Boole, George (1815–64), mathematician, was born 2 November 1815 at Lincoln, the first of four children of John Boole (1777–1848), a cobbler, and Mary Boole (1780–1854).

**Early life and first publications** Boole received only a simple education, but with the encouragement of his father, who was devoted to the study of science and the construction of optical instruments, and of Lincoln friends, he taught himself classical and modern languages, for which he had an unusual aptitude. From 1831, to assist his family's precarious finances, Boole taught at schools in Doncaster, Liverpool, and Waddington, near Lincoln. Then, from 1834 until 1849, he conducted a series of his own schools at Lincoln and in its environs, his entire family assisting in the running of the establishments. Boole's study of higher mathematics also began in 1831, when he embarked on reading books by Lacroix, Lagrange, Laplace, and Poisson. Following his return to Lincoln, Boole was active in the local mechanics’ institute, whose president was Sir Edward Bromhead (1789–1855) of Thurlby Hall, near Lincoln. Bromhead had studied mathematics at Cambridge University, where he had urged the foundation of the undergraduate Analytical Society, established in 1812 with the aim of reforming the Cambridge mathematical curriculum. From the late 1830s Bromhead lent Boole books by leading French mathematicians and Boole corresponded with him in the course of his early mathematical researches.

Boole's first research paper appeared in 1841 in the *Cambridge Mathematical Journal*, whose co-founder and editor, Duncan Gregory (1813–44), advised Boole on mathematical subjects. Between 1841 and 1845 Boole published twelve papers in the journal, one of which, ‘Exposition of a general theory of linear transformations’ (1841), was proclaimed by George Salmon (qv) as the origin of modern algebra for its foundation of invariant theory, a subject linking the algebra of polynomial forms and geometry, which was much extended by Arthur Cayley, J. J. Sylvester, and Salmon himself in the next fifty years. Gregory's development of the algebra of differential and difference operators, as presented in his important textbook *Examples of the processes of the differential and integral calculus* (1841), exerted a strong influence on Boole's subsequent work on differential equations, finite differences, and the algebra of logic. This influence is evident in Boole's paper ‘On a general method in analysis’ (1844), for which he received the Royal Medal of the Royal Society of London; in this prize-winning paper Boole developed formal algebraic rules satisfied by differential operators and then employed the calculus of operations to solve differential equations and to sum series in closed form.

A controversy arose in 1846 between the Scottish philosopher Sir William Hamilton (1788–1856) and the English mathematician Augustus De Morgan (1806–71) concerning a minor refinement to the study of the syllogism, known as the ‘quantification of the predicate’. Boole had been in correspondence with De Morgan since 1842 and, stimulated by the subject matter of the logical dispute, he wrote an 82-page pamphlet entitled *The mathematical analysis of logic* (1847), in which he presented a calculus of deductive reasoning. Characteristic of Boole's analysis was the use of an algebra of so-called ‘elective symbols’, which obey the distributive and commutative rules and an additional idempotent rule, which distinguishes this algebraic system from the traditional algebra of quantities. Writing in 1851, Boole considered his treatise to be ‘a hasty and (for this reason) regretted publication’ (*Studies in logic and probability*, 252), written as it was only a few weeks after the initial conception of a logical calculus.

**Academic appointment and the algebra of logic** The Colleges (Ireland) Bill of 1845 provided for public funding of new colleges in Ireland. Encouraged by support from William Thomson (qv) (later Lord Kelvin), Boole applied in 1846 for a professorship in mathematics or natural philosophy in one of the proposed colleges. He supported his application with testimonials from such accomplished mathematicians as Cayley, De Morgan, Charles Graves (qv), and Thomson, and was eventually appointed in August 1849 to the chair of mathematics at QCC, despite his lack of both formal secondary and university education. His salary was £250 per annum, augmented by yearly tuition fees of around £100. Boole retained the Cork position for the rest of his life, but he looked elsewhere for more congenial employment: in 1860 he submitted his name in candidacy for the Savilian professorship of geometry at Oxford University but as he did not send the necessary testimonials his application was not seriously considered.

In a lecture entitled ‘The claims of science’, given in Cork in 1851, Boole expressed his opinion that ‘It is simply a fact that the ultimate laws of Logic, those alone on which it is possible to construct a Science of Logic, are mathematical in their form and expression, although not belonging to the mathematics of quantity’ (*Studies in logic and probability*, 209). At this time, he was engaged in a new project on logic and probability, which he completed as his book *An investigation of the laws of thought on which are founded the mathematical theories of logic and probabilities* (1854). This was Boole's major contribution to the algebra of logic, but opinions varied as to its value and novelty. Herbert Spencer considered that Boole's ‘application to logic of methods like those of mathematics constituted another step far greater in originality and importance than any taken since Aristotle’ (Spencer, *The study of sociology*, 223). The logician John Venn (1834–1923) wrote in 1876 (*Mind*, i (1876), 479) that on first reading *The laws of thought*, it was as if the key of all knowledge had been placed into his hands, whereby the manipulation of symbols and the application of rules of interpretation led to results beyond those attainable by unassisted thought. Later reflection convinced him that this view was illusory, as by the ordinary deductive processes he could generally reach the same conclusions with less effort. In his *Symbolic logic* (1881) Venn opined that Boole's originality was not as complete as was sometimes claimed, given that Leibniz and Johann Heinrich Lambert had anticipated a symbolic calculus of reasoning; but he still held him to be ‘the indisputable and sole originator of all the higher generalizations of the subject’. He also observed that Boole offered very little direct explanation of his system, especially in his use of the division operation in logic. The economist and logician W. Stanley Jevons (1835–82) believed Boole's logical system to be unnecessarily mathematical. In his *Principles of science* (1874), he articulates conflicting views: first that ‘Boole produced a system which, though wonderful in its results, was not a system of Logic at all’, and second that ‘In spite of several serious errors into which he fell, it will probably be allowed that Boole discovered the true and general form of Logic, and put the science substantially into the form which it must hold for evermore.’

**Probability theory and late mathematical treatises** Boole's work on logic encouraged his interest in probability theory, and he began a series of papers on the subject in 1851. In November of that year, he issued a challenge problem on probability that drew responses from the astronomer William Donkin, the German mathematician Richard Dedekind (1831–1916), and Arthur Cayley. Boole himself published his own solution to the problem in January 1854, using a method that was substantially developed in *The laws of thought*. He devoted over one third of the contents of his book to his idiosyncratic theory of probability, which drew much public criticism. The most serious attack on his solution to the probability problem and the whole theory expounded in his book was that of Henry Wilbraham (1825–83), whose objections appeared a few months after Boole's book was published. Wilbraham, though pursuing a career in the legal profession, was an accomplished mathematician: he had been seventh wrangler at Cambridge University in 1846 and had written a valuable contribution on the so-called ‘Gibbs–Wilbraham phenomenon’ in Fourier series. Wilbraham's main contention was that most of the problems posed by Boole were indeterminate, and that in order to solve them, Boole made tacit assumptions about the data of the problems. Wilbraham argued that in determinate cases it was easier to proceed from first principles, whereas in the indeterminate cases, since Boole gave no explanation of his assumptions, it was not clear how the solution offered was distinguished from any number of similar, equally defensible, solutions. Boole published spirited replies to Wilbraham's strictures in August and September 1854, and challenged Wilbraham to provide an instance of a clearly erroneous deduction from his method.

The dispute on Boole's method resurfaced in 1862, when Cayley once more examined Boole's solution to the challenge problem and professed that he could neither explain nor understand the logical principles on which Boole based his solution. As part of a series of papers entitled ‘The calculus of equivalent statements’, Hugh McColl questioned Boole's method and showed by a simple numerical example how it must be wrong. Finally, in his *Treatise on probability theory* (1921), the economist John Maynard Keynes gave solutions to some of Boole's problems, commenting: ‘Boole's method of solving them is constantly erroneous, and the difficulty of his method is so great that I do not know of anyone but himself who has ever attempted to use it.’ Later writers have examined Boole's theory, but nobody has proposed a plausible interpretation of what Boole intended. After 1854, although he published further substantial papers on probability, in which he employed the notation and ideas of his logical method, Boole avoided the controversial aspects of his theory, and developed an approach exploiting linear inequalities and properties of determinants. While he earned a substantial prize for a paper on probability in 1857, his work has had no impact on the subject and is rarely quoted in the historical literature.

In the later 1850s Boole's attention turned to more traditional mathematical subjects, such as integration and differential equations, and in 1859 he published *A treatise on differential equations*. This was a pioneering work, giving numerous examples of special differential equations and modes of solution. It was based on an intimate knowledge of the research conducted on the subject, and continued to be reprinted well into the twentieth century. In chapters XVI and XVII he gave an exposition of symbolical methods of solution, returning to the subject matter of some of his earliest papers. This work was followed in 1860 by *A treatise on the calculus of finite differences*, another successful production, in which, in addition to employing the symbolical methods, he also exhibited great facility in the theory of infinite series and applications of the integral calculus.

**Marriage, family, and personality** On 11 September 1855 Boole married Mary Everest (1832–1916), daughter of the Rev. Thomas Everest, and niece of both Sir George Everest, surveyor-general of India, and John Ryall, vice-president of QCC. Thomas Everest was a follower of Samuel Hahnemann, the founder of homoeopathy, and his daughter inherited his views on medicine and the treatment of illness. When in November 1864 Boole developed a fever and infection of the lungs, after suffering a saturation in the rain of Cork, his wife supposedly treated him by wrapping him in wet sheets, a remedy suggested by homoeopathy. The treatment failed and Boole died on 8 December 1864 in Cork.

The Booles had five daughters, several of whom were talented and achieved success in their careers. Alicia Boole (qv) (1860–1940), who married the actuary Walter Stott, was a self-taught mathematician and produced original work on the geometry and classification of polytopes. Lucy Boole (qv) (1862–1904) pursued research in chemistry and became a lecturer at the London School of Medicine for Women. Ethel Lilian Boole (qv) (see under Voynich) (1864–1960) was inspired by Russian culture and translated Russian literature; she married the Polish revolutionary Wilfrid Voynich and eventually settled with him in New York. Her suspense novel *The gadfly* (1897), set at the time of the Risorgimento in Italy, became enormously popular in Soviet Russia. Margaret Boole (1858–1935) married the artist Edward Taylor; their son Sir Geoffrey Ingram Taylor (1886–1975) became a famous applied mathematician. Following Boole's death, after which she received a civil list pension of £100, Mary Boole returned to England and devoted herself to explaining the significance of her husband's ideas on logic. She developed eccentric views on education, medicine, philosophy, and religion, and moved in bohemian circles; she also published several books on educational psychology.

John Dowden (qv) was a pupil of Boole at QCC, and his brother Edward (qv) was also acquainted with Boole. In a letter of 1873 Edward Dowden wrote of Boole's strong intellectual simplicity and grave moral energy, as well as his love of cathedral music and hymn singing (he sang out of tune); Dowden also recalled Boole's arguing for the existence of a kind of aesthetics in mathematics – a quality some discerned in his books, by contrast with those of George Salmon. Of Boole's family life, Dowden wrote: ‘His wife was a woman of intellect and attainment in mathematics, with a bright pure face, and they had two or three beautiful little girls. The Cork matrons had stories, partly mythical I am sure, concerning their novel experiments with the children; e.g. that one baby girl was found naked and tethered in a grass plot, chewing pebbles; which stories meant, I suppose, that they did not conform to all domestic conventions.’

**Reputation and influence** Boole received honorary degrees from Dublin University (1851) and Oxford University (1859). He was elected FRS in 1857; in filling out the certificate of candidature he completed the statement ‘Distinguished for his acquaintance with the science of . . .’ with the word ‘psychology’, not ‘mathematics’, suggesting that he considered his logical method as important for the study of the human mind as for deductive reasoning. Boole was not elected to membership of the Royal Irish Academy, a surprising omission given his eminence and wide acquaintance with scientific men of influence. He had little contact with the Irish mathematician William Rowan Hamilton (qv), but in 1855 he sought a testimonial from Hamilton in support of his application for an examinership. Hamilton's lukewarm reply avoided any commitment to a proper assessment of the value of Boole's work or his originality; Boole had sent Hamilton a copy of *The laws of thought*, which was sold at auction with the rest of Hamilton's library in January 1866.

It is difficult to formulate a widely acceptable view of Boole's status in the history of mathematics and logic. He is generally regarded as the founder of mathematical or symbolic logic, but his theory of logic seemed too demanding in its mathematical requirements to be attractive to contemporary logicians, while offering few advantages as a working system for mathematicians. The philosopher A. N. Whitehead saw Boole's algebra of symbolic logic as part of the mid-nineteenth-century discovery of new algebraic systems, such as Hamilton's quaternions and Hermann Grassmann's extensive magnitudes, which broke free from the restrictions of interpretation of symbols in terms of arithmetic, but he doubted if Boole conceived of his achievement in this light (A. N. Whitehead, *A treatise on universal algebra* (1898), 115). At the end of the nineteenth century, Boolean algebra signified the study of the logic of propositions by Boole's method, but in the early twentieth century the related concept of an abstract Boolean algebra was developed. This has ensured Boole's continuing fame, as the 1937 master's thesis of Claude Shannon showed that Boolean algebra provides the best tool to study the digital circuits used in electronic computers. By a concatenation of ideas, Boole himself is sometimes held to be the forefather of modern high-speed computing.

The Boole Papers in the Boole Library, UCC, contain letters, lecture drafts, and material relating to Boole's wife and daughters, while Royal Society, London, MS 782 contains Boole's unpublished mathematical and logical papers. TCD MS 2398 is an incomplete undated ten page paper in Boole's hand, entitled ‘Symbolical logic, being an essay towards a calculus of deductive reasoning’; parts of it are identical with pages 16–18 of *The mathematical analysis of logic*. Cambridge University Library holds correspondence between Boole and William Thomson and G. G. Stokes (qv); Glasgow University Library holds letters from Thomson to Boole (1845–8); and University College London Library has correspondence between Boole and Augustus De Morgan.