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A Comprehensive Exploration of George Boole

By Ernest Davis

The Life and Work of George Boole: A Prelude to the Digital Age. By Desmond MacHale. Cork University Press, Cork, Ireland, November 2014. 360 pages, $25.00.

New Light on George Boole. By Desmond MacHale and Yvonne Cohen. Cork University Press, Cork, Ireland, October 2018. 492 pages, $29.00. 

The Continued Exercise of Reason: Public Addresses by George Boole. By Brendan Dooley (Ed.). The MIT Press, Cambridge, MA, April 2018. 248 pages, $34.00.

Boolean logic and Boolean algebra—the mathematics of true and false, of 1 and 0—are the foundation of essentially all digital computation and digital devices that pervade the modern world. George Boole himself (1815-1864) is less well known than his creations.

Yet Boole was an extraordinary figure in many ways, and his life deserves recognition. Fortunately, the labors of recent biographers have brought his name to the forefront of mathematical discussion. Mathematician Desmond MacHale published the first full-length biography of Boole, entitled George Boole: His Life and Work, in 1985. MacHale released a revised version in 2014 called The Life and Work of George Boole: A Prelude to the Digital Age. In 2018, he collaborated with Yvonne Cohen to publish additional material in New Light on George Boole. Historian Brendan Dooley also edited a collection, called The Continued Exercise of Reason, which contains all of Boole’s surviving public lectures in addition to a 70-page introduction that provides historical and intellectual context.

The Life and Work of George Boole: A Prelude to the Digital Age. By Desmond MacHale. New Light on George Boole. By Desmond MacHale and Yvonne Cohen. The Continued Exercise of Reason: Public Addresses by George Boole. By Brendan Dooley (Ed.). Images courtesy of Cork University Press and the MIT Press.

Boole was born and raised in Lincoln, in the East Midlands of England. He was a conspicuously gifted child, though not especially so at math. At age 14, he published his own translations of three classical Greek poems in the local newspaper. In 1831, when Boole was 16 years old, he became a teacher to support his family after the collapse of his father’s business. He advanced rapidly and opened his own school in 1834.

It was around this time that Boole began to seriously study mathematics. By his own account, he chose the subject because he had almost no money for books, and a math book would take longest to get through. Boole’s mathematical education was entirely self-taught. He started with Sylvestre Lacroix’s Traité du Calcul Différentiel et du Calcul Intégral, then worked through Isaac Newton’s Philosophiae Naturalis Principia Mathematica, Joseph-Louis Lagrange’s Mécanique Analytique, and Pierre-Simon Laplace’s Traité de Mécanique Céleste (teaching himself French, German, and Italian along the way). Boole published his first paper, on the calculus of variations, in an 1838 edition of the Cambridge Mathematics Journal. In 1844, his paper titled “On a General Method for Analysis” won the Royal Society’s first gold prize for mathematics. In 1849, Boole became a professor of mathematics at Queen’s College in Cork, Ireland, where he taught for the rest of his life. He died at age 49 from pneumonia after walking through heavy rain to deliver a lecture.

Boole’s fame rests predominately on his essential creation of mathematical logic in 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). But he made major contributions to other area of mathematics as well. In addition to Boolean logic, The Laws of Thought contains groundbreaking work on the foundations of probability theory. Boole was a major figure in the theory of differential equations and among the first people to formulate the concept of the differential operator. He wrote one of the earliest papers on invariants, thus laying the groundwork for Arthur Cayley’s later development of the field. He also penned influential textbooks on differential equations and difference equations; the latter was not superseded until the 1930s.

Boole did have his gaps and blind spots. As far as I can tell, he made no contribution to geometry beyond teaching it every year. Despite his friendship with Cayley, he was uninterested in group and matrix theory. Most surprisingly, he apparently never realized that one must state the associative law as a separate axiom in Boolean algebra.

Religion was enormously important to Boole, and he struggled all his life to reconcile his rationalism with his Christian faith. In fact, a major underlying motivation for his study of logic was likely to determine the role of human reason in understanding God; The Laws of Thought includes a chapter on the logic pertaining to the theological arguments of Baruch Spinoza and Samuel Clarke. Additionally, Boole so much venerated theologian Frederick Denison Maurice that he had Maurice’s photograph brought to him on his deathbed.

Beyond his teaching, Boole occasionally gave public lectures on a wide variety of subjects. Topics included a celebration of Newton, the origins of ancient mythologies, the social aspect of intellectual culture, the claims and philosophy of science, and the possibility of life on other planets (a prospect in which he believed). When a local social organization’s library introduced a rule against acquiring books on controversial topics, Boole advocated to abolish the regulation. After the passage of a law that limited the work day to 10 hours, he delivered a talk entitled “The Right Use of Leisure” that encouraged people to read about science, history, and moral philosophy; partake in healthful exercise; and enjoy nature. Boole also spoke poignantly about education in a lecture that included insights from his personal experiences as a student and teacher. Some of these are pleasantly idiosyncratic; for instance, he emphasized the importance of good penmanship. Boole clearly had broad interests, was well-read, and possessed confidence in his ability to lecture on many topics.

His public addresses are not milestones in intellectual history by any means, but they do offer a window into the general mindset of Boole’s time and place. Literary scholar George Herbert Palmer wrote that “the tendencies of an age appear more distinctly in its writers of inferior rank than in those of commanding genius.” Boole was slightly younger than Charles Dickens, Charles Darwin, Abraham Lincoln, Benjamin Disraeli, Thomas Babington Macaulay, and Harriet Beecher Stowe, but seems much more “Victorian” in the somewhat pejorative stereotype. He embodied the turgid 19th-century lecture style—stiff, formal, long-winded, humorless, and often pompous—and possessed the smug certainty that 19th-century Christian Europe was the apex of human history. The incessant religiosity—in which any mention of the wonders or regularities of nature is followed by a pious reference to the “Author of Nature,” and any mention of human activities is followed by a reference to man’s “duty to his Creator”—is very characteristic of the era. Also characteristic is the sometimes-stifling level of propriety; in one lecture Boole warns against reading too many novels, and in another he worries about the corruptive dangers of exposing students to polytheist classical literature.

On the other hand, it is worth acknowledging that there is little in these speeches to offend a modern-day reader. It seems that Boole was rarely—if ever—sexist, racist, anti-Semitic, or anti-Catholic. He also does not defend slavery, imperialism, or the inferior status of women; you cannot count on that when reading most lectures from the 1840s.

One unexpected pleasure in Boole’s biography is the figure of his wife, Mary Everest Boole. They married when he was 40 and she was 23, promptly had five daughters, and seemed to have had a happy marriage. Mary was an intelligent woman with a gift for math, and worked through Boole’s entire textbook on differential equations while he was writing it. She was also something of a crackpot, particularly later in life; in fact, she may have inadvertently killed Boole by treating his final illness homeopathically and possibly insisting that he sleep in damp sheets.

Mary wrote a great deal, and in a much more lively and readable style than her husband. At her best she was strikingly insightful and even prescient. In 1868, she wrote that Charles Babbage and William Stanley Jevons “have conclusively proved, by unanswerable logic of facts, that calculation and reasoning, like weaving and ploughing, are work, not for human souls, but for clever combinations of iron and wood.” At other times, she was charmingly nutty. “It is demonstrable that the faculties on which depend the possibility of logic and of algebra must have evolved in connection with an intime and private family life,” she wrote. “Their source was — male and female engaged in peopling the world of the future.”

Boole had no significant teachers and no significant students. He does not have a record in the Mathematics Genealogy Project, but his physical descendants were a remarkable bunch. His daughter Alicia conducted important work on four-dimensional geometry and discovered the six four-dimensional regular polytopes. Boole’s daughter Lucy was the first woman professor of chemistry in England. His daughter Ethel Lilian Voynich wrote The Gadfly, a revolutionary novel that was immensely successful — particularly in the Soviet Union. And Boole’s great-great-grandson Geoffrey Hinton is a leading figure in machine learning, continuing his ancestor’s study of the mathematical analysis of The Laws of Thought

Ernest Davis is a professor of computer science at New York University’s Courant Institute of Mathematical Sciences.

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