When 1+1 Equals 1
In an earlier blog-essay, When 1+1 Equals 0, I explained how 1 + 1 = 0 makes sense in mod 2 arithmetic; today I’ll tell you how the equation 1 + 1 = 1 makes sense in Boolean arithmetic and became a tool for designing the complex digital circuits that power the Information Age.
The two people who deserve the most credit and blame for this state of affairs are George Boole (1815–1864) and Claude Shannon (1916–2001).
While walking across a field in 1833, young George Boole had an epiphany: it was his duty and his destiny to explain the logic of human thought. “Epiphany” is no overstatement; he mentioned the incident many times in later life, sometimes in reverent, almost messianic tones. He took his destiny seriously to the point of sickness – he later told his sister MaryAnn that he had often made himself ill from his struggle to reduce logic to a mathematical science. Even after “the true method flashed upon him” in 1847, he had frequent bouts of poor health.
In his two great works The Mathematical Analysis of Logic (1847) and An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities (1854), Boole repurposed algebraic notation to serve the needs of logic. Classes of things were represented by letters and the symbols + and × were used to represent operations on those classes, though in practice Boole’s algebra omitted × symbols, much as is done in ordinary algebra. If x is the class of all cats and y is the class of all dogs, x + y is the class consisting of all cats and all dogs, while x y is the class of individual animals that are both cats and dogs – an empty class since there is no cat that’s also a dog. Nowadays we refer to Boole’s classes as sets, so that for instance the empty class is called the empty set (the subject of my essay The Null Salad), but I’ll stick to Boole’s terminology for now.
