How did Bertrand Russell and Alfred North Whitehead connect logic and mathematics in Principia Mathematica?

In Principia Mathematica (1910–1913), Bertrand Russell and Alfred North Whitehead sought to demonstrate the school of thought known as logicism, which holds that all mathematics can be deduced from pure logic.12

To connect these fields, they aimed to show that basic mathematical concepts and modes of reasoning are definable in logical terms. For example, they defined the number of a class as the class of all classes equinumerous with it, a definition intended to imply the standard axioms of arithmetic. By reducing arithmetic to logic, they aimed to reduce all of mathematics to logic, as most other mathematical fields had already been reduced to arithmetic.32

Because their initial system encountered Russell’s paradox—a contradiction involving self-referential sets—they implemented a theory of types to restrict how sets could be defined and to avoid "vicious circle" definitions. They also introduced the axiom of reducibility to ensure that standard mathematics could be derived within their complex logical hierarchy.32