How did Bertrand Russell and Alfred North Whitehead promote logicism?

Bertrand Russell and Alfred North Whitehead promoted logicism—the thesis that all mathematics can be deduced from pure logic—primarily through their monumental work, Principia Mathematica (1910–13).123

Their approach involved several key efforts:

• Logical Foundations: They aimed to show that basic mathematical concepts and reasoning are definable in purely logical terms, such as propositions and classes, without requiring specifically mathematical notions like numbers.13
• Systematic Derivation: Building on Russell’s earlier The Principles of Mathematics, they sought to derive the axioms of arithmetic and set theory from a reformed system of formal logic.134
• Addressing Paradoxes: To resolve Russell’s paradox, which threatened the consistency of their system, they developed the theory of types. This included the "ramified" theory of types and the axiom of reducibility to avoid "vicious circle" definitions, where an object is defined by a totality to which it belongs.13

While Principia Mathematica remains a landmark in mathematical logic, its philosophical success is debated, particularly due to the complexity of its type theory and later developments like Kurt Gödel’s incompleteness theorem.3