George Bull as a scientist. George Boole (1815), English mathematician, founder of formal logic ("Investigation of the laws of thought")
Boole is considered the founder of mathematical logic as an independent discipline. In his works, logic found its own alphabet, its own spelling and grammar. No wonder the initial section of mathematical logic is called the algebra of logic, or Boolean algebra.
Shortly after Boole was convinced that his algebra was quite applicable to logic, in 1847 he published a pamphlet "Mathematical Analysis of Logic", in which he expressed the idea that logic is closer to mathematics than to philosophy. This work was highly appreciated by the English mathematician Augustus (August) De Morgan. Thanks to this work, Boole in 1849 received the post of professor of mathematics at Queen's College in County Cork.
In 1854 he published the work "Investigation of the laws of thought, based on mathematical logic and probability theory." The works of 1847-1854 laid the foundation for the algebra of logic, or Boolean algebra. Boole was the first to show that there is an analogy between algebraic and logical operations, since both require only two answers - true or false, zero or one. He came up with a system of notation and rules, using which it was possible to encode any statements, and then manipulate them like ordinary numbers. Boolean algebra had three basic operations - AND, OR, NOT, which allowed for addition, subtraction, multiplication, division, and comparison of characters and numbers. Thus, Boole was able to describe in detail the binary number system. In his work The Laws of Thought (1854), Boole finally formulated the foundations of mathematical logic. He also tried to formulate a general method of probabilities by which, from a given system of probable events, one could determine the probability of a subsequent event logically related to them.
Boole did not consider logic a branch of mathematics, but found a deep analogy between the symbolic method of algebra and the symbolic method of representing logical forms and syllogisms. Boole showed that symbolism of this kind obeys the same laws as algebraic, from which it followed that they can be added, subtracted, multiplied and even divided. In such a symbolism, statements can be reduced to the form of equations, and the conclusion from the two premises of the syllogism can be obtained by eliminating the middle term according to the usual algebraic rules. Even more original and remarkable was the part of his system presented in the "Laws of Thought ...", which forms the general symbolic method of logical inference. Boole showed how, from any number of statements, including any number of terms, to deduce any conclusion that follows from these statements, by purely symbolic manipulations. The second part of "The Laws of Thought..." contains a similar attempt to discover a general method in the calculus of probabilities, which makes it possible, from the given probabilities of a set of events, to determine the probability of any other event logically connected with them.
Boole denoted the universe of conceivable objects, with alphabetic symbols - selections from it, associated with ordinary adjectives and nouns. Boole showed that this kind of symbolism obeys the same laws as algebraic, from which it followed that they can be added, subtracted, multiplied and even divided. In An investigation of the Laws of Thought, Boole showed how, from any number of statements, including any number of terms, to derive any conclusion that follows from these statements, by purely symbolic manipulation. The second part of The Laws of Thought contains a similar attempt to discover a general method in the calculus of probabilities, which makes it possible, from the given probabilities of a set of events, to determine the probability of any other event logically connected with them.
Boole invented a kind of algebra - a system of notation and rules applicable to all kinds of objects, from numbers and letters to sentences. Using this system, Boole could encode propositions—statements that needed to be proven true or false—using the symbols of his language, and then manipulate them in the same way that ordinary numbers are manipulated in mathematics.
The three basic Boolean algebra operations are AND, OR, and NOT. Although Boole's system allows for many other operations—often called logical operations—these three are enough to perform addition, subtraction, multiplication, and division, or to perform operations such as comparing characters and numbers. Logical actions are binary in nature, they operate only with two entities - "true" or "false", "yes" or "no", "open" or "closed", zero or one. Boole hoped that his system, by clearing logical arguments from verbal husks, would facilitate the search for the correct conclusion and make it always achievable.
In 1857, Boole was elected a Fellow of the Royal Society of London. His works Treatise on Differential Equations (1859) and Treatise on the Calculation of Limit Differences (1860) had a tremendous impact on the development of mathematics. They reflected the most important discoveries of Boole.
Most logicians of that time either ignored or sharply criticized Boole's system, but its possibilities turned out to be so great that it could not remain unattended for long.
George Bull
George Boole is considered to be the father of mathematical logic. To process logical expressions in mathematical logic, a propositional algebra, or algebra of logic, was created. Since the foundations of such an algebra were laid in the works of the English mathematician George Boole, the algebra of logic was also called Boolean algebra. The algebra of logic is abstracted from the semantic content of statements and takes into account only the truth or falsity of the statement.
In the twentieth century, scientists combined the mathematical apparatus created by George Boole with the binary number system, thereby laying the foundation for the development of a digital electronic computer.
George Bull was born in Lincoln (England) in the family of a small merchant. The financial situation of his parents was difficult, so George could only finish an elementary school for the children of the poor; he did not study at other educational institutions. This partly explains that, not bound by tradition, he went his own way in science. Buhl independently studied Latin, ancient Greek, German and French, studied philosophical treatises. From an early age, Buhl was looking for a job that left opportunities for self-education. After many unsuccessful attempts, Boole managed to open a small elementary school, where he taught himself. School textbooks in mathematics horrified him with their laxity and illogicality, Boole was forced to turn to the works of the classics of science and independently study the extensive works of Laplass and Lagrange.
In this regard, he had the first independent ideas. Boole reported the results of his research in letters to professors of mathematics (D. Gregory and A. de Morgan) of the famous Cambridge University and soon gained fame as an original thinking mathematician. In 1849, a new higher educational institution, Queen's College, was opened in Cork (Ireland), on the recommendation of fellow mathematicians, Boole received a professorship here, which he retained until his death in 1864. Only here did he get the opportunity not only to provide for his parents, but also calmly, without thinking about his daily bread, to engage in science. Here he married the daughter of a professor of Greek, Mary Everest, who helped Boole in his work and left interesting memories of her husband after his death; she became the mother of four Buhl's daughters, one of whom, Ethel Lilian Buhl, married Voynich, is the author of the popular novel The Gadfly.
The first to attempt to translate the laws of thought (formal logic) from the verbal realm, full of uncertainties, into the realm of mathematics was the German scientist Gottfried Wilhelm Leibniz (in 1666). More than a hundred years later, in 1816, after the death of Leibniz, George Boole picked up his idea of creating a logical universal language that obeys strict mathematical laws. Boole invented a kind of algebra - a system of notation and rules applicable to all kinds of objects, from numbers and letters to sentences.
Boole was probably one of the first mathematicians who turned to logical problems. Boole did not consider logic a branch of mathematics, but found a deep analogy between the symbolic method of algebra and the symbolic method of representing logical forms and syllogisms.
In 1848, George Boole published an article on the principles of mathematical logic - "Mathematical Analysis of Logic, or an Experience in the Calculus of Deductive Inferences", and in 1854 his main work "Investigation of the laws of thought on which the mathematical theories of logic and probability are based" appeared. These works reflected Boole's conviction that it is possible to study the properties of mathematical operations that are not necessarily performed on numbers. The scientist spoke about the symbolic method, which he applied both to the study of differentiation and integration, and to logical inference and to probabilistic reasoning. It was he who built one of the sections of formal logic in the form of a certain "algebra", similar to the algebra of numbers, but not reducible to it.
Boole invented a kind of algebra - a system of notation and rules applicable to all kinds of objects, from numbers to sentences. Using this system, he could encode propositions (statements that he needed to prove true or false) using the symbols of his language, and then manipulate them in the same way that numbers are manipulated in mathematics. The main operations of Boolean algebra are conjunction (AND), disjunction (OR), negation (NOT).
After some time, it became clear that Boole's system is well suited for describing electrical circuit switches. Current in a circuit can either flow or not, just as a statement can be either true or false.
And a few decades later, already in the twentieth century, scientists combined the mathematical apparatus created by George Boole with the binary number system (whose numbers 0 and 1 are also suitable for describing two states: the statement is true - the statement is false, the light is on - the light is off), laying thus the basis for the development of a digital electronic computer.
List of used literature
Kolmykova, E.A. Informatics [Text]: textbook. allowance for students of institutions environments. prof. education / E.A. Kolmykova, I.A. Kumskov. - Moscow: Information Center "Academy", 2011. - 416 p. - [Approved by the Ministry of Defense of Russia].
Project activities of students [Text] / Comp. E. S. Larina. - Volgograd: Publishing House "Teacher", 2009. - 155 p.
(Wikipedia).
(Yandex dictionaries).
George Boole (November 2, 1815 – December 8, 1864) was an English mathematician, educator, philosopher, and logician. He worked in the field of differential equations and algebraic logic, and is best known as the author of the laws of thought (1854), which contains Boolean algebra. Boolean logic is credited with laying the foundations for the information age.
Boole first published a study on the theory of analytic transformations, with a special application to the contraction of a general second-order equation, printed in the Cambridge Mathematical Journal in February 1840 (Vol. 2, pp. 64-73), and it led to a friendship between Boole and Duncan Farcarson Gregory, editor of the magazine. His works are found in about 50 articles and several separate publications.
In 1841 Boole published an important work early in the theory of invariants; he received a Royal Society medal for his memoirs, on a general method of analysis. It was a contribution to the theory of linear differential equations. The innovation in operational methods lies in the fact that operations do not commute. In 1847 Boole published the Mathematical Analysis of Logic, his first work on symbolic logic.
Boole completed two systematic treatises on mathematical subjects during his lifetime. These are the Treatise on Differential Equations, which appeared in 1859, and the Treatise on the Calculus of Finite Differences, a continuation of the former work.
In 1921, the economist John Maynard Keynes published a book on probability theory. Keynes believed that Boole had made a fundamental mistake in his definition of independence which ruined much of his analysis. In his book The Final Problem, David Miller proposes a general method according to the Boolean system and attempts to solve problems previously recognized by Keynes and others.
The works of Boole and later logicians at first seemed to have no purpose. Claude Shannon took part in a philosophy course at Michigan State University which introduced him to Boolean studies. Shannon admits that Boole's work could form the basis of mechanisms and processes in the real world, and therefore he was very relevant. In 1937, Shannon went on to write his master's thesis at the Massachusetts Institute of Technology, in which he showed how Boolean algebra could optimize the design of electromechanical relay systems then be used in routed telephone switches. He also proved that a relay circuit could solve a Boolean algebra problem. Using the properties of electrical switches for a logical process is the basic concept that underlies all modern electronic digital computers. Viktor Shestakov at Moscow State University (1907-1987) proposed a theory of electrical switches based on Boolean logic even earlier than Claude Shannon in 1935 on the testimony of Soviet logicians and mathematicians Sofya Yanovskaya, Dobrushin Roland, Lupanov, Medvedev and Uspensky presented their scientific dissertation in the same year, 1938, but Shestakov's first publication ended up taking place only in 1941 (in Russian). Thus, Boolean algebra became the basis for the practical design of a digital circuit.
George Boole is an English mathematician and logician. Professor of Mathematics at King's College Cork from 1849. One of the founders of mathematical logic.
George Bull was born and raised in the family of a poor artisan John Bull, who was passionate about science. The father, being interested in mathematics and logic, gave the first lessons to his son, but he failed to discover early his outstanding talents in the exact sciences, and classical authors became his first hobby.
Only at the age of seventeen Buhl reached higher mathematics, moving slowly due to the lack of effective help.
From the age of sixteen, Buhl began working as a teacher's assistant at a private school in Doncaster and, one way or another, continued teaching in various positions throughout his life. He was married (since 1855) to Mary Everest (z. Everest-Buhl), the niece of the famous geographer George Everest, who also studied science and taught, and after her husband's death devoted a lot of effort to popularizing his contribution to logic.
Four of their daughters became famous as scientists (geometrist Alicia, chemist Lucy), or members of scientific families (Mary, wife of the mathematician and writer C. G. Hinton, and Margaret, mother of the mathematician J. I. Taylor), and the fifth is Ethel Lilian Voynich - became famous as a writer.
Buhl died at the fiftieth year of life from pneumonia.
Scientific activity
Boole was known to the public mainly as the author of a number of hard-to-understand papers on mathematical topics and three or four monographs that have become classics.
The publication of the first article (Theory of Mathematical Transformations, 1839) led to a friendship between Boole and Duncan F. Gregory (editor of the Cambridge Mathematical Journal, where the article was published), which continued until the latter's death in 1844. Boole submitted twenty-two papers to this journal and its successor, the Cambridge and Dublin Journal of Mathematics.
Sixteen of his articles were published in the Philosophical Magazine, six memoirs in the Philosophical Transactions (Philosophical Magazine). Philosophical Transactions), a number of others in the Proceedings of the Royal Society of Edinburgh and the Royal Irish Academy ( Transactions of the Royal Society of Edinburgh and of the Royal Irish Academy), in the Bulletin of the St. Petersburg Academy ( Bulletin de l'Académie de St-Petersbourg, under the pseudonym G. Boldt, Vol. IV. pp. 198-215) and in Krell's journal ( Journal fur die reine und angewandte Mathematik).
This list is supplemented by an 1848 publication in the Mechanic's Journal ( Mechanic's Magazine) on the mathematical foundations of logic.
mathematical logic
Boole was probably the first mathematician after John Wallis to turn to logical problems. The ideas of applying the symbolic method to logic were first expressed by him in the article "Mathematical Analysis of Logic" (1847). Not satisfied with the results obtained in it, Boole expressed the wish that his views be judged by the extensive treatise "Investigation of the laws of thought on which the mathematical theories of logic and probability are based" (1854). Boole did not consider logic a branch of mathematics, but found a deep analogy between the symbolic method of algebra and the symbolic method of representing logical forms and syllogisms. Boole denoted the universe of conceivable objects with the unit, with alphabetic symbols - selections from it, associated with ordinary adjectives and nouns (for example, if x = "horned" and y = "sheep", the successive selection of x and y from the unit will give the class of horned sheep). Boole showed that this kind of symbolism obeys the same laws as algebraic, from which it followed that they can be added, subtracted, multiplied and even divided. In such symbolism, statements can be reduced to the form of equations, and the conclusion from the two premises of the syllogism can be obtained by eliminating the middle term according to the usual algebraic rules. Even more original and remarkable was the part of his system presented in the "Laws of Thought ...", which forms a general symbolic method of logical inference. Boole showed how, from any number of statements, including any number of terms, to deduce any conclusion that follows from these statements, by purely symbolic manipulation. The second part of the "Laws of Thought ..." contains a similar attempt to discover a general method in the calculus of probabilities, which allows, from the given probabilities of a set of events, to determine the probability of any other event logically connected with them.
Mathematical analysis
During his life, Boole created two systematic treatises on mathematical topics: A Treatise on Differential Equations (1859; the second edition was not completed, materials for it were published posthumously in 1865) and Treatise on Finite Differences, conceived as its continuation (1860). These works made an important contribution to the relevant branches of mathematics and at the same time demonstrated Boole's deep understanding of the philosophy of his subject.
Other writings
Although Buhl published little except for mathematical and logical works, his writings reveal a wide and deep familiarity with literature. His favorite poet was Dante, and he liked Paradise more than Hell.
Boole's constant subjects of study were the metaphysics of Aristotle, the ethics of Spinoza, the philosophical works of Cicero, and many similar works. Reflections on scientific, philosophical and religious questions are contained in four speeches - "The Genius of Sir Isaac Newton", "Worthy Use of Leisure", "The Claims of Science" and "The Social Aspect of Intellectual Culture" - delivered and published by him at different times.
Major works
- "Mathematical Analysis of Logic" (The Mathematical Analysis of Logic, 1847);
- "Logical Calculus" (1848);
- An investigation of the laws of thought (1854).
Memory
- A crater on the Moon was named after George Boole in 1964.
- The Boolean variable type in programming was named after him.
- In 2015, the Irish National University Cork is celebrating the 200th anniversary of the birth of George Boole.
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Today, exactly 200 years ago, November 2, 1815, was born George Boole - English mathematician and logician, professor of mathematics at King's College Cork, one of the founders of mathematical logic.
George's ancestors were yeomen, i.e. farmers who owned a plot of land with an annual income of 40 shillings and therefore had the right to sit in a jury, and in addition to enjoy other rights, as well as small artisans who settled in the east of England, in the city of Lincoln and its environs. Since at least the 16th century, the surname Boole (which is an old spelling of "Bull" - bull) first appears in records in areas southwest of Skegness; a little later in the Newark area they appear as constables in Conton. A branch of George's family has lived northwest of Lincoln at Broxholme since at least the middle of the 17th century. George's father, John Bull, kept a shoe shop. However, shoemaking, which served as a source of subsistence for a family of four children (George was born in 1815, Mary in 1818, William in 1819 and Charles in 1821), he paid much less attention than his main passion for mathematics and logic, as well as the manufacture of various optical devices. The inhabitants of Lincoln knew John Bull well, of course: he not only zealously campaigned for the early wearing of glasses, but often, having completed work on the next telescope, it is worth noting that it was excellent at that time, hung out an announcement in the window of his shop: "Anyone who wishes to observe the creatures of our Lord with a feeling of reverence, I invite you to come and look at them through my telescope." The father of the future scientist was kind, deeply religious and - as they would say today - a social activist. Sacredly believing that vocation and work for the sake of daily bread are different things, he took an active part in the creation of a unique public organization for its time - the Institute of Mechanics, where any city dweller could spend his leisure time doing what he loved. Incredibly, the owners of the city shops and workshops, under the influence of John Bull's agitation, began to close them early in order to enable their employees and workers to attend "interest circles" in this Institute. John's household did not have a very clear idea of the profession of the head of the family. “It seems that he could do everything well,” George’s wife later wrote about her father-in-law, “with the exception of his own business, to be managed in the workshop.” The mother of George Boole, when asked what the father of her famous son did, briefly answered: "He was a philosopher."
Bull Jr. adored his father and from childhood helped him to grind lenses and do other simple mechanical work. The boy received education according to the family's income: he graduated from the local primary school (having learned to write and count). In September 1828 George Bull began attending the Bainbridge Commercial Academy. Of course, education at the Academy at that time no longer met the needs of a talented young man, but his parents could not provide better. The same subjects that were not included in the school curriculum, George studied on his own. Soon the young man decided to put an end to his future stay at the educational institution, since commerce did not seduce the young man. At the same time, he had a strong desire to become a widely educated person. John Bull, who knew only what was necessary in mathematics to calculate lenses and other optics, gave his son his first lessons in geometry and trigonometry, but he failed to discover early his outstanding talents in the exact sciences, and classical authors became his first passion. Of course, no Latin or Greek was taught at the school Buhl attended. Fortunately, the sociable John had many friends in Lincoln, and one of them, the bookseller William Brooke, taught the boy Latin grammar and allowed him to use the book wealth of his shop. Books on history, geography, religious writings, classical and modern fiction, poetry - that's what made up the circle of his reading. Brook had only to be surprised at the young man's industriousness, who did not let the books gather dust on his shelves. He had an almost photographic memory. “My brain is arranged in such a way,” he later wrote, “that any facts or ideas about which I learned were imprinted in it like a well-ordered group of drawings.” An inquisitive young man independently studies ancient Greek, and later German and French and Italian, from books that he borrowed from his friend. At the age of 12 he managed to translate Horace's ode into English. Understanding nothing as a translation technique, Bull's proud father still published it in the local newspaper. Some experts stated that a 12-year-old boy could not make such a translation, others noted serious technical defects in the translation. Deciding to improve his knowledge of Latin and ancient Greek, Buhl spent the next two years in earnest study of these languages, again without any help. Although this knowledge was not enough to transform him into a true gentleman (despite the fact that the Industrial Revolution in England had already taken place, knowledge of ancient languages was an indicator of the level of a gentleman's education), such hard work disciplined him and contributed to the classical style of the maturing Boolean prose. At the age of 14, he translated Meleager's "Ode to Spring" from the ancient Greek, and his father sent the translation to the local newspaper, indicating the age of the translator. The publication of this literary work of George provoked a sharp reaction from a certain teacher, who sent an angry letter to the newspaper, arguing that at such a young age it was impossible to make such a competent translation and the editors were engaged in fraud. There is a blessing in disguise: thanks to this letter, the people of Lincoln learned that an unusually talented young man lives among them.
Self-education went on as usual, but you can’t help your father, who has practically gone bankrupt, to feed his family with talent alone. And as soon as George was 16 years old, he begins to work as a junior teacher (assistant (assistant) teacher) of Latin and mathematics at a Methodist boarding school for boys in Doncaster, Yorkshire, combining the duties of a laboratory assistant and a gatekeeper (one way or another, he continued teaching at different positions throughout life). On cold long nights, when the children fell asleep, he educated himself and thought about the future. How to get out of the cycle of poverty? What place can he take in society? The path to the army was closed for him - money was needed to buy an officer's patent, studying at the university cost a lot, and dragging out a miserable existence of a school teacher under some ignorant and spiteful "Squeers" was not for him. Therefore, George thought about becoming a clergyman (Buhl was deeply religious) and continued to improve in ancient languages, read the classics, studied patristics (the works of the church fathers). But then he became interested in mathematics and soon abandoned the idea of becoming a priest. Wasting no time, the seventeen-year-old laboratory assistant began the systematic study of mathematics, but progressed slowly in this area of \u200b\u200bknowledge due to the lack of effective help, although he was helped (in addition to his father) by his friend D.S. Dixon, who received a mathematical degree from Oxford. According to Mrs. Buhl, her husband told her later that he began to read mathematical books because they were much cheaper than books on classical philology.
Two years later, in 1833, however, he left Doncaster. This happened when the principal of the school learned that the junior teacher belongs to a Unitarian church, does math on Sundays, and even solves math problems in church (what a sin!). George had to look for another job, although some of the students loved him very much and "prayed for his conversion." However, there was another reason for the departure of the young teacher. As one of his colleagues recalled, “it consisted in the fact that Buhl was completely absorbed in his own thoughts, and was “absent” to such an extent that the boys began to cheat. He was an excellent teacher if he saw that the child understood him (he there were two such students)... But for most of the children who did not show zeal in learning and needed continuous coaching, he was the worst teacher I have ever met. and the boy was just waiting for this to leave the lesson.The students slipped him work that was done by others, or showed him the same task several times, and if they said that everything was done correctly for them, he willingly believed this and again delved into his books ... In everything else, he was highly valued, as highly as possible.
George found a job in Liverpool, in an educational institution of a certain Marro. However, after 6 months, unable to withstand, by his own admission, "the chaos that was happening there", he moved to his native city and founded a small boarding school. At this time, George was only 19 years old! The range of Boole's scientific interests at that time was quite wide: he was almost equally interested in mathematics and logic, the ethics of Spinoza, the philosophical works of Aristotle and Cicero. But gradually Boole tends more and more to the problems of applying mathematical methods to the humanities (logic was considered one of these areas at that time). Buhl carefully studies Newton's "Philosophiae Naturalis Principia" and Lagrange's "Mechanics", comparing, along the way, the methods of both scientists. Imagine the difficulties of a young man who is only familiar with the beginnings of mathematics and who is trying to understand statements that are often cited without proof, preceded by the sacramental: "it is easy to see that ..." (especially since he studied the books of the great French in the original). He was amazed at Lagrange's ability to reduce the solution of physical problems to purely mathematical problems. Already here, Boole, apparently, is thinking deeply about the possibility of abstracting from the physical facts and facts of ordinary spoken language and passing to some system of effectively constructed symbols that would have a certain independence and with which one could work according to their inherent laws. Evidence that George did not just leaf through these books, but tried to delve deeply into their content, is his scientific essay "On the genius and discoveries of Sir Isaac Newton" (1835), in which he compared the methodology of Newton and Lagrange: "By the works of Lagrange the question of the motion of perturbed planets, with all its complexity and variety, is reduced to a purely mathematical problem. This eliminates the physical side of the problem; the perturbed and perturbed planets disappear; the ideas of time and force are put an end to; the very elements of the orbits are no longer taken into account, and only exist as variables quantities in mathematical formulas In Newton's investigations, this fortunate transformation does not take place... Disturbing forces are analyzed, their influence is considered for various positions [of the planet] - above and below the elliptical plane and when coinciding with it... The eternal wheels of the Universe rotate before us, and their movements can be traced through a changing variety of causes, conditions, and effects." According to the historian of mathematical logic, this comparison indicates that Buhl was already “thinking about the possibility of abstracting from physical facts. .. and the transition to a certain system of effectively constructed symbols that would have a certain independence and with which one could work according to their inherent laws.
But the school gave too modest income, and in fact the young man was in fact the breadwinner of the family. And in 1838, George Bull readily accepted the offer to head, after the death of the founder and director Robert Hall, the Academy for the children of wealthy farmers in Waddington, a small town near Lincoln, where George moved with his parents, two brothers and sister. The family began to jointly manage the affairs of the school, which helped to solve financial problems. But the young scientist by this time already had his own ideas about what education should be. Even during the existence of his first Lincoln school, he wrote an essay in which he talked about this. Buhl insisted on the need to understand, and not memorize, the material first of all - the idea at that time was not so common. In addition, he argued that in education it is necessary to pay great attention to the formation of moral and ethical values, and he considered this aspect of the teacher's work to be the most difficult, but at the same time the most important. Therefore, as the financial situation of the family improved, George more and more often returned to the idea of creating his own academy.
The publication of the first article (Theory of Mathematical Transformations, 1839) led to a friendship between Boole and Duncan F. Gregory, a young Cambridge algebraist who belonged to a famous Scottish family (who gave the world James Gregory (1638-1675), who invented the refractive telescope and proved the convergence series for the number π, and David Gregory (1659-1708) - mathematician, optician, astronomer, friend of Newton), who headed the newly organized "Cambridge Mathematical Journal", where the article was published. Encouraged by the support, George published papers in the same journal for several years on operator methods of analysis, the theory of differential equations, and algebraic invariants (1841). Perhaps this is the most remarkable achievement of the young Boole: had it not been for the theory of invariants, subsequently developed by Arthur Cayley and James Sylvester, perhaps Albert Einstein's theory of relativity would not have taken place. The creative union continued until Gregory's death in 1844. Buhl submitted 22 papers to this journal and the Cambridge and Dublin Journal of Mathematics, which succeeded it.
In 1840, having saved enough money, Buhl returned at his own risk to Lincoln, where he opened a boarding school. Soon the family joined George and they started working together again. Fortunately, from a commercial point of view, the idea turned out to be successful, and the Bouli did not experience financial problems anymore. It should be noted that having achieved financial independence and position in society, George spent a lot of money and time on charitable activities. He, in particular, became an active member of the Committee that organized the House of Penitent Women. The task of this organization was to help young girls forced into prostitution. In this regard, Lincoln was an extremely unfavorable place, there were about 30 brothels here. Even the mayor of the city admitted that there is no other city in England like this. George also supported the Crafts Institute, read many lectures there, achieved the establishment of a scientific library at the institute. During the day, he taught little boys, and devoted his spare time to reading and ... writing poetry and poems, classical in form, metaphysical and religious in content, such as, for example, "Sonnet No. 3":
Original | Translation |
| When the great Maker, on creation bent Thee from thy brethren chose and framed by thee The world to sense revealed, yet left it free, To those whose intellectual gaze beyond the veil phenomenal is sent, Space diverse systems manifold to see Revealed by thought alone; was it that we In whose mysterious spirits thus are blent Finite of sense and infinite of thought, Should feel how vast how little us our store - As you excelling arch with orbs deep fraught To the light wave that dies along the shore Till from our weakness and our strength may rise One worship unto Him the only wise? | When the great Creator, bowing over his creation, Chose you among your brethren and clothed you, revealing to the world, in a unique form, but leaving accessible for those whose thoughtful gaze seeks to penetrate the curtain of life, to see all the diversity of the universe, subject to mere thought, is it possible, so that we, in whose mysterious soul are united finiteness of feelings and infinity of thought, felt how huge and how small what we own, when, filled with dangers, we rush along a unique arc along with the heavenly bodies to waves of light dying on the shore until faith arises from our weakness and our strength in Him, the only wise? |
As time went on, Boole became more and more interested in mathematics. Pedagogical and organizational activities took a lot of time, only nights were left for independent study of mathematics. But even this genius of Boole was enough to soon declare himself as a serious mathematician. While still at Waddington, George became interested in the work of Laplace and Lagrange. In the margins of their books, he made notes, which later formed the basis of his first research. From 1839, the young scientist began to submit his work to the new "Cambridge Mathematical Journal". His articles were devoted to various questions of mathematics and were distinguished by independent judgments. Gradually, English mathematicians began to pay attention to the self-taught Lincoln. One of the first to appreciate him was the editor of the magazine, Duncan Gregory, who quickly realized that he was dealing with a brilliant scientist. In the future, Gregory corresponded a lot with Buhl and helped him with advice.
But the scientific aspirations of George Boole were not fully satisfied on this. He felt the lack of systematic education and the scientific sphere of communication. At one time, George thought about getting a mathematical degree from Cambridge, but the need to financially support his family forced him to abandon this idea. In addition, Gregory wrote to Boole that in this case he would have to leave his own original research, and they were already beginning to bring glory to the author. In 1842, George sent the eminent mathematician Augustus de Morgan a paper "On a general method of analysis, applying algebraic methods to the solution of differential equations." Morgan got this article published in the proceedings of the Royal Society, and it was awarded the Society's medal for her contribution to the development of mathematical analysis.
Boole enters into correspondence with mathematicians from Cambridge, who note the originality of their correspondent's mathematical ideas and advise him not to keep them under wraps. Heeding the insistence of his new friends, Boole in 1844 was awarded the highest honor for an English mathematician: the Royal Society of London awarded him a gold medal for the article "The General Method of Analysis". In the final paragraph of this work, Boole, as it were, outlines the direction of his future research: “The proposition, the justification of which interests me most, is that any significant progress in higher analysis is unthinkable without increased attention to the laws of combination of symbols. The significance of this proposition can hardly be overestimate, and I only regret that, due to the lack of books, and also due to circumstances unfavorable for the study of mathematics, I cannot give a perfect proof of its validity ... "
To carry out the planned Buhl in the mid-40s. begins to intensively deal with the problems of logic and creates a new calculus: he introduces certain symbols, operations and laws that determine these operations. If Leibniz at one time tried to arithmetize logic, then Buhl algebraizes it, turning it into a mathematical science. In principle, his ideas lay in line with the attempts of English algebraists to create symbolic algebra, i.e. "the science of symbols and their combinations, constructed according to their own rules, which can be applied to arithmetic or to other sciences through interpretation" (D. Peacock ). Rough sketches of the Boolean calculus, which laid the foundation for modern mathematical logic, date back to the summer of 1846.
One of the scientist’s friends recalled: “I remember well the day when Boole wrote the first pages of his first work on logic. This happened during his visit to me in Gainsborough. wandered through them and admired the beautiful scenery, and then he wished to retire.He sat in the shade of a huge bush and remained there until I disturbed him, saying that it was time to return.In the night he read to me what he had written and explained the system, exposition which he published the following year.
The publication referred to in the previous paragraph was a thin book called The mathematical analysis of logic, being an essay towards a calculus of deductive reasoning. In the preface, the author wrote: "Those who are familiar with the current state of symbolic algebra are aware that the validity of the processes of analysis does not depend on the interpretation of the symbols used, but only on the laws of their combination. Each interpretation that preserves the proposed relations is equally admissible, and such a process of analysis may thus, under one interpretation, represent the solution of a question connected with the properties of numbers, under another, the solution of a geometrical problem, and under a third, the solution of a problem of dynamics or optics ... ". Boole's innovation consisted in a clear awareness of the abstract nature of the calculus he created, determined only by those laws that operations are subject to.
Although the "Mathematical Analysis of Logic ..." was essentially a short synopsis of Boole's ideas, it attracted the attention not only of his Cambridge friends, but also of many other famous scientists, including Augustus de Morgan (1806-1871). I have already mentioned him more than once as Lady Lovelace's teacher and admirer of her talent. Now it is worth paying more attention to him, since de Morgan the logician, according to the historian, "paved the way for Boole" and later became an ardent supporter of his ideas.
Boole's studies in logic were largely stimulated by the discussion between A. De Morgan and W. Hamilton, which he followed with interest in the spring of 1847. Boole himself notes this circumstance in the preface to the Mathematical Analysis of Logic, written in October 1847. He also recognizes that A. De Morgan was the first logician who turned to the analysis of quantified sentences. De Morgan enthusiastically welcomed Boole's attempt to apply algebraic methods to solve problems in logic. "I believe," he wrote, "that it was Mr. Boole who established the true connection between algebra and logic." And further: "Boole's system of logic is one of many testimonies of the combined efforts of genius and patience .... Operations on algebraic symbols, invented as a means of numerical calculations, are sufficient to express any movements of thought and provide a grammar and vocabulary of a complete logical system ... When Hobbes Republican times (Commonwealth) published the book "Computation or Logic", he had a vague idea of \u200b\u200bsome of the issues that were illuminated in the days of Mr. Boole. However, the unity of forms of thinking in all the various manifestations of the mind was not achieved and became a subject that aroused general interest The name of Mr. Boole will always be remembered in connection with the fact that he made the most significant steps in this direction.
Along with logical and mathematical studies, Boole continued to compose poetic works, classical in form and philosophical in content. He owns two poems (“Sonnet to the Number Three” and “The Call of the Dead Man”. A letter to Brook dated 1845 was also found in his manuscripts. This letter describes his visit to a meeting of the British Scientific Association, as well as a holiday on the Isle of Wight And in 1847 and 1848, the works "Mathematical Analysis of Logic" and "Logical Calculus" were written, which literally elevated Boole to the top of the scientific Olympus. Interestingly, the first of these works was something like a pamphlet in which the author tried to prove that logic is closer to mathematics than to philosophy. Boole himself later regarded it as a hasty and imperfect demonstration of his ideas. But colleagues, especially Morgan, appreciated the Mathematical Analysis of Logic very highly. In any case, in these works, as well as in written later (in 1854) "Investigation of the laws of thought, based on mathematical logic and the theory of probability" Boole laid the foundations of the so-called "algebra of logic" or "Boolean algebra". showed the analogy between logical and algebraic operations. In other words, the scientist was based on the fact that mathematical operations can be performed not only on numbers. He came up with a system of notation, using which, you can encode any statements. Boole further introduced rules for manipulating propositions as if they were ordinary numbers. Manipulations were reduced to three main operations: And, OR, NOT. With their help, you can perform basic mathematical operations: addition, subtraction, multiplication, division and comparison of symbols and numbers. Thus, the English scientist outlined in detail the basics of the binary number system. I must say that the ideas of George Boole underlie all modern digital devices.
In 1849, Cambridge mathematician friends arranged for Boole a mathematical professorship at the newly opened Queens College (now University College Cork) in Cork (Ireland). The applicant was confirmed in the position despite the fact that he did not have a university education or degree, where he taught for the rest of his life.
Buhl fell in love with wandering around Cork, getting to know and talking with local peasants. He told how one day, caught in the pouring rain, he asked for shelter in a poor house that stood on the edge of a peat bog. Noticing that all the inhabitants of the house were walking barefoot, he took off his shoes and stockings and placed them to dry by the fire. “This exposure (denuding) of the legs,” Buhl recalled, “seems to have contributed to the establishment of friendly relations and aroused general sympathy for me. Children who had previously been shy in front of a stranger joined our circle, followed by a dog; us and slipped his snout between my legs closer to the fire (having received a reprimand from the hostess for this), and, finally, hens and other poultry completed the circle of participants in this secular reception with their presence. One should not look for ridicule or contempt for the "orphans of this world" in these words - having climbed several steps up the social ladder, he remained alien to the social prejudices that were so widespread then in Great Britain. I will cite as confirmation the story of one elderly lady, transmitted by the youngest daughter of a scientist: “On one June day, 1856, she [lady - Yu. Polunov.] went to a slum alley behind the college to hire a chimney sweep to clean the chimney in her house. In the alley she saw her father walking in front of her, who knocked on every door of the houses.As she passed him, she noticed him ardently shaking the hands of a barefoot ragamuffin, saying: "I have come to tell you, dear friend: "I have a child and it's so beautiful!"
The appearance of Bul as a teacher is drawn to us by R. Rice. He cites the recollections of Boole's student R. A. Jemison, who went to teach in Shanghai. Jamison writes that Boole often sought to ensure that his listeners themselves could rediscover some of the results already obtained by other scientists (rather than expounding them all in his lectures). "He taught us, Jamison continues to recall, to feel the 'joy of discovery'." To these remarks by Jemison and Rees, one can only add that, apparently, Boole did not lose hope that someday his students would make an undiscovered discovery.
And here are the memories of other students of Boole.
"The secret of his success, I think, was that he never seemed to repeat or reproduce what he had once learned himself, and always strove to give the impression that he was getting results during a lecture, and that students participate in it with him, and share with him the honor of discovery."
"We never felt that we were in the presence of a person who is a mathematician - rather, in the presence of a person who, like us, is a student who comprehends mathematical truths. He descended to the level of our knowledge, and we moved on at the same time. Although we knew that he was expounding ideas known to him, it seemed that he did not use a pre-prepared and verified set of phrases or problems.The lecture was actually delivered in such a way that it seemed that at that very moment some original ideas visited him.Sometimes, developing them, he seemed to completely forget about our presence ... ".
"He carefully prepared a large list of questions and problems, starting with the basics and ending with the highest sections of mathematics, which he printed from time to time and distributed to students. He liked to repeat that until these examples were solved, one cannot speak of great progress in the study of the subject, and what was learned in lectures will soon be forgotten."
"For lovers of algebraic analysis, it was a real pleasure to watch how some of the fundamental mathematical principles became clear after he covered one board after another with his formulas. Each time he reached a point important for obtaining the final result, his face lit up with a joyful smile of satisfaction, and when he hopefully asked the audience the question: “Can you continue on your own?”, he usually received a positive answer. But if he heard: “We did not understand this or that point,” he never became annoyed, calmly explained again and again, using other means or drawings, or resorting to the help of those who already understood the problem ... ".
The following episode testifies to how students respected and loved their professor. Once he came to the audience long before the start of the lecture, and, turning his face to the blackboard, went deep into thought. The audience was gradually filled with students who were very quiet so as not to disturb the professor. Time passed, and Buhl continued to stand with his back to the students. The lecture hour was over, and the students, as quietly as they entered and sat down, left the audience. When Buhl came home, he said to his wife: "My dear, an extraordinary thing happened today - none of my students came to the lecture."
Around the same time, there were changes in the personal life of George Boole. In 1850, he met Mary Everest, the daughter of Thomas Everest, a professor of Greek at Queen's College, and the niece of the former Governor General of India, the famous surveyor geographer George Everest (the highest peak in the Himalayas, which he first measured, is named after him). In the summer of 1852, Mary again visited Cork, and then Boole visited her family. Despite the big difference in age (17 years), friendships began between Mary and George. They corresponded a lot. At meetings, Bull also gave his young friend lessons in mathematics - it was very difficult for a representative of the weaker sex to get a systematic education in those days. George hid his feelings for Mary for a long time and only in 1855 decided to propose. This happened after the girl's father died, and she was left practically without a livelihood. The marriage was happy. Mary Everest, during her lifetime, became George's muse, believing that her main purpose in life was raising children and creating conditions for the scientific creativity of the great mathematician, which she (rightly) considered her husband, and after his death, having written several essays, in the last of which, Philosophy and Entertainment of Algebra (1909), promoted George's mathematical ideas, popularizing his contributions to logic. True, care for him sometimes took despotic forms. Being engaged in mathematical research, the scientist did not forget about humanitarian subjects. He was interested in linguistics and logic, philosophy, ethics and poetry. This too wide range of interests of the professor of mathematics, his wife, who was distinguished by a strong character, apparently did not approve. One day, seeing that George was engaged in "the painful process of versification," she selected the sheets with the outlines of the sonnet and threw them into the fire of the fireplace, saying that it was not proper for him to use his precious time in this way. Not wanting to quarrel with his wife, Bul decided to urgently end his poetic “career”, believing that the final decision in this matter should belong to his wife, since she knows better. Contemporaries note Boole's democratic habits, his lack of any respect for the social prejudices and barriers established in Britain, point to his principled character and developed sense of humor.
Of his five daughters, three became outstanding personalities. The eldest, Lucy, became the first woman in England to receive the title of professor of chemistry. The third, Alicia, like her father, without having received a special mathematical education, received a number of interesting results in geometry. In particular, she built from cardboard, in a purely Euclidean way, using only compasses and a straightedge, three-dimensional sections of all six regular four-dimensional figures. Her results were only partially published (she photographed some of her models and sent them with explanations to Professor Schout in Groningen; Schout published them along with his article). Like her father, Alice had a highly developed sense of dignity and duty. Unfortunately, she gradually limited her circle of interests to her family (husband-actor Walter Scott and two children), ceasing to engage in scientific work. But the most famous was the youngest daughter - Ethel Lilian, married Voynich, the author of a number of novels, including the popular novel about the liberation struggle of the Italian Carbonari - "The Gadfly". It was followed by several more novels and musical works, as well as an English translation of Taras Shevchenko's poems. Two more daughters are also somehow connected with mathematics. Second, Margaret is the mother of mathematician and physicist Jeffrey Ingram Taylor, a specialist in hydrodynamics and wave theory, a foreign member of the USSR Academy of Sciences. His knowledge came in handy at Los Alamos, where Taylor was sent along with the British delegation of the Manhattan Project of 1944-1945. Fourth, Mary, wife of mathematician, inventor and science fiction writer C.G. Hinton - the author of the well-known story "The Incident in Flatland", which describes some creatures living in a flat two-dimensional world. Of the Hintons' numerous offspring, Joan deserves special attention, she was one of the few female physicists who took part in the work on the atomic project in the United States.
After the publication of An Inquiry into the Laws of Thought, George Boole received honorary degrees from Dublin and Oxford Universities, and in 1857 was elected a Fellow of the Royal Society of London. Subsequently, he published two more important works: Treatise on Differential Equations (1859) and Treatise on the Calculation of Limit Differences (1860), which played an important role in the development of mathematics. In 1861, George Bull was awarded a knighthood.
The death of George Boole was very unexpected. He was full of strength, energy, worked hard, planned to do even more. Only a few lung problems that appeared after moving to Cork, a city with a wetter climate than Lincoln, inspired fear. On November 24, 1864, a seemingly ordinary event happened, which ultimately led to tragic consequences. In the pouring rain, Buhl walked the two miles that separated his house from the college, and although soaked to the skin, the conscientious professor did not cancel the lectures, but spent them in wet clothes, which caused him to catch a bad cold. Soon the cold turned into pneumonia. They say that to care for her husband, Maria Everest used homeopathy, fashionable at that time, which claims that the disease can be cured with the help of the remedy that caused this disease, i.e. "fight fire with fire". As a result, George Bull is wrapped in wet sheets. Therefore, it is not strange that it was not possible to defeat the disease, and on December 8, George Boole died ... 10 years after his main logical work “The Laws of Thought” was published. The manuscripts that remained after him testified to his intentions to continue the development of the logical theory. Beginning in 1854, Boole concentrated his efforts on the application of the calculus he developed to the theory of probability and did not publish papers directly related to logic. However, Boole's work in the field of mathematics was always only a help and was stimulated by his reflections on logic, even when he began to come (in the last period of his creative activity) to the idea that logic is independent of mathematics and should form its basis. Boole began his mathematical research with the development of operator methods of analysis and the theory of differential equations, then took up mathematical logic. In Boole's main works, "the mathematical analysis of logic, which is an experiment in the calculus of deductive reasoning" and "the study of the laws of thought, in which the mathematical theories of logic and probability are based," the foundations of mathematical logic were laid. Boole's mathematical work is characterized by the close attention he devotes to the so-called "symbolic method". The English logician believed that mathematical operations (including such as differentiation and integration) should, first of all, be studied from the point of view of their inherent formal properties, which makes it possible to transform expressions that include these operations, regardless of the internal content such expressions. Boole was known to the public mainly as the author of a number of hard-to-understand papers on mathematical topics and three or four monographs that have become classics. In total, Boole published about fifty articles in various publications and several monographs. Boole's texts have now been collected in two books. Regarding the content of one of them, the German logician G. Scholz notes: “This book combines seventeen lectures: twelve on the theory of probability, a philosophical preface under the heading: “Requirements for science, specially based on its relation to human nature” and four lectures containing idea of logical calculus. I am not in a position to single out the probabilistic lectures for consideration. Boole's ideas in this area seem so unfinished that the question involuntarily arises as to what motivated their reprint in general. However, this bewilderment dissipates as soon as we turn to the consideration of Boole's logical calculus, which he uses as an auxiliary tool for solving probabilistic problems... Among the lectures directly related to the idea of logical calculus, the most significant is the first: "Mathematical Analysis of Logic" ... In another of these books, Boole's manuscripts, which were unpublished during his lifetime, are also collected, which are of significant historical and logical interest. For example, one manuscript contains an anticipation of pure propositional calculus (before Hugh McCall). Boole deals with the philosophical aspects of logic in another manuscript dating back to 1855 or 1856.
mathematical logic
Boole was probably the first mathematician after John Wallis to turn to logical problems. The ideas of applying the symbolic method to logic were first expressed by him in the article "Mathematical Analysis of Logic" (1847). Not satisfied with the results obtained in it, Boole expressed the wish that his views be judged by the extensive treatise "Investigation of the laws of thought on which the mathematical theories of logic and probability are based" (1854). Boole did not consider logic a branch of mathematics, but found a deep analogy between the symbolic method of algebra and the symbolic method of representing logical forms and syllogisms. Boole denoted the universe of conceivable objects with the unit, with alphabetic symbols - selections from it, associated with ordinary adjectives and nouns (for example, if x = "horned" and y = "sheep", the successive selection of x and y from the unit will give the class of horned sheep). Boole showed that this kind of symbolism obeys the same laws as algebraic, from which it followed that they can be added, subtracted, multiplied and even divided. In such symbolism, statements can be reduced to the form of equations, and the conclusion from the two premises of the syllogism can be obtained by eliminating the middle term according to the usual algebraic rules. Even more original and remarkable was the part of his system presented in the "Laws of Thought ...", which forms a general symbolic method of logical inference. Boole showed how, from any number of statements, including any number of terms, to deduce any conclusion that follows from these statements, by purely symbolic manipulation. The second part of the "Laws of Thought ..." contains a similar attempt to discover a general method in the calculus of probabilities, which allows, from the given probabilities of a set of events, to determine the probability of any other event logically connected with them.
Mathematical analysis
During his life, Boole created two systematic treatises on mathematical topics: A Treatise on Differential Equations (1859; the second edition was not completed, materials for it were published posthumously in 1865) and Treatise on Finite Differences, conceived as its continuation (1860). These works made an important contribution to the relevant branches of mathematics and at the same time demonstrated Boole's deep understanding of the philosophy of his subject.
Other writings
Although Buhl published little except for mathematical and logical works, his writings reveal a wide and deep familiarity with literature. His favorite poet was Dante, and he liked Paradise more than Hell. Boole's constant subjects of study were the metaphysics of Aristotle, the ethics of Spinoza, the philosophical works of Cicero, and many similar works. Reflections on scientific, philosophical and religious questions are contained in four speeches - "The Genius of Sir Isaac Newton", "Worthy Use of Leisure", "The Claims of Science" and "The Social Aspect of Intellectual Culture" - delivered and published by him at different times.
Boole's logical ideas were further developed in subsequent years. Logical calculus, constructed in accordance with Boole's ideas, is now widely used in the applications of mathematical logic to technology, in particular to the theory of relay-contact circuits. In modern algebra there are Boolean rings, Boolean algebras are algebraic systems whose composition laws originate from Boole's calculus. In general topology, the Boolean space is known, in the mathematical problems of control systems - Boolean spread, Boolean expansion, Boolean regular point of the kernel. After some time, it became clear that Boole's system is well suited for describing electrical circuit switches. Current in a circuit can either flow or not, just as a statement can be either true or false. And a few decades later, already in the twentieth century, scientists combined the mathematical apparatus created by George Boole with the binary number system, thereby laying the foundation for the development of a digital electronic computer.
It is believed that one of the prototypes of Professor James Moriarty Sir Arthur Conan Doyle was George Bull. Moriarty's story is very similar to Boole's, from his time as a professor at a small university in the fringes to his role in mathematics. Moreover, Conan Doyle knew the scientist's wife, Mary
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