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::Proofs that rely on this assumption are obviously less favored than proofs that do not, and hence the important concept of an [[elementary proof]]. Deny its importance all you want, but the beauty of logic is that it doesn't matter who accepts or denies it. The logic remains as truthful regardless.--[[User:Aschlafly|Andy Schlafly]] 11:26, 27 August 2009 (EDT)
::Proofs that rely on this assumption are obviously less favored than proofs that do not, and hence the important concept of an [[elementary proof]]. Deny its importance all you want, but the beauty of logic is that it doesn't matter who accepts or denies it. The logic remains as truthful regardless.--[[User:Aschlafly|Andy Schlafly]] 11:26, 27 August 2009 (EDT)
== Errors ==
This page is frankly bizarre. I can't imagine a working mathematician agreeing with much of it in its current state.
: 1. The assertion that an elementary proof "uses only real numbers or real analysis rather than the use of complex analysis" agrees with the cited MathWorld stub, but that stub misrepresents its source, Hoffman's book on Erdos. The original source says, "In this context, elementary means that the proof of the formula relies on a restricted set of numbers, the so-called real numbers...", the context being Erdos avoiding the "heavy machinery" including complex analysis employed by Hardy and Ramanujan in an asymptotic estimate of the partition function. An elementary proof is not in general characterized by the avoidance of complex analysis but rather by the avoidance of any "heavy machinery" whatsoever (like, say, the Hahn-Banach theorem, which in its real formulation is very much a part of real analysis). A decent test for the elementary-ness of a proof is whether a bright high-schooler could follow it. The definition needs to be re-written and a better source found. It should also be noted that the definition is inherently vague and different people disagree on the finer points.
: 2. The assertion that non-elementary proofs rely "on less rigorous axioms, such as the Axiom of Choice" is misleading. The axiom of choice is not "less rigorous" than, say, the axiom of infinity--it simply asserts an assumption; rigor does not enter in. The axiom of choice is somewhat controversial, though; "less controversial axioms" in the above would be true.
: 3. "...but in 1949 and 1950 an elementary proof by Paul Erdos and Atle Selberg earned Selberg the highest prize in math" -- it's misleading to imply Selberg's elementary proof was the only thing that got him his Fields medal. "helped earn" would be correct and would avoid a digression into discussing Selberg's work.
: 4. "avoiding the assumption that there is a unique, algebraically manipulable square root of negative one" -- this highlights a fundamental misunderstanding which may underlie my second point. The existence of complex numbers is not an "assumption" in the sense mathematicians use. Typically, one assumes some basic facts about set theory (say the ZF axioms, which includes the axiom of infinity I mentioned before) and uses those to build up algebraic objects. The existence of complex numbers satisfying the usual properties is then a *consequence* of these background assumptions (which are also used in elementary proofs) rather than an assumption itself. The use of complex analysis does not entail the use of a new assumption [nitpick: the axiom of dependent choice is typically added to the usual ZF axioms to do complex analysis, though that's rather uncontroversial and is also used in real analysis, almost always implicitly because it's so intuitive]. Again, the point of an elementary proof is that it uses no "heavy machinery"; I've never actually seen an elementary proof that used fewer assumptions than a non-elementary counterpart even though this page mentions that phenomenon several times. An example (keeping in mind the proper mathematician's idea of an "assumption") would be helpful.
: 5. The article does not discuss this, but elementary proofs are not always preferred. Elementary proofs tend to be very long and rather messy--a famous solution to the Basel problem comes to mind as an example. If one uses non-elementary Fourier analysis, one can solve the problem in less than half a page by computing the Fourier coefficients of f(x) = x and applying Parseval's theorem. The elementary alternative I'm aware of fiddles around with trigonometric identities and polynomial arithmetic for quite a while, is tedious, and is actually much harder to follow. The formulation I'm aware of also uses complex numbers, though not complex analysis.
I'd also like to say that Aschlafly should probably avoid editing this page. The quote, "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" shows a fundamental misunderstanding of the role of assumptions in mathematics and also shows his ignorance about the subject. The construction of complex numbers from real numbers is quite simple: mod the ring of one-variable real polynomials by the ideal generated by (x^2 + 1); this is the standard splitting field construction in this special case. There is no additional assumption necessary, just more machinery (rings, ideals, quotients, etc.).
I haven't edited the article since (from this talk page) doing so seems rather controversial.
[[User:Elproof|Elproof]] 19:45, 7 April 2012 (EDT)