What goal did Alfred North Whitehead and Bertrand Russell pursue in Principia Mathematica?

In Principia Mathematica (1910–1913), Alfred North Whitehead and Bertrand Russell pursued the goal of laying bare the logical foundations of mathematics. Specifically, they sought to demonstrate that the basic concepts and modes of reasoning used in mathematics are entirely definable in logical terms.12

This project was a monumental effort to establish logicism—the thesis that all pure mathematics can be derived from a few simple axioms based on purely logical notions, such as propositions and classes, rather than specifically mathematical ones like numbers. By reducing arithmetic to logic, they aimed to show that the entirety of mathematics could be similarly reduced. To address contradictions like Russell’s paradox, they implemented a complex "ramified" theory of types and the axiom of reducibility to ensure their system could derive standard mathematical results. While the work was a significant influence on mathematical logic, it remains a subject of debate regarding its philosophical success.132