As we all know, this definition of mathematics was put forward by Engels. In fact, Engels' definition has been recognized as the most authoritative definition by mathematical philosophy circles at home and abroad for many years. The latest edition (2005 edition) of Modern Chinese Dictionary still defines mathematics as "a discipline that studies the spatial forms and quantitative relations of the real world". Since the 20th century, new branches of mathematics have emerged. At the same time, applied mathematics is more and more extensive in the real world, and the research object of mathematics seems to be not only the relationship between spatial form and quantity; And many researchers put forward various definitions of mathematics from their own understanding. Therefore, in recent years, some people think that Engels' definition of mathematics is outdated or "far from enough". This understanding is one-sided, because it is not the case. Mr. Kuang Jichang profoundly analyzed "what is mathematics" and thought that "the definition of mathematics should reflect the object of mathematical research and its essential attributes". "Only from the philosophical height of materialist dialectics can we realize that the quantitative relationship and spatial form in the real world are not fixed, but the connotation is deepening and the extension is expanding. So Engels' judgment about what mathematics is is not out of date.
2. Mathematics is systematic common sense.
This is the view of Friedenthal, an internationally renowned mathematician and mathematical educator. He thinks that the root of mathematics is common sense. Mathematics, as common sense, also develops with the continuous progress and development of language from speaking to reading and writing. The concept of "indigenous residence" is mainly supported by the corresponding numerals in spoken language (for example, "1" comes from a person, a pen, ...).
Common sense is hierarchical. After common sense rose from experience to laws, these laws became common sense again, that is, higher-level common sense. Friedenthal once said: "For real mathematics and its progress, common sense must be systematic and organized. As before, the experience of common sense is combined into laws (such as the exchange law of addition), and these laws become common sense, that is, higher-level common sense. As the basis of advanced mathematics.
3. Mathematics is an artificial language and symbol system.
This is the view of some mathematical historians. Although not many people hold this view, it is very representative and gives us a new perspective to understand "what is mathematics". When we open a history of mathematics, we need to explain it with the help of real life facts, but later mathematics paid more and more attention to its "language and symbols". This phenomenon can be traced back to Euclid at the earliest.
Of course, mathematics, as an artificial language and symbol system, must have certain conditions. Generally speaking, this language and symbol system must be self-evident, and elegantly speaking, this system must be complete. For example, if 1+ 1=3 is specified, a set of language and symbol system can be constructed on this basis, which is self-evident. Maybe a new branch of mathematics was born. There are many precedents in the history of mathematics, such as Galois's group theory and Cantor's set theory. When they appeared in front of mathematicians, they were not recognized by everyone. But it turns out that these are all mathematics, and they are very important mathematics. Because Cantor's set theory has some problems in self-proof, it has led to a serious mathematical crisis in history. With this crisis, set theory has become more complete and the foundation of mathematics has become more stable. The establishment of set theory is a great achievement in the history of mathematics, so it is necessary to infiltrate the idea of set theory in today's primary school mathematics teaching, so as to improve students' mathematical cognitive ability.
Mathematics is the absolute truth.
This is the view of some mathematicians and mathematical philosophers. For them, any knowledge can go wrong, but only mathematics can't go wrong. It is the only representative of knowledge. In their view, deduction provides a guarantee that mathematical knowledge is absolute truth. First, the basic statement in mathematical proof is regarded as true, the mathematical axiom is assumed to be true, the mathematical definition makes it true, and the logical axiom recognizes it as true. Second, the rules of logical reasoning keep the truth, that is, only the truth derived from the truth is recognized. Based on the above two facts, we can know that every statement in the deductive proof, including its conclusion, is true. Therefore, "because mathematical theorems are determined by deductive proof, they are all truth." This has formed the basis of many philosophers' assertion that mathematical truth is truth. (Ernest)
This view holds that if there are contradictions or problems in mathematics, it is not the fault of mathematics itself, but that people's understanding has not reached the corresponding level. Mathematicians and philosophers will find ways to solve these contradictions and problems, and the process of solving contradictions and problems itself also promotes the development of mathematics. For example, the appearance of π is like a sunny day, which is unacceptable to ancient Greek mathematicians, so it is called "irrational number". But it is the "unreasonable" that turns into "reasonable", and the concept of number expands from rational number to real number, which promotes the development of mathematics. Later, in order to solve some contradictions between function theory and set theory, mathematical philosophy also developed greatly, forming three schools: logicism, formalism and constructivism (including intuitionism).
5. Mathematics can be wrong and changed.
This is the view of some mathematical philosophers. The main reason why they oppose the absolute truth of mathematics is that the absolute view can be attributed to the method of "hypothesis-deduction". Mathematical truth and proof are based on deduction and logic, but logic itself lacks the basis of reducibility and is also based on the assumption of irreducibility. "But any assumption without a solid foundation, whether it is derived from intuition, convention, meaning or in any other way, is wrong." (Lin