ESHS/HSS Annual Meeting

Boole Before “Boolean”: George Boole, Arthur Cayley, and Plural Mathematics in Debate

• In Event: Rewriting the Histories of Algebra

English Abstract

The term Boolean Algebra is widely recognized by mathematicians, logicians, and engineers. However, far from the clarity of today's Boolean Algebra, Boole's original system was ambiguous and vulnerable. Despite his posthumous fame, Boole was largely misunderstood by his contemporaries. Drawing from unpublished manuscripts, this article unfolds a neglected debate between Boole and mathematician Arthur Cayley. With their disagreement focusing on the valuation of symbols, I argue that this discrepancy was sparked by different epistemic values. Boole preferred unification, while Cayley emphasized quantification and clarity. In the early nineteenth century, when set theory and abstract algebra were yet to be established, quantification remained a principal theme in algebra. Therefore, Boole's symbols, which did not admit numerical values, seemed idiosyncratic to his contemporaries. Despite its mathematical appearance, Boole’s system was not a simple extension of mathematics into logic, but rather a fundamental rupture with both preceding and contemporary mathematics.
Although many of Boole's symbols have been abandoned because of their ambiguities, I reveal the legacy of these ambiguous symbols to counter the Whiggish underestimation of Boole. His symbols envisioned plural possibilities for mathematics and logic, not only foreshadowing multi-value logic but also challenging the distinction between operators and variables. However, these possibilities have long been overlooked due to the premature eclipse of Boole’s original system. The elimination of ambiguity comes at the cost of alternative possibilities. Ambiguity in science is not always a stumbling block but can, in fact, be a scaffold towards plural possibilities.

Author

• Guanyu Shen