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What do we mean when we say that a mathematical proof is elegant? Of course one can say that the proof is beautiful, but what do we precisely mean when we say that a proof is beautiful ? Is there a precise way to measure the elegance of a mathematical proof ? I have thought of some possibilities, but none of them seems to fit properly:

  • A proof is elegant if it has less no of steps when we break up the proof into largest no of pieces possible, i.e. the proof consists of only axioms and modus ponens.

  • A proof is elegant if it based on least no of axioms, but this can't be true because the statement of the proof can itself be treated as an axiom.

  • A proof is elegant if it uses stronger axioms (i.e. with more self evidence than other axioms). But I guess there is some sort of vagueness here, how do we determine whether an axiom is more self evident that other ?

  • A proof is elegant if it uses some results or ideas of a different branch of mathematics which apparently seems to have no connection with the branch under which the proof falls. But this is also not always true.

  • A proof is elegant if it uses simplest arguments, those which can be understood by any bright high school mathematics student.

  • Personally I find a combinatorics or a number theoretic proof more elegant if it uses pure arithmetic and algebra rather than mathematical analysis or calculus tools. But this may be because of my interest in those areas.

So my question is whether there is a measure of the elegance of a mathematical proof ? The answer according to me is most likely to be negative. But if anyone has any idea towards a positive answer or any argument that the answer can't be in positive, please share.

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    "I know it when I see it" - Potter Stewart.2012-08-11
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    http://www.maa.org/mathland/mathtrek_05_24_04.html2012-08-11
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    Gian-Carlo Rota wrote an essay in which, among other things, he said elegance is not at all the same thing as beauty. I don't want to endorse his bottom-line conclusion (which I won't attempt to state here) but I agree with that particular part of what he wrote.2012-08-11

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