Part 2: Chapter 10
A liberal arts history of mathematics
August 16, 2002
The purpose of this chapter (A liberal arts history of mathematics) is to frame some cultural and philosophical issues regarding the nature of mathematics. So references to Foundations will be given, and in some cases specific scholarship will be developed in this chapter. But the aim is to write this material in such a way that a college freshman will be able to have access to a rich set of philosophical and cultural questions.
In many of the freshman mathematics texts we have an excellent history of enumeration. The focus of this curricular material is the provision of a sense of appreciation for the cultural aspects of counting and the role that counting has had in the development of modern society. These textbooks do a fine job at this.
However, the concepts should not be covered in one day and then left to go back to memorizing rules. The concepts are the beginning of a great philosophical debate that needs to be occurring within our democratic social, least we develop into a culture of knowledge haves and knowledge have not.
The author has made a study of the development and evolution of natural language. This study involves cultural anthropology of language evolution and the cognitive neuroscience of first and second language acquisition research. The research literature on these subjects will be made available to the learner, as well as a high level overview of the supporting disciplines.
Several theories about complex systems have developed in the past four decades. One is drawn from the works by John Holland and his students (genetic algorithms). Stu Kauffman draws the other from his works on autocatalytic sets.
The author has used these research areas to develop a sense of how language ability forms in the individual and within a social system. A more general theory of emergence is made available. Work on emergence is described in my Chapter 1, Foundations of Knowledge Science.
A philosophical position is taken in Chapter 2 of Foundations. This position is that formal systems such as categorical abstraction, mathematics and formal logics should not be characterized as a natural system having location, weight and mass. Rather these formal systems have a set of properties that mimic some properties of natural systems, while also having properties not found in natural systems.
These notes are formative towards Prueitt’s new book on learning theory.
We expect that the book will be developed over the next four months.