Subject: Re: [lojban] Algebra

From: Edward Cherlin <[email-masked]>
Date: Sun, 09 Jun 2002 15:25:03 -0700

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 
polynomial equations with integer coefficients. The word comes from 
the title of Al-Khowarizmi's book, Kitab al-jabr w'al muqabalah, 
translated into Latin as Ludus Algebrae et Almucgrabalaeque. The 
great achievement laid out in this book was the solution of the 
general quadratic for real roots.

This type of includes single polynomials (where there are solutions 
in radicals for equations up to the fourth degree), and multiple 
linear equations (where there are solutions for n equations in n 
variables, pseudo-solutions for too many equations using least 
squares, and other pseudo-solutions for too few equations (which I 
have seen, but don't know much about). The solution of mixed linear 
and quadratic systems arises in Newtonian physics, but has no general 
name. There is no general name for the study of sets of polynomials 
of higher degree. When solutions are constrained to the ring of 
integers rather than the field of reals, we get the study of 
Diophantine equations.

Abstract algebra began with Galois's use of group theory in the proof 
that there is no formula in radicals for solving fifth-degree 
polynomial equations. It then proceeded onward to Abelian groups, 
rings, fields, groupoids, semigroups, monoids, lattices, algebras of 
numerous types, and to crown everything, category theory. Another 
direction of expansion was the solution of polynomial equations over 
algebraic structures other than the fields of real and complex 
numbers. Yet another was the study of questions other than 
root-finding for various families of polynomial equations, such as 
elliptic curves. 

Linear algebra is the field dealing with multiple linear equations, 
including vectors, vector spaces, matrices, determinants, matrix 
"multiplication", matrix inversion, and the like, and extending to 
vector spaces and matrices over arbitrary fields, tensors, and much 
more.

Deep connections have been found with other areas of mathematics. 
Knot polynomials arise in von Neumann algebras. A partial resolution 
of the Taniyama-Shimura conjecture connecting elliptic curves and 
modular functions included the proof of Fermat's Last Theorem. 
Diophantine equations turn out to be Turing-complete, resolving 
Hilbert's Tenth problem in the negative (no general method of 
solution is possible). In such cases we tend not to lump the fields 
together. Actually, all of mathematics can be considered branches of 
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" 
uses the metaphor of doing things by hand, which is inappropriate for 
a mental activity. 

What we want here is a term for arithmetic operations on some 
suitable subset of mekso which have been proved to give valid 
results, specifically addition, subtraction, multiplication and 
factoring in a ring of polynomials over some field, or those plus 
division in the field of rational functions (abstract quotients of 
polynomials) over some field. The place structure should allow us to 
define the algebraic "object" in question with some precision.

> 2. Abstract algebra, in one sense, is the study of sujgri,
> piljygri, and dilcygri. So <operation> 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 
obeying various combinations of rules. The determining factors are 
the number of elements in the structure, the number of operations, 
and the set of rules. The essential distinctions are in the rules.

> 3. An algebra is a vector space with multiplication. {farlaili'i
> piljygri}? 

This is a completely different usage. Like every other algebraic 
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 
polynomial equations, we should be able to use it in tanru in just 
this way. If X is the study of solving polynomial equations, 
then the solutions of polynomial equations form precisely the set of 
X-ish numbers (which has of course a natural field structure that has 
no bearing on this discussion). A more precise description, say of 
F-algebraic numbers as the set of numbers which are solutions of 
polynomial equations over some field F , and algebraic numbers as the 
set of numbers which are solutions of polynomial equations over the 
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 
of finding roots/solving sets of equations, a general term for 
solution-preserving operations on mathematical problems, and a 
general term for structures with operations that obey rules, each 
with appropriate ways of specifying the underlying domain. In 
addition, there should be both a general method and several more 
specific terms for restricting the domain of discourse to sets of 
polynomial equations, or 2nd-order partial differential equations, or 
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 
over particular sets (or abstract sets with cardinality in some 
specified set) as domain x3

x1 is a problem-space defined by type of equation set x2 over 
structure x3

x1 is a problem from problem-space x2

x1 is-algebraic (deals with methods of solution for sets of 
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 
and problems, algebraic methods, algebraic structures, and so on. The 
space of traditional Diophantine equation problems can be described 
as 

le problem-space (polynomial equation with integer coefficients) 
(positive integers with addition, multiplication, and limited 
subtraction)

We can extend that to the modern sense simply by specifying the ring 
of integers in x3.

Terms for rings, fields, lattices, and so on should not use the 
English metaphors. 

> phma

Ed Cherlin
Generalist
"A knot!", cried Alice. "Oh, please let me help to undo it."