Talk:Elementary proof
Who uses this concept?
I don't think that it is correct that anyone calls a proof "elementary" just because it does not use the square root of (-1). Nor is it true that an elementary proof is necessarily preferred.
It is not an assumption that (-1) can have a square root. It can be proved. Nor is the square root unique. (-1) has 2 square roots, if it has any.
I suggest killing this article. It just isn't useful. RSchlafly 00:11, 5 February 2007 (EST)
If the test were what is "useful", then most math articles should be deleted!
I'd love to see a proof that the following exists and is unique (plus and minus roots):
--Aschlafly 20:12, 6 February 2007 (EST)
- Existence depends on what system you are operating in. Uniqueness is fairly easy if you are operating in a field. Proposition: in any field F, with c and element of F, the equation
has at most 2 solutions. Proof: Consider . this implies that 
. We may assume that c has at least one square root (call it 
). So we have 
. Now, since fields have no zero divisors (easy excercise), we must have 
or 
which gives us only two choices. Q.E.D. JoshuaZ 20:44, 6 February 2007 (EST)
- I have never heard an "elementary proof" used to refer to a proof that "cannot be broken down into smaller proofs", whatever that is supposed to mean, and moreover, this doesn't clearly have anything to do with the other definition about avoiding complex analysis. Moreover, Selberg's Fields was primarily for his other work on the Riemann zeta function (complex analysis! the horror!) and Selberg formula, which indicated the direction of this proof. The proof itself was not the grounds for the medal, as this article indicates. Some editor here seems to have a bizarre fascination with "elementary proofs", but would surely be hard-pressed to find a serious mathematician who rejected residue calculus on the grounds that it is not an elementary method, as that article claims.
Re: "I'd love to see a proof that the following exists and is unique (plus and minus roots)", the questioner might try to learn a bit about fields and algebraic extensions thereof. By "sqrt(-1)" we mean some symbol which is "adjoined" to a field and which satisfies x^2+1=0. Such a thing exists in a field extension (and one can prove this quite rigorously), and is obviously unique there. Any basic algebra text (Artin, Dummit/Foote, etc) will cover the requisite material better than I could hope to do here.
- jlezzy
This poster was indeed rude, but can we get citations for any of this? I can't find any evidence that there are mathematicians who reject the method of residues for being non-elementary.
Definition of Elementary proof
I would like to remove the statement about elementary proofs being those that cannot be broken down into smaller proofs. This statement doesn't make much sense to me. What so we mean by a 'smaller' proof? What do we mean by breaking a proof down into other proofs? Typically the proof of a long theorem is broken down by proving many of the steps as a lemma. This is done to make the proof of the main theorem easier to comprehend. This can be done with any theorem whose proof has more than one step. Any non-trivial theorem is bound to have many steps so I don't think that this is what the author intends. AndyJM 12:05, 9 January 2009 (EST)
- AndyJM, I really don't understand what your are finding so baffling about the concept of elementary proofs. The line you removed was an important statement, and as you can see, Andy (the owner) reinstated it with some clearer wording. If you do not have experience with elementary proofs, then please do not remove content from the articles -- you've been banned for such actions before. -Foxtrot 22:00, 18 January 2009 (EST)
- Hi Foxtrot. I thought that I had clearly laid out what I find so baffling about one particular statement in this article. Did you read my comment? While I am not delighted with the addition that Schlafly made it is a definite improvement over what had been there. The vast majority (perhaps all) all the proofs that I have published could be classified as elementary according to the first definition, i.e. not using complex analysis. However I have no idea whether or not they would elementary according to the second definition. This suggests to me that the second definition could be a bit suspect or 'loose', so to speak. I am not out to cause trouble here, and despite your beliefs to the contrary I have not made any edits for ideological reasons. In fact to my mind mathematics is refreshingly free of ideology. Dia dhuit (to which the polite response is 'Dia is muire dhuit'). AndyJM 07:22, 19 January 2009 (EST)
I don't think that anyone uses the term "elementary proof" as it is defined here. Here is more typical usage:
- Keep in mind that the word ‘elementary’ is being used in many dif-ferent ways in this discussion. In informal mathematical usage, ‘elemen-tary’ is often used to mean ‘simple’ or ‘easy to understand’. In the Sel-berg/Nathanson sense it is used to mean ‘avoiding the use of infinitary (or analytic) methods’. The fact that Selberg’s proof of Dirichlet’s theorem is harder to understand than the standard analytic one shows that these two senses can be at odds! I have already noted that the use of the word ‘elementary’ in the phrase ‘elementary arithmetic’ is due to the fact that the axiom system is closely related to the class of ‘elementary functions’ in the sense of Kalmar. To complicate matters even further, among logicians the word ‘elementary’ is often used to mean ‘first-order’. [1]
As you can see, none of this matches what is in the article. I suggest that the article be deleted. RSchlafly 15:07, 30 January 2009 (EST)
It certainly seems reasonable that not using AoC would be desirable for calling a proof elementary. However, in my experience, the notion of an elementary proof is closer to "not using mathematics from unrelated areas" than it is in avoiding specific techniques or axioms. In number theory, this usually entails avoiding the methods of complex analysis. But to me, it is perfectly natural to say "To avoid this, we next give an elementary proof, which uses Zorn's lemma in a typical way" in a paper on algebra, where the application is indeed typical. That excerpt is from "Some ring theoretic equivalents to the Axiom of Choice", easily googleable, and I find several other examples of authors citing AoC in "elementary" proofs. It's easy to ignore this as the work of sloppy authors and/or liberals, but clearly the usage defined in this CP is at the very least not universal, and indeed I have never encountered it. Might I modify the page to indicate that some authors use a different characterization of elementary proofs, and can we find a source that actually uses a definition resembling that here? --MarkGall 01:23, 25 August 2009 (EDT)
