Logicism

Logicism 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.

Kurt Gödel's incompleteness theorem is sometimes alleged to undermine the purpose of the project.

Logicism was key in the development of Analytic philosophy in the twentieth century.

Neo-logicism
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. One of the major proponents of neo-logicism is Crispin Wright.