The research methods of logical mathematics include: proof, induction, reverse thinking, construction and reduction to absurdity. Among them, proof is the core of mathematics. By constructing strict logical reasoning, the reason why a proposition is true is expounded. There are different ways of proof, such as direct proof, indirect proof and reduction to absurdity. Induction is to infer general conclusions by observing a series of specific situations. Mathematical induction is often used to prove some propositions about integers or natural numbers. Reverse thinking is to start from the conclusion to be proved, think backwards, and analyze the preconditions and steps to draw a conclusion. Constructivism is to prove some properties or propositions by constructing concrete objects, functions or structures. The reduction to absurdity assumes that the proposition to be proved is false, and then the contradiction is deduced through logical reasoning, thus proving the original proposition to be true.