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: The real objection is that non-elementary proofs rely on additional assumptions. For complex analysis, that additional assumption is that there is a unique square root of negative 1, which can be manipulated algebraically. Maybe true, maybe false, and an elementary proof is desirable because it does not rely on that assumption. The Axiom of Choice also includes a questionable assumption.--[[User:Aschlafly|Andy Schlafly]] 09:50, 25 August 2009 (EDT)
: The real objection is that non-elementary proofs rely on additional assumptions. For complex analysis, that additional assumption is that there is a unique square root of negative 1, which can be manipulated algebraically. Maybe true, maybe false, and an elementary proof is desirable because it does not rely on that assumption. The Axiom of Choice also includes a questionable assumption.--[[User:Aschlafly|Andy Schlafly]] 09:50, 25 August 2009 (EDT)
::I am not questioning that there is a valid objection to using proofs that require additional assumptions, for example choice. This is a very reasonable view. I do question whether such proofs are actually called "elementary" by mathematicians. The case of applying complex analysis to number theory is non-elementary by either proposed definition. I have given one example in the literature which clearly contradicts the characterization of choice as necessarily non-elementary, and can easily find more. Can you please cite someone who uses your definition of an elementary proof? I'm willing to believe that this usage is in circulation, but this is the first I've run into it. --11:00, 25 August 2009 (EDT)