The Mathematical beauty reference article from the English Wikipedia on 24-Jul-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

# Mathematical beauty

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.

## 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.

Hungarian mathematician Paul Erdös imagined that God has a book containing all the theorems of mathematics with their absolutely most beautiful proofs. When Erdös wanted to express particular appreciation of a proof, he would exclaim "This is one from the Book!"

Conversely, results or calculations that are logically correct but involve laborious calculations, over-elaborate methods or very conventional approaches are not considered to be elegant, and may be called ugly or clumsy. For example, proofs which depend on the elimination of many separate cases (the proof by exhaustion method), such as the currently known proofs of the four colour theorem, are definitely not elegant.

## 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 described as deep.

One example of a deep result is Euler's identity e + 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 is a result that can be derived in an obvious and straightforward way from other known results.

## 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 willl 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.

## 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 expressable 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.

Paul Erdös 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."

An extreme tendency to focus on the the elegance or beauty or simplicity of a theory rather than its empirical use in applications is sometimes described as mathematical fetishism or scientism.