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[[File:Pythagorean proof (1).svg|300px|right|thumb|An example of "beauty in method"—a simple and elegant proof of the [[Pythagorean theorem]].]] | |||
'''Mathematical beauty''' describes the notion that some [[mathematician]]s may 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>{{cite book|last=Russell|first=Bertrand|authorlink=Bertrand Russell|title=Mysticism and Logic: And Other Essays|publisher=[[Longman]]|year=1919|page=60|chapter=The Study of Mathematics|url=http://books.google.com/?id=zwMQAAAAYAAJ&pg=PA60&dq=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+inauthor:Russell|accessdate=2008-08-22}}</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."<ref>{{cite book|last=Devlin|first=Keith|title=The Math Gene: How Mathematical Thinking Evolved And Why Numbers Are Like Gossip|publisher=[[Basic Books]]|year=2000|page=140|chapter=Do Mathematicians Have Different Brains?|url=http://books.google.com/?id=AJdmfYEaLG4C&pg=PA140&lpg=PA140&dq=Why+are+numbers+beautiful%3F+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.|accessdate=2008-08-22|isbn=978-0-465-01619-8}}</ref> | |||
==Beauty in method== | |||
Mathematicians describe an especially pleasing method of [[Mathematical proof|proof]] as ''[[Elegance|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 [[axiom]]s 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]] --> | |||
[[File: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<ref>{{citation|last = Rota|year = 1997|title=The phenomenology of mathematical beauty|page = 173}}</ref> see beauty in mathematical results that establish connections between two areas of mathematics that at first sight appear to be 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> | |||
The physicist [[Richard Feynman]] called the equation "our jewel" and "the most remarkable formula in mathematics."<ref>{{cite book|first=Richard P.|last= Feynman|title=The Feynman Lectures on Physics, vol. I|publisher=Addison-Wesley|year=1977|isbn=0-201-02010-6|page=22-10}}</ref> 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 [[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]] (and its vector versions including [[Green's theorem]] and [[Stokes' theorem]]) which is a wonderfully deep and remarkable insight and is breathtaking in its beauty. | |||
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 a beautiful proof or result possesses "inevitability", "unexpectedness", and "economy".<ref>{{cite book | last=Hardy, G.H. | chapter = 18}}</ref> | |||
[[Gian-Carlo Rota|Rota]], however, disagrees with unexpectedness as a condition for beauty and proposes a counterexample: | |||
{{quote|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.<ref>{{citation|last = Rota|title=The phenomenology of mathematical beautyyear = 1997|page = 172}}</ref>}} | |||
Perhaps ironically, Monastyrsky writes: | |||
{{quote|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.<ref>{{citation|last = Monastyrsky|title=Some Trends in Modern Mathematics and the Fields Medal|year = 2001}}</ref>}} | |||
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== | |||
[[File:Compound of five cubes.png|thumb|222px|There is a certain "cold and austere" beauty in this [[compound of five cubes]]]] | |||
{{tone|section|date=March 2013}} | |||
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>{{cite book|last=Phillips|first=George|title=Mathematics Is Not a Spectator Sport|publisher=[[Springer Science+Business Media]]|year=2005|chapter=Preface|isbn=0-387-25528-1|url=http://books.google.com/?id=psFwdN6V6icC&pg=PR7&lpg=PR7&dq=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.|accessdate=2008-08-22|quote="...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.}}</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, for example: | |||
{{quote|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]]. | |||
[[Pythagoras]] (and his entire philosophical school, 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 world view was the [[Limit of a sequence|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. | |||
[[Hungary|Hungarian]] mathematician [[Paul Erdős]]<ref>{{cite book | author=Schechter, Bruce | title=My brain is open: The mathematical journeys of Paul Erdős | publisher=[[Simon & Schuster]] | location=New York | year=2000 | pages=70–71 | isbn = 0-684-85980-7}}</ref> 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 believers. | |||
Twentieth-century French philosopher [[Alain Badiou]] claims that [[ontology]] is mathematics. Badiou also believes in deep connections between mathematics, 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 [[Circumscribed sphere|circumsphere]] of one [[polyhedron]] and the [[Inscribed sphere|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]]. | |||
==Beauty and mathematical information theory== | |||
In the 1970s, [[Abraham Moles]] and [[Frieder Nake]] analyzed links between beauty, [[information processing]], and [[information theory]].<ref>A. Moles: ''Théorie de l'information et perception esthétique'', Paris, Denoël, 1973 ([[Information Theory]] and aesthetical perception)</ref><ref>F Nake (1974). Ästhetik als Informationsverarbeitung. ([[Aesthetics]] as information processing). Grundlagen und Anwendungen der Informatik im Bereich ästhetischer Produktion und Kritik. Springer, 1974, ISBN 3-211-81216-4, ISBN 978-3-211-81216-7</ref> 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.<ref>J. Schmidhuber. [[Low-complexity art]]. Leonardo, Journal of the International Society for the Arts, Sciences, and Technology, 30(2):97–103, 1997. http://www.jstor.org/pss/1576418</ref><ref>J. Schmidhuber. Papers on the theory of beauty and [[low-complexity art]] since 1994: http://www.idsia.ch/~juergen/beauty.html</ref><ref>J. Schmidhuber. Simple Algorithmic Principles of Discovery, Subjective Beauty, Selective Attention, Curiosity & Creativity. Proc. 10th Intl. Conf. on Discovery Science (DS 2007) p. 26-38, LNAI 4755, Springer, 2007. Also in Proc. 18th Intl. Conf. on Algorithmic Learning Theory (ALT 2007) p. 32, LNAI 4754, Springer, 2007. Joint invited lecture for DS 2007 and ALT 2007, Sendai, Japan, 2007. http://arxiv.org/abs/0709.0674</ref> 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 [[Data compression|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 [[bit]]s than before, the temporary interestingness of the data corresponds to the compression progress, and is proportional to the observer's internal curiosity reward<ref>.J. Schmidhuber. Curious model-building control systems. International Joint Conference on Neural Networks, Singapore, vol 2, 1458–1463. IEEE press, 1991</ref><ref>Schmidhuber's theory of beauty and curiosity in a German TV show: http://www.br-online.de/bayerisches-fernsehen/faszination-wissen/schoenheit--aesthetik-wahrnehmung-ID1212005092828.xml</ref>{{Dead link|date=February 2011}} | |||
==Mathematics and art== | |||
The [[psychology]] of the [[aesthetics]] of mathematics is studied post-[[Psychoanalysis|psychoanalytic]]ally in [[psychosynthesis]] (the work of [[Piero Ferrucci]]), in [[cognitive psychology]] ([[illusion]] studies using [[self-similarity]] in [[Shepard tone]]s), and the [[neuropsychology]] of aesthetic appreciation. | |||
Examples of the use of mathematics in the arts include: | |||
===Music=== | |||
{{Main|Mathematics and art|Mathematics and music}} The [[Stochastic music]] of [[Iannis Xenakis]], [[counterpoint]] of [[Johann Sebastian Bach]], [[polyrhythm]]ic 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]]''. | |||
==Visual arts== | |||
Examples include applications of [[chaos theory]] and [[fractal]] [[geometry]] to [[digital art|computer-generated art]], [[symmetry]] studies of [[Leonardo da Vinci]], [[projective geometry|projective geometries]] in development of the [[Perspective (graphical)|perspective]] theory of [[Renaissance]] art, [[grid (page layout)|grids]] in [[Op art]], optical geometry in the [[camera obscura]] of [[Giambattista della Porta]], and multiple perspective in analytic [[cubism]] and [[futurism]]. | |||
The Dutch graphic designer [[M.C. Escher]] created mathematically inspired [[woodcut]]s, [[lithograph]]s, and [[mezzotint]]s. These feature impossible constructions, explorations of [[infinity]], [[architecture]], visual [[paradox]]es and tessellations. British constructionist artist [[John Ernest]] created reliefs and paintings inspired by group theory.<ref>John Ernest’s use of mathematics and especially group theory in his art works is analysed in ‘John Ernest, A Mathematical Artist’ by Paul Ernest in Philosophy of Mathematics Education Journal, No. 24 Dec. 2009 (Special Issue on Mathematics and Art): http://people.exeter.ac.uk/PErnest/pome24/index.htm</ref> A number of other British artists of the constructionist and systems schools also draw on mathematics models and structures as a source of inspiration, including [[Anthony Hill]] and [[Peter Lowe]]. Computer-generated art is based on mathematical [[algorithm]]s. | |||
===Choreography=== | |||
[[Shuffling]] has been applied to [[choreography]] as in the ''Temple of Rudra'' [[opera]].{{citation needed|date=January 2014}} | |||
==See also== | |||
<div style="-moz-column-count:2; column-count:2;"> | |||
* [[Descriptive science]] | |||
* [[Fluency heuristic]] | |||
* [[Golden ratio]] | |||
* [[Mathematics and architecture]] | |||
* [[Normative science]] | |||
* [[Philosophy of mathematics]] | |||
* [[Processing fluency theory of aesthetic pleasure]] | |||
* [[Pythagoreanism]] | |||
</div> | |||
==Notes== | |||
{{reflist|2}} | |||
==References== | |||
{{refbegin}} | |||
* [[Martin Aigner|Aigner, Martin]], and [[Günter 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 (science writer)|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. | |||
* [[Elisha Scott Loomis|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. | |||
* [[Rolf Reber|Reber, R.]], Brun, M., & Mitterndorfer, K. (2008). The use of heuristics in intuitive mathematical judgment. ''Psychonomic Bulletin & Review'', ''15'', 1174-1178. | |||
* Strohmeier, John, and Westbrook, Peter (1999), ''Divine Harmony, The Life and Teachings of Pythagoras'', Berkeley Hills Books, Berkeley, CA. | |||
* {{Cite journal | |||
| last = Rota | |||
| first = Gian-Carlo | |||
| author-link = Gian-Carlo Rota | |||
| title = The phenomenology of mathematical beauty | |||
| year = 1997 | |||
| journal = Synthese | |||
| volume = 111 | |||
| issue = 2 | |||
| pages = 171–182 | |||
| doi = 10.1023/A:1004930722234 | |||
| ref = harv | |||
| postscript = <!--None--> | |||
}} | |||
*{{Cite journal | |||
| 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 | |||
| ref = harv | |||
| postscript = <!--None--> | |||
}} | |||
{{refend}} | |||
==Further reading== | |||
{{refbegin}} | |||
* [[Serge Lang]] (1985). [http://books.google.com/books?id=U_HITkWziLwC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false ''The Beauty of Doing Mathematics: Three Public Dialogues'']. New York: Springer-Verlag. ISBN 0-387-96149-6. | |||
{{refend}} | |||
==External links== | |||
*[http://raharoni.net.technion.ac.il/mathematics-poetry-and-beauty/ Mathematics, Poetry and Beauty] | |||
*[http://www.cut-the-knot.org/manifesto/beauty.shtml Is Mathematics Beautiful?] | |||
*[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''] | |||
*[[Terence Tao]], [http://www.math.ucla.edu/~tao/preprints/Expository/goodmath.dvi ''What is good mathematics?''] | |||
*[http://mathbeauty.wordpress.com/ Mathbeauty Blog] | |||
*[http://www.nybooks.com/articles/archives/2013/dec/05/mathematical-romance/ ''A Mathematical Romance''] [[Jim Holt (philosopher)|Jim Holt]] December 5, 2013 issue of [[The New York Review of Books]] review of ''Love and Math: The Heart of Hidden Reality'' by [[Edward Frenkel]] | |||
{{aesthetics}} | |||
[[Category:Aesthetic beauty]] | |||
[[Category:Elementary mathematics]] | |||
[[Category:Philosophy of mathematics]] | |||
[[Category:Mathematical terminology]] | |||
Revision as of 19:19, 24 January 2014
.svg)
Mathematical beauty describes the notion that some mathematicians may 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.
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.[1]
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."[2]
Beauty in method
Mathematicians describe an especially pleasing method of 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.[3] 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
Some mathematicians[4] see beauty in mathematical results that establish connections between two areas of mathematics that at first sight appear to be 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:
The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics."[5] 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 (and its vector versions including Green's theorem and Stokes' theorem) which is a wonderfully deep and remarkable insight and is breathtaking in its beauty.
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, Hardy suggests that a beautiful proof or result possesses "inevitability", "unexpectedness", and "economy".[6]
Rota, however, disagrees with unexpectedness as a condition for beauty and proposes a counterexample: 31 year-old Systems Analyst Bud from Deep River, spends time with pursuits for instance r/c cars, property developers new condo in singapore singapore and books. Last month just traveled to Orkhon Valley Cultural Landscape. Perhaps ironically, Monastyrsky writes: 31 year-old Systems Analyst Bud from Deep River, spends time with pursuits for instance r/c cars, property developers new condo in singapore singapore and books. Last month just traveled to Orkhon Valley Cultural Landscape. 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
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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.
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.[7] 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, for example: 31 year-old Systems Analyst Bud from Deep River, spends time with pursuits for instance r/c cars, property developers new condo in singapore singapore and books. Last month just traveled to Orkhon Valley Cultural Landscape. 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, 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 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.
Hungarian mathematician Paul Erdős[8] 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 believers.
Twentieth-century French philosopher Alain Badiou claims that ontology is mathematics. Badiou also believes in deep connections between mathematics, 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 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.
Beauty and mathematical information theory
In the 1970s, Abraham Moles and Frieder Nake analyzed links between beauty, information processing, and information theory.[9][10] 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.[11][12][13] 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 interestingness of the data corresponds to the compression progress, and is proportional to the observer's internal curiosity reward[14][15]Template:Dead link
Mathematics and art
The psychology of the aesthetics of mathematics is studied post-psychoanalytically in psychosynthesis (the work of Piero Ferrucci), in cognitive psychology (illusion studies using self-similarity in Shepard tones), and the neuropsychology of aesthetic appreciation. Examples of the use of mathematics in the arts include:
Music
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. The Stochastic music of Iannis Xenakis, 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 Stockhausens Hymnen.
Visual arts
Examples 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.
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. British constructionist artist John Ernest created reliefs and paintings inspired by group theory.[16] A number of other British artists of the constructionist and systems schools also draw on mathematics models and structures as a source of inspiration, including Anthony Hill and Peter Lowe. Computer-generated art is based on mathematical algorithms.
Choreography
Shuffling has been applied to choreography as in the Temple of Rudra opera.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.
See also
Notes
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
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.
- Reber, R., Brun, M., & Mitterndorfer, K. (2008). The use of heuristics in intuitive mathematical judgment. Psychonomic Bulletin & Review, 15, 1174-1178.
- Strohmeier, John, and Westbrook, Peter (1999), Divine Harmony, The Life and Teachings of Pythagoras, Berkeley Hills Books, Berkeley, CA.
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A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
Further reading
- Serge Lang (1985). The Beauty of Doing Mathematics: Three Public Dialogues. New York: Springer-Verlag. ISBN 0-387-96149-6.
External links
- Mathematics, Poetry and Beauty
- Is Mathematics Beautiful?
- The Beauty of Mathematics
- Justin Mullins
- Edna St. Vincent Millay (poet): Euclid alone has looked on beauty bare
- Terence Tao, What is good mathematics?
- Mathbeauty Blog
- A Mathematical Romance Jim Holt December 5, 2013 issue of The New York Review of Books review of Love and Math: The Heart of Hidden Reality by Edward Frenkel
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ Elisha Scott Loomis published over 360 proofs in his book Pythagorean Proposition (ISBN 0873530365).
- ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ A. Moles: Théorie de l'information et perception esthétique, Paris, Denoël, 1973 (Information Theory and aesthetical perception)
- ↑ F Nake (1974). Ästhetik als Informationsverarbeitung. (Aesthetics as information processing). Grundlagen und Anwendungen der Informatik im Bereich ästhetischer Produktion und Kritik. Springer, 1974, ISBN 3-211-81216-4, ISBN 978-3-211-81216-7
- ↑ J. Schmidhuber. Low-complexity art. Leonardo, Journal of the International Society for the Arts, Sciences, and Technology, 30(2):97–103, 1997. http://www.jstor.org/pss/1576418
- ↑ J. Schmidhuber. Papers on the theory of beauty and low-complexity art since 1994: http://www.idsia.ch/~juergen/beauty.html
- ↑ J. Schmidhuber. Simple Algorithmic Principles of Discovery, Subjective Beauty, Selective Attention, Curiosity & Creativity. Proc. 10th Intl. Conf. on Discovery Science (DS 2007) p. 26-38, LNAI 4755, Springer, 2007. Also in Proc. 18th Intl. Conf. on Algorithmic Learning Theory (ALT 2007) p. 32, LNAI 4754, Springer, 2007. Joint invited lecture for DS 2007 and ALT 2007, Sendai, Japan, 2007. http://arxiv.org/abs/0709.0674
- ↑ .J. Schmidhuber. Curious model-building control systems. International Joint Conference on Neural Networks, Singapore, vol 2, 1458–1463. IEEE press, 1991
- ↑ Schmidhuber's theory of beauty and curiosity in a German TV show: http://www.br-online.de/bayerisches-fernsehen/faszination-wissen/schoenheit--aesthetik-wahrnehmung-ID1212005092828.xml
- ↑ John Ernest’s use of mathematics and especially group theory in his art works is analysed in ‘John Ernest, A Mathematical Artist’ by Paul Ernest in Philosophy of Mathematics Education Journal, No. 24 Dec. 2009 (Special Issue on Mathematics and Art): http://people.exeter.ac.uk/PErnest/pome24/index.htm