1997George Boole School Mathematical foundations ofcomputer science Main interests Mathematics, Logic,Philosophy of mathematics Notable ideas Boolean algebra Influenced by[show] Influenced[show] dedicated to George Boole. Plaque beneath Boole's window in Lincoln Cathedral. Professor at Cork [edit] Boole's status as mathematician was recognised by his appointment in 1849 as the first professor of mathematics at Queen's College, Cork inIreland. He met his future wife, Mary Everest, there in 1850 while she was visiting her uncle John Ryall who was Professor of Greek. They married some years later.[11] He maintained his ties with Lincoln, working there with E. R. Larken in a campaign to reduce prostitution.[12] The house in Cork in which Boole lived between 1849 and 1855. Boole was elected Fellow of the Royal Society in 1857;[7] and received honorary degrees ofLL.D. from the University of Dublin and Oxford University. Death [edit] On 8 December 1864, Boole died of an attack of fever, ending in pleural effusion. He was buried in the Church of Ireland cemetery of St Michael's, Church Road, Blackrock (a suburb of Cork City). There is a commemorative plaque inside the adjoining church. Boole's gravestone, Cork, Ireland. Works [edit] Boole's first published paper was Researches in the theory of analytical transformations, with a special application to the reduction of the general equation of the second order, printed in the Cambridge Mathematical Journal in February 1840 (Volume 2, no. 8, pp. 64–73), and it led to a friendship between Boole and Duncan Farquharson Gregory, the editor of the journal. His works are in about 50 articles and a few separate publications.[13] In 1841 Boole published an influential paper in early invariant theory.[7] He received a medal from the Royal Society for his memoir of 1844, On A General Method of Analysis. It was a contribution to the theory of linear differential equations, moving from the case of constant coefficients on which he had already published, to variable coefficients.[14] The innovation in operational methods is to admit that operations may not commute.[15] In 1847 Boole publishedThe Mathematical Analysis of Logic , the first of his works on symbolic logic.[16] Differential equations [edit] Two systematic treatises on mathematical subjects were completed by Boole during his lifetime. The Treatise on Differential Equationsappeared in 1859, and was followed, the next year, by a Treatise on the Calculus of Finite Differences, a sequel to the former work. In the sixteenth and seventeenth chapters of theDifferential Equations is an account of the general symbolic method, and of a general method in analysis, originally described in his memoir printed in the Philosophical Transactions for 1844. During the last few years of his life Boole worked on a second edition of his Differential Equations, and part of his last vacation was spent in the libraries of the Royal Society and the British Museum; but it was left incomplete. Isaac Todhunter printed the manuscripts in 1865, in a supplementary volume. Analysis [edit] In 1857, Boole published the treatise On the Comparison of Transcendents, with Certain Applications to the Theory of Definite Integrals,[17] in which he studied the sum of residues of a rational function. Among other results, he proved what is now called Boole's identity: [18] for any real numbers ak > 0, bk, and t > 0. Generalisations of this identity play an important role in the theory of the Hilbert transform.[18] Symbolic logic [edit] Main article: Boolean algebra In 1847 Boole published the pamphlet Mathematical Analysis of Logic. He later regarded it as a flawed exposition of his logical system, and wanted An Investigation of the Laws of Thought (1854), on Which are Founded the Mathematical Theories of Logic and Probabilities to be seen as the mature statement of his views. Boole's initial involvement in logic was prompted by a current debate onquantification, between Sir William Hamilton who supported the theory of "quantification of the predicate", and Boole's supporterAugustus De Morgan who advanced a version of De Morgan duality, as it is now called. Boole's approach was ultimately much further reaching than either sides' in the controversy.[19] It founded what was first known as the "algebra of logic" tradition.[20] Boole did not regard logic as a branch of mathematics, but he provided a general symbolic method of logical inference. Boole proposed that logical propositions should be expressed by means of algebraic equations. Algebraic manipulation of the symbols in the equations would provide a fail-safe method of logical deduction: i.e. logic is reduced to a type of algebra. By 1 (unity) Boole denoted the "universe of thinkable objects"; literal symbols, such as x, y, z, v, u, etc., were used with the "elective" meaning attaching to adjectives and nouns of natural language. Thus, if x = horned and y = sheep, then the successive acts of election (i.e. choice) represented by x and y, if performed on unity, give the class "horned sheep". Thus, (1 – x) would represent the operation of selecting all things in the world except horned things, that is, all not horned things, and (1 – x) (1 – y) would give all things neither horned nor sheep. Treatment of addition in logic [edit] Boole conceived of "elective symbols" of his kind as an algebraic structure. But this general concept was not available to him: he did not have the segregation standard in abstract algebra of postulated (axiomatic) properties of operations, and deduced properties.[21] His work was a beginning to the algebra of sets, again not a concept available to Boole as a familiar model. His pioneering efforts encountered specific difficulties, and the treatment of addition was an obvious difficulty in the early days. Boole replaced the operation of multiplication by the word 'and' and addition by the word 'or'. But in Boole's original system, + was apartial operation: in the language of set theory it would correspond only to disjoint union of subsets. Later authors changed the interpretation, commonly reading it as exclusive or, or in set theory terms symmetric difference; this step means that addition is always defined.[20][22] In fact there is the other possibility, that + should be read as disjunction,[21] This other possibility extends from the disjoint union case, where exclusive or and non-exclusive or both give the same answer. Handling this ambiguity was an early problem of the theory, reflecting the modern use of both Boolean rings and Boolean algebras (which are simply different aspects of one type of structure). Boole and Jevons struggled over just this issue in 1863, in the form of the correct evaluation of x + x. Jevons argued for the result x, which is correct for + as disjunction. Boole kept the result as something undefined. He argued against the result 0, which is correct for exclusive or, because he saw the equation x + x = 0 as implying x = 0, a false analogy with ordinary algebra.[7] Probability theory [edit] The second part of the Laws of Thought contained a corresponding attempt to discover a general method in probabilities. Here the goal was algorithmic: from the given probabilities of any system of events, to determine the consequent probability of any other event logically connected with the those events. Legacy [edit] Boolean algebra is named after him, as is the crater Boole on the Moon. The keyword Bool represents a Boolean datatype in many programming languages, though Pascal and Java, among others, both use the full name Boolean.[23] The library, underground lecture theatre complex and the Boole Centre for Research in Informatics[24] at University College Cork are named in his honour. 19th century development [edit] Boole's work was extended and refined by a number of writers, beginning with William Stanley Jevons. Augustus De Morgan had worked on the logic of relations, and Charles Sanders Peirce integrated his work with Boole's during the 1870s.[25] Other significant figures were Platon Sergeevich Poretskii, and William Ernest Johnson. The conception of a Boolean algebra structure on equivalent statements of a propositional calculus is credited to Hugh MacColl (1877), in work surveyed 15 years later by Johnson.[25] Surveys of these developments were published by Ernst Schröder, Louis Couturat, and Clarence Irving Lewis. 20th century development [edit] In 1921 the economist John Maynard Keynes published a book on probability theory, A Treatise of Probability. Keynes believed that Boole had made a fundamental error in his definition of independence which vitiated much of his analysis.[26] In his book The Last Challenge Problem, David Miller provides a general method in accord with Boole's system and attempts to solve the problems recognised earlier by Keynes and others.Theodore Hailperin showed much earlier that Boole had used the correct mathematical definition of independence in his worked out problems [27] In modern notation, the free Boolean algebra on basic propositions p and qarranged in a Hasse diagram. The Boolean combinations make up 16 different propositions, and the lines show which are logically related. Boole's work and that of later logicians initially appeared to have no engineering uses.Claude Shannon attended a philosophy class at the University of Michigan which introduced him to Boole's studies. Shannon recognised that Boole's work could form the basis of mechanisms and processes in the real world and that it was therefore highly relevant. In 1937 Shannon went on to write a master's thesis, at the Massachusetts Institute of Technology, in which he showed how Boolean algebra could optimise the design of systems of electromechanical relays then used in telephone routing switches. He also proved that circuits with relays could solve Boolean algebra problems.