Symbolism: As we all know, mathematics is abstract. It studies numbers abstracted from real life and the relationship between numbers. Therefore, the model obtained by mathematical research must be universal, highly abstract and generalized, which can not be said to be applicable to the chicken problem in reality, but not to the duck problem. In order to achieve this goal, when studying the relationship between numbers, it will be very troublesome if you need to compare numbers with numbers. For example, we often say that a frog has a mouth, two eyes and four legs. According to this statement, I can't finish it all my life, but the symbolization of mathematics solves this problem well. Now we all know that frogs have mouths, eyes of 2a and legs of 4a. The relationship between A, 2a and 4a here will not change just because they are frogs or rabbits. This is just a simple example, because this relationship can only be used for quadrupeds, and well-known algorithms are completely used in real life, such as a+b=b+a, so the symbolization of mathematics provides the possibility for better studying the relationship between numbers and lays the foundation for the latter two characteristics.
Axiomatization: In fact, it can be understood as a mathematical proof problem for everyone to learn. You will find that when proving a proposition, we need to get the correct proposition B from the correct proposition A. Finally, after this process several times, we can get whether this proposition is correct or not. In this process, have you considered that every proposition is correct because it has a premise, which leads to its correctness, such as proposition B mentioned above, why is proposition B correct? This is because the correct proposition A deduces that proposition B is correct. Now let's think about why proposition A is correct. Which proposition can be proved? One by one, we will eventually find that there is no end point, but if there is no end point, we cannot prove whether a proposition is correct or not, and not all propositions can be proved. For example, a=b, b=c, then a=c, there is no way to prove this proposition. In view of this, the famous mathematician Euclid put forward five axioms: 1, which means that quantities equal to the same quantity are equal to each other. 2, the same amount plus the same amount makes the sum equal. 3, the same amount minus the same amount, the difference is equal. 4. Objects that can overlap each other are congruent. 5, the whole is greater than the local. It is precisely because of these five axioms that we can start from them and get more relations, thus establishing a mathematical axiom system.
Formalization: Formalization here refers to the formalization of argumentation methods. I have talked about the axiomatization of mathematics before, that is, whether some propositions are correct or not can be obtained through proof, but what should be the process of this proof and how to write it to ensure strict logic and non-redundancy. Therefore, Aristotle put forward the famous "syllogism", namely a general principle (major premise) and a special statement (minor premise) attached to the general principle. For example, animals have thoughts (major premise), and people are animals (minor premise), so people have thoughts (conclusions). This process seems normal now, but this great invention has laid a solid foundation for the establishment of human thinking methods and the improvement of thinking ability.