Bertrand Russell expressed his sense of mathematical beauty in these words:
Paul Erdős expressed 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".
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 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, 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.
Some mathematicians see beauty in mathematical results that establish connections between two areas of mathematics that at first sight appear to be 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 i π + 1 = 0
This is a special case of Euler's formula, which the physicist Richard Feynman called "our jewel" and "the most remarkable formula in mathematics". Modern examples include the modularity 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 Langlands), and "monstrous moonshine", which connects the Monster group to modular functions via string theory for which Richard Borcherds was awarded the Fields Medal.
Other examples of deep results include unexpected insights into mathematical structures. For example, Gauss's Theorema Egregium is a deep theorem which relates a local phenomenon (curvature) to a global phenomenon (area) in a surprising way. In particular, the area of a triangle on a curved surface is proportional to the excess of the triangle and the proportionality is curvature. Another example is the fundamental theorem of calculus (and its vector versions including Green's theorem and Stokes' theorem).
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, Hardy suggests that a beautiful proof or result possesses "inevitability", "unexpectedness", and "economy".