What mathematical definitions are similar from concrete to abstract definition of features?

Characteristics of mathematical abstract definition:

Regarding the characteristics of mathematics, we can compare mathematics with other disciplines, which is very obvious.

Mathematics is more abstract than other subjects. Where is the abstraction of mathematics? That is to put aside the specific content of things for the time being and study only from the abstract figures. For example, a simple calculation, 2+3 can be understood as two trees plus three trees, and can also be understood as two machine tools plus three machine tools. In mathematics, we put aside the specific contents of trees and machine tools, and only study the operation law of 2+3. After mastering this law, trees, machine tools, cars and anything else can be calculated according to the operation law of addition. Operations such as multiplication and division also study abstract numbers, leaving aside specific content.

Many concepts in mathematics are abstracted from the real world. For example, the concept of "straight line" in geometry does not refer to the taut line in the real world, but abandons the quality, elasticity, thickness and other properties of the real line, leaving only the nature of "bi-directional infinite stretching", but there is no bi-directional infinite stretching line in the real world. The concepts of geometry and function are abstract. However, abstraction is not a unique attribute of mathematics, it is a characteristic of any science and even all human thinking. It's just that the abstraction of mathematics is different from other disciplines.

The abstraction of mathematics has the following three characteristics: first, it retains the quantitative relationship or spatial form. Secondly, the abstraction of mathematics is formed through a series of stages, and its degree of abstraction greatly exceeds the general abstraction in natural science. From the most primitive concept to abstract concepts such as function, complex number, differential, integral, functional, n-dimensional space and even infinite-dimensional space, it is a deepening process from simple to complex, from concrete to abstract. Of course, the form is abstract, but the content is very realistic. As Lenin said: "All scientific (correct, solemn and not absurd) abstractions reflect nature more deeply, correctly and completely." (Abstract of Hegel's Logic, Complete Works of Lenin, Volume 38, p. 18 1 page) Third, not only the concept of mathematics is abstract, but also the mathematical method itself is abstract. Physics or chemists always use experimental methods to prove their theories; Mathematicians cannot prove a theorem through experiments, but through reasoning and calculation. For example, although we have accurately measured the two base angles of an isosceles triangle for thousands of times, it cannot be said that the two base angles of an isosceles triangle are equal, but it must be strictly proved by logical reasoning. To prove a theorem in mathematics, we must use the concepts and theorems that have been learned or proved to deduce this new theorem through reasoning. We all know mathematical induction, which is an abstract mathematical proof method. Its principle is to arrange the studied elements into a sequence, and a certain property holds for the first term of this sequence. Suppose that when the term k holds, if it can be proved that the term k+ 1 also holds, then this property holds for any term in this series, even if this series is infinite.

The second characteristic of mathematics is accuracy, or the rigor of logic and the certainty of conclusions.

Mathematical reasoning and its conclusion are indisputable and beyond doubt. The accuracy and certainty of mathematical proof are fully demonstrated in middle school textbooks.

Euclid's classic "Elements of Geometry" can be used as a good example of logical rigor. Starting from several definitions and axioms, it deduces the whole geometric system by using the method of logical reasoning, and arranges rich and scattered geometric materials into a systematic and rigorous whole, which has become one of the scientific masterpieces in human history and is praised by future generations. For more than 2,000 years, all elementary geometry textbooks and works before19th century are based on Geometry Elements. "Euclid" has become synonymous with geometry, and people also call the geometry of this system Euclid geometry.

But the rigor of mathematics is not absolute, and the principle of mathematics is not static, it is also developing. For example, as mentioned earlier, the Elements of Geometry also has some imperfections. Some concepts are not clearly defined, and concepts that should be defined by themselves are adopted. There is still a lack of strict logical basis in the basic propositions. Therefore, a more rigorous Hilbert axiom system was gradually established.

The third feature is the universality of application.

We use mathematics almost all the time in our production and daily life. It is indispensable for measuring land, calculating output, making plans and designing buildings. Without mathematics, the progress of modern technology is impossible. From simple technological innovation to complex satellite launch, mathematics is indispensable.

Moreover, almost all precision sciences, mechanics, astronomy, physics and even chemistry usually use some mathematical formulas to express their laws, and mathematics is widely used as a tool in developing their own theories. Of course, the demand of mechanics, astronomy and physics for mathematics has also promoted the development of mathematics itself. For example, the study of mechanics promoted the establishment and development of calculus.

The abstraction of mathematics is often closely related to the universality of application, and a certain quantitative relationship often represents all practical problems with this quantitative relationship. For example, the vibration of mechanical system and the oscillation of circuit are described by the same differential equation. If we study this formula without considering the meaning in specific physical phenomena, the results obtained can be used for similar physical phenomena. In this way, many similar problems can be solved by mastering a method. Phenomena of different natures have the same mathematical form, that is, the same quantitative relationship, which reflects the unity of the material world. Because the quantitative relationship exists not only in a specific material form or its specific motion form, but also in various material forms and various motion forms, mathematics is widely used.

It is precisely because mathematics comes from the real world and correctly reflects a part of the contact form of the objective world that it can be applied, can guide practice and can show the foresight of mathematics. For example, before a rocket or missile is launched, its flight trajectory and landing point can be predicted by accurate calculation; Before the unknown planet in the celestial body was directly observed, its existence was predicted from astronomical calculations. The same reason makes mathematics an important tool in engineering technology.

Here are some outstanding examples of applied mathematics.

First, the discovery of Neptune. Neptune, one of the planets in the solar system, was discovered on the basis of mathematical calculation in 1846. After the discovery of 178 1, the observation of Uranus' orbit is always quite different from the predicted results. Is the law of gravity incorrect or there are other reasons? Some people suspect that there is another planet around it, which affects its orbit. 1844, British Adams (1819-1892) calculated the orbit of this unknown planet by using the law of universal gravitation and the observation data of Uranus, and spent a long time calculating the position of this unknown planet and its position in the sky. Adams sent the results to Charles, director of the Observatory of Cambridge University, and Airy, director of the Observatory of Greenwich, England, from September of 1845 to September of 10/0, respectively, but Charles and Airy were superstitious about authority and ignored them.

1845, Le Verrier (181-1877), a young French astronomer and mathematician, wrote a letter to Galle (1846), an assistant of the Berlin Observatory, after more than a year of calculation. The letter said: "Please aim your telescope at Aquarius on the ecliptic, that is, longitude 326, and then you will see a star with a brightness of nine degrees in the range of 1 in that place." Galle observed in the direction pointed out by Le Verrier, and sure enough, he found a star that was not on the map, Neptune, within 1 from the pointed position. The discovery of Neptune is not only a great victory of mechanics and astronomy, especially Copernicus and Heliocentrism, but also a great victory of mathematical calculation.

Second, the discovery of Ceres. 180 1 On New Year's Day, Italian astronomer Piazi (1746- 1826) discovered a new asteroid, Ceres. But it soon hid again. Piazi only wrote down the asteroid's motion along an arc of 9, and Piazi and other astronomers could not find out its entire orbit. Gauss, a 24-year-old from Germany, calculated the orbit of this asteroid based on the observation results. On February 7th, 65438, astronomers rediscovered Ceres at the location pointed out by Gauss.

Third, the discovery of electromagnetic waves. Maxwell, a British physicist (1831-1879), summed up the experimental electromagnetic phenomena and expressed it as a second-order differential equation. From the point of view of pure mathematics, he deduced from these equations that electromagnetic waves exist and propagate at the speed of light. Accordingly, he put forward the electromagnetic theory of light, which was fully developed and demonstrated later. Maxwell's conclusion also urges people to look for electromagnetic waves of pure electricity origin, such as electromagnetic waves emitted by vibration discharge. Such electromagnetic waves were discovered by German physicist Hertz (1857- 1894). This is the origin of modern radio technology.

Fourthly, in 1930, the British theoretical physicist Dirac (1902- 1984) predicted the existence of positrons through mathematical deduction and calculation. 1932, American physicist Anderson discovered positrons in cosmic ray experiments. Similar examples are too numerous to mention. In a word, mathematics has made extremely accurate predictions in celestial mechanics, acoustics, fluid mechanics, material mechanics, optics, electromagnetism, engineering science and other fields.