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?
How many different proofs can a theorem have?
10
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soft-question
philosophy
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5How would you define "one way"? You can switch the order a bit, do you then have a new proof? – 2011-05-04
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14Before 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: http://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/ . – 2011-05-04
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0Have a look at this paper: http://mathdl.maa.org/images/upload_library/22/Ford/Wagon601-617.pdf. – 2011-05-04
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3The (many!) proofs of quadratic reciprocity are quite varied, I'm told. – 2011-05-04
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2The 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... – 2011-05-04
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0Prove $1 + 1 = 2$. – 2011-05-05
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0It has been justifiably pointed out that not only is it hard to tell when two proofs are different, indeed, two apparently different proofs sometimes being able to be transformed into one another by a canonical process. Therefore it seems apropos to give at least one specific example of a clearly-different proof. To that end, I hereby mention Ivan Niven’s simple proof that pi is irrational. – 2011-05-07