Changes

We need to work out a careful survey of the things that make some proofs more complicated than other proofs of the same theorem, and how the word "elementary" is used to describe this, and what the various ramifications of this usage are.  I'm willing to work on this.  [[User:PatrickD|PatrickD]] 23:06, 26 August 2009 (EDT)

We need to work out a careful survey of the things that make some proofs more complicated than other proofs of the same theorem, and how the word "elementary" is used to describe this, and what the various ramifications of this usage are.  I'm willing to work on this.  [[User:PatrickD|PatrickD]] 23:06, 26 August 2009 (EDT)

:Beyond specific examples (and I haven't much to add to Patrick's exposition), Mr. Schlafly's general point that "Proofs relying on fewer assumptions are obviously preferable" is an important one, and I think that most mathematicians would agree with this.  (However, as Patrick explains, the use of complex analysis is not an additional assumption in any sense, and it is constructed and verified with complete rigor from nothing beyond ZF.  So I'm mostly interested in AoC as it pertains to the notion of the elementary proof.  More later on the issue of analytic number theory.)  Proofs avoiding choice are to be preferred for their weaker assumptions, as noted: for example, the Schroeder-Bernstein theorem is a theorem for which there are two proofs: a quick trick relying on choice, and a slightly more nuanced proof without it.  In my experience (both in studying and teaching), the latter is generally taught despite its relative difficulty.  My only encounter with the former was as a homework problem introducing the application of AoC.  My issue with the page is that the term "elementary proof" is not actually used to describe such proofs.  There are other ways to characterize such proofs formally, but I'm not sure of any commonly-used term for them.  I'd suggest that discussion of related issues ought to be incorporated into the article on AoC, and that this article be changed to reflect what the term actually means.  

:Beyond specific examples (and I haven't much to add to Patrick's exposition), Mr. Schlafly's general point that "Proofs relying on fewer assumptions are obviously preferable" is an important one, and I think that most mathematicians would agree with this.  However, as Patrick explains, the use of complex analysis is not an additional assumption in any sense, and it is constructed and verified with complete rigor from nothing beyond ZF.  So I'm mostly interested in AoC as it pertains to the notion of the elementary proof -- as RSchlafly had added, this seems to be by far the most important case where we find proofs with extra assumptions.  More later on the issue of analytic number theory.

 

:Proofs avoiding choice are to be preferred for their weaker assumptions, as noted: for example, the Schroeder-Bernstein theorem is a theorem for which there are two proofs: a quick trick relying on choice, and a slightly more nuanced proof without it.  In my experience (both in studying and teaching), the latter is generally taught despite its relative difficulty.  My only encounter with the former was as a homework problem introducing the application of AoC.  My issue with this page is that the term "elementary proof" is not actually used to describe such proofs.  There are other ways to characterize such proofs formally, but I'm not sure of any commonly-used term for them.  I'd suggest that discussion of related issues ought to be incorporated into the article on AoC, and that this article be changed to reflect what the term actually means.  

:I'd also reiterate that Selberg's Fields was awarded largely on the grounds of his contributions to the study of the zeta function, so to imagine that it was awarded for minimizing the use of complex analysis is fantasy.  His proof was important, however, for the new methods that it made available to number theorists. Erdos's dedication to "elementary proofs" arose not from doubts about the validity of complex analysis (in the case of PNT and other problems in number theory), but rather from the intrinsic aesthetic appeal of proofs which don't make use of unnecessarily powerful methods.  A proof which does not crack a nut with a sledgehammer (as older proofs of PNT did) is more beautiful than one that does, and Erdos deeply valued "beautiful" proofs.  It is these proofs, which expose the profound and elegant nature of the theorems they establish, that are to be found "in the book".  

:I'd also reiterate that Selberg's Fields was awarded largely on the grounds of his contributions to the study of the zeta function, so to imagine that it was awarded for minimizing the use of complex analysis is fantasy.  His proof was important, however, for the new methods that it made available to number theorists. Erdos's dedication to "elementary proofs" arose not from doubts about the validity of complex analysis (in the case of PNT and other problems in number theory), but rather from the intrinsic aesthetic appeal of proofs which don't make use of unnecessarily powerful methods.  A proof which does not crack a nut with a sledgehammer (as older proofs of PNT did) is more beautiful than one that does, and Erdos deeply valued "beautiful" proofs.  It is these proofs, which expose the profound and elegant nature of the theorems they establish, that are to be found "in the book".  

:An unrelated issue is that there seem to be two notions of elementary proof used in this article, and I'm not sure how to reconcile them.  Selberg's proof of PNT is magnificently complicated, though it avoids the methods of analytic number theory.  It's not a proof that I would say "cannot be improved by expressing it in simpler form" -- indeed the analytic proofs are far simpler.  It's true that at heart these have a different method, but it's my understanding that a touch of complex analysis can simplify the Selberg argument while leaving its fundamental insights intact. --[[User:MarkGall|MarkGall]] 00:24, 27 August 2009 (EDT)

:An unrelated issue is that there seem to be two notions of elementary proof used in this article, and I'm not sure how to reconcile them.  Selberg's proof of PNT is magnificently complicated, though it avoids the methods of analytic number theory.  It's not a proof that I would say "cannot be improved by expressing it in simpler form" -- indeed the analytic proofs are far simpler.  It's true that at heart these have a different method, but it's my understanding that a touch of complex analysis can simplify the Selberg argument while leaving its fundamental insights intact. --[[User:MarkGall|MarkGall]] 00:24, 27 August 2009 (EDT)