An example of "beauty in method" - a simple and elegant proof of the Pythagorean theorem.

Some mathematicians derive aesthetic pleasure from their work, and from mathematics in general. They express this pleasure by describing mathematics (or, at least, some aspect of mathematics) as beautiful. Sometimes mathematicians describe mathematics as an art form or, at a minimum, as a creative activity. Comparisons are often made with music and poetry. Image File history File links Pythagorean_proof. ... Image File history File links Pythagorean_proof. ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... The Parthenons facade showing an interpretation of golden rectangles in its proportions. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... This article is about the philosophical concept of Art. ... For other uses, see Music (disambiguation). ... This article is about the art form. ...

Bertrand Russell expressed his sense of mathematical beauty in these words: Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS, (18 May 1872 – 2 February 1970), was a British philosopher, logician, mathematician, advocate for social reform, and pacifist. ...

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. (The Study of Mathematics, in Mysticism and Logic, and Other Essays, ch. 4, London: Longmans, Green, 1918.)

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." Paul ErdÅ‘s, also ErdÅ‘s Pál, in English Paul Erdos or Paul Erdös (March 26, 1913 – September 20, 1996), was an immensely prolific (and famously eccentric) Hungarian-born mathematician. ... To say that something is ineffable means that it cannot or should not, for overwhelming reasons, be expressed in spoken words. ...

Beauty in method

Mathematicians describe an especially pleasing method of proof as elegant. Depending on context, this may mean: In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ...

• A proof that uses a minimum of additional assumptions or previous results.
• A proof that is unusually short.
• A proof that derives a result in a surprising way (e.g. from an apparently unrelated theorem or collection of theorems.)
• A proof that is based on new and original insights.
• A method of proof that can be easily generalised to solve a family of similar problems.

In the search for an elegant proof, mathematicians often look for different independent ways to prove a result — the first proof that is found may not be the best. The theorem for which the greatest number of different proofs have been discovered is possibly the Pythagorean theorem with hundreds of proofs having been published.¹ Another theorem that has been proved in many different ways is the theorem of quadratic reciprocity — Carl Friedrich Gauss alone published eight different proofs of this theorem. In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... In number theory, the law of quadratic reciprocity connects the solvability of two related quadratic equations in modular arithmetic. ... Johann Carl Friedrich Gauss or Gauß ( ; Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ...

Conversely, results that are logically correct but involve laborious calculations, over-elaborate methods, very conventional approaches, or that rely on a large number of particularly powerful axioms or previous results are not usually considered to be elegant, and may be called ugly or clumsy. This is perhaps related to the notion of Occam's Razor. For the House episode, see Occams Razor (House episode) Occams razor (sometimes spelled Ockhams razor) is a principle attributed to the 14th-century English logician and Franciscan friar William of Ockham. ...

Beauty in results

Starting at e⁰ = 1, travelling at the velocity i relative to one's position for the length of time π, and adding 1, one arrives at 0. (The diagram is an argand diagram)

Some mathematicians see beauty in mathematical results which establish connections between two areas of mathematics that at first sight appear to be totally unrelated. These results are often described as deep. Image File history File links EulerIdentity2. ... Image File history File links EulerIdentity2. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...

While it is difficult to find universal agreement on whether a result is deep, some examples are often cited. One is Euler's identity e^iπ + 1 = 0. This has been called "the most remarkable formula in mathematics" by Richard Feynman. 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 connected the Monster group to modular functions via a string theory for which Richard Borcherds was awarded the Fields medal. For other meanings, see List of topics named after Leonhard Euler In mathematical analysis, Eulers identity, named after Leonhard Euler, is the equation where is Eulers number, the base of the natural logarithm, is the imaginary unit, one of the two complex numbers whose square is negative one... This article is about the physicist. ... In mathematics, the modularity theorem establishes an important connection between elliptic curves over the field of rational numbers and modular forms, certain analytic functions introduced in 19th century mathematics. ... In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ... Modular form - Wikipedia /**/ @import /skins-1. ... The Wolf Prize has been awarded annually since 1978 to living scientists and artists for achievements in the interest of mankind and friendly relations among peoples . ... For the French mathematician with work in the area of elliptic curves, see André Weil. ... Robert Langlands (born 1936 in Canada) is one of the most significant mathematicians of the 20th century, with profound insights in number theory and representation theory. ... In mathematics, monstrous moonshine is a term devised by John Horton Conway and Simon P. Norton in 1979, used to describe the (then totally unexpected) connection between the monster group M and modular functions (particularly, the j function). ... In mathematics, the Monster group M is a group of order    246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71 = 808017424794512875886459904961710757005754368000000000 ≈ 8 · 1053. ... In mathematics, modular functions are certain kinds of mathematical functions mapping complex numbers to complex numbers. ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects called strings, rather than the zero-dimensional point... Richard Ewen Borcherds (born November 29, 1959) is a British mathematician specializing in lattices, number theory, group theory, and infinite-dimensional algebras. ... The Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, a meeting that takes place every four years. ...

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. Sometimes, however, a statement of a theorem can be original enough to be considered deep, even though its proof is fairly obvious.

Beauty in experience

Some degree of delight in the manipulation of numbers and symbols is probably required to engage in any mathematics. Given the utility of mathematics in science and engineering, it is likely that any technological society will actively cultivate these aesthetics, certainly in its philosophy of science if nowhere else. For other uses, see Number (disambiguation). ... Part of a scientific laboratory at the University of Cologne. ... Engineering is the applied science of acquiring and applying knowledge to design, analysis, and/or construction of works for practical purposes. ... The Parthenons facade showing an interpretation of golden rectangles in its proportions. ... Philosophy of science is the study of assumptions, foundations, and implications of science, especially in the natural sciences and social sciences. ...

The most intense experience of mathematical beauty for most mathematicians comes from actively engaging in mathematics. It is very difficult to enjoy or appreciate mathematics in a purely passive way - in mathematics there is no real analogy of the role of the spectator, audience, or viewer.^[1] Bertrand Russell referred to the austere beauty of mathematics. Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS, (18 May 1872 – 2 February 1970), was a British philosopher, logician, mathematician, advocate for social reform, and pacifist. ...

Beauty and philosophy

Some mathematicians are of the opinion that the doing of mathematics is closer to discovery than invention. 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. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... This article does not cite any references or sources. ...

Pythagoras (and his entire philosophical school of the Pythagoreans) believed in the literal reality of numbers. The discovery of the existence of irrational numbers was a shock to them - they considered the existence of numbers not expressible as the ratio of two natural numbers to be a flaw in nature. From the modern perspective, Pythagoras' mystical treatment of numbers was that of a numerologist rather than a mathematician. It turns out that what Pythagoras had missed in his insufficiently sophisticated world view was the limits of infinite sequences of ratios of natural numbers - the modern notion of a real number. Pythagoras of Samos (Greek: ; between 580 and 572 BC–between 500 and 490 BC) was an Ionian (Greek) philosopher[1] and founder of the religious movement called Pythagoreanism. ... The Pythagoreans were a Hellenic organization of astronomers, musicians, mathematicians, and philosophers who believed that all things are, essentially, numeric. ... In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... Numerology is an arcane study of the purported mystical relationship between numbers and the character or action of physical objects and living things. ...

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. PLATO was one of the first generalized Computer assisted instruction systems, originally built by the University of Illinois (U of I) and later taken over by Control Data Corporation (CDC), who provided the machines it ran on. ...

Galileo Galilei is reported to have said "Mathematics is the language with which God wrote the universe", a statement which (apart from the implicit deism) is consistent with the mathematical basis of all modern physics. Galileo Galilei (15 February 1564 – 8 January 1642) was an Italian physicist, mathematician, astronomer, and philosopher who is closely associated with the scientific revolution. ... For other uses, see Ceremonial Deism. ... This is a discussion of a present category of science. ...

Hungarian mathematician Paul Erdős, although an atheist, 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!". This viewpoint expresses the idea that mathematics, as the intrinsically true foundation on which the laws of our universe are built, is a natural candidate for what has been personified as God by different religious mystics. Paul ErdÅ‘s, also ErdÅ‘s Pál, in English Paul Erdos or Paul Erdös (March 26, 1913 – September 20, 1996), was an immensely prolific (and famously eccentric) Hungarian-born mathematician. ... For information about the band, see Atheist (band). ... For other uses, see Universe (disambiguation). ... This article discusses the term God in the context of monotheism and henotheism. ...

Twentieth-century French philosopher Alain Badiou claims that ontology is mathematics. Badiou also believes in deep connections between math, poetry and philosophy. This article does not cite any references or sources. ... This article is about ontology in philosophy. ...

In some 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 had 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 theory could only accommodate six planetary orbits, and was disproved by the subsequent discovery of Uranus. Johannes Kepler (December 27, 1571 – November 15, 1630) was a German mathematician, astronomer and astrologer, and a key figure in the 17th century astronomical revolution. ... This article is about the Solar System. ... In geometry, a Platonic solid is a convex regular polyhedron. ... For other uses, see Uranus (disambiguation). ...

References

• Aigner, Martin, and Ziegler, Gunter M. (2003), Proofs from THE BOOK, 3rd edition, Springer-Verlag.

• Chandrasekhar, Subrahmanyan (1987), Truth and Beauty. Aesthetics and Motivations in Science, University of Chicago Press, Chicago, IL.

• Hadamard, Jacques (1949), The Psychology of Invention in the Mathematical Field, 1st edition, Princeton University Press, Princeton, NJ. 2nd edition, 1949. Reprinted, Dover Publications, New York, NY, 1954.

• Hardy, G.H. (1940), A Mathematician's Apology, 1st published, 1940. Reprinted, C.P. Snow (foreword), 1967. Reprinted, Cambridge University Press, Cambridge, UK, 1992.

• Hoffman, Paul (1992), The Man Who Loved Only Numbers, Hyperion.

• Huntley, H.E. (1970), The Divine Proportion: A Study in Mathematical Beauty, Dover Publications, New York, NY.

• Loomis, Elisha Scott (1968), The Pythagorean Proposition, The National Council of Teachers of Mathematics. Contains 365 proofs of the Pythagorean Theorem.

• Peitgen, H.-O., and Richter, P.H. (1986), The Beauty of Fractals, Springer-Verlag.

• Strohmeier, John, and Westbrook, Peter (1999), Divine Harmony, The Life and Teachings of Pythagoras, Berkeley Hills Books, Berkeley, CA.

Proofs from THE BOOK is a book of mathematical proofs by Martin Aigner and Günter M Ziegler. ... Chandrasekhar redirects here. ... This page is a candidate for speedy deletion. ... G. H. Hardy Godfrey Harold Hardy (February 7, 1877 – December 1, 1947) was a prominent British mathematician, known for his achievements in number theory and mathematical analysis. ... C. P. Snow, born Charles Percy Snow, (1905-1980) was a scientist and novelist. ... Paul Hoffman has been publisher of Encyclopædia Britannica since June 1997. ... To meet Wikipedias quality standards, this article may require cleanup. ...

See also

• Aesthetics
• Descriptive science
• Elegance
• Mathematics and art
• Fractal
• Mathematics and architecture
• Normative science
• Philosophy of mathematics
• Truth

The Parthenons facade showing an interpretation of golden rectangles in its proportions. ... A descriptive science, also called a special science, is a form of inquiry, typically involving a community of inquiry and its accumulated body of provisional knowledge, that seeks to discover what is true about a recognized domain of phenomena. ... Elegance is the attribute of being unusually effective and simple. ... Mathematics and art have a long historical relationship. ... The boundary of the Mandelbrot set is a famous example of a fractal. ... Mathematics and architecture have always enjoyed a close association with each other, not only in the sense that the latter is informed by the former, but also in that both share the search for order and beauty, the former in nature and the latter in buildings. ... The three normative sciences, according to traditional conceptions in philosophy, are aesthetics, ethics, and logic. ... // Philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. ... A common dictionary definition of truth is agreement with fact or reality.[1] There is no single definition of truth about which the majority of philosophers agree. ...

External links

• Is Mathematics Beautiful?

• Links Concerning Beauty and Mathematics

• Mathematics and Beauty

• The Beauty of Mathematics

• Justin Mullins

• Edna St. Vincent Millay (poet): Euclid alone has looked on beauty bare

• The method for transformation of music into an image through numbers and geometrical proportions

• Terence Tao, What is good mathematics?