Logicism's role in modern mathematics: A formal verification approach

This study investigates the applicability of logicism in contemporary mathematics, focusing on the explanatory power of Whitehead and Russell’s Principia Mathematica through formal verification techniques. By analyzing key theorems in elementary number theory, real analysis, and combinatorics, the research evaluates logicism’s ability to derive mathematical truths using purely logical axioms. Results show significant variability across domains: while the Four Color Theorem in combinatorics achieves a high logical score of 0.81, indicating strong alignment with logical principles, the Fundamental Theorem of Arithmetic (0.63) and the Intermediate Value Theorem (0.43) demonstrate greater dependence on mathematical axioms. These findings highlight logicism’s enduring influence in discrete mathematics but also reveal its limitations in continuous structures. The study concludes that logicism remains a valuable framework for understanding mathematical foundations, offering insights for the development of formal verification tools and guiding future mathematical practices in the digital age. This research underscores the need for tailored proof assistants and highlights the importance of balancing logical purity with mathematical rigor in an increasingly automated world.