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→Definition of Elementary proof: All of those proofs rest on the same set of assumptions
:The view that "all proofs are equivalent no matter what theories they use" may be appealing to some, but it does not withstand scrutiny. Proofs relying on fewer assumptions are obviously preferable.--[[User:Aschlafly|Andy Schlafly]] 22:44, 26 August 2009 (EDT)
:The view that "all proofs are equivalent no matter what theories they use" may be appealing to some, but it does not withstand scrutiny. Proofs relying on fewer assumptions are obviously preferable.--[[User:Aschlafly|Andy Schlafly]] 22:44, 26 August 2009 (EDT)
:: Selberg did get a Fields Medal for an elementary proof of the prime number theorem, but I am not sure that anyone says that Selberg's proof is any better than any other. All of those proofs rest on the same set of assumptions. Can you give an example of an elementary proof that is better than a non-elementary proof because it relies on fewer assumptions? I agree that a proof not using the axiom of choice might be preferable, but can you give another example? [[User:RSchlafly|RSchlafly]] 02:43, 27 August 2009 (EDT)
The notion that complex analysis depends on a purported uniqueness of the square root of -1 needs to be put to rest. There is no assumption that the square root of -1 has to be unique. And a good thing, too, because it isn't unique. No one denies that -1, like all nonzero numbers, has two square roots, that are negatives of each other. All nonzero numbers also have three cube roots, related by de Moivre's formula, and so on. Complex analysis does not rely on any such uniqueness. The complex numbers (or complex plane, or Argand diagram) is constructed from a special formal element "i", which simply serves as the second basis element of the vector space being constructed. The "square root of -1" and "i" are not the same thing. There are two square roots of -1, but only one "i". The square root of -1 doesn't even exist until the complex plane has been constructed, so it couldn't have been used in the construction.
The notion that complex analysis depends on a purported uniqueness of the square root of -1 needs to be put to rest. There is no assumption that the square root of -1 has to be unique. And a good thing, too, because it isn't unique. No one denies that -1, like all nonzero numbers, has two square roots, that are negatives of each other. All nonzero numbers also have three cube roots, related by de Moivre's formula, and so on. Complex analysis does not rely on any such uniqueness. The complex numbers (or complex plane, or Argand diagram) is constructed from a special formal element "i", which simply serves as the second basis element of the vector space being constructed. The "square root of -1" and "i" are not the same thing. There are two square roots of -1, but only one "i". The square root of -1 doesn't even exist until the complex plane has been constructed, so it couldn't have been used in the construction.