From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5986 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= Newsgroups: gmane.science.mathematics.categories Subject: Are mathematical proofs incomparable with proofs in other disciplines? Date: Fri, 9 Jul 2010 15:55:53 -0400 Message-ID: References: Reply-To: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1278765866 3721 80.91.229.12 (10 Jul 2010 12:44:26 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 10 Jul 2010 12:44:26 +0000 (UTC) To: "Vaughan Pratt" , "categories list" Original-X-From: categories@mta.ca Sat Jul 10 14:44:21 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OXZPl-0001nn-Ie for gsmc-categories@m.gmane.org; Sat, 10 Jul 2010 14:44:17 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OXZ0p-0003sC-Ch for categories-list@mta.ca; Sat, 10 Jul 2010 09:18:31 -0300 Thread-Index: Acsfbr3HP/YBbYuJQnGV9Z+x04gAHQAA65rl Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5986 Archived-At: Dear Vaughan, I agree with your definition: "A proof is sufficient evidence for the truth of a proposition," The article http://en.wikipedia.org/wiki/Proof does not discuss the idea (of Paul Lorenzen) that a mathematical proof=20 is essentially a winning strategy in a formal game. I first learned the idea from Andreas Blass who introduced the game semantic of linear logic, http://arxiv.org/abs/math/9310211 A proof can be viewed as an argumentation to convince others of the = validity of a statement. In mathematics, the argumentation must be solid enough to resist any conter-argumentation by an ideal opponent. It can be compared to a winning strategy in a game with two players,=20 one defending a statement and the other attacking it. Lorenzen associates to every mathematical statement S a formal game with = two players G(S), the defender and the attacker. The defender has a winning strategy iff = the statement has a formal proof. The rules of the games for a proof in intuitinistic logic differ from the rules for a proof in classical logic. In other words, the rules of the games are determining the logic and = vice versa. I believe that game semantic is putting some light on the origin of = logic. I guess that logic was discovered by peoples debating in a democratic = manner. All communities need to choose between different courses of actions.=20 There are many answers to the question: how should this choice made?=20 One was given by Plato who favored a government=20 by the "philosopher king" who "loves the sight of truth": http://en.wikipedia.org/wiki/Plato#The_State Plato does not like Athenian democracy because it is imperfect.=20 He observes that its political debates are manipulated by sophists.=20 I agree with Plato that democracy is imperfect. But it should be improved, not condemned. Logic is anti-authoritarian since it wishes to convince, not to coerce. Best, Andr=E9 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]