Mathematicians and the mathematically erudite alike love unanimity; and for some the quest to have truly resolvable arguments plays a role in their attraction to the subject. I suspect that is the reason we frequently encounter statements like: (1) "Gauss is widely recognized as the greatest mathematician of all time." (2) "Euclid's proofs of the infinitude of primes and of the irrationality of [itex]\sqrt(2)[/itex] are beautiful yet accessible gems of pure mathematics". But is seems that (1) originated with ET Bell's 1926 Men of Mathematics, and that (2) originated with Hardy's 1940 Mathematician's Apology. It seems that the opinions expressed in these texts form a disproportionate amount of our current Mathematical culture. To make this topic truly disputable I would need to cite examples of (1) and (2) being parroted, but have encountered so many that I rely on the readers to relate to a common experience. As a point of discussion, I disagree that the proofs in (2) are characteristic of mathematical beauty in general, as is often claim "if you don't find these proofs beautiful, consider switching subjects away from pure mathematics".
another jerry springer moment. unfortunately this is likely to generate merely argumentation. how about a more positive approach, such as "what arguments do you think beautiful?"
Perception of mathematical beauty unfortunately depends on the philosophy of the student of mathematics. As a formalist who hasn't gone outside mathematics described by predicate logic and set theory, I find proofs by contradiction to be quite beautiful because they are usually quite short and IMO powerful (and sometimes quite subtle). (The, "yeah, imagine that it were this way instead. But then you'd have that this would happen... and that's exactly what we said can't happen, so obviously, what we just imagined can't happen at all") I usually find proof of existence by construction to be ugly, especially when the object being constructed is very complicated, unless it actually matters that you know the form of the object. A constructionist though would find proof by contradiction to be ugly (if acceptable at all) and proof by construction to be beautiful. Likewise, if working in a different system of logic (or in want of proofs that are applicable to some other system of logic), a formalist might find proof by contradiction to be ugly or unacceptable.
(2) "Euclid's proofs of the infinitude of primes and of the irrationality of are beautiful yet accessible gems of pure mathematics". Hardy was a good writer and so was Bell, reference (1), As for (2) I don't see any reason to dispute it. What's wrong with saying that? After all, it's a take it or leave it matter, but in every class I had where the proof of primes or square root of 2 comes up, everyone gets very interested, wondering if they can follow it, and the teacher glows. As far as Bell, he did write a chapter on Gauss, "The Prince of Mathematicians." I am not aware that he or for that matter anybody else unequivocally ranked Gauss as greater than everybody else.
As far as switching out of pure math, I have read Hardy and understand that he was making an argument on the side of pure rather than applied math. Now, if you want to work in industry or the military, naturally they much prefer applied math. So much for ugly versus beautiful math! Pure math is mostly restricted to the universities. However, if you want to applaud that side of things, ivory tower existence, so what! What is the point of knocking it? Give Hardy credit, he was a very good, and apparently influential, writer. But it's still a take it or leave it matter. After all, when you come down to it, there is no Noble prize in Mathematics, you know. Very few have written in favor of pure math, and that may explain why Hardy was so influential.
LukeD: I appreciate a good contradiction proof; I think they are there to remind us that, however complicated our definitions may be, the basic principles of logic still apply. On the other hand, my favorite proofs are the ones that manage to be both concise and constructive. Existence proofs can sometimes leave me feeling empty, knowing that something is true, but not knowing, in some sense, "why" it is true. Granted, there are few proofs which are both concise and constructive...
It makes plenty of sense to call irony differentiable. It's just false, as irony tends to appear in sudden sharp spikes, which are not generally smooth. You should hang around my University's math club. I still remember the conversation about how people are Lie Groups, because they clearly are embedded manifolds, and they also obviously act on other embedded manifolds (usually smoothly).