Mathematical Beauty - Beauty in Method

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 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.