A few mathematicians obtain artistic delight from their work, and from mathematics overall. 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 artistic action. 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.” (The Study of Mathematics, in Mysticism and Logic, and Other Essays, ch. 4, London: Longmans, Green, 1918.)
Paul Erdős articulated 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."Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were initiate in commerce, land measurement and later astronomy; these days, all sciences propose problems studied by mathematicians, and a lot of problems arise within mathematics itself.
Newton was one of the insignificant calculus inventors, Feynman invented the Feynman path integral using a mixture of reasoning and physical insight, and today's string premise also inspires new mathematics.
Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. The remarkable fact that even the
"purest" mathematics often turns out to have useful applications is what
Eugene Wigner has called "the unreasonable effectiveness of mathematics."
As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One most important difference is between pure mathematics and applied mathematics. Some areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science.
For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the grace of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty in a simple and elegant proof, such as Euclid's proof that there are infinitely many prime numbers, and in an elegant numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in A Mathematician's Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Mathematicians often try hard to find proofs of theorems that are mostly elegant, a quest Paul Erdős often referred to as finding proofs from
"The Book" in which God had written down his favorite proofs. The popularity of leisure mathematics is another symbol of the pleasure many find in solving mathematical questions.
Beauty in methodMathematicians describe an especially pleasing method of proof as elegant. Depending on situation, this may mean:
* A proof that uses a minimum of additional assumptions or previous results.
* A proof that is unusually short.
* 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 generalised to solve a family of similar problems.
Beauty in resultsSome mathematicians see beauty in mathematical results which establish connections between two areas of mathematics that at first sight emerge to be completely unconnected. These results are often described as deep.
While it is hard to find total agreement on whether a result is deep, some examples are often cited. One is Euler's identity eiπ + 1 = 0. This has been called "the most extraordinary formula in mathematics" by Richard Feynman. Modern examples take in the modularity theorem which establishes an significant 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 connected the Monster group to modular 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 exacting objects such as the empty set. From time to time, however, a proclamation of a theorem can be unique enough to be considered deep, even though its proof is comparatively obvious.
Beauty in experienceSome degree of enchantment in the exploitation of numbers and symbols is probably necessary to engage in any mathematics. Given the efficacy of mathematics in science and engineering, it is likely that any technical society will actively cultivate these aesthetics, certainly in its beliefs 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 like or appreciate mathematics in a simply inactive approach - in mathematics there is no real correspondence of the role of the watcher, listeners, or observer. Bertrand Russell referred to the rigorous beauty of mathematics.
Source : www.wikipedia.org