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Boole, George


Not to be confused with George Boolos.
"Boole" redirects here. For other uses, see Boole (disambiguation).
George Boole

George Boole
Full name George Boole
Born 2 November 1815
Lincoln, Lincolnshire, England
Died 8 December 1864(1864-12-08) (aged 49)
Ballintemple, County Cork, Ireland
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 (pronounced /ˈbuːl/) (2 November 1815 – 8 December 1864) was an English mathematician and philosopher.

As the inventor of Boolean logic—the basis of modern digital computer logic—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 ...[1]

Contents

[edit] Biography

George Boole's father, John Boole(1779–1848), was a tradesman of limited means, but of "studious character and active mind".[2] Being especially interested in mathematical science and logic, the father gave his son his first lessons; but the extraordinary mathematical talents of George Boole did not manifest themselves in early life. At first, his favorite subject was classics.

It was not until his successful establishment of a school at Lincoln, its removal to Waddington, and later his appointment in 1849 as the first professor of mathematics of then Queen's College, Cork in Ireland (now University College Cork, where the library, underground lecture theatre complex and the Boole Centre for Research in Informatics[3] are named in his honour) that his mathematical skills were fully realized. In 1855 he married Mary Everest (niece of George Everest), who later, as Mrs. Boole, wrote several useful educational works on her husband's principles.

To the broader public Boole was known only as the author of numerous abstruse papers on mathematical topics, and of three or four distinct publications that have become standard works. His earliest published paper was the "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 D.F. Gregory, the editor of the journal, which lasted until the premature death of the latter in 1844. A long list of Boole's memoirs and detached papers, both on logical and mathematical topics, are found in the Catalogue of Scientific Memoirs published by the Royal Society, and in the supplementary volume on Differential Equations, edited by Isaac Todhunter. To the Cambridge Mathematical Journal and its successor, the Cambridge and Dublin Mathematical Journal, Boole contributed twenty-two articles in all. In the third and fourth series of the Philosophical Magazine are found sixteen papers. The Royal Society printed six important memoirs in the Philosophical Transactions, and a few other memoirs are to be found in the Transactions of the Royal Society of Edinburgh and of the Royal Irish Academy, in the Bulletin de l'Académie de St-Pétersbourg for 1862 (under the name G Boldt, vol. iv. pp. 198–215), and in Crelle's Journal. Also included is a paper on the mathematical basis of logic, published in the Mechanic's Magazine in 1848. The works of Boole are thus contained in about fifty scattered articles and a few separate publications.

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

Only two systematic treatises on mathematical subjects were completed by Boole during his lifetime. The well-known Treatise on Differential Equations appeared in 1859, and was followed, the next year, by a Treatise on the Calculus of Finite Differences, designed to serve as a sequel to the former work. These treatises are valuable contributions to the important branches of mathematics in question. To a certain extent these works embody the more important discoveries of their author. In the sixteenth and seventeenth chapters of the Differential Equations we find, for instance, an account of the general symbolic method, the bold and skilful employment of which led to Boole's chief discoveries, and of a general method in analysis, originally described in his famous memoir printed in the Philosophical Transactions for 1844. Boole was one of the most eminent of those who perceived that the symbols of operation could be separated from those of quantity and treated as distinct objects of calculation. His principal characteristic was perfect confidence in any result obtained by the treatment of symbols in accordance with their primary laws and conditions, and an almost unrivalled skill and power in tracing out these results.

During the last few years of his life Boole was constantly engaged in extending his researches with the object of producing a second edition of his Differential Equations much more complete than the first edition, and part of his last vacation was spent in the libraries of the Royal Society and the British Museum; but this new edition was never completed. Even the manuscripts left at his death were so incomplete that Todhunter, into whose hands they were put, found it impossible to use them in the publication of a second edition of the original treatise, and printed them, in 1865, in a supplementary volume.

With the exception of Augustus De Morgan, Boole was probably the first English mathematician since the time of John Wallis who had also written upon logic. His novel views of logical method were due to the same profound confidence in symbolic reasoning to which he had successfully trusted in mathematical investigation. Speculations concerning a calculus of reasoning had at different times occupied Boole's thoughts, but it was not till the spring of 1847 that he put his ideas into the pamphlet called Mathematical Analysis of Logic. Boole afterwards regarded this as a hasty and imperfect exposition of his logical system, and he desired that his much larger work, An Investigation of the Laws of Thought (1854), on Which are Founded the Mathematical Theories of Logic and Probabilities, should alone be considered as containing a mature statement of his views. This ushered in a new focus on the nature of evidence, argument, and proof. Nevertheless, there is a charm of originality about his earlier logical work that is easy to appreciate.

Plaque beneath Boole's window in Lincoln Cathedral

He did not regard logic as a branch of mathematics, as the title of his earlier pamphlet might be taken to imply, but he pointed out such a deep analogy between the symbols of algebra and those that can be made, in his opinion, to represent logical forms and syllogisms, that we can hardly help saying that (especially his) formal logic is mathematics restricted to the two quantities, 0 and 1. By 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 common adjectives and substantives. Thus, if x = horned and y = sheep, then the successive acts of election represented by x and y, if performed on unity, give the whole of the class horned sheep. Boole showed that elective symbols of this kind obey the same primary laws of combination as algebraic symbols, whence it followed that they could be added, subtracted, multiplied and even divided, almost exactly in the same manner as numbers. 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 us all things neither horned nor sheep. By the use of such symbols propositions could be reduced to the form of equations, and the syllogistic conclusion from two premises was obtained by eliminating the middle term according to ordinary algebraic rules.

Boole's House and School in Lincoln

Still more original and remarkable, however, was that part of his system, fully stated in his Laws of Thought, formed a general symbolic method of logical inference. Given any propositions involving any number of terms, Boole showed how, by the purely symbolic treatment of the premises, to draw any conclusion logically contained in those premises. The second part of the Laws of Thought contained a corresponding attempt to discover a general method in probabilities, which should enable us from the given probabilities of any system of events to determine the consequent probability of any other event logically connected with the given events.

In 1921 the economist John Maynard Keynes published a book that has become a classic on probability theory, "A Treatise of Probability." Keynes's comments about Boole's theory of probability were generally taken to be the definitive statement on the subject. Keynes believed that Boole had made a fundamental error that vitiated much of his analysis.[4] In a recent book, "The Last Challenge Problem," David Miller provides a general method in accord with Boole's system, and attempts to solve the problems recognized earlier by Keynes and others.[5]

Though Boole published little except his mathematical and logical works, his acquaintance with general literature was wide and deep. Dante was his favourite poet, and he preferred the Paradiso to the Inferno. The metaphysics of Aristotle, the ethics of Spinoza, the philosophical works of Cicero, and many kindred works, were also frequent subjects of study. His reflections upon scientific, philosophical and religious questions are contained in four addresses upon The Genius of Sir Isaac Newton, The Right Use of Leisure, The Claims of Science and The Social Aspect of Intellectual Culture, which he delivered and printed at different times.

The personal character of Boole inspired all his friends with the deepest esteem. He was marked by true modesty, and his life was given to the single-minded pursuit of truth. Though he received a medal from the Royal Society for his memoir of 1844, and the honorary degree of LL.D. from the University of Dublin, he neither sought nor received the ordinary rewards to which his discoveries would entitle him. On 8 December 1864, in the full vigour of his intellectual powers, he died of an attack of fever, ending in pleural effusion, an accumulation of fluid around the lungs. He is 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 proposed that logical propositions should be expressed as algebraic equations. The algebraic manipulation of the symbols in the equations provides a fail-safe method of logical deduction, i.e. logic is reduced to algebra. Boole replaced the operation of multiplication by the word 'and' and addition by the word 'or'. The symbols in the equations can stand for collections of objects (sets) or statements in logic. For example, if x is the set of all brown cows and y is the set of all fat cows, then x+y is the set of all cows that are brown or fat, and xy is the set of all cows that are brown and fat.

Let z = the set of all Irish cows. Then z(x+y) = zx+zy; in other words 'the set of Irish cows that are either brown or fat is the same as the collection of cows that are Irish and brown or Irish and fat'.

[edit] Family

The Booles had five daughters:

  • Mary Ellen, (1856 – ?) who married the mathematician and author Charles Howard Hinton and had two children: George and Sebastian. Sebastian had three children:
    • William H. Hinton 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, who made important contributions to four-dimensional geometry
  • Lucy Everest (1862–1905), who was first female Professor of Chemistry in England
  • Ethel Lilian, who married the Polish scientist and revolutionary Wilfrid Michael Voynich and was the author of the novel The Gadfly.

[edit] Legacy

Boole's work was extended and refined by William Stanley Jevons, Augustus De Morgan, Charles Sanders Peirce, and William Ernest Johnson. This work was summarized by Ernst Schröder, Louis Couturat, and Clarence Irving Lewis.

Plaque commemorating Boole on his house in Lincoln

Boole's work (as well as that of his intellectual progeny) was relatively obscure, except among logicians. At the time, it appeared to have no practical uses. However, approximately seventy years after Boole's death, Claude Shannon attended a philosophy class at the University of Michigan that 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 optimize 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 S.A. Yanovskaya, Gaaze-Rapoport, Dobrushin, Lupanov, Medvedev, and Uspensky, though they presented their academic theses in the same year, 1938[clarification needed]. 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.[6]

The crater Boole on the Moon is named in his honour.

[edit] References

  1. http://www.kerryr.net/pioneers/boole.htm
  2.  Chisholm, Hugh, ed (1911). "Boole, George". Encyclopædia Britannica (Eleventh ed.). Cambridge University Press. 
  3. Boole Centre for Research in Informatics
  4. Chapter XVI, page 167, section 6 of A treatise on probability, volume 4: "The central error in his system of probability arises out of his giving two inconsistent definitions of 'independence' (2). He first wins the reader's acquiescence by giving a perfectly correct definition: "Two events are said to be independent when the probability of either of them is unaffected by our expectation of the occurrence or failure of the other." (3) But a moment later he interprets the term in quite a different sense; for, according to Boole's second definition, we must regard the events as independent unless we are told either that they must concur or that they cannot concur. That is to say, they are independent unless we know for certain that there is, in fact, an invariable connection between them. "The simple events, x, y, z, will be said to be conditioned when they are not free to occur in every possible combination; in other words, when some compound event depending upon them is precluded from occurring. ... Simple unconditioned events are by definition independent." (1) In fact as long as xz is possible, x and z are independent. This is plainly inconsistent with Boole's first definition, with which he makes no attempt to reconcile it. The consequences of his employing the term independence in a double sense are far-reaching. For he uses a method of reduction which is only valid when the arguments to which it is applied are independent in the first sense, and assumes that it is valid if they are independent in second sense. While his theorems are true if all propositions or events involved are independent in the first sense, they are not true, as he supposes them to be, if the events are independent only in the second sense."
  5. http://zeteticgleanings.com/boole.html
  6. "That dissertation has since been hailed as one of the most significant master's theses of the 20th century. To all intents and purposes, its use of binary code and Boolean algebra paved the way for the digital circuitry that is crucial to the operation of modern computers and telecommunications equipment."Andrew Emerson (2001-03-08). "Claude Shannon". The Guardian. http://www.guardian.co.uk/science/2001/mar/08/obituaries.news. 
  •  Chisholm, Hugh, ed (1911). "Boole, George". Encyclopædia Britannica (Eleventh ed.). Cambridge University Press. 
  • Ivor Grattan-Guinness, The Search for Mathematical Roots 1870–1940. Princeton University Press. 2000.
  • Des MacHale, George Boole: His Life and Work. Boole Press. 1985.

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