545546043860 . . .
0.000000000000100004534237 . . .
0.999999999961045674732017 . . .
0.222955600333054564501179 . . .
0.141592653589793238462643 . . .
0.777777777777777777777777 . . .
0.421047542507075505555001 . . .
0.777777771777777777777777 . . .
0.777777777177777777777777 . . .
0.010010001000010000010000 . . .
0.099999999999999900000000 . . .
The diagonal marker is 0.600952741109 . . . , the nth digit of the nth number. If you change each digit of the diagonal number, the result cannot be in any row; therefore it is not in the list. A beautiful proof! Why? Because it proves universally that there are different kinds of infinity, the infinity of positive whole numbers 1, 2, 3, . . . and the infinity of the continuum 0_______1. Intuitively it demonstrates that a discrete infinity exists that is different from the infinity of the continuum such as a straight line. Cantor’s diagonal argument is not limited to this theorem; it has become a powerful tool as well in mathematical logic, on the nature of infinity, and in computer science, on the nature of complexity. Think of the staggering mathematical consequences of Cantor’s theorem for euclidean geometry. The philosophical consequences irrevocably have shaken the foundations of analysis and any a priori synthetic judgment refuting Hume’s denial that true knowledge of any metaphysics is possible.
Cantor showed that the countable infinity of positive whole numbers is smaller than the infinity of the continuum. The famous continuum hypothesis is the as- sertion that there is no infinity larger than the countable infinity and smaller than the infinity of the continuum. The technical situation is this: Gödel proved that the continuum hypothesis cannot be disproved from the axioms of set theory. Paul Cohen proved that it cannot be proved. Thus the continuum hypothesis is inde- pendent from the axioms of set theory; it can be neither proved or disproved. The philosophical consequence is that the truth value of the continuum hypothesis is uncertain, or at best, undefined. Kant believed that the axioms of euclidean geometry were true. But now we know that there are noneuclidean geometries as well. Furthermore, John Conway has suggested that there are unsettleable simple mathematical assertions that do not follow from the axioms of set theory. What, then, is truth in mathematics? Is there any fundamental difference between knowing whether the axioms of euclidean geometry are true, and knowing whether the continuum hypothesis is true? In each case, we see that truth is intrinsically fragile and is not to be identified with the absence of contradiction.
Axiomatic reasoning and construction each possess a different kind of fragility. Constructivists and intuitionists put strong limits on what can be done, whereas ax- iomatic reasoning may only demonstrate the existence of an object with no method for constructing it. That is, axiomatic reasoning may show that the hypothesis that the object does not exist leads to a contradiction. Cantor’s diagonal argument proves that the continuum is not listable; it says nothing, however, about the structure of the continuum. Also, constructivism maintains that constructive reasoning is correct. But the practice of the great majority of mathematicians is to use nonconstructive axiomatic reasoning and then to investigate the possibilities of construction that may simply be a matter of taste.
Hume was the skeptical prophet of taste heralding the role of the subject – the self – in the perception of beauty as the highest good of aesthetic pleasure. Long before Freud made the pleasure principle the fixed norm of human intention, Hume had made aesthetic pleasure the standard of beauty verified on the evidence of the senses. But Kant’s aesthetic of objects prior to perception of beauty – and of truth – revised that relation between physical and nonphysical reality. Kant the platonist made beauty in itself the revealed emanation of mathematical truth and of pure metaphysical forms or objects. How is truth as beauty to be verified if it is a revelation from Plato’s realm of pure forms? On what evidence of the senses do you verify a revelation? And are the senses reliably verifiable? Is the perception of beauty a mere projection of the wish for pleasure to avert or suppress pain? Is a real mathematical object physical or not? What is a real object? Does the mathematical verification of truth participate in physical reality because it is discovered, or because it is invented from the culture of human works and the fragile processes of cultural selection? Does a proof truly save the appearance of pure forms discovered in a mystical realm of beautiful objects?
Analogous questions in art, poetry, music, and history inform the question of beauty because of its deeply personal role in the formation of human works and human values. Mathematicians work like poets or painters: the difference is that a room of mathematicians looking at a problem get the same answer within a community that values and requires consensus. The construction of mathematical objects by individuals is fragile; these objects in themselves, however, are robust because of durable social norms of peer review and consensus. Like poets, painters, and composers of music, mathematicians have their own style and technique. But mathematical truth is not just a collection of theorems, any more than painting is a mere collection of pigments. For mathematicians, as Tarski proved, theorems are established truths, obtained by a fragile construction of proofs leading to consensus of verification. The process from fragile construction to robust verification not only describes the social norms of the mathematical community; it also points to the gradual discernment of relations among the mathematical objects of a proof.
Mathematicians, sometimes involuntarily, tend to accept Wittgenstein’s concept of aspect – the temporal perception of an object’s internal relations – as an essential part of mathematics. This is because any given aspect, element, or property of relations among objects or proofs unfolds indeterminately in the changing light of perception, much like the relational aspect of objects tends to change as you look at a painting. A poem, a symphony, a painting, or a written narrative fixed in time never changes, but the way you read a text or listen to music or look at art does change with temporal shifts in emotion, taste, or angles of light and space that make consensus immaterial. Truth-bearing history is done regardless of the beauty or goodness of events of scholarly representation, even though historians value robust consensus upon fragile historical truths as the objects and relations of factual analysis. Poetry rarely is written now to be beautiful or even necessarily true, but rather to satisfy a powerful pre-verbal consciousness of the way things are changed by the poem apart from any formal analysis of aspect. Quoting Keats,
“Beauty is truth, truth beauty, – that is all
Ye know on earth, and all ye need to know."