background
The third mathematical crisis occurred at the end of 19 and the beginning of the 20th century, when mathematics was in an unprecedented period of prosperity. The first is the mathematization of logic, which promotes the birth of mathematical logic.
The theory of * * * founded by Cantor in 1970s is the foundation of modern mathematics and the direct source of crisis. /kloc-At the end of 0/9th century, Dedeking and piano axiomatized arithmetic and real number theory, which promoted the axiomatic movement. The greatest achievement of axiomatic movement is Hilbert's axiomatization of elementary geometry in 1899.
In order to find out the ins and outs of the third mathematical crisis, we must first explain what a mathematical crisis is. Generally speaking, the crisis is an intensified and unsolvable contradiction. From a philosophical point of view, contradictions are ubiquitous and inevitable, even in mathematics, which is famous for its certainty.
There are many contradictions in mathematics, such as positive and negative, addition and subtraction, differentiation and integration, rational and irrational numbers, real and imaginary numbers and so on. However, in the whole process of mathematics development, there are still many profound contradictions, such as infinity and infinity, continuity and discreteness, even existence and construction, logic and intuition, concrete objects and abstract objects, concepts and calculations, and so on. Throughout the history of mathematics development, the struggle and solution of contradictions runs through. When the contradiction intensifies to involve the whole mathematical foundation, the mathematical crisis arises.
The elimination of contradictions and the resolution of crises usually bring new contents, new progress and even revolutionary changes to mathematics, which also embodies the basic principle that contradictions and struggles are the historical driving force for the development of things. The whole history of mathematics development is the history of contradiction and struggle, and the result of struggle is the development of mathematics field.
Man first knew natural numbers. There has been a struggle since the introduction of zero and negative numbers: either these numbers are introduced or the subtraction of a large number is not possible; Similarly, the introduction of fraction makes multiplication have an inverse operation-division, otherwise many practical problems cannot be solved. But then the question comes, can all quantities be expressed by rational numbers? So the discovery of irrational numbers led to the first mathematical crisis, and the solution of the crisis also promoted the development of logic and the systematization of geometry.
The solution of the equation led to the emergence of imaginary numbers, which were considered "unreal" from the beginning. And this pseudo number can solve the problem that real number can't solve, thus winning the right to exist for itself.
The development of geometry has developed from Euclidean geometry to various non-Euclidean geometries. /kloc-many problems that cannot be solved by traditional methods were discovered in the 0/9th century. For example, algebraic equations of quintic or above cannot be solved by addition, subtraction, multiplication, division, multiplication and division. The three major problems in ancient Greek geometry, that is, bisecting any angle, doubling a cube and turning a circle into a square, can not be solved by drawing with compasses and rulers, and so on.
These negative results show the limitations of traditional methods and also reflect the deepening of human understanding. This discovery has brought great impact to these disciplines and almost completely changed their direction. For example, algebra has developed from that time to abstract algebra, and solving the roots of equations has become a subject of analytical mathematics and computational mathematics. In the third mathematical crisis, this situation also appeared many times, especially the incompleteness of formal system including integer operation and the uncertainty of many problems greatly improved people's understanding and promoted the great development of mathematical logic.
This contradiction and crisis caused development, changed the face, and even caused revolution, which is not uncommon in the history of mathematical development. The second mathematical crisis is caused by infinitesimal contradiction, which reflects the contradiction between finiteness and infinity in mathematics. Mathematics has always been a contradiction between calculation method and analysis method, with clear concept and strict logic. In this respect, practical mathematicians pay more attention to blind application. And more mathematicians and philosophers who pay attention to it criticize it. Only when these two aspects are well coordinated can the contradiction be solved. Later, operator calculus and δ function also repeated this process, starting with formal calculus and arbitrary application, and it was not until Schwal that the strict system of generalized function theory was established.
For the third mathematical crisis, some people think that it is only the crisis of mathematical foundation and has nothing to do with mathematics. This view is one-sided. It is true that the problem involves mathematical logic and * * * theory, but it involves infinity from the beginning. Modern mathematics can be said that it is difficult to move without infinity. Because if we only consider finite * * * or at most countable * * *, most mathematics will cease to exist. And even with these limited mathematical contents, there are many problems involving infinite methods, such as using analytical methods to solve many problems in number theory. From this point of view, the third mathematical crisis is a profound mathematical crisis.