Summary: On basis of that mathematics take the axiom deduction system as the elementary theory, this paper elaborates the significance of mathematical rigor, and explains that the mathematics abstract characteristic and mathematical proof must take the mathematical rigor as the first necessity. Taking the contemporary mathematics in two classical cases as examples, this paper discusses the importance of rigor in mathematics research and popularization. Some mathematicians also make mistakes. In the mathematical treatment, some just stress that the methods are novel and the topis are interesting, but do not check the proof in detail. The mathematical articles for the public are more likely carelessness and causes the wide negative influence. It points out that the science rigors is the first essential factor for mathematics education and the mathematics popular. If we neglect the rigor and pursue the fun blindly, it will be counterproductive and harm the younger generation.