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Logicism is the theory that mathematics is an extension of logic and therefore all mathematics is reducible to logic. Modern philosophers believed that proof of this theory was the means of banishing the befuddlement of natural language and metaphysics from philosophical argumentation. 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 with success except for the paradox of trying to formulate a logical definition of natural numbers in terms of classes.
Kurt Gödel's incompleteness theoremIn mathematical logic, Godel's incompleteness theorems are two celebrated theorems proved by Kurt Godel in 1930. Somewhat simplified, the first theorem states: In any consistent formalization of mathematics that is sufficiently strong to define the concep ultimately undermined the purpose of the project. The attempted resurrection of this theory is styled neo-logicism.