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. 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."
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 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 Pythagoras' 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.
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, 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. Another example is the Taniyama-Shimura 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; however, sometimes a statement of a theorem can be original enough to be considered deep, even though its proof is fairly obvious.
Beauty in experience
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.
Beauty and mysticism
Some mathematicians express beliefs about mathematics that are close to mysticism.
Pythagoras (and his entire philosophical school) 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.
Galileo Galilei is reported to have said "Mathematics is the language with which God wrote the universe".
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 insphere of another.
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!"
- G. H. Hardy, A Mathematician's Apology ISBN 0521427061
- H.-O. Peitgen and P.H. Richter, The Beauty of Fractals ISBN 0387158510
- Martin Aigner, Karl Heinrich Hofmann, Gunter M. Ziegler, Proofs from the Book ISBN 3540404600
- Paul Hoffman, The man who loved only numbers, Hyperion (May 12 1992) ISBN 0786884061