Logicism is an early 20th Century philosophical and mathematical movement, initially developed in the late 19th Century by the German mathematician and logician Gottlob Frege. It is based on the premise that mathematics is just an extension of Logic, and therefore that some or all mathematics is reducible to logic. It effectively holds that mathematical theorems and truths are logically necessary or analytic. For more details, see the section on the doctrine of Logicism.
Although the movement was fathered by Gottlob Frege, he later abandoned it after Bertrand Russell pointed out a paradox exposing an inconsistency in Frege's naive set theory. The Incompleteness Theorems of Kurt Gödel (1906 - 1978), which point out the limitations of all but the most trivial formal mathematical systems, also impacted on the credibility of Logicism to some extent.
Russell and Alfred North Whitehead, however, continued to champion the theory in their ground-breaking "Principia Mathematica", which was published in 1910 - 1913. None of these early proponents actually used the term "logicism", which was only applied retroactively.
There were subsequent attempts, known as Neo-Logicism, to resurrect Frege's theory through the use of Frege's own Hume's Principle. The British philosopher Crispin Wright (1942 - ) was a strong advoicate.
Logicism, along with Logical Positivism, was key in the development of the Analytic Philosophy movement later in the 20th Century.