Frege, Russell, and Logicism

Logicism holds that mathematics, or at least arithmetic, reduces entirely to logic: mathematical concepts are definable in purely logical terms, and mathematical theorems are logical truths derivable from logical axioms and definitions alone, by logical inference, without any further mathematical assumption. Gottlob Frege inaugurated the modern program in the Begriffsschrift (1879), inventing modern quantificational logic expressly as a tool for this reduction, then pursued it programmatically in Die Grundlagen der Arithmetik (1884), and finally formally, with full technical apparatus, across the two volumes of Grundgesetze der Arithmetik (1893, 1903).