From edward.cherlin.sy.67@aya.yale.edu Sun Jun 09 15:25:04 2002 Return-Path: X-Sender: cherlin@pacbell.net X-Apparently-To: lojban@yahoogroups.com Received: (EGP: mail-8_0_3_2); 9 Jun 2002 22:25:04 -0000 Received: (qmail 86892 invoked from network); 9 Jun 2002 22:25:04 -0000 Received: from unknown (66.218.66.216) by m15.grp.scd.yahoo.com with QMQP; 9 Jun 2002 22:25:04 -0000 Received: from unknown (HELO mta6.snfc21.pbi.net) (206.13.28.240) by mta1.grp.scd.yahoo.com with SMTP; 9 Jun 2002 22:25:04 -0000 Received: from there ([216.102.199.245]) by mta6.snfc21.pbi.net (iPlanet Messaging Server 5.1 (built May 7 2001)) with SMTP id <0GXG00FGKM9RZZ@mta6.snfc21.pbi.net> for lojban@yahoogroups.com; Sun, 09 Jun 2002 15:25:04 -0700 (PDT) Date: Sun, 09 Jun 2002 15:25:03 -0700 Subject: Re: [lojban] Algebra In-reply-to: <02053122153303.02104@neofelis> To: lojban@yahoogroups.com Message-id: <0GXG00FGMM9RZZ@mta6.snfc21.pbi.net> Organization: Web for Humans MIME-version: 1.0 X-Mailer: KMail [version 1.3.2] Content-type: text/plain; charset=iso-8859-1 Content-transfer-encoding: quoted-printable References: <02053122153303.02104@neofelis> X-eGroups-From: Edward Cherlin From: Edward Cherlin Reply-To: edward@webforhumans.com X-Yahoo-Group-Post: member; u=31895329 X-Yahoo-Profile: echerlin On Friday 31 May 2002 19:15, Pierre Abbat wrote: > There are at least four different meanings of "algebra, algebraic" > in mathematics. I don't think it's right to call them all > {aljebra}, so I try to come up with terms for them: 0. Algebra is originally the study of methods for solving sets of=20 polynomial equations with integer coefficients. The word comes from=20 the title of Al-Khowarizmi's book, Kitab al-jabr w'al muqabalah,=20 translated into Latin as Ludus Algebrae et Almucgrabalaeque. The=20 great achievement laid out in this book was the solution of the=20 general quadratic for real roots. This type of includes single polynomials (where there are solutions=20 in radicals for equations up to the fourth degree), and multiple=20 linear equations (where there are solutions for n equations in n=20 variables, pseudo-solutions for too many equations using least=20 squares, and other pseudo-solutions for too few equations (which I=20 have seen, but don't know much about). The solution of mixed linear=20 and quadratic systems arises in Newtonian physics, but has no general=20 name. There is no general name for the study of sets of polynomials=20 of higher degree. When solutions are constrained to the ring of=20 integers rather than the field of reals, we get the study of=20 Diophantine equations. Abstract algebra began with Galois's use of group theory in the proof=20 that there is no formula in radicals for solving fifth-degree=20 polynomial equations. It then proceeded onward to Abelian groups,=20 rings, fields, groupoids, semigroups, monoids, lattices, algebras of=20 numerous types, and to crown everything, category theory. Another=20 direction of expansion was the solution of polynomial equations over=20 algebraic structures other than the fields of real and complex=20 numbers. Yet another was the study of questions other than=20 root-finding for various families of polynomial equations, such as=20 elliptic curves.=20 Linear algebra is the field dealing with multiple linear equations,=20 including vectors, vector spaces, matrices, determinants, matrix=20 "multiplication", matrix inversion, and the like, and extending to=20 vector spaces and matrices over arbitrary fields, tensors, and much=20 more. Deep connections have been found with other areas of mathematics.=20 Knot polynomials arise in von Neumann algebras. A partial resolution=20 of the Taniyama-Shimura conjecture connecting elliptic curves and=20 modular functions included the proof of Fermat's Last Theorem.=20 Diophantine equations turn out to be Turing-complete, resolving=20 Hilbert's Tenth problem in the negative (no general method of=20 solution is possible). In such cases we tend not to lump the fields=20 together. Actually, all of mathematics can be considered branches of=20 algebra, but this is not really a helpful point of view. > 1. The basic meaning of "algebra" is manipulation of mekso. What's > the word for "manipulate"? First, this is a derivative meaning. Second, the word "manipulate"=20 uses the metaphor of doing things by hand, which is inappropriate for=20 a mental activity.=20 What we want here is a term for arithmetic operations on some=20 suitable subset of mekso which have been proved to give valid=20 results, specifically addition, subtraction, multiplication and=20 factoring in a ring of polynomials over some field, or those plus=20 division in the field of rational functions (abstract quotients of=20 polynomials) over some field. The place structure should allow us to=20 define the algebraic "object" in question with some precision. > 2. Abstract algebra, in one sense, is the study of sujgri, > piljygri, and dilcygri. So girzu saske. But what's the > word for "operation"? (The other sense is algebra, in the first > sense, in fields other than Q, R, or C. For instance, one can > compute an elliptic curve sum by taking the formal derivative of a > polynomial in a finite field and get sensible results, even though > taking a derivative of a function in a finite field makes no > sense.) Abstract algebra is the study of structures with various operations=20 obeying various combinations of rules. The determining factors are=20 the number of elements in the structure, the number of operations,=20 and the set of rules. The essential distinctions are in the rules. > 3. An algebra is a vector space with multiplication. {farlaili'i > piljygri}?=20 This is a completely different usage. Like every other algebraic=20 structure, it should have a descriptive name. >4. An algebraic number is a number which is a solution > to a polynomial equation. Whatever we do about the term for the study of solutions of=20 polynomial equations, we should be able to use it in tanru in just=20 this way. If X is the study of solving polynomial equations,=20 then the solutions of polynomial equations form precisely the set of=20 X-ish numbers (which has of course a natural field structure that has=20 no bearing on this discussion). A more precise description, say of=20 F-algebraic numbers as the set of numbers which are solutions of=20 polynomial equations over some field F , and algebraic numbers as the=20 set of numbers which are solutions of polynomial equations over the=20 reals, will of course be available also. > Any more suggestions? In sum, I would like to see a general term for the study of methods=20 of finding roots/solving sets of equations, a general term for=20 solution-preserving operations on mathematical problems, and a=20 general term for structures with operations that obey rules, each=20 with appropriate ways of specifying the underlying domain. In=20 addition, there should be both a general method and several more=20 specific terms for restricting the domain of discourse to sets of=20 polynomial equations, or 2nd-order partial differential equations, or=20 ruler-and-compass constructions, or whatever. Here is a suggestion for some subdivisions of the semantic space: x1 is a structure with combinations of rulesets and operations x2=20 over particular sets (or abstract sets with cardinality in some=20 specified set) as domain x3 x1 is a problem-space defined by type of equation set x2 over=20 structure x3 x1 is a problem from problem-space x2 x1 is-algebraic (deals with methods of solution for sets of=20 polynomial equations) x1 is a solution-preserving-operation (method) on a problem-space Further precision is possible, and might be desirable. Then we can speak of algebraic equations, algebraic problem-spaces=20 and problems, algebraic methods, algebraic structures, and so on. The=20 space of traditional Diophantine equation problems can be described=20 as=20 le problem-space (polynomial equation with integer coefficients)=20 (positive integers with addition, multiplication, and limited=20 subtraction) We can extend that to the modern sense simply by specifying the ring=20 of integers in x3. Terms for rings, fields, lattices, and so on should not use the=20 English metaphors.=20 > phma Ed Cherlin Generalist "A knot!", cried Alice. "Oh, please let me help to undo it."