Notes on Algebraic Methods
Introduction
These are some thoughts about algebraic methods which arose while considering the formalisation of geometric algebra.
The topic here involves the interplay in methods used in abstract mathematics and theoretical computer science. The methods in mathematics originate in the solution of arithmetical problems (I'm guessing that this is the origin of "algebraic" methods in general), and progress through various stages of abstraction as follows:
  • arithmetic and algebra
  • other algebras
  • abstract algebras
  • universal algebra and category theory

It is not my aim to tell the story of this progression as a purely mathematical affair. It is firstly of interest here primarily as a story of the evolution of methods, and little attention will be given to the details of the mathematics. Algebraic methods are of interest as representing the progress of abstraction, though initially they concern particular areas of mathematics, (arithmetic rather than geometry) they lead ultimately to category theory which transcends such limits.

It seems to me that in transition from the algebra of number systems to abstract algebra, the principle characteristic of algebra is completely transformed. Algebra appears first as a feature of numerical rather than geometric mathematics, but ends as a feature of abstract mathematics (concerned with classes of structures, e.g. groups, rings, categories) rather than of concrete mathematics (concerned with particular structures, such as the natural numbers or the reals)..

It is of interest methodologically as one line in the development of methods which bear upon the formalisation and mechanisation of mathematics.

Other historical threads which contribute to this broader topic include:

  • the axiomatic method
  • the formalisation of mathematics
  • formal methods in computer science and information engineering
  • the automation of reason

Other Algebras

The development of number systems provides for the evolution of the concept of number and with it of the techniques of algebra, but leaves algebra as a single growing body of theory and technique. Eventually however, this linear development is complicated by the study of structures which have a weaker claim to be considered natural extensions to a single number system, which may represent multiple distinct lines of extension, or completely new domains for algebraic methods.

Abstract Algebra
Universal Algebra and Category Theory
Arithmetic and Algebra

Though some authors place the beginning of mathematics with the Greeks, because no mathematical results are known to have been proven before them, the practical use of algebraic methods pre-dates any rigorously developed theory and is found in Babylonia.

Arithmetic in its most elementary practical form consists in having a notation for numbers together with methods for doing computations using that notation.

Algebra begins as methods extending the applications of arithmetic in the following ways. It involves the use of symbols distinct from the numerals for unknown or variable values. By manipulation of expressions involving unknowns, possibly using transformations which may be described using variables (because they apply regardless of the particular numbers involved) it is possible to solve practical problems which might be soluble by straightforward computation.

Algebraic techniques broadly of this kind continued to be developed over millenia. In this the main milestones were, first of all the instigation of mathematics as a deductive theoretical discipline, which results in proven general theorems which legitimate applicable algebraic manipulations, and then the occasional elaboration of the concept of number, most notably that which was necessary for the differential and integral calculus and the enormous expansion of applicable mathematics which flowed from it under the stimulus of demand from science and engineering.

Peacock

Algebra up to, let us say, 1830, was a fruitful but not very rigorously founded enterprise, in which algebraic laws known to be true for the natural numbers were applied to a much larger number system, elements of which were controversial. At this time there was still controversy even over the use of negative numbers, but also, particularly of complex numbers. The status of transcendental numbers and of the number system as a whole was not well defined.

George Peacock, a professor of mathematics at Cambridge, attempted to put together a rationale for the algebra of the time which is illuminating. He divided algebra into two areas:

  • arithmetical algebra
  • symbolic algebra
The first deals with symbols representing the positive integers and is on solid ground, subject to the constraint that only operations yielding positive integers are allowed. "symbolic algebra" adopts the rules of "arithmetical algebra" but removes the restriction to positive integers. All results deduced in arithmetical algebra which are general in form, though restricted in the application to positive integer values, are deemed to be valid in symbolic algebra without the restriction on the range of the variables. This is known as the "principle of permanence of form".

Axioms of Algebra

As of the middle of the 19th century the following were the generally accepted laws of algebra:

  1. equal quantities added to a third yield equal quantities
  2. associativity of addition
  3. commutativity of addition
  4. equals added to equals give equals
  5. equals added to unequals give unequals
  6. associativity of multiplication
  7. commutativity of multiplication
  8. right distribution of multiplication over addition
The principle of permanence of form is said to have rested on these axioms.


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