What do we mean by an "Elegant Proof"? 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. 
 A: Elegance is certainly subjectives and people will view the term differently.  I think a proof that is easy to follow and shows the essential ideas is elegant. It can use fancy tools or proceed in unexpected ways but that is not needed.  Simple, brief and very clear, those are the trademarks of an elegant proof to me. 
A: In my opinion, many mathematicians define elegant a proof which is not "mainstream". Such a proof should not be a trivial or a brute force application of previous results. It should possibly mix different branches of mathematics, and contain some surprise. 
Of course it should also rather short, but this is not really necessary for a proof to be elegant.
There are cases in which an elegant proof could be described as "too abstract" or "unnecessarily based on advanced theories". But this is natural, since the concept of elegance is linked to our education and to our preferences.
A: First, note that when you prove something, you usually stated that a statement is proveably from something. Suppose $\Lambda$ is a set of axioms. $\Lambda \vdash \varphi$ denotes a proof $\varphi$ using $\Lambda$. You are looking the most elegant proof of $\varphi$ from $\Lambda$. Therefore, your second point is not very meaningful since a proof of the statement from itself (thought may be short and elegant) is not a proof using $\Lambda$ is $\varphi$ is not a statement in $\lambda$. 
However, I can try to address some ways that you may be able to measure some of the characteristics that you brought up. 
1) It is possible to measure the length of the proof. Assuming you are working in some formals system (like propositional logic or first order logic) and trying to prove something from a set of axioms $\Lambda$, then you can count the length of the proof. A proof is really a sequence of statements each of which is axioms or follows from previous statements using the logical deduction rule of your formal system. The length of the proof would be the number of steps used in the proof. Note that a proof can get arbitrarily long since you can always add unnessary steps to a proof. 
2) If you are trying to prove a statement using some axioms $\Lambda$, you can also make sense of using the least number of axiom. Of course the statement $\varphi$, you are proving is already a statement in $\Lambda$, then that single axiom sufficies. However, given a proof, you can always analyze the proof to see exactly which axioms were necessary. This sort of idea is used very often by logician, especially set theory. You must have seen statement like $ZF$ proves cartesian product of sets exists, $ZFC$ proves every set can be well ordered; $ZF$ + the axiom of determinacy proves every subset of $\mathbb{R}$ is measureable.  Moreover using technique of logic, you may be able to determine if some of your axions of redundant. A well known result is that the axiom of choice is not proveable from $ZF$. Knowing which axioms are necessary for a proof can be helpful for understanding the limits of provability; however, when all the axioms of well-accepted, the proof using less axioms may be more difficult. For example, every vector has a basis can be proven using $ZF$ plus the well-ordering principle and without the power set axiom; however, the more common Zorn Lemma approach require the power set axiom. 
Your other points are somewhat subjective. These other aspect are somewhat phycological. Some people may find that a proof of a result that most people would consider to be part of Algebra or Analysis is easier to understand if it is proven using results of algebra or analysis, respectively. It would be reasonable to expect that if bunch of statement are equivalent, the form closest to statement you are trying to prove would give the easiest proof. Regarding this, there is a program in logic called reverse math that attempts to classify theorem of mathematics over very weak base system according to their logical equivalence. Some result in this areas have shown that combinatorial result, topological theorem, and algebriac theorem are equivalent over weak system of arithmetics. Though these results may be equivalent, the most evident proof would likely use the result closest to field you are working in.
Again other people may like proof that applies techniques from other areas. These results may be surprising and yield new and potentially useful connections to other fields. 
Also it is hard to say that a proof is elegant if it is understandable by particular people. Depending on background and point of view proof may be more understandable or more appealing. For instance, some results may have less general forms that are understandable by high schoolers but the proof is very long or intricate. A good example may be the intermediate and extreme value theorem. The statement of the result and proof of the result using general topology ideas like continuous function, connectedness, and compactness is much cleaner after the appropriate definition and lemmas are given. These results are then more applicable for other areas of mathematics. 
