Pure mathematics and Applied Mathematics

A proof that uses a minimum of additional assumptions or previous results.
A proof that derives a result in a surprising way 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. 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 or very conventional approaches are not usually considered to be elegant, and may be called ugly or clumsy. 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, here are some examples that are often cited. One is Euler's identity eiπ + 1 = 0. This has been called "the most remarkable formula in mathematics" by Richard Feynman. Modern examples include the Taniyama-Shimura theorem which establishes an important connection between elliptic curves and modular forms (work on which led to the awarding of the Wolf Prize to Andrew Wiles and Robert Langland's), 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; however, sometimes a statement of a theorem can be original enough to be considered deep, even though its proof is fairly obvious. 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 its philosophy 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. Bertrand Russell referred to the austere beauty of mathematics.
Some mathematicians are of the opinion that the doing of mathematics is closer to discovery than invention. 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 of the Pythagoreans) believed in the literal reality of numbers. The discovery of the existence of irrational numbers was a shock to them - they considered 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. 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", a statement which (apart from the implicit deism) is consistent with the mathematical basis of all modern physics. Hungarian mathematician Paul Erdős, although not a believer, 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.

In some cases, mathematicians have made leaps of inference between beauty and physical truth in ways that turned out not to be confirmed. 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 had 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 in sphere of another. As there are exactly five Platonic solids, Kepler's theory could only accommodate six planetary orbits, and was disproved by the subsequent discovery of Uranus. James Watson made a similar error when he originally postulated that each of the four bases of DNA connected to a base of the same type in the opposite strand (thymine linking to thymine, etc.) based on the belief that "it is so beautiful it must be true."