Many mathematicians A mathematician is a person whose primary area of study or research, or both, is the field of mathematics. Mathematicians are concerned with particular problems related to logic, space, transformations, numbers and more general ideas which encompass these concepts. Some notable mathematicians include Sir Isaac Newton, Muhammad ibn Mūsā al-Khwā derive aesthetic Aesthetics is a branch of philosophy dealing with the nature of beauty, art, and taste, and with the creation and appreciation of beauty. It is more scientifically defined as the study of sensory or sensori-emotional values, sometimes called judgments of sentiment and taste. More broadly, scholars in the field define aesthetics as "critical pleasure from their work, and from mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions 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 Art is the process or product of deliberately arranging elements in a way to affect the senses or emotions. It encompasses a diverse range of human activities, creations, and modes of expression, including music, literature, film, photography, sculpture, and paintings. The meaning of art is explored in a branch of philosophy known as aesthetics form or, at a minimum, as a creative activity. Comparisons are often made with music Music is an art form whose medium is sound. Common elements of music are pitch , rhythm (and its associated concepts tempo, meter, and articulation), dynamics, and the sonic qualities of timbre and texture. The word derives from Greek μουσική (mousike), "(art) of the Muses." and poetry Poetry is a form of literary art in which language is used for its aesthetic and evocative qualities in addition to, or in lieu of, its apparent meaning. Poetry may be written independently, as discrete poems, or may occur in conjunction with other arts, as in poetic drama, hymns, lyrics, or prose poetry. It is published in dedicated magazines (. Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, atheist, socialist, pacifist, and social critic. Although he spent most of his life in England, he was born in Wales where he also died, aged 97 expressed his sense of mathematical beauty in these words:

Mathematics, rightly viewed, possesses not only truth Truth can have a variety of meanings, from the state of being the case, being in accord with a particular fact or reality, being in accord with the body of real things, events, actuality, or fidelity to an original or to a standard, truth "behind" everything, the ontological truth. In archaic usage it could be fidelity, constancy or, but supreme beauty — a beauty cold and austere, like that of sculpture Sculpture is three-dimensional artwork created by shaping or combining hard materials, typically stone such as marble, metal, glass, or wood, or plastic materials such as clay, textiles, polymers and softer metals. The term has been extended to works including sound, text and light, without appeal to any part of our weaker nature, without the gorgeous trappings of painting Painting is the practice of applying paint, pigment, color or other medium to a surface . The application of the medium is commonly applied to the base with a brush but other objects may be used. In art the term describes both the act and the result which is called a painting. Paintings may have for their support such surfaces as walls, paper, 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 Paul Erdős was an immensely prolific and notably eccentric Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory expressed his views on the ineffability Ineffability is concerned with ideas that cannot or should not be expressed in spoken words , often being in the form of a taboo or incomprehensible term. This property is commonly associated with philosophy, aspects of existence, and similar concepts that are inherently "too great", complex, or abstract to be adequately communicated. In of mathematics when he said, "Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony The Symphony No. 9 in D minor, Op. 125 "Choral" is the final complete symphony of Ludwig van Beethoven. Completed in 1824, the symphony is one of the best known works of the Western classical repertoire. It is considered one of Beethoven's most highly regarded masterpieces 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 In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single exception. An unproved proposition that is believed to 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 In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle . In terms of areas, it states:, with hundreds of proofs having been published.^[3] Another theorem that has been proved in many different ways is the theorem of quadratic reciprocity The law of quadratic reciprocity is a theorem from modular arithmetic, a branch of number theory, which gives conditions for the solvability of quadratic equations modulo prime numbers. There are a number of equivalent statements of the theorem, which consists of two "supplements" and the reciprocity law:—Carl Friedrich Gauss Johann Carl Friedrich Gauss (pronounced /ˈɡaʊs/; German: Gauß listen , Latin: Carolus Fridericus Gauss) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy 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 (Rota 1977, 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 Euler's identity is considered by many to be remarkable for its mathematical beauty. Three basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:

Richard Feynman Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics (he proposed the parton model). For his contributions to the development of quantum called this "the most remarkable formula in mathematics". Modern examples include the modularity theorem, which establishes an important connection between elliptic curves In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity element. Often the curve itself, without O and modular forms In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other (work on which led to the awarding of the Wolf Prize The Wolf Prize is an international award, that has been presented annually since 1978 to living scientists and artists for "achievements in the interest of mankind and friendly relations among peoples ... irrespective of nationality, race, colour, religion, sex or political views." to Andrew Wiles Sir Andrew John Wiles KBE FRS is a British mathematician and a professor at Princeton University, specializing in number theory. He is most famous for proving Fermat's Last Theorem and Robert Langlands), and "monstrous moonshine," which connects the Monster group It is a simple group, meaning it does not have any normal subgroups except for the subgroup consisting only of the identity element, and M itself to modular functions via a string theory String theory is a developing theory in particle physics which attempts to reconcile quantum mechanics and general relativity. String theory posits that the electrons and quarks within an atom are not 0-dimensional objects, but rather 1-dimensional oscillating lines , possessing only the dimension of length, but not height or width. The theory for which Richard Borcherds was awarded the Fields medal 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 Fields Medal is often viewed as the top honor a mathematician can receive. It comes with a monetary award, which in 2006 was C$15,.

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 A Mathematician's Apology is a 1940 essay by British mathematician G. H. Hardy. It concerns the aesthetics of mathematics with some personal content, and gives the layman an insight into the mind of a working mathematician, Hardy Godfrey Harold Hardy FRS, known as G. H. Hardy was a prominent English mathematician, known for his achievements in number theory and mathematical analysis suggests that mathematical beauty arises from an element of surprise. 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 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." (Rota 1977, p. 172)

Perhaps ironically, Monastyrsky (2001) 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 numbers A number is a mathematical object used in counting and measuring. A notational symbol which represents a number is called a numeral, but in common usage the word number is used for both the abstract object and the symbol, as well as for the word for the number. In addition to their use in counting and measuring, numerals are often used for labels , and symbols A symbol is something such as an object, picture, written word, sound, or particular mark that represents something else by association, resemblance, or convention. For example, a red octagon may be a symbol for "STOP". On maps, crossed sabres may indicate a battlefield. Numerals are symbols for numbers . All language consists of symbols is probably required to engage in any mathematics. Given the utility of mathematics in science Science is a systematic enterprise of gathering knowledge about nature and organizing and condensing that knowledge into testable laws and theories. As knowledge has increased, some methods have proved more reliable than others, and today the scientific method is the standard for science. It includes the use of careful observation, experimentation, and engineering Engineering is the discipline, art and profession of acquiring and applying technical, scientific, and mathematical knowledge to design and implement materials, structures, machines, devices, systems, and processes that safely realize a desired objective or invention, it is likely that any technological society will actively cultivate these aesthetics Aesthetics is a branch of philosophy dealing with the nature of beauty, art, and taste, and with the creation and appreciation of beauty. It is more scientifically defined as the study of sensory or sensori-emotional values, sometimes called judgments of sentiment and taste. More broadly, scholars in the field define aesthetics as "critical, certainly in its philosophy of science The philosophy of science is concerned with the assumptions, foundations, methods and implications of science. In addition to these central problems for science as a whole, many philosophers of science consider these problems as they apply to particular sciences . Some philosophers of science also use contemporary results in science to draw 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.^[4] Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, atheist, socialist, pacifist, and social critic. Although he spent most of his life in England, he was born in Wales where he also died, aged 97 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:

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 William Kingdon Clifford FRS was an English mathematician and philosopher. Along with Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his honour, with interesting applications in contemporary mathematical physics and geometry. He was the first to suggest that gravitation might, 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 In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {1, 2, 3, ...} according to the traditional definition; or the set of non-negative integers{0, 1, 2, ...} according to a definition first appearing in the nineteenth century 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 Mysticism is the pursuit of communion with, identity with, or conscious awareness of an ultimate reality, divinity, spiritual truth, or God through direct experience, intuition, instinct or insight. Mysticism usually centers on a practice or practices intended to nurture those experiences or awareness. Mysticism may be dualistic, maintaining a.

Pythagoras Pythagoras of Samos was an Ionian Greek philosopher and founder of the religious movement called Pythagoreanism. Most of our information about Pythagoras was written down centuries after he lived, thus very little reliable information is known about him. He was born on the island of Samos, and may have travelled widely in his youth, visiting Egypt (and his entire philosophical school of the Pythagoreans Pythagoreanism is a term used for the esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were considerably influenced by mathematics. Pythagoreanism greatly influenced Platonism. Later revivals of Pythagorean doctrines led to what is now called Neopythagoreanism) believed in the literal reality of numbers. The discovery of the existence of irrational numbers In mathematics, an irrational number is any real number which cannot be expressed as a fraction p/q, where p and q are integers, with q non-zero and is therefore not a rational number. Informally, this means that an irrational number cannot be represented as a simple fraction. It can be proven that irrational numbers are precisely those real was a shock to them—they considered the existence of numbers not expressible as the ratio of two natural numbers In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {1, 2, 3, ...} according to the traditional definition; or the set of non-negative integers{0, 1, 2, ...} according to a definition first appearing in the nineteenth century 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 Plato , was a Classical Greek philosopher, mathematician, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the foundations of Western philosophy and science. Plato was originally a'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 Galileo Galilei was an Italian physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations, and support for Copernicanism. Galileo has been called the "father of modern observational astronomy," is reported to have said, "Mathematics is the language with which God wrote the universe," a statement which (apart from the implicit theism Theism in the broadest sense is the belief that at least one deity exists. In a more specific sense, theism refers to a doctrine concerning the nature of a monotheistic God and his relationship to the universe. Theism, in this specific sense, conceives of God as personal, present and active in the governance and organization of the world and the) is consistent with the mathematical basis of all modern physics Physics is a natural science that involves the study of matter and its motion through space-time, as well as all applicable concepts, such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.

Hungarian Hungary /ˈhʌŋɡəri/ (Hungarian: Magyarország [ˈmɒɟɒrorsaːɡ] ( listen)), officially the Republic of Hungary (Magyar Köztársaság listen (help·info)), is a landlocked country in the Carpathian Basin in Central Europe, bordered by Austria, Slovakia, Ukraine, Romania, Serbia, Croatia, and Slovenia. Its capital is Budapest. Hungary is a mathematician Paul Erdős Paul Erdős was an immensely prolific and notably eccentric Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory, although an atheist Atheism, in a broad sense, is the rejection of belief in the existence of deities. In a narrower sense, atheism is specifically the position that there are no deities. Most inclusively, atheism is simply the absence of belief that any deities exist. Atheism is contrasted with theism, which in its most general form is the belief that at least one^[5], 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 Proofs from THE BOOK is a book of mathematical proofs by Martin Aigner and Günter M. Ziegler. The book is dedicated to the mathematician Paul Erdős, who often referred to "The Book" in which God keeps the most elegant proof of each mathematical theorem. During a lecture in 1985, Erdős said, "You don't have to believe in God, but!" 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 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.^[6][7] 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 relative to what the observer already knows.^[8][9][10] 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^[11][12]

Mathematics and art

The psychology of the aesthetics of mathematics is studied post-psychoanalytically in psychosynthesis (in the work of Piero Ferrucci), in cognitive 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'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.
• 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 of Leonardo da Vinci, projective geometrys 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.

See also

• Descriptive science
• Elegance
• Fluency heuristic
• Golden ratio
• Mathematics and architecture
• Normative science
• Philosophy of mathematics

Notes

1. ^ Russell, Bertrand (1919). "The Study of Mathematics". Mysticism and Logic: And Other Essays. Longman. pp. 60. http://books.google.com/books?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&lr=&as_brr=0&client=opera. 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. pp. 140. http://books.google.com/books?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.&source=web&ots=eRoQLJAgG-&sig=hBu19Oddbjz9zw_koj6-dJkyxM0&hl=en&sa=X&oi=book_result&resnum=2&ct=result. Retrieved 2008-08-22.
3. ^ Elisha Scott Loomis published over 360 proofs in his book Pythagorean Proposition (ISBN 0873530365).
4. ^ Phillips, George (2005). "Preface". Mathematics Is Not a Spectator Sport. Springer Science+Business Media. ISBN 0387255281. http://books.google.com/books?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.&source=web&ots=GBNplD5Kl9&sig=k3_W0DQ5LM7UCD_6Xw7RBdYuSto&hl=en&sa=X&oi=book_result&resnum=1&ct=result. 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."
5. ^ 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.
6. ^ A. Moles: Théorie de l'information et perception esthétique, Paris, Denoël, 1973 (Information Theory and aesthetical perception)
7. ^ F Nake (1974). Ästhetik als Informationsverarbeitung. (Aesthetics as information processing). Grundlagen und Anwendungen der Informatik im Bereich ästhetischer Produktion und Kritik. Springer, 1974, ISBN 3211812164, ISBN 9783211812167
8. ^ 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
9. ^ J. Schmidhuber. Papers on the theory of beauty and low-complexity art since 1994: http://www.idsia.ch/~juergen/beauty.html
10. ^ 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
11. ^ .J. Schmidhuber. Curious model-building control systems. International Joint Conference on Neural Networks, Singapore, vol 2, 1458–1463. IEEE press, 1991
12. ^ 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

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.

• 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), http://www.fields.utoronto.ca/aboutus/FieldsMedal_Monastyrsky.pdf

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?

Categories: Aesthetics | Elementary mathematics | Philosophy of mathematics

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