Mathematical beauty is the aesthetic pleasure derived from the abstractness, purity, simplicity, depth or orderliness of mathematics. Mathematicians may express this pleasure by describing mathematics (or, at least, some aspect of mathematics) as beautiful or describe mathematics as an art form, (a position taken by G. H. Hardy [1] ) or, at a minimum, as a creative activity.
Mathematicians [ who? ][ when? ] describe an especially pleasing method of proof as elegant . Depending on context, this may mean:
In the search for an elegant proof, mathematicians[ who? ] often[ how 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 the Pythagorean theorem, with hundreds of proofs being published up to date. [2] Another theorem that has been proved in many different ways is the theorem of quadratic reciprocity. In fact, Carl Friedrich Gauss alone had eight different proofs of this theorem, six of which he published. [3]
Conversely, results that are logically correct but involve laborious calculations, over-elaborate methods, highly conventional approaches or a large number of powerful axioms or previous results are usually not considered to be elegant, and may be even referred to as ugly or clumsy.
Some mathematicians see beauty in mathematical results that establish connections between two areas of mathematics that at first sight appear to be unrelated. [4] These results are often described as deep . While it is difficult to find universal agreement on whether a result is deep, some examples are more commonly cited than others. One such example is Euler's identity: [5]
This elegant expression ties together arguably the five most important mathematical constants (e, i, π, 1, and 0) with the two most common mathematical symbols (+, =). Euler's identity is a special case of Euler's formula, which the physicist Richard Feynman called "our jewel" and "the most remarkable formula in mathematics". [6] Modern examples include the modularity theorem, which establishes an important connection between elliptic curves and modular forms (work on which led to the awarding of the Wolf Prize to Andrew Wiles and Robert Langlands), and "monstrous moonshine", which connects the Monster group to modular functions via string theory (for which Richard Borcherds was awarded the Fields Medal).
Other examples of deep results include unexpected insights into mathematical structures. For example, Gauss's Theorema Egregium is a deep theorem which relates a local phenomenon (curvature) to a global phenomenon (area) in a surprising way. In particular, the area of a triangle on a curved surface is proportional to the excess of the triangle and the proportionality is curvature. Another example is the fundamental theorem of calculus [7] (and its vector versions including Green's theorem and Stokes' theorem).
The opposite of deep is trivial. A trivial theorem may be a result that can be derived in an obvious and straightforward way from other known results, or which applies only to a specific set of particular objects such as the empty set. In some occasions, a statement of a theorem can be original enough to be considered deep, though its proof is fairly obvious.
In his 1940 essay A Mathematician's Apology , G. H. Hardy suggested that a beautiful proof or result possesses "inevitability", "unexpectedness", and "economy". [8]
In 1997, Gian-Carlo Rota, disagreed with unexpectedness as a sufficient 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 of non-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. [9]
In contrast, Monastyrsky wrote in 2001:
It is very difficult to find an analogous invention in the past to Milnor'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. Brieskorn showed that these differential structures can be described in an extremely explicit and beautiful form. [10]
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.
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Interest in pure mathematics that is separate from empirical study has been part of the experience of various civilizations, including that of the ancient Greeks, who "did mathematics for the beauty of it". [11] The aesthetic pleasure that mathematical physicists tend to experience in Einstein's theory of general relativity has been attributed (by Paul Dirac, among others) to its "great mathematical beauty". [12] The beauty of mathematics is experienced when the physical reality of objects are represented by mathematical models. Group theory, developed in the early 1800s for the sole purpose of solving polynomial equations, became a fruitful way of categorizing elementary particles—the building blocks of matter. Similarly, the study of knots provides important insights into string theory and loop quantum gravity.[ citation needed ]
Some[ who? ] believe that in order to appreciate mathematics, one must engage in doing mathematics. [13]
For example, Math Circles are after-school enrichment programs where students engage with mathematics through lectures and activities; there are also some teachers who encourage student engagement by teaching mathematics in kinesthetic learning. In a general Math Circle lesson, students use pattern finding, observation, and exploration to make their own mathematical discoveries. For example, mathematical beauty arises in a Math Circle activity on symmetry designed for 2nd and 3rd graders, where students create their own snowflakes by folding a square piece of paper and cutting out designs of their choice along the edges of the folded paper. When the paper is unfolded, a symmetrical design reveals itself. In a day to day elementary school mathematics class, symmetry can be presented as such in an artistic manner where students see aesthetically pleasing results in mathematics.[ citation needed ]
Some[ who? ] teachers prefer to use mathematical manipulatives to present mathematics in an aesthetically pleasing way. Examples of a manipulative include algebra tiles, cuisenaire rods, and pattern blocks. For example, one can teach the method of completing the square by using algebra tiles. Cuisenaire rods can be used to teach fractions, and pattern blocks can be used to teach geometry. Using mathematical manipulatives helps students gain a conceptual understanding that might not be seen immediately in written mathematical formulas. [14]
Another example of beauty in experience involves the use of origami. Origami, the art of paper folding, has aesthetic qualities and many mathematical connections. One can study the mathematics of paper folding by observing the crease pattern on unfolded origami pieces. [15]
Combinatorics, the study of counting, has artistic representations which some[ who? ] find mathematically beautiful. There are many visual examples which illustrate combinatorial concepts. Some of the topics and objects seen in combinatorics courses with visual representations include, among others Four color theorem, Young tableau, Permutohedron, Graph theory, Partition of a set. [16]
Brain imaging experiments conducted by Semir Zeki and his colleagues [17] show that the experience of mathematical beauty has, as a neural correlate, activity in field A1 of the medial orbito-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 music or joy or sorrow. Moreover, mathematicians seem resistant to revising their judgment of the beauty of a mathematical formula in light of contradictory opinion given by their peers. [18]
Some[ who? ] mathematicians are of the opinion that the doing of mathematics is closer to discovery than invention, for example:
There is no scientific discoverer, no poet, no painter, no musician, who will not tell you that he found ready made his discovery or poem or picture—that it came to him from outside, and that he did not consciously create it from within.
— William Kingdon Clifford, from a lecture to the Royal Institution titled "Some of the conditions of mental development"
These mathematicians believe that the detailed and precise results of mathematics may be reasonably taken to be true without any dependence on the universe in which we live. For example, they would argue that the theory of the natural numbers is fundamentally valid, in a way that does not require any specific context. Some mathematicians have extrapolated this viewpoint that mathematical beauty is truth further, in some cases becoming mysticism.
In Plato's philosophy there were two worlds, the physical one in which we live and another abstract world which contained unchanging truth, including mathematics. He believed that the physical world was a mere reflection of the more perfect abstract world. [19]
Hungarian mathematician Paul Erdős [20] spoke of an imaginary 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 exclaim "This one's from The Book!"
Twentieth-century French philosopher Alain Badiou claimed that ontology is mathematics. [21] Badiou also believes in deep connections between mathematics, poetry and philosophy.
In many cases, natural philosophers and other scientists who have made extensive use of mathematics have made leaps of inference between beauty and physical truth in ways that turned out to be erroneous. For example, at one stage in his life, Johannes Kepler believed that the proportions of the orbits of the then-known planets in the Solar System have been arranged by God to correspond to a concentric arrangement of the five Platonic solids, each orbit lying on the circumsphere of one polyhedron and the insphere of another. As there are exactly five Platonic solids, Kepler's hypothesis could only accommodate six planetary orbits and was disproved by the subsequent discovery of Uranus.
In the 1970s, Abraham Moles and Frieder Nake analyzed links between beauty, information processing, and information theory. [22] [23] In the 1990s, Jürgen Schmidhuber formulated a mathematical theory of observer-dependent subjective beauty based on algorithmic information theory: the most beautiful objects among subjectively comparable objects have short algorithmic descriptions (i.e., Kolmogorov complexity) relative to what the observer already knows. [24] [25] [26] Schmidhuber explicitly distinguishes between beautiful and interesting. The latter corresponds to the first derivative of subjectively perceived beauty: the observer continually tries to improve the predictability and compressibility of the observations by discovering regularities such as repetitions and symmetries and fractal self-similarity. Whenever the observer's learning process (possibly a predictive artificial neural network) leads to improved data compression such that the observation sequence can be described by fewer bits than before, the temporary interesting-ness of the data corresponds to the compression progress, and is proportional to the observer's internal curiosity reward. [27] [28]
Examples of the use of mathematics in music include the stochastic music of Iannis Xenakis, the Fibonacci sequence in Tool's Lateralus, counterpoint of Johann Sebastian Bach, polyrhythmic structures (as in Igor Stravinsky's The Rite of Spring ), the Metric modulation of Elliott Carter, permutation theory in serialism beginning with Arnold Schoenberg, and application of Shepard tones in Karlheinz Stockhausen's Hymnen . They also include the application of Group theory to transformations in music in the theoretical writings of David Lewin.
Examples of the use of mathematics in the visual arts include applications of chaos theory and fractal geometry to computer-generated art, symmetry studies of Leonardo da Vinci, projective geometries in development of the perspective theory of Renaissance art, grids in Op art, optical geometry in the camera obscura of Giambattista della Porta, and multiple perspective in analytic cubism and futurism.
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 in Islamic architecture. It also provides a means of meditation and comtemplation, for example the study of the Kaballah Sefirot (Tree Of Life) and Metatron's Cube; and also the act of drawing itself.
The Dutch graphic designer M. C. Escher created mathematically inspired woodcuts, lithographs, and mezzotints. These feature impossible constructions, explorations of infinity, architecture, visual paradoxes and tessellations.
Some painters and sculptors create work distorted with the mathematical principles of anamorphosis, including South African sculptor Jonty Hurwitz.
British constructionist artist John Ernest created reliefs and paintings inspired by group theory. [29] 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, including Anthony Hill and Peter Lowe. [30] Computer-generated art is based on mathematical algorithms.
Bertrand Russell expressed his sense of mathematical beauty in these words:
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. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry. [31]
Paul Erdős expressed his views on the ineffability of mathematics when he said, "Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is". [32]
"...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 all doers, not spectators.
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.
Aesthetics is the branch of philosophy concerned with the nature of beauty and the nature of taste; and functions as the philosophy of art. Aesthetics examines the philosophy of aesthetic value, which is determined by critical judgements of artistic taste; thus, the function of aesthetics is the "critical reflection on art, culture and nature".
Beauty is commonly described as a feature of objects that makes them pleasurable to perceive. Such objects include landscapes, sunsets, humans and works of art. Beauty, art and taste are the main subjects of aesthetics, one of the major branches of philosophy. As a positive aesthetic value, it is contrasted with ugliness as its negative counterpart.
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline.
In mathematics, a theorem is a statement that has been proved, or can be proved. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's lives.
A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.
Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of the mathematical field of combinatorics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in Ramsey theory typically ask a question of the form: "how big must some structure be to guarantee that a particular property holds?"
In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or an object which possesses a simple structure. The noun triviality usually refers to a simple technical aspect of some proof or definition. The origin of the term in mathematical language comes from the medieval trivium curriculum, which distinguishes from the more difficult quadrivium curriculum. The opposite of trivial is nontrivial, which is commonly used to indicate that an example or a solution is not simple, or that a statement or a theorem is not easy to prove.
Ronald Lewis Graham was an American mathematician credited by the American Mathematical Society as "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years". He was president of both the American Mathematical Society and the Mathematical Association of America, and his honors included the Leroy P. Steele Prize for lifetime achievement and election to the National Academy of Sciences.
Low-complexity art, first described by Jürgen Schmidhuber in 1997 and now established as a seminal topic within the larger field of computer science, is art that can be described by a short computer program.
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles.
Paul Erdős was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. Erdős pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered around discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics.
Neuroesthetics is a relatively recent sub-discipline of applied aesthetics. Empirical aesthetics takes a scientific approach to the study of aesthetic experience of art, music, or any object that can give rise to aesthetic judgments. Neuroesthetics is a term coined by Semir Zeki in 1999 and received its formal definition in 2002 as the scientific study of the neural bases for the contemplation and creation of a work of art. Neuroesthetics uses neuroscience to explain and understand the aesthetic experiences at the neurological level. The topic attracts scholars from many disciplines including neuroscientists, art historians, artists, art therapists and psychologists.
The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in lectures, and sometimes in print, as informal shorthand for rigorous arguments or precise ideas. Much of this is common English, but with a specific non-obvious meaning when used in a mathematical sense.
In physics and cosmology, the mathematical universe hypothesis (MUH), also known as the ultimate ensemble theory, is a speculative "theory of everything" (TOE) proposed by cosmologist Max Tegmark. According to the hypothesis, the universe is a mathematical object in and of itself. Furthermore, Tegmark suggests that not only is the universe mathematical, but it is also computable.
Applied aesthetics is the application of the branch of philosophy of aesthetics to cultural constructs. In a variety of fields, artifacts are created that have both practical functionality and aesthetic affectation. In some cases, aesthetics is primary, and in others, functionality is primary. At best, the two needs are synergistic, in which "beauty" makes an artifact work better, or in which more functional artifacts are appreciated as aesthetically pleasing. This achievement of form and function, of art and science, of beauty and usefulness, is the primary goal of design, in all of its domains.
Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.
Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned in arts such as music, dance, painting, architecture, sculpture, and textiles. This article focuses, however, on mathematics in the visual arts.
A Pythagorean tiling or two squares tessellation is a tiling of a Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides. Many proofs of the Pythagorean theorem are based on it, explaining its name. It is commonly used as a pattern for floor tiles. When used for this, it is also known as a hopscotch pattern or pinwheel pattern, but it should not be confused with the mathematical pinwheel tiling, an unrelated pattern.
