From njs@uclink4.berkeley.edu Tue Jun 12 17:33:36 2001 Return-Path: X-Sender: njs@has-96-90.reshall.berkeley.edu X-Apparently-To: lojban@yahoogroups.com Received: (EGP: mail-7_1_3); 13 Jun 2001 00:33:36 -0000 Received: (qmail 67253 invoked from network); 13 Jun 2001 00:10:06 -0000 Received: from unknown (10.1.10.142) by m8.onelist.org with QMQP; 13 Jun 2001 00:10:06 -0000 Received: from unknown (HELO ostrich.mail.pas.earthlink.net) (207.217.120.14) by mta3 with SMTP; 13 Jun 2001 00:10:06 -0000 Received: from falcon.mail.pas.earthlink.net (falcon.mail.pas.earthlink.net [207.217.120.74]) by ostrich.mail.pas.earthlink.net (8.9.3/8.9.3) with ESMTP id RAA10633 for ; Tue, 12 Jun 2001 17:05:09 -0700 (PDT) Received: from WyrmWeyr.frop.org (pool0058.cvx21-bradley.dialup.earthlink.net [209.179.192.58]) by falcon.mail.pas.earthlink.net (EL-8_9_3_3/8.9.3) with ESMTP id RAA21340 for ; Tue, 12 Jun 2001 17:04:35 -0700 (PDT) Received: from njs by WyrmWeyr.frop.org with local (Exim 3.22 #1 (Debian)) id 159y9J-0004eH-00 for ; Tue, 12 Jun 2001 17:04:41 -0700 Date: Tue, 12 Jun 2001 17:04:41 -0700 To: lojban@yahoogroups.com Subject: Re: [lojban] Sapir-Whorf Hypothesis Message-ID: <20010612170441.A17190@njs.dhis.org> Reply-To: njs@uclink4.berkeley.edu Mail-Followup-To: lojban@yahoogroups.com Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline In-Reply-To: <20010612152214.L14438@digitalkingdom.org> User-Agent: Mutt/1.3.18i Sender: Nathaniel Smith From: Nathaniel Smith On Tue, Jun 12, 2001 at 03:22:14PM -0700, Robin Lee Powell wrote: > On Tue, Jun 12, 2001 at 12:50:20PM -0700, Edward Cherlin wrote: > > The best recent example is non-standard arithmetic, which comes in two > > forms, one from Robinson's model theory, and the other from Conway's > > advances in game theory. Both provide consistent but significantly > > different arithmetics with actual infinitesimals, and both can be > > extended to analysis. Without the appropriate definitions of terms and > > proofs of theorems, there is no way anybody outside the field can > > understand what either form is talking about, since mathematicians had > > previously "proved" that arithmetic with infinitesimals was > > impossible, and in particular Peano thought that he had proved the > > impossibility of any non-standard models of the natural numbers. > > As someone who almost has a math degree , I'm intrigued. > > What are infinitesimals, and where do I find out about game-theory based > arithmetic? Infinitesimals are numbers which are larger than zero, but smaller than any positive real number. Conway's game theoretic version is purportedly described best in "Winning Ways" by Berlekamp and Conway, but that's a two-volume several thousand page tome in which I have yet to find them. The book I read is Knuth's "Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness". It's a fun book, but not as detailed mathematically as one might want -- great if you want to explore them on your own, though. "On Numbers and Games" by Conway might also be good. I don't know much about Robinson's infinitesmals; they're interesting partly because he (and others) went through and redeveloped a whole theory of calculus and analysis where dx/dy really does mean dividing an infinitesmal by another infinitesmal... the book at hand mentions: Robinson, Abraham. (1966). Non-Standard Analysis. Amsterdam: North-Holland. Robinson, Abraham. (1979). Selected Papers of Abraham Robinson. Vol. 2: Nonstandard Analysis and Philosophy. New Haven: Yale University Press. Incidentally, I don't see how mathematical discourse is relevant to S-W. Mathematical discourse has a very different character, notation even more so, and refusing to accept ideas for philosophical reasons is different from being unable to understand them because of a restrictive language. -- Nathaniel -- But in Middle-earth, the distinct accusative case disappeared from the speech of the Noldor (such things happen when you are busy fighting Orcs, Balrogs, and Dragons).