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Mathematical beauty From Wikipedia, the free encyclopedia

An example of "beauty in method"—a simple and elegant geometrical proof that the Pythagorean theorem is true for a

particular right-angled triangle.

Many mathematicians derive aesthetic pleasure from their work, and frommathematics 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 Symphonybeautiful. If you don't see why, someone can't

tell you. I know numbers are beautiful. If they aren't beautiful, nothing is."[2] 

Contents

[hide] 

1 Beauty in method 

2 Beauty in results 

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  3 Beauty in experience 

4 Beauty and philosophy 

5 Beauty and mathematical information theory 

6 Mathematics and art 

7 See also 

8 Notes 

9 References 

10 External links 

[edit]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 orclumsy .

[edit]Beauty in results

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Starting at e 0 = 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 (Rota (1977), p. 173) see beauty in mathematical results that 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: 

Physicist Richard Feynman called this "the most remarkable formula in mathematics". Modern

examples include the modularity theorem, which establishes an important connection betweenelliptic

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 tomodular functions 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 , Hardy suggests that a beautiful proof or result possesses

"inevitability", "unexpectedness", and "economy".[4] 

Rota, however, disagrees with unexpectedness as a condition for beauty 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  non-equivalent

differentiable structures on spheres of high dimension was thought to be surprising, but it didnot occur to anyone to call such a fact beautiful, then or now.[5] 

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Perhaps ironically, Monastyrsky 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 structurescan be described in an extremely explicit and beautiful form.[6] 

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.

[edit]Beauty in experience

There is a certain "cold and austere" beauty in this compound of five cubes 

Infinite tesselation by M.C. Escher

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Polyhedrons and impossible constructions in Escher's "Waterfall"

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 itsphilosophy 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.

[edit]Beauty and philosophy

Some 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 wil l 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 conditionsof 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 ofnumbers. The discovery of the existence of irrational numbers was a shock to them—they considered

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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.

Galileo Galilei is reported to have said, "Mathematics is the language with which God wrote the

universe."

Hungarian mathematician Paul Erdős, although an atheist,[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 mystics.

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 byGod 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. 

[edit]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 andcompressibility of the observations by discovering regularities such as repetitions

and symmetries and fractal self-similarity. Whenever the observer's learning process (possibly apredictive artificial neural network) leads to improved data compression such that the observation

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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][dead link ] 

[edit]Mathematics and artMain articles:  Mathematics and art  and  Mathematics and music  

The psychology of the aesthetics of mathematics is studied post-

psychoanalytically in psychosynthesis (in the work of Piero Ferrucci), incognitive

psychology (in 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 – the Stochastic music of Iannis Xenakis, counterpoint of Johann Sebastian

Bach, polyrhythmic structures (as in Igor Stravinsky'sThe 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 . 

  Choreography – shuffling has been applied to choreography as in the Temple of Rudra  opera. 

  Visual arts – examples include applications of chaos theory and fractal geometry to computer-

generated art, symmetry studies ofLeonardo 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 symmetries of two dimensional tesselations and three dimensional mathematical objects, can

evoke feelings of "mathematical beauty" as expressed by Bertrand Russell in the first paragraphs of

this article. This may apply to polyhedrons (three dimensional geometric solids), many of which show

perfect symmetries that, combined with the use of colours, result in a visual experience that many

consider attractive. The use in art of such objects or tesselations is limited though, as this beauty is

often considered soulless and does not evoke feelings of emotion. 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.

Currently, also computer generated art is based on mathematical  algorithms. 

[edit]See also

  Descriptive science 

  Fluency heuristic 

  Golden ratio 

  Mathematics and architecture 

  Normative science 

  Philosophy of mathematics 

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  Processing fluency theory of aesthetic pleasure 

  Pythagoreanism 

[edit]Notes

1. ^ Russell, Bertrand (1919). "The Study of Mathematics".Mysticism and Logic: And Other 

Essays . Longman. p. 60. Retrieved 2008-08-22.

2. ^ Devlin, Keith (2000). "Do Mathematicians Have Different Brains?". The Math Gene: How 

Mathematical Thinking Evolved And Why Numbers Are Like Gossip . Basic Books. 

p. 140.ISBN 9780465016198. Retrieved 2008-08-22.

3. ^ Elisha Scott Loomis published over 360 proofs in his book Pythagorean Proposition (ISBN

0873530365).

4. ^ Hardy, G.H.. "18".

5. ^ Rota (1977), p. 172

6. ^ Monastyrsky (2001)

7. ^ Phillips, George (2005). "Preface". Mathematics Is Not a Spectator Sport . Springer

Science+Business Media.ISBN 0387255281. Retrieved 2008-08-22. ""...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."

8. ^ Schechter, Bruce (2000). My brain is open: The mathematical journeys of Paul Erdős . New

York: Simon & Schuster. pp. 70 –71.ISBN 0-684-85980-7. 

9. ^ A. Moles: Théorie de l'information et perception esthétique , Paris, Denoël, 1973 (Information

Theory and aesthetical perception)

10. ^ F Nake (1974). Ästhetik als Informationsverarbeitung. (Aestheticsas information processing).

Grundlagen und Anwendungen der Informatik im Bereich ästhetischer Produktion und Kritik.

Springer, 1974, ISBN 3211812164, ISBN 9783211812167 

11. ^ 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 

12. ^ J. Schmidhuber. Papers on the theory of beauty and low-complexity art since

1994: http://www.idsia.ch/~juergen/beauty.html 

13. ^ 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 

14. ^ .J. Schmidhuber. Curious model-building control systems. International Joint Conference on

Neural Networks, Singapore, vol 2, 1458 –1463. IEEE press, 1991

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15. ^ 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 

[edit]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.

  Rota, Gian-Carlo (1977). "The phenomenology of mathematical beauty". Synthese  111 (2): 171 –

182. doi:10.1023/A:1004930722234 

  Monastyrsky, Michael (2001). "Some Trends in Modern Mathematics and the Fields Medal". Can.

Math. Soc. Notes  33 (2 and 3)

[edit]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  

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

proportions 

  Terence Tao, What is good mathematics?  

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