Logicism is the position in the Philosophy of Mathematics that mathematics, or at least a significant portion of it, is reducible to logic-that mathematical truths are logical truths and mathematical concepts can be defined in purely logical terms. The term is most closely associated with the program pursued by Gottlob Frege and later by Alfred North Whitehead and Bertrand Russell in Principia Mathematica, though what counts as “logic” for purposes of the reduction, and whether such a reduction succeeds, are themselves contested questions; see logicism-viewpoint and Principia Mathematica - Logicism Refuted Viewpoint.

Classical logicism, as developed by Frege and later by Whitehead and Russell, aimed to derive the truths of arithmetic and, by extension, much of mathematics from purely logical axioms and definitions, without appeal to specifically mathematical primitives or intuition. Frege's project foundered when Russell identified a contradiction (Russell's Paradox) in Frege's system of unrestricted set formation. Whitehead and Russell's subsequent attempt in Principia Mathematica avoided this contradiction through type theory, but required additional axioms-notably the Axiom of Reducibility, the axiom of infinity, and the axiom of choice-whose status as purely logical truths remains disputed.

Whether these supplementary axioms count as “logical” bears directly on whether the Principia Mathematica program achieved a reduction of mathematics to logic in the sense originally intended, or instead reduced mathematics to a system that smuggles in mathematical or empirical assumptions under a logical guise. This question remains unsettled and is treated at length in Principia Mathematica - Logicism Refuted Viewpoint.

A distinct contemporary program, Neo-Logicism, associated with Crispin Wright and Bob Hale, attempts to recover a logicist reduction of arithmetic using Hume's Principle-a single abstraction principle-within second-order logic, building on a theorem (Frege's Theorem) showing that the Peano axioms are derivable from Hume's Principle in second-order logic. Whether Hume's Principle and second-order logic itself qualify as logical, analytic, or definitional in the relevant sense is debated. For historical background on the development of logicism from Frege through Principia Mathematica and beyond, see logicism-history.

• logicism-classical-viewpoint - Holds that the core program of Frege, Whitehead, and Russell succeeded, or nearly succeeded, in showing mathematics to be a branch of logic, and that residual difficulties (such as the axiom of reducibility) are repairable without abandoning the logicist thesis.
• Principia Mathematica - Logicism Refuted Viewpoint - Holds that Principia Mathematica's reliance on the axiom of reducibility and other non-logical axioms shows that the logicist reduction was not achieved, and that mathematics was instead reduced to a system with covert mathematical or empirical content.
• neo-logicism-viewpoint - Holds that a logicist reduction of arithmetic (and potentially other branches of mathematics) can be recovered through abstraction principles such as Hume's Principle within second-order logic, even if the original Fregean or Russellian programs failed.
• logicism-structuralist-alternative-viewpoint - Holds that mathematics is better understood through structuralism, formalism, or other non-logicist foundational programs, and that the logicist reduction-classical or neo-is either unnecessary or unsuccessful.
• logicism-godelian-skeptical-viewpoint - Holds that Gödel's incompleteness theorems undermine the broader logicist ambition of grounding all of mathematics in a single complete logical system, regardless of the status of any particular axiom.

• Logical status of the axiom of reducibility - Whether the axiom of reducibility, and similar supplementary axioms in Principia Mathematica, qualify as logical truths or instead smuggle in non-logical content remains disputed; see axiom-of-reducibility-logical-status-debate.

• logicism-history - History
• logicism-classical-vs-principia-mathematica-logicism-refuted-viewpoint-debate - Viewpoint Debate
• axiom-of-reducibility-logical-status-debate - Debate
• Philosophy of Mathematics - Related Main Topic
• Russell's Paradox - Related Main Topic
• Neo-Logicism - Related Main Topic

1. Gottlob Frege, *Die Grundlagen der Arithmetik* (Breslau: Wilhelm Koebner, 1884).
2. Alfred North Whitehead and Bertrand Russell, *Principia Mathematica*, 3 vols. (Cambridge: Cambridge University Press, 1910-1913).
3. Bertrand Russell, *Introduction to Mathematical Philosophy* (London: George Allen & Unwin, 1919).
4. Crispin Wright, *Frege's Conception of Numbers as Objects* (Aberdeen: Aberdeen University Press, 1983).
5. Bob Hale and Crispin Wright, *The Reason's Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics* (Oxford: Clarendon Press, 2001).
6. Kurt Gödel, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I," *Monatshefte für Mathematik und Physik* 38 (1931): 173-198.