From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6634 Path: news.gmane.org!not-for-mail From: David Yetter Newsgroups: gmane.science.mathematics.categories Subject: Re: Explanations Date: Sat, 23 Apr 2011 15:27:31 -0500 Message-ID: References: Reply-To: David Yetter NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v1084) Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1303689807 5220 80.91.229.12 (25 Apr 2011 00:03:27 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 25 Apr 2011 00:03:27 +0000 (UTC) Cc: categories@mta.ca To: Graham White Original-X-From: majordomo@mlist.mta.ca Mon Apr 25 02:03:23 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1QE9Gs-0003w5-S9 for gsmc-categories@m.gmane.org; Mon, 25 Apr 2011 02:03:23 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:38305) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QE9GT-0004g6-Rf; Sun, 24 Apr 2011 21:02:57 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QE9GQ-0006za-TM for categories-list@mlist.mta.ca; Sun, 24 Apr 2011 21:02:55 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6634 Archived-At: My private reply to the original query from Jean-Pierre Marquis pointed to the style of combinatorial proof you refer to: they are called = "bijective" or "combinatorial" proofs depending on the author, and rely on giving interpretations of "big ugly formula_1" and "big ugly formula_2" as = enumerating the same thing by different means. For instance on can prove that n*2^{n-1} =3D \sum_{k=3D1}^n k*C(n,k) (writing C(n,k) for the binomial coefficient "n chose k") by = differentiating the binomial theorem and evaluating at 1, but this hardly seems to explain = it. Better is to observe that both sides count the number of ways to select = a=20 subset with a distinguished element from an n element set, the LHS by selecting the distinguished element, then the rest of the subset, the = RHS by choosing a cardinality k for the subset, selecting the subset then = selecting the distinguished element from the subset. David Y. On 22 Apr 2011, at 08:55, Graham White wrote: > And the folklore is (I haven't checked this in a proper history book) > that Gauss proved quadratic reciprocity numerous times because he = didn't > consider the proofs sufficiently explanatory. It's certainly true that > modern proofs (i.e. those using the methods of algebraic number = theory) > generalise it, and thereby explain, for example, what it is about the > rationals, and the number two, that makes primes in the rationals obey > quadratic reciprocity. I think one conclusion here is that, if you say > "explanatory", I am entitled to answer "so what do you want = explained?" >=20 > Another point is this: there are lots of combinatorial > identities of the form >=20 > big ugly formula_1 =3D big ugly formula_2 >=20 > which can be proved directly (for example, by induction > and a lot of algebra), but where the proof is utterly unilluminating. > And in many cases there are more conceptual proofs which people > generally find more illuminating (depending on taste, of course). >=20 > Graham >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]