Frege thought in 1884 that each number in the "arithmetic basis" is an independent object. He believes that arithmetic rules are analytical judgments, so they are transcendental. Accordingly, arithmetic is only the form of further development of logic, and every arithmetic theorem is a logical law. The application of arithmetic to the interpretation of natural phenomena is only the logical processing of observed facts, and calculation is reasoning. The law of number can be applied to the outside world without practical test, but there is no concept or number in the outside world, spatial population and its content. Therefore, the law of numbers cannot be applied to the outside world. These laws are not natural laws. However, what they can apply to the world are true judgments, and these judgments are the laws of nature. They reflect not the relationship between natural phenomena, but the relationship between judgments about natural phenomena.

Long before Russell discovered the paradox, he tried to simplify mathematics into logic when he wrote Principles of Mathematics. The plan met with difficulties because of the discovery of paradox. After he found a way to eliminate the paradox, he began to realize his plan concretely, which is the Principles of Mathematics co-authored by him and Whitehead.

Because the original purpose of Russell and Whitehead's Principles of Mathematics is to try to establish mathematics on the basis of logic, several undefined concepts and some logical axioms are put forward at the beginning of the book, from which logical rules and mathematical qualitative are deduced.

Undefined concepts include basic proposition, propositional function, assertion, or, no; A proposition here refers to a statement stating a fact or describing a relationship, such as "Zhang San is a person" and "Apple is red". These concepts can define the most important concept "implication" in logic.