Mathematical beauty 298416 224349267 2008-07-08T13:36:54Z Mhaitham.shammaa 1608031 [[Image:Pythagorean proof (1).svg|300px|right|thumb|An example of "beauty in method"—a simple and elegant proof of the [[Pythagorean theorem]].]] Many [[mathematician]]s derive [[aesthetics|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]]. [[Bertrand Russell]] expressed his sense of mathematical beauty in these words: <blockquote> 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.<ref>(''The Study of Mathematics'', in ''Mysticism and Logic, and Other Essays'', ch. 4, London: Longmans, Green, 1918.)</ref> </blockquote> [[Paul Erdős]] expressed his views on the [[ineffability]] of mathematics when he said, "Why are numbers beautiful? It's like asking why is [[Symphony No. 9 (Beethoven)|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." ==Beauty in method== Mathematicians describe an especially pleasing method of [[Mathematical proof|proof]] as ''elegant.'' Depending on context, this may mean: * A proof that uses a minimum of additional assumptions or previous results. * A proof that is unusually succinct. * 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 generalized 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.<ref>Elisha Scott Loomis published over 360 proofs in his book Pythagorean Proposition (ISBN 0873530365).</ref> 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. 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''. ==Beauty in results==<!-- This section is linked from [[Fermat's last theorem]] --> [[Image:EulerIdentity2.svg|thumb|right|Starting at ''e''<sup>0</sup> = 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 {{Harvard citations|last = Rota|year = 1977|loc = p. 173}} 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.'' While it is difficult to find universal agreement on whether a result is deep, some examples are often cited. One is [[Euler's identity]] :<math>\displaystyle e^{i \pi} + 1 = 0.</math> [[Richard Feynman]] called this "the most remarkable formula in mathematics". Modern examples include the [[modularity theorem]], which establishes an important connection between [[elliptic curve]]s and [[modular form]]s (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 function]]s via a [[string theory]] for which [[Richard Borcherds]] was awarded the [[Fields medal]]. 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. In his ''[[A Mathematician's Apology]]'', [[G. H. Hardy|Hardy]] suggests that mathematical beauty arises from an element of surprise. [[Gian-Carlo Rota|Rota]], however, disagrees and proposes 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 [[Exotic sphere|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." {{Harvard citations|last = Rota|year = 1977|loc = p. 172}} Perhaps ironically, {{Harvard citations|last = Monastyrsky|year = 2001|txt = yes}} writes: :"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. Briscorn showed that these differential structures can be described in an extremely explicit and beautiful form." 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. ==Beauty in experience== Some degree of delight in the manipulation of [[number]]s and [[symbol]]s 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. 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.<ref> "...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." {{cite book | last = Phillips | first = George M. | title = Mathematics Is Not A Spectator Sport | publisher = Springer | date = 2005 | isbn = 0387255281 }} </ref> [[Bertrand Russell]] referred to the ''austere beauty'' of mathematics. ==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]]. [[Pythagoras]] (and his entire philosophical school of the [[Pythagoreans]]) believed in the literal reality of numbers. The discovery of the existence of [[irrational number]]s 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. 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. [[Galileo Galilei]] is reported to have said, "Mathematics is the language with which God wrote the universe," a statement which (apart from the implicit [[theism]]) is consistent with the mathematical basis of all modern [[physics]]. [[Hungary|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 [[Proofs from THE BOOK|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. Twentieth-century French philosopher [[Alain Badiou]] claims that [[ontology]] is mathematics. Badiou also believes in deep connections between math, poetry and 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]] have been arranged by God to correspond to a concentric arrangement of the five [[Platonic solid]]s, 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]]. ==References== {{reflist}} * [[Martin Aigner|Aigner, Martin]], and [[Gunter M. Ziegler|Ziegler, Gunter M.]] (2003), ''[[Proofs from THE BOOK]],'' 3rd edition, Springer-Verlag. * [[Subrahmanyan Chandrasekhar|Chandrasekhar, Subrahmanyan]] (1987), ''Truth and Beauty: Aesthetics and Motivations in Science,'' University of Chicago Press, Chicago, IL. * [[Jacques Hadamard|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. * [[G.H. Hardy|Hardy, G.H.]] (1940), ''A Mathematician's Apology'', 1st published, 1940. Reprinted, [[C.P. Snow]] (foreword), 1967. Reprinted, Cambridge University Press, Cambridge, UK, 1992. * [[Paul Hoffman|Hoffman, Paul]] (1992), ''[[The Man Who Loved Only Numbers]]'', Hyperion. * [[H.E. Huntley|Huntley, H.E.]] (1970), ''The Divine Proportion: A Study in Mathematical Beauty'', Dover Publications, New York, NY. * [[Elisha Scott Loomis|Loomis, Elisha Scott]] (1968), ''The Pythagorean Proposition'', The National Council of Teachers of Mathematics. Contains 365 proofs of the Pythagorean Theorem. * [[H.-O. Peitgen|Peitgen, H.-O.]], and [[P.H. Richter|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. * {{citation | last = Rota | first = Gian-Carlo | author-link = Gian-Carlo Rota | title = The phenomenology of mathematical beauty | year = 1977 | journal = Synthese | volume = 111 | issue = 2 | pages = 171&ndash;182 | issn = 0039-7857 }} *{{citation | last1 = Monastyrsky | first1 = Michael | title = Some Trends in Modern Mathematics and the Fields Medal | journal = Can. Math. Soc. Notes | year = 2001 | volume = 33 | issue = 2 and 3 | url = http://www.fields.utoronto.ca/aboutus/FieldsMedal_Monastyrsky.pdf }} ==See also== <div style="-moz-column-count:3; column-count:3;"> * [[Aesthetics]] * [[Descriptive science]] * [[Elegance]] * [[Fluency heuristic]] * [[Fractal]] * [[Golden ratio]] * [[Mathematics and architecture]] * [[Mathematics and art]] * [[Normative science]] * [[Philosophy of mathematics]] * [[Truth]] </div> ==External links== *[http://www.cut-the-knot.org/manifesto/beauty.shtml Is Mathematics Beautiful?] *[http://www.madras.fife.sch.uk/maths/linksbeauty.html Links Concerning Beauty and Mathematics] *[http://www.chemistrycoach.com/science_mathematics_and_beauty.htm Mathematics and Beauty] *[http://users.forthnet.gr/ath/kimon/ The Beauty of Mathematics] *[http://www.justinmullins.com/ Justin Mullins] *[http://www.the-athenaeum.org/poetry/detail.php?id=80 Edna St. Vincent Millay (poet): ''Euclid alone has looked on beauty bare''] *[http://creativelab.kiev.ua/eng/index_eng.htm The method for transformation of music into an image through numbers and geometrical proportions] *[[Terence Tao]], [http://www.math.ucla.edu/~tao/preprints/Expository/goodmath.dvi ''What is good mathematics?''] [[Category:Aesthetics]] [[Category:Elementary mathematics]] [[Category:Philosophy of mathematics]] [[ar:جمال رياضياتي]] [[bn:গাণিতিক সৌন্দর্য]] [[da:Matematisk skønhed]] [[es:Belleza matemática]] [[fr:Beauté mathématique]] [[pl:Matematyka a estetyka]] [[vi:Vẻ đẹp của toán học]] [[zh:數學之美]]