From Quaternions and Octaves to Circuit Design

An Exploration of Modern Boolean Algebra via its Origins in British Symbolic Algebra

Abstract

This Primary Source Project (PSP) is designed for an introductory or intermediate course in discrete or finite mathematics or in an "introduction to proof" course. Its primary goal is to introduce students to a more abstract view of algebra, one focused on the structure and underlying axioms of a system, through an examination of the specific structure known today as a boolean algebra. It does this through historical examples, including select readings from two important primary sources: George Boole’s 1847 Laws of Thought and Claude Shannon’s 1938 A Symbolic Analysis of Relay and Switching Circuits. Both sources also raise questions related to mathematical proof which are further and more explicitly addressed within the final non-historical section of the project on modern boolean algebra.

Although Boole’s writing is essentially focused on what would now be called 'introductory set theory,' it also lays the ground work for a more abstract discussion of boolean algebra as a discrete axiomatized structure. Importantly for students making the transition to more abstract mathematical studies, it does so in a very concrete fashion, while simultaneously exposing what seem like strange algebraic rules such as idempotency. Shannon’s paper is similarly quite concrete, and has the advantage of revealing the important applied side of boolean algebra. Accordingly, this project may also be used as a complete introduction to the study of elementary boolean algebra in any course that considers that structure from either a mathematical or computer science perspective.