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There is a famous saying in mathematics from Paul Erdős: "You don't have to believe in God, but you should believe in The Book." "The Book" is an imaginary book in which God had written down the best and most elegant proofs for mathematical theorems.

If there is a book written down by God, why not a book from the Devil? I mean, a book with the most ugly proofs, but yet the best ones we have as accepted proofs. I don't mean to make a horrible proof on purpose, but sometimes ugly proofs is all you have.

I wish if you could share some theorem from the Ugly Book, some theorem proved by a real ugly proof (and yet the only one that there is). I'm asking this not for fun only, but I'm curious about how ugly proofs can be.

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First proofs of a result are frequently ugly. – André Nicolas Jun 30 '13 at 21:58
Perhaps the four-color theorem qualifies? – Alex Becker Jun 30 '13 at 21:58
The devil can be quite elegant at times. – Lord Soth Jun 30 '13 at 21:58
classification of finite simple groups ? – Amr Jun 30 '13 at 21:59
I would say Carleson's theorem on a.e. convergence of Fourier series doesn't have an elegant proof yet. – icurays1 Jun 30 '13 at 22:05
up vote 5 down vote accepted

Beauty is quite subjective. One may prove something in 2 lines using the newly developed supersymmetric coffee spaces, and that may be cool to some. Suppose you have another proof of the same result that uses only some primitive set of axioms. This proof may potentially be 1000 pages long, but it will be more beautiful to some (for example me), as it is a demonstration of the fact that all that mind blogging complexity is actually the result of addition, multiplication, etc, and some first order logic.

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+1 for supersymmetric coffee spaces. – James S. Cook Jun 30 '13 at 22:19

There exists irrational numbers $x$ and $y$ such that $x^y$ is rational.

Proof: If $\sqrt{2}^\sqrt{2}$ is rational, we can take $x=y=\sqrt{2}$. If $\sqrt{2}^\sqrt{2}$ is irrational, we take $x=\sqrt{2}^\sqrt{2}$ and $y=\sqrt{2}$.

The proof is based on a case distinction in which only one case is true, without telling us which one. The proof is discused at this wikipedia page.

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I always thought that this proof was beautiful, do you think its ugly ? – Amr Jun 30 '13 at 22:12
@Amr Yes, a proof that acknowledges ignorance so fully and leaves so many open questions makes me feel empty inside. – Michael Greinecker Jun 30 '13 at 22:13
I think you have a very different notion of proof beauty from mine. – Amr Jun 30 '13 at 22:18
@EricTressler How does your continuity argument work? – Michael Greinecker Jul 1 '13 at 1:42
I’d say that it’s beautiful, delightfully clever, and utterly uninformative. – Brian M. Scott Jul 1 '13 at 6:06

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