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George Boole - One of the founders of mathematical logic. Professor of Mathematics at King's College Cork (now University College Cork) from 1849.

Extending the general method of Leibniz, formulated 188 years earlier, in which all true causes were reduced to the form of calculations, D. Buhl in 1854 laid the foundation for what we know today as mathematical logic by publishing the work “Investigation of the Laws of Thought”.

In this work, published when he was 39, Boole reduced logic to an extremely simple type of algebra, the algebra of propositional logic, which was a system of symbols and rules applied to various objects (numbers, letters, sentences).

His theory of logic, based on three basic actions - AND (and), OR (or), NOT (not), - was to become the basis for the development of switching telephone lines and the computer project in the 20th century. Just like the ideas of Leibniz, Boolean algebra was neglected for many years after it was created.

The importance of the work, recognized by the logician de Morgan, a contemporary of Boole, was as follows: “The symbolic processes of algebra, designed as instruments of numerical calculation, competently express every law of thought and possess the grammar and vocabulary of all that contains a system of logic. We did not assume this until it was proven in the Laws of Thought.

An English mathematician and logician was born on November 2, 1815. He grew up in the family of a poor artisan John Bull, who was passionate about science. It is known that his father left school after three years of study, and at the same time it is surprising that Buhl received his early mathematical education from his father, who was self-taught in this field. Dad, 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.

At the age of 16, it became necessary for Buhl to start working life to help his parents. After getting a job as a "junior teacher", or assistant teacher at an elementary school, Buhl had to spend 4 years teaching at two different schools.

Always thinking about the perspective of his place in life, Buhl began to consider several paths open to him. His initial teaching was always on the level, but he did not consider it a profession, although it was an honorary one. Bull became a clergyman.

When not teaching, he spent his time in serious study of French, German, and Italian, in preparation for church life. Failures, the poverty of his family once again ruined Buhl's plans; his parents urged him to give up religious life because of their deteriorating financial situation.

Responsive, as always, to the advice of his parents, Buhl decided to open his own school. He was 20 years old. While teaching, Buhl considered himself also a student and began to study the full course of higher mathematics. He studied Newton's Principia Mathematica, Lagrange's Analytical Mechanics, the works of Laplace and other authors.

Boole began his mathematical research with the development of operator methods of analysis and the theory of differential equations, and then, like De Morgan, with whom he became friends by this time, he took up mathematical logic.

In his first major work "Mathematical analysis of logic, which is an experiment in the calculus of deductive reasoning" in 1847, Boole clearly showed the so-called quantitative interpretation of the objects of logic and the need for a new approach to solving problems of logic. This approach required a change and expansion of the symbolic language of algebra: the choice of symbols, operations and laws that define these operations and reflect the specifics of the objects of study, i.e., in essence, the creation of a new calculus. Buhl 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 valid, and such a process of analysis may thus, in one interpretation, represent a solution to a question related to the properties of numbers, in another, a solution to a geometric problem, and in a third, a solution to a problem of dynamics or statics. It is necessary to emphasize the fundamental nature of this principle.” With the publication of "Mathematical Analysis ..." the views and brilliant intuition of this quiet, simple man became clear to his friends - mathematicians, who advised him to go to Cambridge for a generally accepted mathematical education.

Buhl reluctantly rejected these proposals, because his relatives completely subsisted on his earnings. Without complaining about the peculiarities of his training from time to time, Boole finally got a little break in 1849, when he was appointed professor of mathematics at the newly opened King's College.

This appointment allowed him to devote more time to "The Laws of Thought ..." - his second major work, which he continuously honed and improved for another 5 years, until publication in 1854. As Boole wrote in the first paragraph of the book: “The purpose of this treatise:

explore the fundamental laws of those operations of the mind by which reasoning is carried out;
express them in the symbolic language of calculus and, on this basis, create the science of logic and construct a method;
to make this method directly the basis of a general method for expressing the theory of probability;
finally, get the various elements of truth;
assess some likely message within the framework of addressing these issues.”
And further: “Now, in fact, the studies of the following pages show logic, in a practical aspect, as a system of processes carried out with the help of symbols that have a certain interpretation and are subject to laws based on this single interpretation. But at the same time they show these laws as being identical in form to those of the general symbols of algebra, with one single addition, viz.”

In other words, in general algebra it does not hold, for example: that each x is identically equal to its square - but this is true in Boolean algebra. According to Boole, x2 = x for any x in his system. In a numerical system, this equation has a unique solution "O" and "1?". This is the importance of the binary system for modern computers, whose logical parts effectively implement binary operations.

Besides logic, Boolean algebra has two other important uses. Boolean algebra is used in natural algebra. Taking into account also the idea of ​​“the number of elements” in a set, Boolean algebra became the basis for the theory of probability.

Despite the great importance of Boolean algebra in many other areas of mathematics, Boole's extraordinary work was considered an oddity for many years. Like Babbage, Buhl was a man ahead of his time. This happened before Alfred Whitehead and Bertrand Russell published their three-volume Principia Mathematica (1910-1913), which dealt with formal logic.

It is also noteworthy that Boole's achievements partially relied on mathematical discoveries that had appeared in England by that time, including Babbage's ideas. Mathematicians drew attention to Babbage's idea of ​​mathematical operations and the quantities used in them. The idea was made possible by a group of British algebraists to which Buhl belonged.

Boole demonstrated,” that logic can be reduced to very simple algebraic systems, after which it became possible for Babbage and his followers to create mechanical devices that could solve the necessary logical problems.

A year after The Laws of Thought was published, Boole married Mary Everest, niece of a professor of Greek at King's College. The happy marriage lasted for nine years, until the untimely death of George Bull. On December 8, 1864, at the age of 49, revered and famous, he died of pneumonia.

Buhl was a consistent and disciplined person, however, he widely demonstrated his own vision of the world in his statements. This powerful combination of intelligence and intuition in George Boole was embodied in various mathematical ideas. In conclusion of the essay on the father of Boolean algebra, I would like to briefly talk about the Boole family.

As already mentioned, Boole's wife was the niece of George Everest, who in 1841 completed grandiose works in India in 1841.

In honor of his merits, the highest peak of the world, Chomolungma in the Himalayas, at one time was even called Everest. Mary herself, unlike the wives of many other mathematicians, understood the scientific ideas of her husband and, with her attention and participation, encouraged him to continue research. After his death, she wrote several essays and in the last of them - "Philosophy and entertainments of algebra", - published in 1909, promoted George's mathematical ideas.

The Buleys had five daughters. The eldest, Mary, married C. Hinton, a mathematician, inventor and science fiction writer, the author of the well-known story “The Incident in Flatland”, which describes some creatures living in a flat two-dimensional world. Of the numerous offspring of the Hintons, three grandchildren became scientists: Howard - an entomologist, and William and Joan - physicists. The latter was one of the few female physicists who took part in the work on the atomic project in the United States.

Booley's second daughter, Margaret, went down in history as the mother of Geoffrey Taylor, a prominent English mechanic and mathematician, a foreign member of the USSR Academy of Sciences. The third, Alicia, specialized in the study of high-dimensional spaces and received an honorary degree from the University of Groningen. The fourth, Lucy, became the first female professor in England to head a department of chemistry.

But the most famous of all the Buley daughters was the youngest, Ethel Lilian, who married a scientist - an emigrant from Poland, Voynich. Having entered the revolutionary emigrant environment, she wrote the novel Gadfly that glorified her throughout the world. It was followed by several more novels and musical works, as well as an English translation of Taras Shevchenko's poems. Voynich died in New York at the age of 95, short of the centenary of the death of her famous mathematician father George Boole.

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Boole's father, George Boole, was a merchant in London and it was he who gave his son his first mathematics lessons. He also taught his son how to make optical measuring instruments. Buhl was rather a self-taught mathematician, although both his father and school gave him some knowledge of mathematics. He had to work to help his family after his father's business went into decline.

Career

Buhl worked as a teacher's assistant in Doncaster and also briefly taught in Liverpool. For some time he was associated with the Lincoln Mechanics Institute, which opened in 1833. And in 1834 he opened his school in Lincoln.

During this time, he devoted much of his time to social work and adult education. He founded the Repentant Women's Shelter, whose purpose was to rehabilitate prostitutes. In order to educate the poor, Buhl also worked at the Institute of Mechanics. Buhl became the owner of Hall's Academy in Waddington, near Lincoln, four years later. In 1839 he submitted several papers, among which was The Theory of Mathematical Transformations for the Cambridge Journal of Mathematics.

These papers dealt with differential equations and the algebraic problem of linear transformation by highlighting the idea of ​​an invariant linear transformation through highlighting the idea of ​​invariance.

In 1840 he returned to Lincoln to direct the boarding school.

In 1841 he discovered invariant theory, a new branch of mathematics. This branch of mathematics was later the source of inspiration for Einstein.

In 1844 he analyzed the combined methods of algebra and calculus in a publication entitled Philosophical Transactions of the Royal Society.

In 1847, together with E. R. Larken, he founded a housing society. In the same year, in the pamphlet "Mathematical Analysis of Logic", he expressed the opinion that logic should be connected with mathematics.

Boole's innovative contribution to mathematics was truly effective in creating the digital computer and electronic circuits.

In 1849 he became the first professor of mathematics at King's College in Cork, Ireland.

In 1854, he studied algebra and logic, and his work in this area is better known as Boolean algebra (algebra of logic). In the same year, he introduced the concept of the symbolic method of inference in the publication "Laws of Thought".

Boolean algebra serves as the basis for the analysis of the validity of logical judgments, since it is binary in nature of statements that can turn out to be either positive or false.

The binary method and logic elements of Boolean logic are used in telephone switching and in electronic computers during their creation and operation.

In the second part of The Law of Thought, Boole tried to discover a general method in calculating probabilities.

In 1857 Boole presented the publication "On the Comparison of Transcendental Functions" with certain overlays on the theory of definite integrals. In the publication, he studies the sum of the remainders of a rational function. And part of the study was the proof of the Boolean identity.

In 1859 Boole publishes a "Treatise on Differential Equations" in which he reports on the general symbolic method; in 1860 he publishes a sequel with the title Treatise on the Calculus of Finite Differences.

Buhl made contributions to such sciences as: electronics, mathematics, information theory, logic, cybernetics and computer science.

Awards and achievements

The first gold medal of the Royal Society, 1844.
Member of the Royal Society in London, 1857.
Honorary Doctor of Laws from Dublin and Oxford Universities, 1857.

Personal life and legacy

George Bull married Mary Everest in 1854. The couple had five daughters. Buhl died in 1864 due to pneumonia.

Boolean algebra and the crater Boole on the Moon are named after George Boole.

In many programming languages, a "boolean type" is a boolean data type (where a value can either be true or not true).

The library, the underground lecture hall complex and the Boole Center for Research in Informatics at the Irish National University in Cork are named after George Boole.

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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 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 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 Boolean 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

Father of boolean algebra

Pure mathematics was discovered by Boole in a work he called The Laws of Thought.

Bertrand Russell

George Bull

All mechanisms, gears, vacuum tubes and printed circuit boards - all this is not yet a computer.

Also important are the developments of Pascal and Leibniz, which we have already told you about, and Babbage, whose achievements we will tell you in the next chapter. These developments required an initial theory of logic in order to eventually breathe life into machines that "think".

Extending the general method of Leibniz, formulated 188 years earlier, in which all true causes were reduced to the form of calculations, the English mathematician D. Buhl in 1854 laid the foundation for what we know today as mathematical logic by publishing the work “Investigation of the Laws of Thought”.

In this work, published when he was 39, Boole reduced logic to an extremely simple type of algebra, the algebra of propositional logic, which was a system of symbols and rules applied to various objects (numbers, letters, sentences).

His theory of logic, based on three basic actions - AND (and), OR (or), NOT (not), - was to become the basis for the development of switching telephone lines and the computer project in the 20th century. Just like the ideas of Leibniz, Boolean algebra was neglected for many years after it was created.

The importance of the work, recognized by the logician de Morgan, a contemporary of Boole, was as follows: “The symbolic processes of algebra, designed as instruments of numerical calculation, competently express every law of thought and possess the grammar and vocabulary of all that contains a system of logic. We did not assume this until it was proven in The Laws of Thought.

George Bull was born November 2, 1815 in Lincoln (England), the son of a poor shoemaker. Although he was a contemporary of C. Babbage, he did not come from a privileged class, like Babbage.

Coming from a stratum of society whose children were effectively deprived of attending the university, George had to study on his own.

Although the Industrial Revolution had already taken place in England, knowledge of ancient languages ​​was an indicator of a gentleman's level of education. Of course, no Latin or Greek was taught at the school Buhl attended. Buhl himself studied Greek and Latin, with the support of a poorly educated father, and 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 Greek, Buhl spent the next two years studying these languages ​​in earnest, again without any help.

Although this knowledge was not enough to turn into a true gentleman, such hard work disciplined him and contributed to the classical style of the maturing Boolean prose.

It is known that his father left school after three years of study, and at the same time it is surprising that Buhl received his early mathematical education from his father, who was self-taught in this field.

At the age of 16, it became necessary for Buhl to start working life to help his parents. Given a job as a "junior teacher", or primary school teacher's assistant, Buhl had to spend 4 years teaching at two different schools.

Always thinking about the perspective of his place in life, Buhl began to consider several paths open to him. His initial teaching was always on the level, but he did not consider it a profession, although it was an honorary one. Bull became a clergyman.

When not teaching, he spent his time in serious study of French, German, and Italian, in preparation for church life. Failures, the poverty of his family once again ruined Buhl's plans; his parents urged him to give up religious life because of their deteriorating financial situation.

Responsive, as always, to the advice of his parents, Buhl decided to open his own school. He was 20 years old. While teaching, Buhl considered himself also a student and began to study the full course of higher mathematics. He studied Newton's Principia Mathematica, Lagrange's Analytical Mechanics, the works of Laplace and other authors.

Boole began his mathematical research with the development of operator methods of analysis and the theory of differential equations, and then, like de Morgan, with whom he became friends by this time, he took up mathematical logic.

In his first major work, "The Mathematical Analysis of Logic, which is an Experiment in the Calculus of Deductive Reasoning" in 1847, Boole clearly showed the so-called quantitative interpretation of the objects of logic and the need for a new approach to solving the problems of logic.

This approach required a change and expansion of the symbolic language of algebra: the choice of symbols, operations and laws that define these operations and reflect the specifics of the objects of study, i.e., in essence, the creation of a new calculus. Buhl wrote: “Those who are familiar with the present 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 valid, and such a process of analysis may thus, in one interpretation, represent a solution to a question related to the properties of numbers, in another, a solution to a geometric problem, and in a third, a solution to a problem of dynamics or statics. It is necessary to emphasize the fundamental nature of this principle.”

With the publication of "Mathematical Analysis ..." the views and brilliant intuition of this quiet, simple man became clear to his friends - mathematicians, who advised him to go to Cambridge for a generally accepted mathematical education.

Buhl reluctantly rejected these proposals, because his relatives completely subsisted on his earnings. Without complaining about the peculiarities of his training from time to time, Boole finally got a little break in 1849, when he was appointed professor of mathematics at the newly opened King's College.

This appointment allowed him to devote more time to "The Laws of Thought ..." - his second major work, which he continuously honed and improved for another 5 years, until publication in 1854.

As Boole wrote in the first paragraph of the book: “The purpose of this treatise:

Investigate the fundamental laws of those operations of the mind by which reasoning is carried out;

Express them in the symbolic language of calculus and, on this basis, create a science of logic and construct a method;

To make this method directly the basis of a general method for expressing the theory of probability;

Finally, get the various elements of truth;

Evaluate some probable message within the framework of resolving these issues.

And further: “Now, in fact, the studies of the following pages show logic, in a practical aspect, as a system of processes carried out with the help of symbols that have a certain interpretation and are subject to laws based on this single interpretation. But at the same time they show these laws as being identical in form to those of the general symbols of algebra, with one single addition, viz."

In other words, in general algebra does not hold, for example: that each X is identically equal to its square - but this is true in Boolean algebra. According to Boole, x 2 = X for anyone X in his system. In a number system, this equation has a single solution "0" and "1". This is the importance of the binary system for modern computers, whose logical parts effectively implement binary operations.

Besides logic, Boolean algebra has two other important uses. Boolean algebra is used in natural algebra. Taking into account also the idea of ​​"the number of elements" in a set, Boolean algebra became the basis for the theory of probability.

Despite the great importance of Boolean algebra in many other areas of mathematics, Boole's extraordinary work was considered an oddity for many years. Like Babbage, Buhl was a man ahead of his time. This happened before Alfred Whitehead and Bertrand Russell published their three-volume Principia Mathematica (1910–1913), which dealt with formal logic.

It is also noteworthy that Boole's achievements partially relied on mathematical discoveries that had appeared in England by that time, including Babbage's ideas. Mathematicians drew attention to Babbage's idea of ​​mathematical operations and the quantities used in them. The idea was made possible by a group of British algebraists to which Buhl belonged.

Boole demonstrated that logic could be reduced to very simple algebraic systems, after which it became possible for Babbage and his followers to create mechanical devices that could solve the necessary logical problems.

A year after The Laws of Thought was published, Boole married Mary Everest, niece of a professor of Greek at King's College. The happy marriage lasted for nine years, until the untimely death of George Bull. On December 8, 1864, at the age of 49, revered and famous, he died of pneumonia.

Buhl was a consistent and disciplined person, however, he widely demonstrated his own vision of the world in his statements. This powerful combination of intelligence and intuition in George Boole was embodied in various mathematical ideas. In conclusion of the essay on the father of Boolean algebra, I would like to briefly talk about the Boole family.

As already mentioned, Boole's wife was the niece of George Everest, who in 1841 completed grandiose works in India in 1841.

In honor of his merits, the highest peak of the world, Chomolungma in the Himalayas, at one time was even called Everest. Mary herself, unlike the wives of many other mathematicians, understood the scientific ideas of her husband and, with her attention and participation, encouraged him to continue research. After his death, she wrote several essays, and in the last of them - "Philosophy and entertainments of algebra", - published in 1909, she promoted George's mathematical ideas.

The Buleys had five daughters. The eldest, Mary, married C. Hinton - a mathematician, inventor and science fiction writer - the author of the well-known story "The Incident in Flatland", which describes some creatures living in a flat two-dimensional world. Of the numerous offspring of the Hintons, three grandchildren became scientists: Howard - an entomologist, and William and Joan - physicists. The latter was one of the few female physicists who took part in the work on the atomic project in the United States.

Booley's second daughter, Margaret, went down in history as the mother of Geoffrey Taylor, a prominent English mechanic and mathematician, a foreign member of the USSR Academy of Sciences. The third, Alicia, specialized in the study of high-dimensional spaces and received an honorary degree from the University of Groningen. The fourth, Lucy, became the first female professor in England to head a department of chemistry.

But the most famous of all the Buley daughters was the youngest, Ethel Lilian, who married a scientist - an emigrant from Poland, Voynich. Having entered the revolutionary emigrant environment, she wrote the novel The Gadfly that glorified her throughout the world. It was followed by several more novels and musical works, as well as an English translation of Taras Shevchenko's poems. Voynich died in New York at the age of 95, short of the centenary of the death of her famous mathematician father George Boole.

George On a freighter bound for Africa, I met and befriended a little boy named George, who was traveling with his mother and young aunt. One afternoon, on deck, the boy left his mother and came up to me; they followed him with their eyes. He announced that tomorrow is his day