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::::: Perhaps there are some new mathematicians who are in denial about the importance of elementary proofs. There are also new physicists who deny the importance of [[falsifiability]]. But self-described experts do not get to be the only watchdogs for their fields. Some accountability can and should come from outside the fields of expertise.--[[User:Aschlafly|Andy Schlafly]] 12:23, 25 August 2009 (EDT)
::::: Perhaps there are some new mathematicians who are in denial about the importance of elementary proofs. There are also new physicists who deny the importance of [[falsifiability]]. But self-described experts do not get to be the only watchdogs for their fields. Some accountability can and should come from outside the fields of expertise.--[[User:Aschlafly|Andy Schlafly]] 12:23, 25 August 2009 (EDT)
::::::If that is indeed the traditional definition (i.e., if the traditional definition would necessarily exclude proofs using AoC), you should have no difficulty providing a relevant citation, and I won't argue any more! It is true that proofs involving choice are generally dispreferred, but as far as I know they have never been called "non-elementary". I have given a citation which contradicts your characterization, and can produce more -- please provide just one in favor of it! If your definition were indeed in use at all, I would expect that searching for "Tychonoff theorem non-elementary" would find some page describing this result as "non-elementary" since it is one of the most famous proofs involving choice. However, it does not, but it does find an "elementary" proof of the theorem (employing choice, of course).
::::::Of course some accountability from non-mathematicians is important, but this discussion isn't about whether proofs using AoC should be recognized as such and pointed out (such oversight is of course welcome). The question is whether the term "elementary proof" excludes all proofs using choice, and it seems to me that it does not.
::::::Historically this term originated to describe proofs in number theory which did not use complex analysis, a usage which now falls under either umbrella. Selberg's Fields Medal was for a variety of contributions (like the Nobel prize, it is given not for a single result, but for overall contributions to mathematics), the primary of which (according to the official citation) was his extension of Brun's work on the zeta function (showing that a positive proportion of zeroes lie on the critical line), decidedly non-elementary work! --[[User:MarkGall|MarkGall]] 12:55, 25 August 2009 (EDT)