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The contribution of British mathematicians to the development of mathematics is in terms of originality. Boolean is a typical example. Boolean logical algebra has rapidly developed into a major branch of pure mathematics. Mathematicians all over the world have extended it to all fields of mathematics. Bertrand Russell said:

Pure mathematics was discovered by Bull in his book the Law of thinking published in 1854.

This shows the importance of mathematical logic and its branches today. Others before Boer, especially Leibniz and de Morgan, had dreamed of adding logic itself to the field of algebra. Bull made it a reality.

Born in Lincoln, England, on November 2, 1815, Bull was the son of a small shopkeeper and belonged to the bottom of society. At that time, the goal of students in British schools was to serve as foremen in factories and mines that became popular at that time. These schools are not for people like Bull. The main purpose of Booljin's "state school" is to keep the poor in a humble position suitable for them. Bull lived in a time when knowing a little bit of Latin, or a little bit of Greek, was the mark of a gentleman. Oddly enough, memorizing Latin syntax by rote is considered to be the most useful mental exercise.

Bull also flocked to make a sad misjudgment, deciding that to get out of trouble, he had to learn Latin and Greek. The fact is that Latin and Greek have nothing to do with the cause of his poverty. By the age of 12, Bull had mastered enough Latin to translate a poem by Horace into English and publish it in the local newspaper. This caused a cultural quarrel, partly in praise of Bull and partly in humiliation.

At the age of 14, Bull translated the local Greek poem Ode to Spring. Bull's earliest math education came from his father, who went far beyond his own education through self-study. But Bull insists that classical literature is the key to life. By the age of 16, he was forced to support his parents and found a job as a primary school teacher.

Bull taught in two primary schools for four years. He had nothing in capital, and every penny he earned was the minimum need to support his parents and maintain his poor life. He was unable to join the army at that time, because he could not afford a certificate of appointment. As a lawyer, he has obvious requirements in terms of property and education, and it is impossible for him to meet them. What else is there? Only the church, Bull decided to become a priest. But under the torment of poverty, Bull gave up all the idea of priesthood. But his four years of private preparation for his ideal career have not been in vain; he is proficient in French, German and Italian.

After several twists and turns, Bull opened a private school of his own at the age of 20. In the process of teaching mathematics to his students, Bull became interested in mathematics. The mediocre and obnoxious textbooks of the time first surprised him and then aroused his contempt. Are these things math? Unbelievable. Like Abel and Galois, Bull went directly to the primitive continent of mathematics in search of real mathematics. He received only preliminary mathematical training, but through his own efforts, he mastered Laplace's "astromechanics", which is one of Laplace's most esoteric masterpieces. He also made a thorough study of Lagrange's very abstract "analytical mechanics". He even made his first contribution to mathematics by his own efforts, without anyone's guidance-- writing a paper on variational methods.

In 1831, Bull began an ambitious self-study program in mathematics. He read the works of higher mathematics by Lagrange and Laplace in French. He studied and mastered Sir Isaac Newton's great book the principles of Mathematics. Bull made another achievement from his lonely study, and he found invariants. Without the mathematical theory of invariants, the theory of relativity would be impossible. The reason why Bull can see what others ignore is undoubtedly because he has a strong sense of the symmetry and beauty of algebraic relations.

It can be pointed out that the modern concept of algebra began in England. Peacock published his Algebra thesis in 1830, which was regarded as something of a heresy at that time, and it has become common sense in any textbook today. Peacock has completely abandoned the relationships we see in elementary algebra, such as x _ ray _ y _ y _ x _ x _ (x _ z) = xy+xz, etc. The concept of "representing numbers" is inevitable. They don't have to represent numbers, which is the most important thing about algebra and its applications. XMagna yjinz, … An arbitrary symbol that is simply combined according to some operations.

If you do not understand that algebra itself is merely an abstract system, then algebra may still be firmly stuck in the arithmetic quagmire of the 18th century, rather than moving towards its extremely useful modern variants under the guidance of Hamilton. This innovation in algebra provided Bull with his first chance. He creatively pointed out that the symbols of mathematical operations were separated from the things on which they were operated, and studied the operations themselves. How do they combine? Are they also dominated by some kind of symbolic algebra? His research in this area is extremely meaningful, but his other great contribution, that is, the creation of a simple and feasible symbolic system or mathematical logic system, dwarfs this work.

In order to introduce Bull's outstanding findings, let's deviate a little from the subject. In the 19th century, there were two famous Hamiltons, one was the Irish mathematician William Ron Hamilton, and the other was the Scottish philosopher William Hamilton. The rhetorical philosopher Hamilton eventually became a professor of logic and metaphysics at the University of Edinburgh. Hamilton of Ireland became an original mathematician in the 19th century.

Hamilton in Scotland was too stupid to learn more than elementary mathematics at school, and his weakness was that he thought he knew everything. When he began to teach and write about philosophy, he told the world how worthless mathematics was.

Mathematics stiffens and dries up the mind; the overstudy of mathematics completely deprives the mind of the intelligence needed for philosophy and life; mathematics does not help to develop logical habits at all; in mathematics, dullness is promoted to talent, and ability is reduced to incompetence; mathematics can distort the mind, but it can never correct it.

De Morgan, an important figure in the history of British mathematics, appeared. He was one of the most sophisticated debaters of all time, an energetic mathematician, and a great logician who opened the way for Boolean.

Demo is caught up in an argument with Hamilton about his "quantification of predicates" (there is no need to explain what this mysterious thing is or used to be). De Morgan made a real contribution to deduction, but the philosopher Hamilton publicly accused de Morgan of stealing his work, and the battle began. In de Morgan's case, debate is a pleasant play. De Morgan never lost his temper; Hamilton never learned not to lose his temper.

If this were just one of the squabbles over priorities, it would not be worth mentioning. Its historical importance is that Bull was a staunch supporter of de Morgan at the time. When Bull was still teaching in primary school, he knew de Morgan was right and Hamilton was wrong. So, in 1848, Bull published a thin book, Mathematical Analysis of Logic.

The pamphlet was strongly praised by Demo. This pamphlet is just a preview of something greater to emerge in six years' time. At the same time, Bull rejected the proposal to go to Cambridge for orthodox math training. He continues to teach in primary school because his parents are completely dependent on him.

In 1849, he was finally appointed professor of mathematics at the newly established Queen's College. He has done all kinds of noteworthy mathematical work, but his main effort is to continue to perfect his masterpiece (symbolic logic). In 1854, the 39-year-old Bull published this masterpiece, the study of the laws of thinking that lay the mathematical theoretical foundation of logic and probability.

The following excerpts will give us some idea of Boer's style and his field of work.

The purpose of this paper is to study the basic laws of mental arithmetic on which reasoning is carried out, to express them in calculus language, to establish logical science on this basis, and to construct its method; make this method itself the basis of the general method applied to the mathematical principles of probability. Finally, from the various truths discovered in these explorations, collect some hints that may be related to the composition of nature and human thinking.

It is true that there are some general principles based on the characteristics of language itself, which determine the use of symbols as elements of scientific language. To some extent, these elements are arbitrary. Their interpretation is purely conventional ∶ and we can apply them in any sense we like. But this permission is limited by two indispensable conditions-first, once this meaning is routinely established, we must not deviate from it in the same process of reasoning; second, the law that guides the process should be based entirely on the above-mentioned fixed meaning of the symbols used. Consistent with these principles, any consistency established between the symbolic law of logic and the symbolic law of algebra can only lead to a process-consistent result. The two fields of interpretation are still separate and independent, and each field is subject to its own laws and conditions.

The practical research on the following pages, in its practical aspect, shows logic as a process system implemented with the help of symbols with a definite interpretation, and is only subject to the laws based on that interpretation. But at the same time, they show that those symbols are formally the same as the general symbols of algebra, adding only one point, that is, logical symbols have to be subject to a special law, and in this regard, the symbols of quantity do not have to obey this law.

Boolean reduced logic to an extremely easy kind of algebra. In this algebra, proper reasoning becomes the elementary operation of the formula. Therefore, logic itself is dominated by mathematics.

Since Bull's groundbreaking work, his great discoveries have been improved and popularized in many ways. Today, symbols or mathematical logic are indispensable in understanding the nature of mathematics. If we can only use the logical methods before Boolean, then human reason can not deal with the intricate difficulties of symbolic reasoning. Bull's bold ideas are a milestone.

Since Hilbert published his masterpiece on the basis of geometry in 1899, people have begun to pay attention to the systematic exposition of the postulates of several branches of mathematics. As a result of Hilbert's work, the public method was recognized. This trend of abstraction was popular for a time, in which the symbols and rules of operations in a particular topic completely lost their meaning, but were discussed from the point of view of pure form, thus ignoring the application. And application is the ultimate pursuit of any scientific activity. However, abstract methods do provide irreplaceable insight, especially when it is easy to see the simplicity of Boolean logical algebra.

Therefore, we will describe the axiom of Boolean algebra (logical algebra). The following set of postulations is extracted from an article published by Huntington in the Journal of the American Mathematical Society (1933).

This set of axioms is denoted by K _ quotient, where K is a class of uncertain (completely arbitrary, without any pre-specified meaning or beyond the properties given by the axiom) elements a ~. , while aqb and a × b are the results of two uncertain binary operations +, ×. A total of 10 postulates

Ⅰ a: if an and b are in class K, then axib is in class K.

Ⅰ b: if an and b are in class K, then ab is in class K.

Ⅱ a: there is an element Z such that there is an a+Z=a for each element a.

Ⅱ b: there is an element U such that there is an aU=a for each element a.

Ⅲ a:a+b=b+a .

Ⅲ b:ab=ba .

IVa:a+bc= (aqb) (aqc).

Ⅳ BRV a (Benzc) = ab+ac.

V: for each element a, there is an element called axiom, such that it is possible to use the same element as a result.

Ⅵ: there are at least two different elements in class K.

It is easy to see that the following explanation satisfies these postulates ∶ a dint bje c, … Are classes; ab is a class made up of all those things that are at least in one of the classes an and b; Z is an "empty class"-a class without elements; U is a "whole class"-a class that contains everything in all the classes under discussion. Then the axiom V states that for any known class a, there is a class a 'that contains all the things that are not in a. Note that Ⅵ indicates that U _ Z is not the same class.

From such a set of simple and obvious statements, the whole classical logic can be established by using symbols of algebra generated by postulates. From these postulates, the theoretical problems in ∶ logic, which can be called "logical equations", are transformed into such equations, and then these equations are "solved" by algebraic methods, and then the solution is reinterpreted according to the logical data, giving the answer to the original problem.

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