Inspiration, pure and applied mathematics, and aesthetics.

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton was one of the infinitesimal calculus inventors, Feynman invented the Feynman path integral using a combination of reasoning and physical insight, and today's string theory 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 practical 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 major distinction is between pure mathematics and applied mathematics. Several 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.

Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a 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.

Most mathematicians 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.

Mathematical beauty.

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 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.

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. 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.

Mathematicians 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 e^iπ + 1 = 0. This has been called “the most remarkable formula in mathematics” by Richard Feynman. Modern examples include the modularity theorem which establishes an important connection between elliptic curves and modular forms.

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.