A broadly applicable method for modelling and analysis method exploiting symbolic logic.

These are some thoughts about algebraic methods which arose while considering the formalisation of geometric algebra.

Leibniz on the Universal Characteristic.

Key features of the method are its applicability to deductive reasoning in any domain through the use of abstract models defined using formal notations supported by appropriate computern software.

When deduction is possible, conditions for soundness.

The universality of abstract models. Their advantages.

The advantages of abstract modelling in conceptual clarification.

The use of formal notations is made feasible by the availability of software support.

This method of analysis is a central feature of the philosophy of Metaphysical Positivism, which provides a conceptual framework and rationale for the method and its applications.

This work in progress is intended to provide a history of the developments over a period of 2500 years which lead to the establishment of this method, and applies the method to historical exegesis.

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

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.