VI.43 George Boole
b. Lincoln, England, 1815; d. Cork, Republic of Ireland, 1864
Boolean algebra; logic; operator theory; differential equations;
difference equations
Boole, who never attended secondary school, college, or university, was almost entirely self-taught. His father was a poor shoemaker who was more interested in building telescopes and scientific instruments than making shoes—the result being that his business failed and Boole had to leave school at the age of fourteen and take a job as a junior teacher to support his parents, sister, and two brothers. By the age of ten he had mastered Latin and Ancient Greek, and by the age of sixteen he could read and speak French, Italian, Spanish, and German fluently. From his father he got a love of mechanics, physics, geometry, and astronomy, and together they built functioning scientific instruments. Boole then turned to mathematics and by the age of twenty he was publishing original research in calculus and linear systems. He wrote two seminal papers on linear transformations (1841, 1843), which provided the starting point for invariant theory, but he left it to others such as CAYLEY [VI.46] and SYLVESTER [VI.42] to develop the subject. In 1844 he was awarded the Royal Society’s Gold Medal for his paper on operators in analysis, the first gold medal for mathematics to be presented by the society. The paper was important not only because it contained (arguably for the first time) a clear definition of the concept of an OPERATOR [III.50], but also because of the influence it had on Boole’s subsequent ideas. An operator, for Boole, was an operation of the calculus, such as differentiation (which he denoted by D), considered as an object in its own right. There was an explicit similarity between the laws he derived for functions of D and the laws of his algebra of logic, which we shall discuss below.
At one time Boole had hoped to become a clergyman but family circumstances prevented this. His reverence for creation made him interested in the workings of the human mind, which he regarded as God’s greatest accomplishment. He longed, as Aristotle and LEIBNIZ [VI.15] had before him, to explain how the brain processes information and to express this information in mathematical form. In 1847 he published a book entitled A Mathematical Analysis of Logic in which he took the first steps toward achieving his goal, but the book did not have a wide circulation and so made very little impact on the mathematical world.
In 1849 Boole was appointed Professor of Mathematics at Queen’s College, Cork. It was there that he rewrote and expanded his ideas in a book entitled An Investigation of the Laws of Thought (1854), in which he introduced a new type of algebra, an algebra of logic, which evolved into what we now call Boolean algebra. From his earlier study of languages, he realized that there were mathematical structures concealed in everyday speech. For example, the class of European men, together with (i.e., union) the class of European women, is the same as the class of European men and women. By using letters to represent a class, or set, of objects, he could write the above as z(x + y) = zx + zy, where the letters x,y, and z represent the class of men, the class of women, and the class of all Europeans, respectively. Here addition is to be understood as union, at least for disjoint classes like men and women, and multiplication is to be understood as intersection.
The principal laws of Boole’s algebra are commutativity, distributivity, and the law which he called the “fundamental law of duality” and which is represented by x2 = x. This law can be interpreted by observing that the class of all white sheep intersected with the class of all white sheep is still the class of all white sheep. Unlike his other laws, all of which apply to ordinary numerical algebra, this law applies to numerical algebra only when x is 0 or 1.
Boole broke with traditional mathematics by showing that the study of well-defined classes or sets of objects is capable of precise mathematical interpretation and is indeed fundamental to mathematical analysis. In simple cases, his approach also reduces classical logic to symbolic mathematical form. Using the symbols 0 and 1 to denote “nothing” and “universe” respectively, and denoting the complement of the class x by 1 - x, he derived (from the law of duality) the law x(1 - x) = 0, which represents the impossibility of an object simultaneously possessing and not possessing a given property, otherwise known as the principle of contradiction. Boole also applied his calculus to the theory of probability.
Boole’s algebra lay dormant until 1939, when Shannon discovered that it was the appropriate language for describing digital switching circuits. Boole’s work thus became an essential tool in the modern development of electronics and digital computer technology.
Boole also made several other contributions to mathematics: differential equations, difference equations, operator theory, calculus of integrals, etc. His textbooks on differential equations (1859) and finite differences (1860) include much of his original research and are still in print today, but he is best remembered as the father of symbolic logic and one of the founders of computer science.
Further Reading
MacHale, D. 1983. George Boole, His Life and Work. Dublin: Boole Press.