Mathematical beauty is a type ofaesthetic value that is experienced in doing or contemplatingmathematics. The testimonies of mathematicians indicate that various aspects of mathematics—including results, formulae, proofs and theories—can trigger subjective responses similar to the beauty of art, music, or nature. The pleasure in this experience can serve as a motivation for doing mathematics, and some mathematicians, such asG.H. Hardy, have characterized mathematics as anart form that seeks beauty. The logician and philosopherBertrand Russell made a now-famous statement of this position:
Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.[1]
Beauty in mathematics has been subject to examination by mathematicians themselves and by philosophers, psychologists, and neuroscientists. Understandingbeauty in general can be difficult because it is a subjective response to sense-experience but is perceived as a property of an external object, and because it may be shaped by cultural influence or personal experience. Mathematical beauty presents additional problems, since the aesthetic response is evoked by abstractideas which can be communicated symbolically, and which may only be available to a minority of people with mathematical ability and training. The appreciation of mathematics may also be less passive than (for example) listening to music.[2] Furthermore, beauty in mathematics may be connected to other aesthetic or non-aesthetic values. Some authors identify mathematical elegance with mathematical beauty; others distinguish elegance as a separate aesthetic value, or as being, for instance, limited to the form mathematical exposition.[3]: 177–178 Beauty itself is often linked to, or thought to be dependent on, the abstractness, purity, simplicity, depth or order of mathematics.
Euler's identity is often given as an example of a beautiful result:[4]: 1–3 [5]: 835–836 [6]
This expression ties together arguably the five most importantmathematical constants (e,i, π, 1, and 0) with the two most common mathematical symbols (+, =). Euler's identity is a special case ofEuler's formula, which the physicistRichard Feynman called "our jewel" and "the most remarkable formula in mathematics".[7]
Another example isFermat's theorem on sums of two squares, which says that anyprime number such that can be written as a sum of two square numbers (for example,,,), which bothG.H. Hardy[8]: §12 andE.T. Bell[9]: ch.4 thought was a beautiful result.
In a survey in which mathematicians were asked to evaluate 24 theorems for their beauty, the top-rated three theorems were:[10] Euler's equation;Euler's polyhedron formula, which asserts that for a polyhedron withV vertices,E edges, andF faces,; andEuclid's theorem that there are infinitely many prime numbers, which was also given byHardy as an example of a beautiful theorem.[8]: §12
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Cantor's diagonal argument, which establishes that there areinfinite sets which cannot be put intoone-to-one correspondence with the infinite set ofnatural numbers, has been cited by both mathematicians[11] and philosophers[12] as an example of a beautiful proof.
Visual proofs, such as the illustrated proof of thePythagorean theorem, and otherproofs without words generally, such as the shown proof that the sum of all positiveodd numbers up to 2n − 1 is aperfect square, have been thought beautiful.[13]
The mathematicianPaul Erdős spoke ofThe Book, an imaginary infinite book in which God has written down all the most beautiful mathematical proofs. When Erdős wanted to express particular appreciation of a proof, he would proclaim it "straight from The Book!".[14]: 35 His rhetorical device inspired the creation ofProofs from THE BOOK, a collection of such proofs, including many suggested by Erdős himself.[15]: v
InPlato'sTimaeus, the fiveregular convex polyhedra, called thePlatonic solids for their role in this dialogue, are called the "most beautiful" ("κάλλιστα") bodies.[16]: 53e In theTimaeus, they are described as having been used by thedemiurge, or creator-craftsman who builds the cosmos, for the fourclassical elements plus the heavens, because of their beauty.[16]: 54e–55e
In his 1596 bookMysterium Cosmographicum,Johannes Kepler argued that the orbits of the then-known planets in theSolar System have been arranged byGod to correspond to a concentric arrangement of the fivePlatonic solids, each orbit lying on thecircumsphere of onepolyhedron and theinsphere of another. For Kepler, God had wanted to shape the universe according to the five regular solids because of their beauty, and this explainedwhy there were six planets (according to the knowledge of the time).[17]: ch.3 [5]: 280--285
A more modern example is the exceptionalsimple Lie group, which has been called "perhaps the most beautiful structure in all of mathematics".[18]
The mathematical statements of scientific theories, especially in physics, are sometimes considered to be mathematically beautiful. For example,Roger Penrose thought there was a "special beauty" inMaxwell's equations of electromagnetism:[19]: 268
Einstein's theory ofgeneral relativity has been characterized as a work of art, and, among other aesthetic praise,[20]: 148 was described byPaul Dirac as having "great mathematical beauty"[21]: 123 and by Penrose as having "supreme mathematical beauty".[22]: 1038
(There can be more to the beauty of a scientific theory than just its mathematical statement. For example, whether a theory is visualizable or deterministic might have an influence on whether it is seen as beautiful.[23]: 53 [5]: 873–877 )
Many mathematicians and philosophers who have written about mathematical beauty have tried to identify properties or criteria that are conducive to the perception of beauty in a piece of mathematics. It is debated whether beauty can be clarified or explained by such properties:Paul Erdős thought that it was no more possible to convince someone of the beauty of a piece of mathematics than to convince them of the beauty ofBeethoven's Ninth Symphony, if they couldn't see it for themselves.[24]
In his 1940 essayA Mathematician's Apology,G. H. Hardy said that a beautiful result, including its proof, possesses three "purely aesthetic qualities", namely "inevitability", "unexpectedness", and "economy". He particularly excluded enumeration of cases as "one of the duller formsof mathematical argument".[8]: §18
In 1997,Gian-Carlo Rota, disagreed with unexpectedness as a necessary condition for beauty and proposed a counterexample:
A great many theorems of mathematics, when first published, appear to be surprising; thus for example some twenty years ago [from 1977] the proof of the existence ofnon-equivalent differentiable structures on spheres of high dimension was thought to be surprising, but it did not occur to anyone to call such a fact beautiful, then or now.[3]: 172
In contrast, Monastyrsky wrote in 2001:
It is very difficult to find an analogous invention in the past toMilnor's beautiful construction of the different differential structures on the seven-dimensional sphere... The original proof of Milnor was not very constructive, but later E. Briscorn showed that these differential structures can be described in an extremely explicit and beautiful form.[25]: 44
This disagreement illustrates both the subjective nature of mathematical beauty and its connection with mathematical results: in this case, not only the existence of exotic spheres, but also a particular realization of them.
Besides Hardy's properties of "unexpectedness", "inevitability", "economy", which he applied to proofs as well as results, mathematicians have customarily thought beautiful proofs that are short and simple.[26]: 22
In the search for an elegant proof, mathematicians often look for different independent ways to prove a result—as the first proof that is found can often be improved. The theorem for which the greatest number of different proofs have been discovered is possibly thePythagorean theorem, with hundreds of proofs having being published.[27] Another theorem that has been proved in many different ways is the theorem ofquadratic reciprocity. In fact,Carl Friedrich Gauss alone had eight different proofs of this theorem, six of which he published.[28]
In contrast, results that are logically correct but involve laborious calculations or consideration of many cases, are not usually considered beautiful, and may be even referred to asugly orclumsy. For example,Kenneth Appel andWolfgang Haken's proof of thefour color theorem made use of computer checking of over a thousand cases.Philip J. Davis andReuben Hersh said that when they first heard that about the proof, they hoped it contained a new insight "whose beauty would transform my day", and were disheartened when informed the proof was by case enumeration and computer verification.[29]: 384 Paul Erdős said it was "not beautiful" because it gave no insight into why the theorem was true.[30]: 44
Aristotle thought that beauty was found especially in mathematics, writing in theMetaphysics that
those who assert that the mathematical sciences say nothing of the beautiful or the good are in error. For these sciences say and prove a very great deal about them; for if they do not expressly mention them, but prove attributes which are their results or their formulae, it is not true to say that they tell us nothing about them. The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree.[31]: 1078a32–35
In the twentieth century, some philosophers questioned whether there was genuinely beauty in mathematics. The philosopher of scienceRom Harré argued that there were no true aesthetic appraisals of mathematics, but onlyquasi-aesthetic appraisals. Any mathematical success described by an aesthetic term was a second-order success besides understanding and correctness. In contrast, aesthetic appraisal of a work of art was first-order. Harré considered this to be the difference between a quasi-aesthetic and a genuinely aesthetic appraisal.[32]
Nick Zangwill thought that there were no true aesthetic experiences of mathematics and that a proofs or theories could only bemetaphorically beautiful. His argument had two bases. First, he thought that aesthetic properties depended on sensory properties, and so abstract entities could not have aesthetic properties. Second, he thought that proofs, theorems, theories, and so on had purposes such as demonstrating correctness or granting understanding, and that any praise of them reflectedonly how well they achieved their purpose.[33]: 140–142
In the 1970s,Abraham Moles andFrieder Nake analyzed links between beauty,information processing, andinformation theory.[34][35] In the 1990s,Jürgen Schmidhuber formulated a mathematical theory of observer-dependent subjective beauty based onalgorithmic information theory: the most beautiful objects among subjectively comparable objects have shortalgorithmic descriptions (i.e.,Kolmogorov complexity) relative to what the observer already knows.[36][37][38] Schmidhuber explicitly distinguishes between beautiful and interesting. The latter corresponds to thefirst derivative of subjectively perceived beauty:the observer continually tries to improve thepredictability andcompressibility of the observations by discovering regularities such as repetitions andsymmetries andfractalself-similarity. Whenever the observer's learning process (possibly a predictive artificialneural network) leads to improved data compression such that the observation sequence can be described by fewerbits than before, the temporary interesting-ness of the data corresponds to the compression progress, and is proportional to the observer's internal curiosity reward.[39][40]
Brain imaging experiments conducted bySemir Zeki,Michael Atiyah and collaborators show that the experience of mathematical beauty has, as a neural correlate, activity in field A1 of the medialorbito-frontal cortex (mOFC) of the brain and that this activity is parametrically related to the declared intensity of beauty. The location of the activity is similar to the location of the activity that correlates with the experience of beauty from other sources, such as visual art or music.[41] Moreover, mathematicians seem resistant to revising their judgment of the beauty of a mathematical formula in light of contradictory opinion given by their peers.[42]
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Examples of the use of mathematics in music include thestochastic music ofIannis Xenakis, theFibonacci sequence inTool'sLateralus, counterpoint ofJohann Sebastian Bach,polyrhythmic structures (as inIgor Stravinsky'sThe Rite of Spring), theMetric modulation ofElliott Carter,permutation theory inserialism beginning withArnold Schoenberg, and application of Shepard tones inKarlheinz Stockhausen'sHymnen. They also include the application ofGroup theory to transformations in music in the theoretical writings ofDavid Lewin.
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Examples of the use of mathematics in the visual arts include applications ofchaos theory andfractal geometry tocomputer-generated art, symmetry studies ofLeonardo da Vinci,projective geometries in development of theperspective theory ofRenaissance art,grids inOp art, optical geometry in thecamera obscura ofGiambattista della Porta, and multiple perspective in analyticcubism andfuturism.
Sacred geometry is a field of its own, giving rise to countless art forms including some of the best known mystic symbols and religious motifs, and has a particularly rich history inIslamic architecture. It also provides a means of meditation and comtemplation, for example the study of theKaballahSefirot (Tree Of Life) andMetatron's Cube; and also the act of drawing itself.
The Dutch graphic designerM. C. Escher created mathematically inspiredwoodcuts,lithographs, andmezzotints. These feature impossible constructions, explorations ofinfinity, architecture, visualparadoxes andtessellations.
Some painters and sculptors create work distorted with the mathematical principles ofanamorphosis, including South African sculptorJonty Hurwitz.
Origami, the art of paper folding, has aesthetic qualities and many mathematical connections. One can study themathematics of paper folding by observing thecrease pattern on unfolded origami pieces.[43]
British constructionist artistJohn Ernest created reliefs and paintings inspired by group theory.[44] A number of other British artists of the constructionist and systems schools of thought also draw on mathematics models and structures as a source of inspiration, includingAnthony Hill andPeter Lowe.[45] Computer-generated art is based on mathematicalalgorithms.
Mathematics rightly viewed possesses not only truth but supreme beauty a beauty cold and austere like that of sculpture without appeal to any part of our weaker nature without the gorgeous trappings Russell.
"...there is nothing in the world of mathematics that corresponds to an audience in a concert hall, where the passive listen to the active. Happily, mathematicians are alldoers, not spectators.
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