by I. Grattan-Guinness
© Oxford University Press 2004 All rights reserved
Boole, George (1815-1864), mathematician and logician, was born on 2 November 1815 in Lincoln, the eldest of four children of a local tradesman, John Boole (1777-1848), and his wife, Mary Ann (1780-1854). He adopted a love of learning from his father, who was active in the local mechanics' institute, to such an extent that in his teens he was employed as a teacher at schools in Lincoln and nearby; in his twentieth year he opened his own school. He also learned classical and modern languages and wrote poetry, and taught himself mathematics to a level which permitted him to engage in research. He began publishing in 1841, first and frequently in the Cambridge Mathematical Journal, founded two years earlier by D. F. Gregory, his initial mathematical mentor and inspiration.
Although his contributions quickly gained attention, Boole continued as a private teacher until the Queen's University in Ireland was set up in 1845. After delays caused by the potato famine, he took up appointment in 1849 as founder professor of mathematics at Queen's College, Cork, a post which he filled for the rest of his life. Often he spent the summer vacations in England, at Lincoln and studying in London libraries. While he was unable to generate a school of research students at Cork, his work was recognized by honorary doctorates from Trinity College, Dublin, in 1851 and Oxford University eight years later, and by fellowship of the Royal Society in 1857, although he was not elected to the Royal Irish Academy. Among other duties, between 1847 and at least 1852 he was an examiner of mathematics for the College of Preceptors, which was founded in 1846 as an institution to raise the reputation and calibre of the profession of school teachers.
Marriage and family
On 11 September 1855 Boole married Mary Everest (1832-1916) [see Boole, Mary], daughter of the Revd Thomas Everest and niece of Sir George Everest of mountain fame. Brought up in France, she became greatly interested in mathematics; she had met Boole in 1850 when she was visiting her uncle John Ryall, vice-president of Queen's College. She assisted her husband with his two textbooks, as the first reader for intelligibility and also as checker of some of the exercises. After his death she moved her family to London, where Frederick Denison Maurice found her a post at his Queen's College in Harley Street. Later she was secretary to the philosopher James Hinton, and became interested in Judaism, and also in psychical research. She continued to proselytize Boole's philosophy, especially its application to educational questions, both in discourse and extensively in print. She gained a reputation as a crank, but her understanding of his ideas on logic and education were basically sound. She presented the bulk of his manuscripts to the Royal Society in 1873.
George and Mary produced five daughters with repetitive two-year regularity. Mary Ellen Boole (b. 1856) married the mathematician Charles Howard Hinton, the eldest son of James Hinton. Margaret Boole (1858-1935) married Edward Ingram Taylor, an artist; their son Geoffrey Ingram Taylor (1886-1975) was to become one of the finest British mathematical physicists of the twentieth century. Alicia Stott, née Boole (1860-1940) [see under Boole, Mary], was led into mathematics by her brother-in-law Charles Hinton, and produced fine work in solid geometry (a startling contrast to her father's totally algebraic style). From 1889 to 1896 she worked with friends on organizing her father's manuscripts, with an intention (not realized) of producing a new edition of Laws. Lucy Everest Boole (1862-1904) [see under Boole, Mary] made a career in chemistry, teaching at the London School of Medicine for Women. She was the first woman professor of chemistry in Britain. Ethel Lilian Boole (1864-1960) [see Voynich, Ethel Lilian] studied in Germany in her teens, and became associated with revolutionary causes in central Europe. She became a radical socialist and freethinker, and consorted with the activist W. M. Voynich, whom she married in 1906; however, within four years she ran away with the remarkable spy Sydney Reilly. In 1897 she produced a novel called The Gadfly, seemingly based upon Reilly, which was to become very well known in the Soviet Union.
Contributions to mathematics
Boole's contributions to mathematics were both characteristic of and important for the development of English mathematics in his time, for they were entirely guided by algebras of new kinds. His earliest work extended the notion of the invariant, which was to become a major English industry in the hands of Arthur Cayley and J. J. Sylvester. His own main effort was directed to the algebraic form of the differential and integral calculus. Born at the start of the century out of certain ideas of J. L. Lagrange, it had been taken up in England from the 1810s by Charles Babbage and especially John Herschel. Taking much inspiration from Gregory, Boole specialized in differential operators, in which the operation of differentiation of a mathematical function was represented by the letter 'D', and an algebra constructed in which second differentiation was given by the power D2, the inverse operation of integration was D-1, and so on. He became a leading practitioner of this theory for solving differential and difference equations, and summing series. His main paper, 'On a general method in analysis', was submitted to the Royal Society; after initial rejection, it was published in the Philosophical Transactions in 1844, and Boole later received the society's gold medal. This operator algebra became the largest single concern of English mathematicians in Boole's time, and his contributions gained him most of his attention. He examined operator functions F and G which did not commute (FG ≠ GF), and applied his methods and solutions especially to linear differential equations. He also wrote two well-received general textbooks: A Treatise on Differential Equations (1859) and A Treatise on the Calculus of Finite Differences (1860).
The algebra of differential operators has a close bearing upon Boole's contributions to logic, which was then only an intriguing curiosity to his contemporaries but for which he is best remembered today. A dispute developed in the mid-1840s between the Scottish philosopher Sir William Hamilton and the English mathematician Augustus De Morgan over a certain extension of syllogistic logic which became called 'quantification of the predicate'; it inspired Boole to write up his ideas on logic in a short book entitled The Mathematical Analysis of Logic (1847). Seven years later, he presented a much longer account as An Investigation of the Laws of Thought; today it is recognized as the more authoritative and substantial book, but at the time it gained even less attention than had his Analysis.
Boole's logic
Boole's logic was much more revolutionary relative to syllogistic logic than De Morgan's algebraic treatment; although they corresponded regularly, they communicated their respective ideas more than discussed them. However, Boole seems not to have recognized the full consequences of his own ideas when writing the Analysis, in that he referred there frequently to syllogistic logic. By the time of the Laws, however, it was treated only in the last of the fifteen chapters on logic.
Boole's basic method may be explained as follows, using the version in Laws where Boole gave prime place to classes, rather than the psychological interpretation concerning mental acts ('elective symbols') of choosing properties, which he preferred in Analysis. From a given universal class 1 (of men, say) the mind picks out some class x (of Englishmen, say), leaving the complementary class (1 - x) of non-English men. The basic laws of these classes, which grounded his algebra of logic, imitated as closely as possible the laws that he had found for differential operators in the 1844 paper: commutativity, distributivity in both theories, and a third law which he called in each case the 'index law'. For differential operators the index law followed the usual property of powers, DmDn = Dn+m, but in logic it asserted that x2 = x, that is, that the class x taken together with the class x gives the class x (or that to choose the property x and to choose x again is the same as choosing x once). This law distinguished this algebra from all others of his time. As consequences of it he formed equations expressing the laws of contradiction and of excluded middle, which were taken as basic assumptions in traditional logics.
To the operations of subtraction and multiplication Boole appended that of addition, where two classes were adjoined as long as they had no parts in common. He also worked with 0, denoting 'Nothing'--but none too clearly, since it was a class of some sort.
One main purpose of Boole's algebra was to take one or several propositions as premisses, express them as algebraic equations, and then to use various expansion theorems and processes of elimination of letters to relate a selected class to the others as derived equations; sometimes further equations arose which stipulated conditions (of no parts in common, say) on the classes in hand in order that any solution be found at all. As logic, these new equations expressed logical consequences of the premisses; syllogistic logic provided many cases, but only of special kinds. However, Boole's methods often did not find special solutions, and he did not always distinguish some of the modes used in quantification of the predicate.
Boole always understood his algebra of logic 'to investigate the fundamental laws of those operations of the mind by which reasoning is performed', as he put it in the opening of Laws, but he changed position on the philosophical foundations of his theory. The preference of classes over mental acts between the Analysis and Laws arose from a rejection of syllogistic logic as a foundation in favour of a different tradition stemming from John Locke and continued by the Scottish common-sense philosophers in some ways. In this view signs were taken as the primary cognitive notion, with language as an essential means of expressing them. In Boole's hands, nouns and adjectives were principal components, denoting classes: thus, say, 'men' were formed from a universe of humans, and 'good men' similarly from 'men'; taking 'good good men' made no difference by the index law. Prepositions expressed the means of combination: 'and' for conjunction, exclusive 'or' for disjunction, and 'except' for subtraction.
Boole tried several times to write a successor to the Laws which would explain its principles and philosophy in a general manner, but he never succeeded. His manuscripts show that he tried to ground his logic in a philosophical procedure of his time, though he paid better attention than normal to the difference between a mental act and its product. For him the act of conception by the mind produced a concept, such as 'man' (or the class of them); then the act of judgement of the copresence of concepts produced a proposition, such as 'this is a wise man'; finally, the act of reasoning produced a conclusion, perhaps 'wise men are ...', as delivered by his algebra of logic. However, he still found no commensurably comprehensive philosophy of mathematics to explain how his logic had its mathematical basis.
Both in logic and in his educational theory, Boole thought that the mind was capable of original action, such as grasping general laws from particular cases. He would have hated the modern association of his logic with computing, and he had no particular congress with Babbage or concern to apply his logic to calculating machines. The religious aspects were also important; his 'universe 1' corresponded to the ecumenism of his day, in that it stood over and above the factions into which the Christian church was split. He was especially admiring of F. D. Maurice, who advocated ecumenism with great force in mid-century and so was dismissed from his chair at King's College, London: in his last years Boole attended Maurice's services when researching in London. He discreetly revealed his adherence right at the end of the Laws, with a reference to 'the Father of Lights'.
Chapters 16 to 21 of Laws were concerned with probability theory. One of Boole's main insights was to interpret compound events as Boolean combinations of simple ones. He also considered probabilistic inference, and in estimating probability values of logical consequences he manipulated inequalities in ways which place him among the precursors of linear programming. He also queried subjective interpretations of inverse probability (that is, given some event, calculating the probability of the circumstances which could have caused it to happen). After Laws he produced important papers in this area; one won the Keith prize of the Royal Society of Edinburgh in 1857. Later his principles and calculations were criticized by Jevons, among others.
Character and reputation
Boole comes over as honest and straightforward, even naive (as when he told people in the street about the birth of his first daughter). He was also rather serious, showing little sign of a sense of humour. The circumstances of his upbringing and career isolated him from the mathematical and scientific communities; perhaps in compensation he corresponded quite intensively, having substantial exchanges with William Thomson on operator methods and with De Morgan on algebra and logic. He was also on good terms with fellow mathematicians such as Arthur Cayley and Robert Harley.
Boole died from pleuro-pneumonia, the result of foolishly walking the 3 miles from home to university without proper protection from a rain storm--so probable an event in Cork. He died at his home, Lichfield Cottage, Ballintemple, on 8 December 1864, with Maurice's portrait set by his bedside at his request, and was buried at St Michael's Church of Ireland, Blackrock, on 12 December.
After Boole's death, his reputation declined for some decades. Differential operator methods became generally eclipsed by other techniques in the calculus, although his textbooks of differential and difference equations, which contained all basic techniques, continued to be well used after his death and are still in print. His work on logic remained marginal; John Venn was to remain its sole close adherent. Shortly after his death the chemist Benjamin Collins Brodie unsuccessfully tried to develop a similar algebra for chemical combination. The psychological and religious aspects of his logic disappeared completely, and the development of mathematical logic by men such as Giuseppe Peano and Bertrand Russell drew on quite different principles. His work on probability also made little impact.
The main single issue for followers of Boole's logic was his restriction of (x+y) to classes with no parts in common. His arguments for this were not convincing, least of all to his first serious reader, Stanley Jevons, who declared that + should be definable for all classes. Jevons outlined his position in his book Pure Logic (1864), and corresponded at the time with Boole. Their test case was the expression (x+x); for Jevons (x+x)=x, whereas for Boole it could not be 'interpreted', although when expanded the premiss (x+x)=0 led to the consequence x=0. A few years later the American logician C. S. Peirce came independently to the same modification; it altered Boole's algebra in quite a substantial way, which distanced it from the algebraic logic to come later in the century. The modern 'Boolean algebra' has x2=x among its axioms, but in technical and philosophical ways it differs from Boole's algebra of logic.
Boolean logic is the basis for the design of all modern computers since the ultimate components of these devices were capable of storing just two values (equated with true and false) and their circuitry calculates the basic Boolean operators over these two values. These domains of application have added to the importance of Boole's work already evident in the relationship between logic and mathematics.
I. GRATTAN-GUINNESS
Sources
D. MacHale, George Boole: his life and work (1985)
M. Panteki, 'Relationships between algebra, logic and differential equations in England, 1800-1860', PhD diss., Middlesex University (CNAA), 1992
G. Boole, The mathematical analysis of logic (1847)
G. Boole, An investigation of the laws of thought (1854)
G. Boole, A treatise on differential equations (1859)
G. Boole, A treatise on the calculus of finite differences (1860)
G. Boole, Studies in logic and probability, ed. R. Rhees (1952)
M. E. Boole, Collected works, 4 vols. (1931)
T. Hailperin, Boole's logic and probability, 2nd edn (1986)
S. Neil, 'Modern logicians: the late George Boole', British Controversialist, 3rd ser., 14 (1865), 81-94, 161-74
I. Grattan-Guinness, 'The correspondence between George Boole and Stanley Jevons', History and Philosophy of Logic, 12 (1991), 15-35
George Boole: selected manuscripts on logic and its philosophy, ed. I. Grattan-Guinness and G. Bornet (1997)
L. M. Laita, 'Boolean algebra and its extra-logical sources', History and Philosophy of Logic, 1 (1980), 37-60
Archives
RS, scientific MSS
TCD, MSS on symbolic logic
University College, Cork, corresp. and papers | CUL, letters to Lord Kelvin; letters to Sir George Stokes; letters to W. Thomson
U. Glas. L., letters to W. Thomson
UCL, corresp. with Augustus De Morgan
Likenesses
pencil drawing, 1847, NPG [see illus.]
stained-glass window, University College, Cork
wood-engraving (after photograph), NPG; repro. in ILN (21 Jan 1865)
Wealth at death
under £3000: probate, 15 Feb 1865, CGPLA Ire.
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