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.^{[1]} Bertrand Russell and Alfred North Whitehead championed this theory fathered by Richard Dedekind and Gottlob Frege. Dedekind's path to logicism had a turning point when he was able to reduce the theory of real numbers to the rational number system by means of set theory. This and related ideas convinced him that arithmetic, algebra and analysis were reducible to the natural numbers plus a "logic" of sets; furthermore by 1872 he had concluded that the naturals themselves were reducible to sets and mappings. It is likely that other logicists, most importantly Frege, were also guided by the new theories of the real numbers published in the year 1872. This started a period of expansion of logicism, with Dedekind and Frege as its mains exponents, which however was brought to a deep crisis with the discovery of the classical paradoxes of set theory (Cantor 1896, Zermelo and Russell 19001901). Frege gave up on the project after Russell recognized and communicated his paradox exposing an inconsistency in naive set theory. Russell and Whitehead continued on with the project in their Principia Mathematica.^{[2]}
Today, the bulk of modern mathematics is believed to be reducible to a logical foundation using the axioms of ZermeloFraenkel 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). Thus to some extent Dedekind's project was proved viable, but in the process the theory of sets and mappings came to be regarded as transcending pure logic.
Kurt Gödel's incompleteness theorem is sometimes alleged to undermine Logicism because it shows that if mathematics is consistent, some of its theorems are not derivable; since firstorder logic is both consistent and all of its theorems are derivable, mathematics cannot be a simple extension of firstorder logic. 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.
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Neologicism describes a range of views claiming to be the successor of the original logicist program. ^{[3]} More narrowly, it is defined as attempts to resurrect Frege's programme through the use of Hume's Principle.^{[4]} Two of the major proponents of neologicism are Crispin Wright and Bob Hale.^{[5]}
