On the Algebra in Boole's Laws of Thought

This article explores the ideas that went into George Boole’s development of an algebra for logic in his book The Laws of Thought. The many theories that have been proposed to explain the origins of his algebra have ignored his wife Mary Boole’s claim that he was deeply influenced by Indian logic. This paper investigates this claim and argues that Boole’s focus was more than a framework for propositions and that he was trying to mathematize cognitions as is assumed in Indian logic and to achieve this he believed an algebraic approach was the most reasonable. By exploring parallels between his work and Indian logic, we are able to explain several peculiarities of his algebraic system. Introduction There is continuing interest in the antecedents to George Boole’s The Laws of Thought [1] and an ongoing discussion on how he created a system in which the algebraic and logical calculi are not in perfect accord [2]. The sum and difference operations that Boole denotes by + and − are neither the standard set-theoretical union (between arbitrary sets) nor the set-theoretical difference. The discrepancy between algebra and logic seen in Boole’s system is problematic given that it was a period where these questions were much in discussion and his friend Augustus De Morgan (1806-1871) had also presented a formal framework for logic [3][4]. Boole (1815-1864) was the younger colleague of De Morgan, and the two of them carried on an extensive correspondence for years that was only published in 1982 [5]. However, this correspondence, which was only published in 1982, shows they ignored each other’s work suggesting that they were still in the process of developing their ideas and they saw their work as somewhat tentative. Another interesting perspective related to Boole’s work is provided by his wife Mary Boole (1832-1916), who, during her times, was a well-known writer on mathematical subjects. She claims [6] that her husband as well as De Morgan and Charles Babbage were influenced deeply by Indian logic and her uncle George Everest (1790-1866), who lived for a long time in India and whose name was eventually given to the world’s highest peak, was the intermediary of these ideas. She adds [6]: “Think what must have been the effect of the intense Hinduizing of three such men as Babbage, De Morgan, and George Boole on the mathematical atmosphere of 1830–65,” further speculating that these ideas also influenced the development of vector analysis and modern mathematics. Although the statement of Mary Boole is well known, I know