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I notice some problems has many different proofs, do all theorems have multiple proofs, is there some theorems which has only 1 way to prove it? $n$ ways? infinite?

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How would you define "one way"? You can switch the order a bit, do you then have a new proof? –  Jonas Teuwen May 4 '11 at 16:48
Before you can ask how many different proofs a theorem have, you have to decide what it means to be different. There is a discussion of this at Gowers' blog: gowers.wordpress.com/2007/10/04/… . –  Raeder May 4 '11 at 16:49
Have a look at this paper: mathdl.maa.org/images/upload_library/22/Ford/Wagon601-617.pdf. –  Yuval Filmus May 4 '11 at 17:06
The (many!) proofs of quadratic reciprocity are quite varied, I'm told. –  J. M. May 4 '11 at 17:16
The collection of all proofs of a theorem does not form a set. It at least forms a category with nontrivial isomorphisms, possibly an $\infty$-category... –  Qiaochu Yuan May 4 '11 at 18:43

2 Answers 2

If one looks at any of the formal definitions of proof in books on logic, there is a countable infinity of proofs for any theorem (if, as usual, the language has a finite or countably infinite set of symbols).

This may not answer the question, which was not explicitly about formal proof. Informally, there are examples of famous results (like the Arithmetic Mean/Geometric Mean Inequality) that have quite a number of proofs based on essentially different ideas.

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I guess one way to interpret the question is how many proofs of a given theorem can be published, under the (somewhat dubious) assumption that two proofs have to be significantly different in order for both of them to get published. So I direct your attention to Murray Gerstenhaber's paper, The 152nd proof of the law of quadratic reciprocity, Amer. Math. Monthly 70 (1963) 397–398, MR0150097 (27 #100) (but I warn you that the title was somewhat tounge-in-cheek). See also Elisha Loomis, The Pythagorean Proposition, which has 365 proofs of that well-known result.

There have been many "proofs" of P = NP, and roughly an equal number of "proofs" of P $\ne$ NP. GJ Woeginger keeps track of them at http://www.win.tue.nl/~gwoegi/P-versus-NP.htm

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