1. Why is it so general?
First of all, theoretically speaking, the essence of mathematics is an important issue of mathematical view. Mathematical view and mathematical methodology are unified, and mathematical view can be analyzed through methodology. The particularity of mathematical cognitive objects determines the particularity of mathematical cognitive methods. This particularity lies in that mathematical research, like natural science, must adopt deduction, observation, experiment and induction. Therefore, we can reflect the essence of mathematical cognition by studying the method of mathematical cognition.
Secondly, in fact, the experience of mathematics knowledge shows that mathematics is produced to meet the needs of social practice and is the accumulation of experience in solving practical problems. Mathematical questions raised by social practice all require quantitative answers, and concrete calculations are needed to make quantitative answers, so calculations represent the characteristics of mathematical experience and knowledge. As for various specific calculation methods and their generalized "algorithms" (including formulas, principles and rules), "calculation" can also be used to summarize and reflect the empirical calculation or algorithm characteristics of mathematical knowledge in methodology. At the same time, the deductive nature of mathematical knowledge reflects the deductive characteristics of mathematical knowledge in methodology, so the deductive nature of mathematical knowledge can be embodied by "performance". So calculus can be used to embody the experience and deduction of mathematics.
Third, in order to avoid the one-sidedness of generalizing the essence of mathematics. Since mathematics has been divided into applied mathematics and pure mathematics, many mathematicians think that mathematics came from experience a long time ago, but now it is not, but has become a deductive science. And most people accept this view. However, this emphasis on the deductive characteristics of mathematics ignores the empirical side of mathematics. In order to avoid this one-sidedness, the essence of mathematics is summarized and reflected here, especially through mathematical methodology.
2. Calculus embodies the characteristics of mathematical research.
The particularity of mathematical research object has produced a unique problem of mathematical research: calculation and proof. They have become two main tasks in mathematical research. About "proof" Because of the particularity of mathematical objects, mathematical achievements cannot be proved by experiments like natural science achievements, but must be proved by logical deduction, otherwise mathematicians will not recognize them. Therefore, how mathematicians express their achievements into a series of deductive reasoning (that is, proof) has become an important job. Proof has become an important feature of mathematical research. About "calculation". Mathematics itself originated from calculation, and even if mathematics develops into a highly abstract theory today, it cannot be without calculation. Before proving a theorem, mathematicians must go through a lot of concrete calculations, carry out various experiments or experiments, analyze and summarize, and then form the ideas and methods of proof. Only then can we make a comprehensive demonstration logically and express it as a series of deductive reasoning processes, that is, proof. From the perspective of applied mathematics, it needs a lot of calculations, so people invented various computers. Today, with the wide application of electronic computers, the scale of calculation is even larger, so that numerical experiments appear in mathematics. Therefore, calculation has become another important work in mathematical research.
Since "calculation and proof" are the two main tasks and characteristics of mathematical research, does the generalization that "mathematics is the science of calculus" reflect this characteristic? "Proof" is a deductive reasoning based on certain premises (basic concepts and axioms) and logical rules. And "deduction" can better reflect the characteristics of "proof". And "calculation" can obviously directly reflect "calculation" or "algorithm" and its characteristics. It can be seen that "calculus" embodies the calculation and proof of mathematical research and its characteristics.
3. The unity of opposites between "doing" and "calculating" embodies the dialectical essence of mathematics.
First of all, from the macro perspective of mathematical development. The history of mathematics tells us that mathematics originated from "calculation", that is, from the calculation of the number of objects, the area of fields and the length of objects. To calculate, there must be a calculation method. When various calculation methods accumulate to a certain amount, mathematicians classify them and summarize the calculation formulas, rules and principles suitable for a certain kind of problems, which are collectively called algorithms. Therefore, the childhood of mathematics is called arithmetic, which is manifested as an empirical knowledge. When Euclid established the first axiomatic system in the history of mathematics, "deductive method" appeared. Since then, "expression" and "calculation" have constituted a pair of basic contradictions in the development of mathematics, which has promoted the development of mathematics. This is the most prominent in the history of western mathematical thought. Generally speaking, before Euclid, mathematical thoughts were mainly algorithms; In the pre-Alexandria period of Euclid's life, the main idea of mathematics has changed from algorithm to deduction; From the late Alexander to the18th century, the main idea of mathematics once again turned from deduction to algorithm; From19th century to the first half of 20th century, the main idea of mathematics changed from algorithm to deduction. The application of computer promotes the development of computational mathematics and the emergence of interdisciplinary subjects such as computational fluid dynamics and computational geometry, as well as the emergence of mathematical experiments. All these make the algorithm idea develop again and become an idea that keeps pace with deduction. It can be predicted that with the establishment of computer as a mathematical research tool, the idea of algorithm will become the main idea of mathematics for a long time to come. This alternation and substitution of algorithmic thought and deductive thought in the development of mathematics reflects the mutual transformation of the contradiction between "expression" and "calculation" under certain conditions. Therefore, some mathematical historians summarize the development of mathematics as a process of alternating spiral rise of algorithmic tendency and deductive tendency from the perspective of methodology.
Secondly, from the microscopic point of view of mathematical research. There is "calculation" in "acting", which fully shows that the "certificate" we analyzed above contains "calculation" and the transformation from "calculation" to "acting". There is "play" in "calculation", which is fully manifested in arithmetic and algebra. Both arithmetic and algebra are expressed as "calculation", but the "calculation" of arithmetic and algebra is not free calculation, but should follow the basic four operations and their laws, that is, calculation should follow certain calculation rules, just like proof should follow reasoning rules. So "calculation" includes "performance" and the transformation from "performance" to "calculation". This unity of opposites between "deduction" and "calculation" has been more fully reflected in the numerical calculation and theorem proving by computer. This unity of opposites between "calculation" and "expression" reflects the dialectical relationship between mathematical experience and deduction from one side and embodies the dialectical essence of mathematics.
To sum up, since "calculus" summarizes the characteristics of mathematical research and reflects the empirical and deductive nature of mathematics and their dialectical relationship, we have reason to take it as a summary of the essence of mathematics and say that "mathematics is the science of calculus".