More than a hundred years ago, Engels said: "The application of mathematics in chemistry is only a linear equation, and its application in biology is equal to zero." Today, the situation is very different. It is difficult to find a field of human knowledge that has nothing to do with mathematics now. Take the most basic number in mathematics as an example! It is wrong to put forward the view that "everything counts". Because number is a concept, not a thing, it is a reflection of the quantitative characteristics of things in people's minds, rather than an objective number transformed into things. Pythagoras reversed things. The collapse of the idea that "everything is important" is also a mistake. The mistake is that numbers are not enough to express everything; This mistake was caused by a great progress: the discovery of irrational numbers. People find irrational numbers, but they dare not admit that they are numbers. This is the first mathematical crisis. If we discard the idealistic component of the view that everything is a number, we can understand that everything is related to numbers. Isn't it? Everything is tangible, and the shape can be described by numbers. Movement and change are accompanied by the exchange and transformation of energy, which can be expressed by numbers. Human knowledge is essentially information, and information can be memorized by numbers. For example, computers use binary counting algorithm. Everything has a qualitative change, but quality can be described by numbers. The plane intersects the cone, and the geometric characteristics of the notch change with the included angle between the plane and the cone axis. When the intersection angle is a right angle, the cross section is a circle. With a slight change, the circle becomes an ellipse, then changes, and then changes. At the key point, the ellipse becomes a parabola, and then it becomes a hyperbola. Quadratic equation with real coefficients has discriminant. Discriminate whether the formula is positive, negative or zero, so that the equation has different real roots, multiple roots or the same real roots respectively. Most methods for judging the convergence of infinite series depend on a certain parameter. When the parameters reach a certain limit, the convergence and divergence of the series will change. Dialectics has three basic laws: the law of unity of opposites, the law of mutual change of quality and the law of negation of negation. If you want to ask, why are there such three laws? What would philosophers say? Is there a more basic principle behind these three rules? Perhaps, it can be explained by mathematics. Dialectics holds that everything contains contradictions, that is, "one divides into two" and unity of opposites. Why does everything contain the face of unity of opposites? Why is it "one in two" instead of "one in three"? Philosophers have no further research and answers about this. But we can understand that the essence of a thing can be described by a series of numbers, and it can be regarded as a vector-valued function with finite or infinite dimensions, with time as the independent variable. The change of things is nothing more than the change of each component of a vector-increase or decrease. Because numbers can only change so much. Increase and decrease are exactly two aspects of the unity of opposites, which are manifested as "one divided into two" rather than one divided into more than three. Functions have continuous points and discontinuous points. Generally speaking, everything in nature can often be described by analytic functions. The analytic function is continuous except for a few points. When the change and connection of things can't keep the continuity of functions, that is, they reach the discontinuous point, people say that things have changed qualitatively. There are two basic forms of function change: monotonic increase or decrease and periodic change. The two basic forms are spiral motion. Every link of spiral motion can be regarded as a process of negation of negation. The deeper people know about the world, the deeper they will understand the importance of mathematics. Philosophy is a science about the universal laws of nature, society and thinking. This universal law can only become a universally applicable law if it is divorced from the specific content. Because of its abstract ability, mathematics undertakes the task of describing this universal law, which is inextricably linked with philosophy and Marxist philosophy. In spite of his busy schedule, Marx also studied mathematics seriously and wrote a mathematical manuscript with important ideological value. At that time, as Engels said, the application of mathematics in social sciences was almost zero. If Marx lived to this day and saw mathematics permeate almost all disciplines, the sequel to Mathematical Manuscripts would probably involve the application of mathematics in philosophy. Mathematics is the tool of all sciences, and it can and should be the tool of philosophy. Marxism originates from practice, while mathematics is a knowledge system established by people in the process of solving practical problems for a long time, and their roots are the same. The development of Marxism is inseparable from mathematics, and the development of mathematics needs the guidance of Marxism. The world built by mathematics and Marxism is refreshing and reassuring! ① Note: See Dialectics of Nature translated by Yu Guangyuan, People's Publishing House (1984), P 172 References: Mathematics and Philosophy, Mathematics and Thinking, Dalian University of Technology Press.