Logicism

Logicism is the thesis that mathematics can be derived from logic, with mathematical concepts defined in logical terms. Its major historical programs sought to reduce arithmetic and other mathematics to logical principles.

Gottlob Frege: German philosopher and logician who developed modern predicate logic and sought to derive arithmetic from logic. His work established the first systematic logicist foundation for arithmetic.
First-order logic: A formal logic with quantifiers over objects, predicates, variables, and rules of inference. Fregean logicism uses quantified predicates to express mathematical relations and definitions.
Intuitionism: A philosophy of mathematics that treats mathematical objects as constructions of the mind and rejects some classical inferences. It denies that classical logic alone captures legitimate mathematical reasoning.
Russell's paradox: A contradiction arising from the collection of all sets that do not contain themselves. It exposed inconsistency in Frege’s system and forced logicists to revise their foundations.
The Foundations of Arithmetic: Frege’s 1884 book arguing that arithmetic is analytic and grounded in logic. It states the philosophical case for deriving arithmetic from logical definitions and principles.
Second-order logic: A logic that quantifies over predicates or properties as well as individual objects. Frege’s arithmetic depends on second-order principles stronger than ordinary first-order logic.
Formalism: A philosophy of mathematics that emphasizes formal symbol systems and rule-governed proofs. It explains mathematics through formal manipulation rather than reduction to logical meaning.
Type theory: A family of formal systems that organizes expressions into types to prevent certain forms of paradox. Russell and Whitehead used a type hierarchy to block the paradox that damaged Frege’s system.
The Basic Laws of Arithmetic: Frege’s two-volume work presenting his formal system and derivation of arithmetic. It carries Frege’s logicist program into a detailed formal theory.
Hume's Principle: The principle that the number of F objects equals the number of G objects exactly when F and G are equinumerous. It supports a reconstruction of arithmetic from abstraction over concepts and logical equivalence.

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