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一、公理方法为数学这門科学带来的特点在数学中,有些命題不加証明,直接用作邏輯推理时的依据,这类命題叫作公理;用巳知概念說明使用概念的意义的命題叫作定义。任何一个数学命題,不管直观看来多么明显,但是只要它不是公理,都要求严格地証明。一个命题,只有当它作为定义、公理的邏輯結果,才算是被証明了的定理。这种用定义、公理作为選輯推理的基础、以建立科学体系的方法叫作公理方法。公理方法是建立数学这门科学体系的重要方法,尤其自20世纪以来,它已成为現在数学各个分支建立科学体系的基本方法。使用这种方法的結果,就为数学带来两个明显的特点。
First, the axiom method brings the characteristics of mathematics to science. In mathematics, some propositions are not proved, and are directly used as the basis for logical reasoning. Such propositions are called axioms; they use prognostic concepts to explain propositions that use the meaning of concepts. This is called definition. Any mathematical proposition, no matter how intuitive it may appear, but as long as it is not axiom, requires strict proof. A proposition, only if it is the logical result of the definition, the axiom, can be regarded as a proved theorem. This method of using definitions, axioms as the basis for selecting inferences, and methods for establishing scientific systems are called axiomatic methods. The axiomatic method is an important method for establishing the scientific system of mathematics. Especially since the 20th century, it has become the basic method for establishing the scientific system in all branches of mathematics. The use of this method results in two distinct features for mathematics.