George Boole

Not to be confused with George Boolos.

"Boole" redirects here. For other uses, see Boole (disambiguation).

George Boole
George Boole
Born	2 November 1815 Lincoln, Lincolnshire, England
Died	8 December 1864(1864-12-08) (aged 49) Ballintemple, County Cork, Ireland
Nationality	English
Era	19th-century philosophy
Region	Western Philosophy
School	Mathematical foundations of computer science
Main interests	Mathematics, Logic, Philosophy of mathematics
Notable ideas	Boolean algebra

George Boole (/ˈbuːl/; 2 November 1815 – 8 December 1864) was an English mathematician, philosopher and logician. He worked in the fields of differential equations and algebraic logic, and is now best known as the author of The Laws of Thought. As the inventor of the prototype of what is now called Boolean logic, which became the basis of the modern digital computer, Boole is regarded in hindsight as a founder of the field of computer science. Boole said,

... no general method for the solution of questions in the theory of probabilities can be established which does not explicitly recognise ... those universal laws of thought which are the basis of all reasoning ...

Early life

George Boole's father, John Boole (1779–1848), was a tradesman in Lincoln and gave him lessons. He had an elementary school education, but little further formal and academic teaching. William Brooke, a bookseller in Lincoln, may have helped him with Latin; which he may also have learned at the school of Thomas Bainbridge. He was self-taught in modern languages. At age 16 Boole became the breadwinner for his parents and three younger siblings, taking up a junior teaching position in Doncaster, at Heigham's School. He taught briefly in Liverpool.

Boole participated in the local Mechanics Institute, the Lincoln Mechanics' Institution, which was founded in 1833. Edward Bromhead, who knew John Boole through the Institution, helped George Boole with mathematics books; and he was given the calculus text of Sylvestre François Lacroix by Rev. George Stevens Dickson, of St Swithin Lincoln. Without a teacher, it took him many years to master calculus.

Boole's Lincoln House
Boole's House and School at 3 Pottergate in Lincoln. Plaque from the house in Lincoln.

At age 19 Boole successfully established his own school at Lincoln. Four years later he took over Hall's Academy, at Waddington, outside Lincoln, following the death of Robert Hall. In 1840 he moved back to Lincoln, where he ran a boarding school.

Boole became a prominent local figure, an admirer of John Kaye, the bishop. He took part in the local campaign for early closing. With E. R. Larken and others he set up a building society in 1847. He associated also with the Chartist Thomas Cooper, whose wife was a relation.

From 1838 onwards Boole was making contacts with sympathetic British academic mathematicians, and reading more widely. He studied algebra in the form of symbolic methods, as these were understood at the time, and began to publish research papers.

Professor at Cork

File:Boole House Cork.jpg

The house in Cork in which Boole lived between 1849 and 1855.

Boole's status as mathematician was recognised by his appointment in 1849 as the first professor of mathematics at Queen's College, Cork in Ireland. 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. He maintained his ties with Lincoln, working there with E. R. Larken in a campaign to reduce prostitution.

Boole was elected Fellow of the Royal Society in 1857; and received honorary degrees of LL.D. from the University of Dublin and Oxford University.

Death

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.

Detail of stained glass window in Lincoln Cathedral dedicated to George Boole.

Plaque beneath Boole's window in Lincoln Cathedral.

Works

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.

In 1841 Boole published an influential paper in early invariant theory. 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. The innovation in operational methods is to admit that operations may not commute. In 1847 Boole published The Mathematical Analysis of Logic , the first of his works on symbolic logic.

Differential equations

Two systematic treatises on mathematical subjects were completed by Boole during his lifetime. The Treatise on Differential Equations appeared 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 the Differential 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

In 1857, Boole published the treatise On the Comparison of Transcendents, with Certain Applications to the Theory of Definite Integrals, in which he studied the sum of residues of a rational function. Among other results, he proved what is now called Boole's identity:

$\mathrm {mes} \left\{x\in \mathbb {R} \,\mid \,\Re {\frac {1}{\pi }}\sum {\frac {a_{k}}{x-b_{k}}}\geq t\right\}={\frac {\sum a_{k}}{\pi t}}$

for any real numbers a_k > 0, b_k, and t > 0. Generalisations of this identity play an important role in the theory of the Hilbert transform.

Symbolic logic

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. Contrary to widespread belief, Boole never intended to criticize or disagree with the main principles of Aristotle’s logic. Rather he intended to systematize it, to provide it with a foundation, and to extend its range of applicability. Boole's initial involvement in logic was prompted by a current debate on quantification, between Sir William Hamilton who supported the theory of "quantification of the predicate", and Boole's supporter Augustus 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. It founded what was first known as the "algebra of logic" tradition.

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.Among his many innovations is his principle of wholistic reference, which was later, and probably independently, adopted by Gottlob Frege and by logicians who subscribe to standard first-order logic.A 2003 article provides a systematic comparison and critical evaluation of Aristotelian logic and Boolean logic; it also reveals the centrality of wholistic reference in Boole's philosophy of logic.

Boole’s 1854 Definition of Universe of Discourse

In every discourse, whether of the mind conversing with its own thoughts, or of the individual in his intercourse with others, there is an assumed or expressed limit within which the subjects of its operation are confined. The most unfettered discourse is that in which the words we use are understood in the widest possible application, and for them the limits of discourse are co-extensive with those of the universe itself. But more usually we confine ourselves to a less spacious field. Sometimes, in discoursing of men we imply (without expressing the limitation) that it is of men only under certain circumstances and conditions that we speak, as of civilized men, or of men in the vigour of life, or of men under some other condition or relation. Now, whatever may be the extent of the field within which all the objects of our discourse are found, that field may properly be termed the universe of discourse. Furthermore, this universe of discourse is in the strictest sense the ultimate subject of the discourse.

Treatment of addition in logic

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. 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 a partial 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.

In fact there is the other possibility, that + should be read as disjunction, 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.

Probability theory

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

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. The library, underground lecture theatre complex and the Boole Centre for Research in Informatics at University College Cork are named in his honour.

19th-century development

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. 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. Surveys of these developments were published by Ernst Schröder, Louis Couturat, and Clarence Irving Lewis.

20th-century development

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. 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

In modern notation, the free Boolean algebra on basic propositions p and q arranged 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. Employing the properties of electrical switches to process logic is the basic concept that underlies all modern electronic digital computers. Victor Shestakov at Moscow State University (1907–1987) proposed a theory of electric switches based on Boolean logic even earlier than Claude Shannon in 1935 on the testimony of Soviet logicians and mathematicians Yanovskaya, Gaaze-Rapoport, Dobrushin, Lupanov, Medvedev and Uspensky, though they presented their academic theses in the same year, 1938. But the first publication of Shestakov's result took place only in 1941 (in Russian). Hence Boolean algebra became the foundation of practical digital circuit design; and Boole, via Shannon and Shestakov, provided the theoretical grounding for the Digital Age.

Views

Boole's views were given in four published addresses: The Genius of Sir Isaac Newton; The Right Use of Leisure; The Claims of Science; and The Social Aspect of Intellectual Culture. The first of these was from 1835, when Charles Anderson-Pelham, 2nd Baron Yarborough gave a bust of Newton to the Mechanics' Institute in Lincoln. The second justified and celebrated in 1847 the outcome of the successful campaign for early closing in Lincoln, headed by Alexander Leslie-Melville, of Branston Hall. The Claims of Science was given in 1851 at Queen's College, Cork. The Social Aspect of Intellectual Culture was also given in Cork, in 1855 to the Cuvierian Society.

Boole read a wide variety of Christian theology. Combining his interests in mathematics and theology, he compared the Christian trinity of Father, Son, and Holy Ghost with the three dimensions of space, and was attracted to the Hebrew conception of God as an absolute unity. Boole considered converting to Judaism but in the end was said to have chosen Unitarianism. However, his biographer, Des MacHale, describes him as an "agnostic deist".

Two influences on Boole were later claimed by his wife, Mary Everest Boole: a universal mysticism tempered by Jewish thought, and Indian logic. Mary Boole stated that an adolescent mystical experience provided for his life's work:

My husband told me that when he was a lad of seventeen a thought struck him suddenly, which became the foundation of all his future discoveries. It was a flash of psychological insight into the conditions under which a mind most readily accumulates knowledge [...] For a few years he supposed himself to be convinced of the truth of "the Bible" as a whole, and even intended to take orders as a clergyman of the English Church. But by the help of a learned Jew in Lincoln he found out the true nature of the discovery which had dawned on him. This was that man's mind works by means of some mechanism which "functions normally towards Monism."

In Ch. 13 of Laws of Thought Boole used examples of propositions from Benedict Spinoza and Samuel Clarke. The work contains some remarks on the relationship of logic to religion, but they are slight and cryptic. Boole was apparently disconcerted at the book's reception just as a mathematical toolset:

George afterwards learned, to his great joy, that the same conception of the basis of Logic was held by Leibnitz, the contemporary of Newton. De Morgan, of course, understood the formula in its true sense; he was Boole's collaborator all along. Herbert Spencer, Jowett, and Leslie Ellis understood, I feel sure; and a few others, but nearly all the logicians and mathematicians ignored [953] the statement that the book was meant to throw light on the nature of the human mind; and treated the formula entirely as a wonderful new method of reducing to logical order masses of evidence about external fact.

Mary Boole claimed that there was profound influence — via her uncle George Everest — of Indian thought on George Boole, as well as on Augustus De Morgan and Charles Babbage:

Think what must have been the effect of the intense Hinduizing of three such men as Babbage, De Morgan, and George Boole on the mathematical atmosphere of 1830–65. What share had it in generating the Vector Analysis and the mathematics by which investigations in physical science are now conducted?

Family

In 1855 he married Mary Everest (niece of George Everest), who later wrote several educational works on her husband's principles.

The Booles had five daughters:

• Mary Lucy Margret (1856–1908) who married the mathematician and author Charles Howard Hinton and had four children: George (1882–1943), Eric (*1884), William (1886–1909) and Sebastian (1887–1923) inventor of the Jungle gym. Sebastian had three children:
  • William H. Hinton (1919–2004) visited China in the 1930s and 40s and wrote an influential account of the Communist land reform.
  • Joan Hinton (1921–2010) worked for the Manhattan Project and lived in China from 1948 until her death on 8 June 2010; she was married to Sid Engst.
  • Jean Hinton (married name Rosner) (1917–2002) peace activist.
• Margaret, (1858 – ?) married Edward Ingram Taylor, an artist.
  • Their elder son Geoffrey Ingram Taylor became a mathematician and a Fellow of the Royal Society.
  • Their younger son Julian was a professor of surgery.
• Alicia (1860–1940), who made important contributions to four-dimensional geometry
• Lucy Everest (1862–1905), who was first female professor of chemistry in England
• Ethel Lilian (1864–1960), who married the Polish scientist and revolutionary Wilfrid Michael Voynich and was the author of the novel The Gadfly.

References

• Chisholm, Hugh, ed. (1911). "Boole, George" . Encyclopædia Britannica (11th ed.). Cambridge University Press.
• Ivor Grattan-Guinness, The Search for Mathematical Roots 1870–1940. Princeton University Press. 2000.
• Francis Hill (1974), Victorian Lincoln; Google Books.
• Des MacHale, George Boole: His Life and Work. Boole Press. 1985.
• Stephen Hawking, God Created the Integers. Running Press, Philadelphia. 2007.

External links

• Roger Parsons' article on Boole
• Works by George Boole at Project Gutenberg
• George Boole's work as first Professor of Mathematics in University College, Cork, Ireland