From edward.cherlin.sy.67@aya.yale.edu Mon Feb 18 23:02:04 2002 Return-Path: X-Sender: cherlin@pacbell.net X-Apparently-To: lojban@yahoogroups.com Received: (EGP: mail-8_0_2); 19 Feb 2002 07:02:03 -0000 Received: (qmail 84457 invoked from network); 19 Feb 2002 07:02:03 -0000 Received: from unknown (216.115.97.167) by m11.grp.snv.yahoo.com with QMQP; 19 Feb 2002 07:02:03 -0000 Received: from unknown (HELO mta5.snfc21.pbi.net) (206.13.28.241) by mta1.grp.snv.yahoo.com with SMTP; 19 Feb 2002 07:02:03 -0000 Return-path: Received: from there ([216.102.199.245]) by mta5.snfc21.pbi.net (iPlanet Messaging Server 5.1 (built May 7 2001)) with SMTP id <0GRR00JONQ7EGY@mta5.snfc21.pbi.net> for lojban@yahoogroups.com; Mon, 18 Feb 2002 23:02:03 -0800 (PST) Received: from mta3.snfc21.pbi.net (mta3-pr.snfc21.pbi.net) by sims1.snfc21.pbi.net (Sun Internet Mail Server sims.3.5.2000.01.05.12.18.p9) with ESMTP id <0GRN00D5ESUJ2I@sims1.snfc21.pbi.net> for cherlin@sims-ms-daemon; Sat, 16 Feb 2002 20:08:46 -0800 (PST) Received: from mailusa.dvtnet.com ([209.220.160.4]) by mta3.snfc21.pbi.net (Sun Internet Mail Server sims.3.5.2000.01.05.12.18.p9) with ESMTP id <0GRN00CIFSUA5T@mta3.snfc21.pbi.net> for cherlin@sims1.snfc21.pbi.net; Sat, 16 Feb 2002 20:08:34 -0800 (PST) Received: from SMTP32-FWD by webforhumans.com (SMTP32) id A00000CEC; Sat, 16 Feb 2002 20:06:05 -0800 Date: Mon, 18 Feb 2002 23:02:02 -0800 Subject: Constant functions Sender: postmaster@mailusa.dvtnet.com To: lojban@yahoogroups.com Message-id: <0GRR00JOOQ7EGY@mta5.snfc21.pbi.net> Organization: Web for Humans MIME-version: 1.0 X-Mailer: KMail [version 1.3.1] Content-type: text/plain; charset=iso-8859-1 Content-transfer-encoding: quoted-printable X-eGroups-From: Edward Cherlin From: Edward Cherlin Reply-To: edward@webforhumans.com X-Yahoo-Group-Post: member; u=31895329 X-Yahoo-Profile: echerlin From: Invent Yourself Date: Wed, 13 Feb 2002 19:53:00 -0500 (EST) >On Thu, 14 Feb 2002, Jorge Llambias wrote: > >> >> la xod cusku di'e >> >> >Well, I wasn't using "1" as an analog for "f", but as a=20 polynomial. >> >In other words, not its name, but its expression. So I don't see=20 how the >> >confusion concerning f(x) is relevant. >> >> The expression "x^2+x+1" is sometimes used to refer to the=20 polynomial, >> and other times to refer to the value that the polynomial takes on >> for a given value in its domain. When we write f(x)=3Dx^2+x+1, we >> mean sometimes that the function f(x) is the funtion x^2+x+1, >> and sometimes we mean that the value f(x) is equal to the value >> x^2+x+1. In other words, we use the same expression to refer to >> {lo te fancu} and to {lo ve fancu}. You are doing the same with >> {li pa}. It's a very widespread practice in mathematics, and it >> hardly ever leads to confusion, but it is the type of thing that >> Lojban takes pride in differentiating. > >Yet in this case, I don't yet see how I could express the function=20 "1" as >distinct from the value "1". Do you? TCLL 18.6, p. 438 How do we express "z =3D f(x)"? The answer is: 6.3) li zy du li ma'o fy.boi xy. The construct "ma'o fy.boi" is the equivalent of an operator... TCLL 18.21, p. 460 To change an operand into an operator, we use the cmavo "ma'o"...In=20 fact, "ma'o" can be followed by any mekso operand, using the elidable=20 terminator "te'u" if necessary. That means we can turn an expression into a function with "ma'o".=20 So any function with constant value 1 from any non-empty domain to=20 any range containing 1 is evaluated by "li ma'o li pa", and we can=20 say things like zoi .kuot lambda(x).1 .kuot fancu lo'i namcu lo'i namcu li ma'o=20 li pa ko'a goi le fancu be jo'ite'u be'i lo'i li pa be'i li ma'o li pa=20 be'o... where "jo'ite'u" is the empty vector, the vector of length 0. Note: The use of the terms "operand" and "operator" in the RefGrammar=20 is not always consistent with the practice of mathematicians.=20 An expression is an operand only if a operator is being applied to=20 it.=20 Most programming languages describe functions as operators, except=20 for the two families founded by mathematicians, namely APL and LISP.=20 LISP does not distinguish data objects from function objects (both=20 are just lists), while APL has a three-level type hierarchy: arrays=20 (data), functions (data-->data), and operators=20 (functions-->functions, or as an option in some dialects,=20 functions-->data)