Abstract: Based on the standpoint of realism, the reality of mathematical objects referred to by mathematical concepts constitutes the premise and foundation of the strictness of initial axioms of mathematics; The semantic space of initial axioms of mathematics contains a epistemological choice with truthfulness of mathematicians as the knowing subject regarding the possible world of mathematical objects with reality; Possible world semantics can provide a semantic analysis support on the basis of realism for the strictness argument of initial axiom of mathematics at the methodological level; The generation of initial axioms of mathematics is a process of methodology construction of hermeneutics, in which mathematicians engage in an overall thinking mechanism of “understanding-explanation(psychology)-explanation(text)” about mathematical objects with reality; The “hermeneutic cycle” methodological construction feature of mathematical axiom system, which starts from initial axioms of mathematics with reality, not only reflects the stability of mathematical axiom system at the epistemological level, but also further demonstrates and confirms the strictness of mathematical initial axiom.

Key Words: Initial axioms of mathematics; Realism; Strictness; Possible world semantics; Hermeneutics

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