logicism

Logicism is the view that mathematics can be reduced to logic, meaning that mathematical truths are logical truths derivable from the laws of logic and basic definitions rather than from a separate mathematical ontology. It arose in the late 19th and early 20th centuries as an attempt to secure a rigorous, unified foundations for mathematics.

Gottlob Frege argued that arithmetical truths could be grounded in predicate logic, defining numbers and numerical relations in purely logical terms. His program faced a fatal obstacle when Bertrand Russell discovered a paradox in Frege’s system, revealing an inconsistency that undermined the project of a straightforward logical foundation.

In response, Bertrand Russell and Alfred North Whitehead attempted to rebuild mathematics on a formal logical basis in Principia Mathematica (1910–1913). They sought to derive much of mathematics from a relatively small set of logical axioms and definitions, using a sophisticated type-theoretic framework to avoid known paradoxes and to illuminate how mathematical objects could be constructed from logic.

The logicist project faced a decisive challenge with Kurt Gödel’s incompleteness theorems (1931). Gödel showed that any sufficiently powerful, consistent formal system capable of expressing basic arithmetic cannot be both complete and provable within itself, and cannot prove its own consistency. This result undermined the aspiration of a complete, purely logical foundation for all of mathematics.

Today, most foundational work favors set theory or type theory, and mathematics is typically formalized within those frameworks. Nevertheless, logicism influenced the development of formal logic and the philosophy of mathematics, contributing to ongoing discussions about the nature of mathematical truth and the prospects for reductionism.