Introduction | History of Logicism
Logicism is the theory in Logic and Philosophy of Mathematics that mathematics is 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. It does not make any claims about whether or not logical system are formalizable, nor about how mathematical truths are discovered or applied.
Logicism was key in the development of the Analytic Philosophy movement in the 20th Century.
Although it is generally agreed that set theory is required for modern mathematics, it has been argued that the existence of sets is, however, not logically necessary, and the truths of mathematics can therefore be considered contingent. The Incompleteness Theorems of Kurt Gödel (1906 - 1978), which point out the limitations of all but the most trivial formal mathematical systems, are sometimes alleged to have impacted on the credibility of Logicism.
The theory was fathered by Gottlob Frege, although he later abandoned it after Bertrand Russell pointed out a paradox exposing an inconsistency in Frege's naive set theory. 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 proponents actually used the term "logicism", which was applied retroactively.
There were subsequent attempts, known as Neo-Logicism, particularly by the British philospher Crispin Wright (1942 - ), to resurrect Frege's theory through the use of Frege's own Hume's Principle.