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In mathematics, a proof's validity is based upon whether it consists of a sequence of logical steps that can be deduced from one another. This is an objective process. However, mathematicians, surprisingly to some, have a second criterion by which they judge proofs: elegance.

An elegant proof is one which you read and then surprises with its simplicity. For example, there is a diagrammatic proof to Pythagorean Theorem that works by drawing physical squares on each of the sides, and showing that if you chop up the squares on the opposite two sides, cut them into triangles, and paste them into the square on the hypotenuse, then it actually fits perfectly, proving it. This is an elegant proof because it's nice, simple, takes a few lines and anyone can understand it.

The proof that the square root of 2 is not rational, for example, is also considered elegant. It is a proof done using reductio ad absurdum and it's only about 10 lines long.

An example of an inelegant proof is, say, the computational proof of the four-color theorem. They just got a computer to go through all the possibilities and show that each one worked. Ugh.

Of course, there is room for subjective judgement in the issue. For example, the proof of Fermat's Last Theorem has been done, but it's a 100-page+ monster that ventures into elliptic curve theory and is seriously difficult to understand for non-mathematicians. To give you some idea, the first version of the proof had a divide-by-zero error in it that was only discovered a year (I think) after it was first published. To a mathematician, this might be elegant, but for most people it would just be WTF?

To say that Wiles's proof of FLT is `seriously difficult to understand for non-mathematicians' is an understatement. It is seriously difficult to understand for any mathematician who has not spent a great deal of time studying certain very narrow bits of mathematics.

One thing that makes a proof elegant is generality. Another, as ymelup pointed out, is brevity and conciseness. There is also parsimony or conceptual simplicity, which is not the same thing as conciseness: a ten-line proof that involves four special cases may be brief, but is probably not elegant.

Of course, elegance, as an aesthetical judgement, is very subjective. Though mathematicians may agree that a particular proof is elegant (though such a concurrence of opinion is not by any means universal; I, for example, being an algebraist at heart, do not find geometrical proofs of the Pythagorean theorem very elegant), they will often have a hard time explaining why the proof is elegant—­especially to non-mathematicians.

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