Mathematical beauty


July 5, 2022

Mathematical beauty is a sense of beauty that some people experience when dealing with mathematics. Some mathematicians seek aesthetic pleasure in their work or in mathematics in general. They express this pleasure by describing “beautiful” parts of mathematics. They may see math as an art or as a creative activity. Comparisons are often made with music and poetry. For Bertrand Russell, mathematical beauty is "cold and austere, like that of a sculpture without reference to some part of our fragile nature, without the magnificent illusions of painting or music, and yet pure and sublime, capable of a strict perfection that only the greatest arts can show. » Paul Erdős evoked the ineffable character of the beauty of mathematics by declaring: “Why are the numbers beautiful? It comes down to asking why Beethoven's Ninth Symphony is beautiful. If you don't see why, no one can explain it to you. I know numbers are beautiful. If they are not beautiful, nothing is. »

In formulas

A formula is considered "beautiful" if it brings an essential and surprising result by its simplicity compared to the apparent complexity (therefore in particular an equality of which one of the members is very simple while the other member is very complicated). An example of a beautiful formula is that of Leonhard Euler e I π + 1 0 {\displaystyle \mathrm {e} ^{\mathrm {i} \pi }+10} , which Euler himself said showed the presence of the hand of God. In Robert Harris' novel Enigma, fictional mathematician Tom Jericho calls the beauty of Leibniz's formula "crystalline" 1 1 − 1 3 + 1 5 − 1 7 + … + ( − 1 ) k 2 k + 1 + … π 4 . {\displaystyle {\dfrac {1}{1}}-{\dfrac {1}{3}}+{\dfrac {1}{5}}-{\dfrac {1}{7}}+\ldots + {\dfrac {(-1)^{k}}{2k+1}}+\ldots {\dfrac {\pi }{4}}.}

In methods

Mathematicians can call a method in a proof "elegant" when: it uses few preliminary results; it is exceptionally short; it establishes a result in a surprising way (eg from theorems which are apparently unrelated to it); it is based on original concepts; it uses a method that can be generalized to easily solve a family of similar problems. In the quest for an elegant proof, mathematicians often look for different independent ways to establish a theorem; the first demonstration found may not be the best. The theorem for which the greatest number of different proofs has been