There are two standpoints on relationship between mathematic and logic:logicalism and intuitionism.

On logicism in critical thinking teaching;

According to logicism,truth can be recognized by mere process of logic or operational methods,and the recognition of truth has nothing to do with the spirituality of knower.

Based on the essence of Frege\'s logicism program and inspired by his articles,neo-logicism advocates that mathematics focusing on arithmetic can be reduced to the subsystem of second-order logic through consensus abstract principles.

Second-level Ontology Based On Mutually-inversistic Logic;

For single logical theorem, a new way to prove is proposed: second level explicit inductive composition in totally abstract fact space in mutually-inversistic logic.

Likewise,in mutually-inversistic logic,the two conjuncts of the antecedent of a mutually inverse implicational proposition are two curved surfaces, and their conjunction is a spatial curve.

The logical atomism discussed in this article is the major philosophical ideology of Bertrand Russell after the year of 1899 and of Ludwig Wittgenstein s early stage.