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, created by mathematicians 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 main exponents, which however was brought to a deep crisis with the discovery of the classical paradoxes of set theory (Cantor 1896, Zermelo and Russell 1900–1901). Frege gave up on the project after Russell recognized and communicated his paradox exposing an inconsistency in naive set theory. On the other hand, Russell wrote The Principles of Mathematics in 1903 using the paradox and developments of Giuseppe Peano's school of geometry. Since he treated the subject of primitive notions in geometry and set theory, this text is a watershed in the development of logicism. Evidence of the assertion of logicism was collected by Russell and Whitehead in their Principia Mathematica.Today, the bulk of modern mathematics is believed to be reducible to a logical foundation using the axioms of Zermelo-Fraenkel 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 no particular axiomatization of mathematics can decide all statements. However, the basic spirit of logicism remains valid, as that theorem is proved with logic just like other theorems.Logicism was key in the development of analytic philosophy in the twentieth century.