Represented by Russell and A.N. Whitehead. They believe that all mathematical concepts come down to the concept of natural number arithmetic, and the concept of arithmetic can be given by definition with the help of logic. They tried to establish a logical axiom system including all mathematics, from which all mathematics was derived. According to logicism, mathematics is an extension of logic, and Russell's axiomatic system has to quote illogical axiom of choice and infinite axioms. Without these two axioms, it is impossible to deduce all arithmetic, let alone all mathematics. Of course, Russell's axiom system fully developed the axiom system of mathematical logic, and on this basis, it showed rich mathematical content, greatly promoted the research of mathematical logic and mathematical foundation, and made great contributions.

② Intuitionism.

Also known as constructivism. Its representative figure is L.E.J Brouwer. Intuitionists believe that mathematics comes from intuition, and the argument can only be made by construction method. They think that natural numbers are the basis of mathematics. When a mathematical proposition is proved to be correct, its construction method must be given, otherwise it is meaningless. Intuitionism holds that classical logic is abstracted from finite sets and their subsets, and applying it to infinite mathematics will inevitably lead to contradictions. They object to the use of law of excluded middle in infinite sets. They don't recognize the real infinite body, and think that infinity is potential, just the possibility of infinite growth. Constructibility plays an important role in the development of mathematical logic and computing technology. But intuitionism makes mathematics very complicated. Lost the beauty of mathematics, so it is not accepted by most mathematicians.

③ Formalism.

Represented by D. Hilbert, it can be said that it is Hilbert's mathematical viewpoint and basic mathematical viewpoint. Hilbert advocates defending law of excluded middle. He believes that to avoid the paradox in mathematics, we only need to formalize mathematics and standardize the proof. In order to make the formal mathematical system not contain contradictions, he founded the proof theory (meta-mathematics). He tried to prove the coordination of various branches of mathematics by finite methods. In 193 1, K. Godel proved the incompleteness theorem, which showed that the Hilbert scheme could not be successful. Later, many people improved the Hilbert scheme. Keeling proved the contradiction of arithmetic by transfinite induction. In the research of mathematical foundation, Robinson and P.J. Cohen call themselves formalists (Hilbert himself does not consider himself a formalist). They think that mathematics studies only symbolic systems without content, and "infinite set" and "infinite whole" do not exist objectively. Although Hilbert's idea was not realized, he founded the proof theory and promoted the development of recursive theory, so he made great contributions to the research of mathematical foundation. bale