Boolean Algebra
At the beginning of the 19th century, the stagnation of British mathematics contrasted sharply with continental ferment. A dispute between Newton's and Leibniz's disciples had paralyzed work across the Channel. While Europe calculated using Leibnizian differential notation, the British clung to Newtonian fluxions, which were less practical and less fruitful.
Around 1810, a few innovative minds founded the Analytical Society at Cambridge. Babbage and his colleagues broke the intellectual isolation by importing methods from the continent. This called into question the very foundations of algebra: what did negative numbers and imaginary quantities truly mean? An answer emerged in symbolic algebra. It no longer derived its legitimacy from operations on numbers, but established formal laws applicable to any symbols whatsoever.
It was in this intellectual atmosphere that George Boole forged his work. Born in 1815 in Lincoln into modest circumstances, never a university graduate, he taught himself mathematics with astonishing rigor. Initially a schoolmaster, he became professor at Queen's College Cork in 1849. A controversy between Augustus De Morgan and William Hamilton on quantification of the predicate prompted him to publish The Mathematical Analysis of Logic in 1847. His insight was to apply algebra to logical reasoning.
Boole transformed logic into a formal system with his algebraic notation. Literal symbols like x or y represented classes of objects, combinable through operators (+, ×). These operations followed precise rules: commutativity (xy = yx), distributivity (x(u + v) = xu + xv), and this singular property: x2 = x. This last rule marked a break with ordinary numerical algebra, where only 0 and 1 satisfy it.
His system translated logical propositions into equations. Thus, all X are Y became x(1 − y) = 0. The use of symbol 1 for the universe of discourse and 0 for the empty class constituted another discovery.
Augustus De Morgan, Boole's friend and contemporary, was developing his own logical theories in parallel. His Formal Logic appeared on the same day as Boole's work in 1847. De Morgan introduced the notion of variable universe of discourse, breaking with the fixed Aristotelian universe. He also formulated the famous laws that bear his name: the negation of a conjunction equals the disjunction of negations, and vice versa.
Boolean notation suffered from limitations. Boole required that addition apply only to disjoint classes, complicating the expression of certain relations. His definition of subtraction required that the subtracted class be included in the initial one.
In subsequent decades, mathematicians refined this system. Charles Sanders Peirce made decisive contributions in the 1880s, notably demonstrating that all Boolean operations could be reduced to a single one: NAND (not-and) or NOR (not-or).
Modern notation took shape gradually. Bertrand Russell introduced the symbol ∨ (or) in 1906, while Arend Heyting proposed ∧ (and) in 1930. The expression "Boolean algebra" was first used by Henry Maurice Sheffer in 1913.
In 1936, Marshall Harvey Stone took a step toward abstraction by unifying earlier work under the concept of Boolean ring. He established the isomorphism between Boolean algebra and this structure, creating a fundamental theoretical bridge.
Practical applications of Boolean algebra exploded in the 20th century. In 1938, Claude Shannon demonstrated in his master's thesis at MIT that these principles enabled the analysis and design of electrical switching circuits. This discovery created a link between mathematical logic and electronic design, founding modern computer science.
Today, Boolean algebra permeates all of computer science. It structures the design of logic circuits, formal verification of programs, and query optimization in databases. Every microprocessor relies on optimization techniques derived from this theory.
The history of Boolean algebra shows how an abstract theory, born from questions about logic, transforms an entire technological domain. It underscores the importance of mathematical notations in which an adapted symbolism reveals previously invisible relations. George Boole dreamed of a "calculus of thought" mechanizing logical reasoning. While this philosophical project did not succeed as such, his mathematical tools became the secret language of the machines that shape our world.