**Logicism** is a program in the philosophy of mathematics that suggests mathematics is either an extension of logic, can be reduced to logic, or can be represented in logical terms. Bertrand Russell and Alfred North Whitehead were key figures in promoting this idea, which was initially proposed by Gottlob Frege and later expanded upon by Richard Dedekind and Giuseppe Peano.

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Dedekind and Frege played key roles in the development of logicism. Dedekind's breakthrough came when he constructed a model representing real numbers using sets of rational numbers. This led him to believe that arithmetic, algebra, and analysis could be reduced to natural numbers plus a "logic" of classes. By 1872, he concluded that natural numbers themselves could be reduced to sets and mappings. Frege was similarly motivated by dissatisfaction with existing accounts of natural numbers and a rejection of Kant's view of them as synthetic a priori truths.

However, logicism faced challenges with the discovery of set theory paradoxes, which led Frege to abandon his project after Russell identified an inconsistency in Frege's system. Russell, on the other hand, continued to develop logicism in his book "The Principles of Mathematics" in 1903, using Peano's geometry developments and addressing set theory paradoxes. Russell and Whitehead further collected evidence for logicism in "Principia Mathematica."

Despite Gödel's incompleteness theorems challenging some aspects of logicism, it has remained influential in analytic philosophy. Today, while much of mathematics is believed to be logically derivable from axioms like those in set theory, logicism has evolved to recognize some aspects of mathematics as extralogical. However, debates continue regarding the extent to which logicism remains a valid program in light of Gödel's results. Nevertheless, logicism's influence on analytic philosophy has been significant throughout the twentieth century.

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