The beauty of mathematics illustrates the assumption that some mathematicians can gain an aesthetic pleasure from their work, and from mathematics in general. They express this pleasure by describing mathematics (or, at least, some aspects of mathematics) as beautiful . Mathematicians describe mathematics as an art form or, at least, as a creative activity. Comparisons are often made with music and poetry.
Bertrand Russell expressed his mathematical beauty in these words:
Mathematics, rightly seen, not only possesses the truth, but the ultimate beauty - cool and hard beauty, like a statue, without appeal to any part of our weaker realm, without beautiful or exquisite paintings or music, decisive as only the greatest art can be displayed. The spirit of true joy, exaltation, the sense of being more than Man, which is the touchstone of supreme excellence, can be found in mathematics as well as poetry.
Paul Erd expressed his view of the ineffectiveness of mathematics when he said, "Why are these numbers beautiful? It's like asking why Beethoven's Ninth Symphony is beautiful If you do not see why, somebody can not tell you I know > the numbers are beautiful If they are not beautiful, no one ".
Video Mathematical beauty
Beauty in the
methodThe mathematicians describe a very pleasing method of proof as elegant . Depending on the context, this may mean:
- Evidence that uses additional minimum assumptions or previous results.
- Unusual evidence.
- Evidence that produces results in a surprising way (for example, from a seemingly unrelated theorem or set of theorems).
- Evidence based on new and original insights.
- Proven methods that can be easily generalized to resolve similar family issues.
In search of elegant evidence, mathematicians often seek different independent ways to prove the results - the first evidence found may not be the best. The theorem with the most different evidence has been found possible Pythagoras theorem, with hundreds of evidences published. Another theorem that has been proven in many ways is the quadratic reciprocity theorem - Carl Friedrich Gauss himself published eight different proofs of this theorem.
Conversely, a logically correct result but involving exhausting calculations, overly complex methods, a very conventional approach, or that depends on a large number of very strong axioms or previous results is usually not considered elegant, and can be called ugly or clumsy .
Maps Mathematical beauty
Beauty in results
Some mathematicians see beauty in mathematical results that build relationships between the two mathematical fields that at first glance seem unrelated. These results are often described as within .
Meskipun sulit untuk menemukan kesepakatan universal tentang apakah hasilnya mendalam, beberapa contoh sering dikutip. Salah satunya adalah identitas Euler:
This is a special case of Euler's formula, which by physicist Richard Feynman called "our gem" and "the most remarkable formula in mathematics". Modern examples include the modularity theorem, which forms an important link between the elliptical curve and the modular form (the work that leads to the awarding of the Wolves Prize to Andrew Wiles and Robert Langlands), and "mononshine monstrous", which links the Monster group to the modular function through Richard's string theory Borcherds was awarded the Fields Medal.
Other examples of deep results include unexpected insights into the structure of mathematics. For example, Gauss's Theorema Egregium is a profound theorem that connects local phenomena (curvature) with global phenomena (areas) in a surprising way. In particular, the area of ​​the triangle on the curved surface is proportional to the excess triangle and the proportionality is curvature. Another example is the fundamental theorem of calculus (and its vector version includes Green's theorem and Stokes's theorem).
The opposite of in is trivial . Trivial theorems may be derivable results in a clear and straightforward way from other known results, or that apply only to specific sets of specific objects such as empty sets. Sometimes, however, the theorem statement can be original enough to be considered in, although the evidence is quite clear.
In his book A Mathematician's Apology, Hardy points out that beautiful evidence or results have "inevitable", "unexpected", and "economy".
Rota, however, disagrees with disorder as a condition sufficient for beauty and suggests the opposite:
Much of the mathematical theorem, when first published, seems surprising; so for example some twenty years ago [from 1977] evidence of unequal differentiated structures in high-dimensional fields is surprising, but it does not occur to anyone to call such a beautiful, past or present fact..
Perhaps ironically, Monastyrsky wrote:
It is very difficult to find analog discovery in the past for Milnor's beautiful construction of different differential structures on a seven-dimensional scope... The original proof of Milnor is not very constructive, but then E. Briscorn suggests that this differential structure can be described in a very explicit form and beautiful.
This disagreement illustrates the subjective nature of the beauty of mathematics and its relation to mathematical results: in this case, not only the existence of exotic fields, but also certain realizations of them.
Beauty in experience
Interest in pure mathematics that is separate from empirical studies has become part of the experience of various civilizations, including from the ancient Greeks, who "do the math for its beauty". The aesthetic pleasures which tend to be experienced by mathematical physicists in Einstein's general theory of relativity have been recognized (by Paul Dirac, among others) on "the beauty of great mathematics". The beauty of mathematics is experienced when the physical reality of an object is represented by a mathematical model. The group theory, developed in the early 1800s for the sole purpose of solving polynomial equations, became a useful way of categorizing elementary particles - the building blocks of matter. Similarly, the study of vertices provides important insights into string theory and loop quantum gravity.
Some believe that to appreciate mathematics, one must be involved in doing mathematics. There are some teachers who encourage student involvement by teaching maths in a kinesthetic way (see kinesthetic learning). For example, the Mathematical Circle is an afterschool enrichment program in which students perform mathematics through games and activities; in the Mathematics course of the General Circle, students use the invention of patterns, observations, and explorations to make their own mathematical discoveries. For example, the beauty of mathematics appears in the activities of Mathematical Circles in symmetry designed for 2nd and 3rd graders. In this activity, students create their own snowflakes by folding a piece of square paper and cutting their design of choice along the edges of the folded paper. When the paper is folded, the symmetrical design reveals itself. In the elementary school math class everyday, symmetry can be presented in such an artistic way in which students see results that are aesthetically satisfying in mathematics.
Some teachers prefer to use mathematical manipulatives to present mathematics in a fun way. Manipulative examples include algebraic tiles, cuisenaire rods, and pattern blocks. For example, one can teach a box finishing method by using an algebraic tile. The Cuisenaire rod can be used to teach fractions, and block patterns can be used to teach geometry. Using mathematical manipulatives helps students gain a conceptual understanding that may not be immediately discernible in a written mathematical formula.
Another example involves origami. Origami, the art of paper folding, has aesthetic qualities and many mathematical connections. One can learn the mathematics from folding paper by observing the fold patterns on the unfolded origami pieces.
Combinatorics (the study of counting) has an artistic representation that some people consider very mathematical. There are many visual examples illustrating combinatorial concepts. Here are some topics and objects that are seen in combinatorics courses with visual representations:
- Four color theorems
- Young tablo
- Permutohedron
- Graph theory
- Partition from set
Beauty and philosophy
Some mathematicians argue that doing mathematics is closer to the invention than the invention, for example:
There is no scientific inventor, no poet, no painter, no musician, who will not tell you that he finds ready to make a discovery or poetry or image - that it comes to him from the outside, and that he does not consciously create it from within.
This mathematician believes that detailed and precise mathematical results can be regarded as true without any dependence on the universe in which we live. For example, they would argue that the theory of natural numbers is fundamentally valid, in a way that does not require a specific context. Some mathematicians have extrapolated this point of view that the beauty of mathematics is a further truth, in some cases becoming mysticism.
Pythagoras mathematicians believe in the reality of literal numbers. The discovery of the existence of irrational numbers is a surprise to them, since they assume the existence of an irreversible number as the ratio of two natural numbers to be defective in nature (the Pythagoras world view does not reflect on the boundaries of the infinite series of the ratio of the natural numbers - the modern notion of real numbers). From a modern perspective, their mystical approach to numbers can be seen as numerology.
In Plato's philosophy there are two worlds, the physical one in which we live and another abstract world containing unchanging truths, including mathematics. He believed that the physical world was merely a reflection of a more perfect abstract world.
Hungarian mathematician Paul Erd speaks of an imaginary book, in which God has written all the most beautiful mathematical proofs. When Erd wants to express a special appreciation of the evidence, he will exclaim "This one from the Book!"
The French philosopher of the twentieth century, Alain Badiou claimed that ontology was mathematics. Badiou also believes in the profound relationship between mathematics, poetry, and philosophy.
In some cases, natural philosophers and other scientists who have used mathematics extensively have made a leap of conclusion between beauty and physical truth in a way that turns out to be wrong. For example, at one stage in his life, Johannes Kepler believed that the proportion of planets' orbits known in the Solar System was governed by God to conform to the concentric arrangements of the five Platonic solids, each orbit lying above the circumsphere of a polyhedron and the insphere another. Because there are exactly five Platoon solids, the Kepler hypothesis can hold only six orbits of planets and is denied by later Uranus discovery.
Theory of beauty and mathematical information
In the 1970s, Abraham Moles and Frieder Nake analyzed the relationship between beauty, information processing, and information theory. In the 1990s, JÃÆ'¼rgen Schmidhuber formulated a mathematical theory of subjective beauty that relies on observers based on algorithmic information theory: the most beautiful object among comparable subjective objects has a short algorithmic description (ie, Kolmogorov complexity) relative to what already known by the observer. Schmidhuber explicitly distinguishes between beautiful and attractive. The latter corresponds to the first derivative of the subjective perceived beauty: observers continue to try to increase the predictability and compressibility of observation by finding regularities such as repetition and symmetry and fractal self-similarities. Whenever the observer's learning process (perhaps a predictive neural network) leads to an increase in data compression so that the observed sequence can be explained by fewer bits than before, the temporary interest of the data corresponds to the compression progress, and is proportional to the observer. awareness of internal curiosity.
Math and art
Music
Examples of mathematical use in music include stochastic music Iannis Xenakis, Fibonacci in Tool's Lateralus, rival Johann Sebastian Bach, polyrhythmic structure (as in Igor Stravinsky The Rite of Spring), Elliott Carter's Metric Modulation, permutation theory in serialism which began with Arnold Schoenberg, and the application of the Shepard tone in Karlheinz Stockhausen Hymnen .
Visual art
Examples of the use of mathematics in the visual arts include the application of chaos theory and fractal geometry to computer-generated arts, the study of symmetry of Leonardo da Vinci, projective geometry in the development of Renaissance art perspective theories, the grids in Op art, optical geometry in the obscura camera Giambattista della Porta, perspective in analytic cubism and futurism.
The Dutch graphic designer M. C. Escher created wood carvings, lithographs, and mezzotints that were mathematically inspired. It features impossible constructions, infinity exploration, architecture, visual paradoxes and tesselations. British construction artist John Ernest made reliefs and paintings inspired by group theory. A number of other British artists from construction and systems schools also draw on mathematical models and structures as a source of inspiration, including Anthony Hill and Peter Lowe. Computer-generated art is based on mathematical algorithms.
See also
Note
References
Further reading
- Zeki, S.; Romaya, J. P.; Benincasa, D. M. T.; Atiyah, MF (2014), "Experience of mathematical beauty and neurral correlation", Boundary in Human Neuroscience , 8 , doi: 10.3389/fnhum.2014.00068 < span title = "ctx_ver = Z39.88-2004 & amp; rft_val_fmt = info% 3Aofi% 2Ffmt% 3Akev% 3Amtx% 3Ajournal & amp; rft.genre = articles & amp; rft.jtitle = Frontier in Human Neuroscience & amp; rft.atitle = Experience the mathematics of beauty and nerves correlated & amp; rft.volume = 8 & amp; rft.date = 2014 & amp; rft_id = info% 3Adoi% 2F10.3389% 2Ffnhum.2014.00068 & amp; rft.aulast = Zeki & amp; rft.aufirst = S. & amp; rft.au = Romaya% 2C JP & amp; rft.au = Benincasa% 2C DMT & amp; rft.au = Atiyah% 2C MF & amp; rfr_id = info% 3Asid % 2Fen.wikipedia.org% 3AMathematical beauty "> .
External links
- Math, Poetry, and Beauty
- Is Math Beautiful?
- Justin Mullins
- Edna St. Vincent Millay (poet): Euclid himself has seen the beauty of naked
- Terence Tao, What is good math?
- Blog Mathbeauty
- Aesthetic Appeal collection in the Internet Archive
- Mathematical romance Jim Holt December 5, 2013 issue of The New York Review of Books reviews about Love and Math: The Heart of Hidden Reality by Edward Frenkel li >
Source of the article : Wikipedia
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