George Boole

George Boole (/buːl/) (Lincoln, Lincolnshire, England Born November 2, 1815-Ballintemple, County Cork, Ireland, December 8, 1864) was a British mathematician and logician.

As the inventor of Boolean algebra, which lays the foundations of modern computational arithmetic, Boole is considered one of the founders of the field of computer science. In 1854 he published An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, where he developed a system of rules that allowed him to express, manipulate, and simplify logical and philosophical problems whose arguments admit two states (true or false) by mathematical procedures. It could be said that he is the father of symbolic logical operators and that thanks to his algebra today it is possible to operate symbolically to perform logical operations.

Biography

Boole House and school in No. 3 Pottergate (Lincoln).

Commemorative plaque at your Lincoln house.

George Boole's father, John Boole (1779-1848), was a poor merchant. He was especially interested in mathematics and logic. John gave his son his first lessons, but George Boole's extraordinary mathematical talents did not manifest themselves during his youth, as he initially showed more interest in the humanities.

The combination of his interests in theology and mathematics led him to compare the Christian Trinity of Father, Son, and Holy Spirit to the three dimensions of space, and he was drawn to the Hebrew concept of God as an absolute unity. Boole considered adopting Judaism, but ultimately opted for Unitarianism.

It was not until his successful establishment of a school in Lincoln, his move to Waddington, and later his appointment in 1849 as the first Professor of Mathematics at the then Queen's College Cork (now University College Cork) that his mathematical abilities were fully realized.

In 1855, he married Mary Everest, niece of George Everest, who later, as Mrs. Boole, wrote several useful educational works in her husband's early days.

Although Boole published little except his Logic and mathematical works, his knowledge of literature in general was wide and deep. Dante was his favorite poet, preferring Paradise to Hell . Aristotle's Metaphysics, Spinoza's Ethics, Cicero's philosophical works and many other related works were also frequent subjects of study. His reflections on philosophical and religious questions of a scientific nature were oriented in four directions: the genius of Sir Isaac Newton; the correct use of leisure; the demands of Science; and the social aspect of intellectual culture.

Boole's stone, Cork, Ireland.

Boole's personal character inspired all his friends in the highest esteem. She was characterized by modesty, and gave her life to the search for the truth. Although he received a Royal Society medal for his 1844 memoirs, and an honorary degree of Doctor of Laws from the University of Dublin, he neither sought nor received the ordinary benefits that his discoveries would have brought. entitled.

On 8 December 1864, in the full vigor of his intellectual faculties, he died of a fit of fever, which ended in a pleural effusion. He was buried in the churchyard of St. Michael's Church, Church Road, Blackrock (a neighborhood in the city of Cork, in Ireland). There is a commemorative plaque in the adjoining church.

Work

To the general public, Boole is known primarily as the author of numerous abstruse works in the field of mathematics, and of various publications that have become treatises. His first published work was "Investigations into the Theory of Transformations of Analysis, 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 which led to a friendship between Boole and D. F. Gregory, the magazine's editor, which lasted until the latter's premature death in 1844.

A long list of Boole's memoirs and papers, both on subjects of logic and mathematics, are found in the Catalog of Memoirs of Science published by the Royal Society, and in volume Supplementary on Differential Equations, edited by Isaac Todhunter.

In 1841 Boole published an influential paper on the nascent theory of invariants. He received a Royal Society medal for his 1844 memoir entitled On a General Method of Analysis, a contribution to the equations linear differentials, starting from the case of constant coefficients on which he had already worked, to address the case of variable coefficients. His main innovation in operational methods consisted in admitting that operations could not be commutative. In 1847 Boole published The Mathematical Analysis of Logic, the first of his works on symbolic logic.

Boole would publish twenty-two articles in The Cambridge Mathematical Journal and its successor, The Cambridge and Dublin Mathematical Journal. Likewise, he would publish sixteen articles in the third and fourth series of the Philosophical Magazine. The Royal Society has six important memoirs in print in the Philosophical Transactions, and memoirs of some other works are found in the Transactions of the Royal Society of Edinburgh and of the Royal Academy. of Ireland, in the Bulletin de l'Académie de St-Pétersbourg of 1862 (under the name of G. Boldt, vol. iv, pp. 198-215), and in the Crelle Magazine. Also included is a paper on the mathematical basis of logic, published in the Mechanic's Magazine in 1848.

Boole's works are scattered among some fifty articles and other independent publications. 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 following year, by a Treatise on the Calculus of Finite Differences, designed to serve as a sequel to the previous work. These treatises are valuable contributions to the important branches of mathematics that are dealt with in them. To a certain extent, these works represent the most relevant discoveries of their author in the field of calculus. In the sixteenth and seventeenth chapters of Differential Equations may be found, for example, the development of the general symbolic method, with the skilful and daring use of the procedure which led Boole to his other discoveries, and in a general method of analysis, originally described in his famous printed memoir in the Philosophical Transactions of 1844. Boole was one of the first and most eminent mathematicians to perceive that symbols of operations could be separated from quantities on which they operate, and be treated as objects other than the calculation itself. Boole's chief characteristic was his absolute confidence in any results obtained by the treatment of symbols in accordance with his primary laws and conditions, and an almost unequaled ability to locate applications for these results.

During the last years of his life, Boole devoted himself constantly to extending his investigations 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. The manuscripts left at his death were so incomplete that even Isaac Todhunter, in charge of whom they were left, was unable to complete a second edition of the original treatise, and he published 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 to write on logic. His views on the application of logical method stemmed from the same deep reliance on symbolic reasoning with which he had successfully broken into mathematical research. Speculations about a calculus of reasoning occupied Boole's thoughts, but it was not until the spring of 1847 that he expressed his ideas in the pamphlet entitled Mathematical Analysis of Logic . He regarded this publication as a hasty and imperfect exposition of his logical system. Subsequently, Boole stated that his most important work, his Inquiry into the Laws of Thought (1854), in which his mathematical theories on Logic and Probability are based, should only be considered as a statement matured from his points of view. This work marked the beginning of a new approach to the nature of validation of arguments and proofs. Yet it is easy to see an undeniable charm in the originality of his earlier logical work.

In modern notation, boolean algebra free of two basic propositions p and q organized in a Hasse diagram. Bonoleaan combinations generate 16 different propositions, and the lines show how they are logically related.

Boole did not consider logic to be a branch of mathematics, as might be interpreted from the title of his previous pamphlet, but he noted a deep analogy between the symbols of algebra and the symbolic representation, in his opinion, necessary to represent logical forms. and syllogisms, matching formal logic with mathematics limited to the use of operations with zeros and ones. To unify different systems of logical operators, Boole organized the universe of all these imaginable objects; creating a symbolic notation suitable for his purposes, with symbols such as x, y, z, v, u, etc., which he uses to characterize the attributes corresponding to common adjectives and nouns. He proposed that logical propositions should be expressed in the form of algebraic equations, so that algebraic manipulation of the symbols in the equations provides a fail-safe method of logical deduction, that is, logic reduces to algebra. Through the use of symbols, such propositions could be reduced to the form of equations, and the syllogistic conclusion from two premises is obtained by eliminating the middle term according to ordinary algebraic rules.

Even more original and remarkable, however, was that part of his system, totally based on his Laws of Thought, allowed to structure a general symbolic method of inference logic. Given a proposition involving any number of terms, Boole showed how, by the purely symbolic treatment of these premises, any logical conclusion contained in those premises could be deduced. The second part of his Laws of Thought contains his corresponding attempt to discover a general method of probabilities, which, as a consequence, should make it possible to determine the probability of any event logically related to a given system of events, from the probabilities of the aforementioned system of given events.

In 1921 economist John Maynard Keynes published a book that has become a classic in probability theory, A Treatise of Probability.). In his book, Keynes commented on Boole's theory of probability, arguing that Boole had made a fundamental mistake about the concept of stochastic independence which in his view flawed most of the work. of his predecessor. In his book, The Last Challenge Problem: George Boole's Theory of Probability (2009), David Miller provides a general method according to Boole's system, and attempts to solve previously recognized problems. by Keynes and others.

Mathematical analysis

In 1857, Boole published his treatise On the Comparison of Transcendents, with Certain Applications to the Theory of Definite Integrals. /i>), where he studied the sum of residues of a rational function. Among other results, he proved the so-called Boolean identity:

mes{x한 한 R R R 1π π ␡ ␡ akx− − bk≥ ≥ t!=␡ ␡ akπ π t{displaystyle mathrm {mes} left{xin mathbb {R} ,mid ,Re {frac {1}{pi }sum {frac {a_{k}}}{x-b_{k}}}}}}}{geq tright}={frac {sum a_{k}{pi t}}}}}{

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

Cover The Mathematical Analysis of Logic, 1847 edition.

Main publications

• 1847: The Mathematical Analysis of Logic (Mathematical Analysis of Logic)

As a statement of intent, on its cover includes the phrase of Aristotle (Anal. post., lib. I, chap. XI) "All sciences are associated with others regarding common elements. (And I call common to all that they use in their demonstrations, not to that which may be or may not be proved)». Boole considered this brief publication (80 pages) as an imperfect sketch of his logical system, although it contained most of the principles in which he founded his later work.

• 1854: An investigation of the Laws of Thought (An investigation of the laws of thought)

An extension of the previous work can be considered. It contains a complete explanation of the procedures of inference and logical-mathmatic deduction.

• 1859: A Treatise on Differential Equations (Treaty on differential equations)

Its main contribution is the development of the general symbolic method, and of a general method of analysis that applies to the study of different types of differential equations. A second completely renovated edition of this work was truncated by the death of Boole. It was a textbook at the University of Cambridge (it was edited uninterruptedly until 1923, and in 2014 a revised edition has been published)

• 1860: A Treatise on the Calculus of Finite Differences (Treaty on the Calculation of Finite Differences)

Complement of the previous book, it treats the differential and integral calculation, the series, and the equations of differential functions; and contains more than two hundred problems and their solutions discussed by Boole himself.

Family

Boole had five daughters:

• Mary Lucy Margaret (1856-1908), who married mathematician and writer Charles Howard Hinton and had four children:
  • George (1882-1943)
  • Eric (1884-?)
  • William (1886-1909)
  • Sebastian (1887-1923), inventor of the “monkey bars” (installation for children’s games). Sebastian had three children:
    • Jean Hinton (Rosner married name) (1917-2002), who was a peace activist.
    • William H. Hinton (1919-2004), who visited China in the 1930s and 1940s, and wrote an influential account on communist agrarian reform.
    • Joan Hinton (1921-2010), who worked at the Manhattan Project and lived in China from 1948 until his death on 8 June 2010, married Sid Engst.
• Margaret (1858-1935), who married the artist Edward Ingram Taylor, with whom he had two children:
  • His older son Geoffrey Ingram Taylor became a mathematician and became a member of the Royal Society.
  • His younger son, Julian was a professor of surgery.
• Alicia (1860-1940), who made important contributions to the four-dimensional geometry.
• Lucy Everest (1862-1905), who was the first female chemistry teacher in England.
• Ethel Lilian (1864-1960), who married Polish scientist and revolutionary Wilfrid Michael Voynich, and who was the author of the novel The Gadfly.

Legacy

Glass shop dedicated to Boole in Lincoln Cathedral.

• Boole's algebra carries his name.
• The key word Bool represents a type of data in many programming languages. For example, Pascal and Java, among others, use the full name Boolean.
• Boole's work was expanded and perfected by William Stanley Jevons, Augustus De Morgan, Charles Sanders Peirce and William Ernest Johnson. This work was summed up by Ernst Schröder, Louis Couturat, and Clarence Irving Lewis.
• Boole's work (as well as that of his intellectual offspring) was relatively dark, except among the logical ones. At the time it seemed to have no practical use. However, about 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 recognized that Boole's work could be the basis of mechanisms and processes in the real world and that it was therefore of great relevance. In 1937 Shannon was dedicated to writing a bachelor's thesis at the Massachusetts Institute of Technology, in which he demonstrated how Boole's algebra can optimize the design of the electromechanical relay systems, then used in telephone routing switches. He also showed that relay circuits could solve boolean algebra problems. The use of electrical switch properties to process logic is the basic concept underlying all modern electronic systems in digital equipment. Victor Shestakov, of the Moscow State University (1907-1987), proposed a theory of electric switches based on Boolean logic in 1935 (even before Claude Shannon), according to the testimony of the Soviet logics and mathematicians Sofia Yanovskaya, Gaaze-Rapoport, Dobrushin, Lupanov, Dmitri Medvédev and Uspensky, even though they presented their academic year.38 But the first publication of Shestakov's results took place only in 1941 (in Russian). Therefore, Boole's algebra became the foundation of the practice of digital circuit design, and Boole, through Shannon and Shestakov, in the theoretical basis for the digital era.

Acknowledgments and Honors

Commemorative plaque dedicated to Boole in Lincoln Cathedral.

• Boole was awarded the Keith Medal of the Royal Edinburgh Society in 1855
• He was elected a member of the Royal Society in 1857.
• Doctor honoris causes the University of Dublin and the University of Oxford.
• The bookstore, the underground reading room and the Boole Centre for Research in InformaticsAt the University of Cork they bear this name in their honor.
• A street from Bracknell, Berkshire, takes its name (Boole Heights).
• Boole's crater on the moon carries that name in his honor.
• The asteroid (17734) Boole also commemorates its name.
• In 2015 the 200th anniversary of George Boole's birth was celebrated in 1815. To commemorate the bicentennial, the University of Cork brought together Boole fans from all countries of the world to celebrate their life and legacy with various activities, including a new edition of the biography The Life and Work of George Boole: A Prelude to the Digital Age (Cork University Press, 2014), originally published in 1985 by Desmond MacHale.
• The Google search engine redesigned the 200th anniversary of its birth on November 2, 2015 with an algebraic inspiration image on its entrance screen (Google Doodle).

Boole's legacy resonates everywhere: on computers, storage and access to information, on electronic circuits and controls that support life, teaching and communications of the 21st century. Its key advances in mathematics, logic and probability are the substrate of modern mathematics, microelectronic engineering and computer sciences.

University College Cork