is one of the schools of thought in the philosophy of mathematics
, putting forth the theory that mathematics
is an extension of logic
and therefore some or all mathematics is reducible
to logic. Bertrand Russell
and Alfred North Whitehead
championed this theory fathered by Gottlob Frege
. Frege gave up on the project after Russell recognized a paradox
exposing an inconsistency in naive set theory
. Russell and Whitehead continued on with the project in their Principia Mathematica
. Today, the bulk of modern mathematics is believed to be reducible to a logical foundation using the axioms of Zermelo-Fraenkel set theory
(or one of its extensions, such as ZFC
), which has no known inconsistencies (although it remains possible that inconsistencies in it may still be discovered).
Kurt Gödel's incompleteness theorem is sometimes alleged to undermine the purpose of the project. (However, one can argue that the basic spirit of Logicism remains valid, though in a somewhat less powerful sense than was originally thought.)
Logicism was key in the development of Analytic philosophy in the twentieth century.
Neo-logicism describes a range of views claiming to be the successor of the original logicist program. More narrowly, it is defined as attempts to resurrect Frege's programme through the use of Hume's Principle. Two of the major proponents of neo-logicism are Crispin Wright and Bob Hale.