Neo-logicism is a position in the philosophy of mathematics that attempts to revive and rehabilitate the logicist program originally advanced by Gottlob Frege in the late nineteenth century. Logicism held that mathematics - particularly arithmetic - could be derived from purely logical principles alone, without appeal to intuition, geometric construction, or empirical observation. Frege's original program collapsed when Bertrand Russell identified a contradiction in Frege's Basic Law V in 1902, now known as Russell's Paradox. Neo-logicism, developed primarily in the late twentieth century, seeks to recover the logicist ambition by identifying which of Frege's principles can be retained or modified to yield a consistent and mathematically adequate foundation.
Frege's Grundgesetze der Arithmetik (1893, 1903) attempted to derive arithmetic from a small set of logical axioms and inference rules. The fatal flaw was Basic Law V, which asserts that two concepts have the same extension if and only if they apply to exactly the same objects. Russell's Paradox showed this generates a contradiction. In the decades following, alternative foundational programs - set theory (Zermelo-Fraenkel), intuitionism (Brouwer), and formalism (Hilbert) - largely displaced logicism in technical foundations research.
The neo-logicist revival centers on a result noted by Charles Parsons and developed by Crispin Wright and Bob Hale beginning in the 1980s.(1) Wright's Frege's Conception of Numbers as Objects (1983) argued that arithmetic can be derived from Hume's Principle - the claim that the number of Fs equals the number of Gs if and only if the Fs and Gs can be put into one-to-one correspondence - together with standard second-order logic. This derivation is sometimes called Frege's Theorem. Unlike Basic Law V, Hume's Principle has not been shown to be inconsistent, and it suffices to derive the Dedekind-Peano axioms of arithmetic.
The program associated primarily with Wright and Hale is sometimes called abstractionism or the Scottish school of neo-logicism, and it has generated substantial technical and philosophical literature. Related but distinct neo-logicist proposals have been advanced by George Boolos, Kit Fine, and others, some more sympathetic to the program's ambitions and some more skeptical.
Several contested questions structure the neo-logicist debate:
The status of Hume's Principle. Whether Hume's Principle qualifies as a logical truth, an analytic truth, or merely a stipulative definition is disputed. Wright and Hale argue it is an abstraction principle whose truth can be grasped through understanding the concept of number. Critics, including Boolos and Michael Dummett, have questioned whether a principle with such substantial mathematical consequences can plausibly count as logical or analytic. See Neo-Logicism - Debate.
The Bad Company problem. Hume's Principle is not the only abstraction principle one might propose; some analogously structured principles are inconsistent (Basic Law V) or mutually incompatible. The question of what distinguishes acceptable from unacceptable abstraction principles - the so-called bad company problem - remains active in the literature. See Neo-Logicism - Abstraction Principles Debate.
Ontological commitments. Neo-logicism, following Frege, treats numbers as objects - specifically, as the extensions (or value-ranges) of concepts. Whether this platonist commitment is defensible, and whether it is genuinely derived from logic alone, is a core philosophical dispute. Nominalist and structuralist philosophers of mathematics contest the ontological picture. See Philosophy of Mathematics - Ontology Viewpoint.
Scope beyond arithmetic. Frege's original program aimed to cover all of mathematics, including real analysis. Extending neo-logicism beyond arithmetic to real and complex analysis, and eventually to set theory, has proven technically difficult and remains an open research program. Extensions by Wright, Hale, and others have been proposed but are not universally regarded as successful.
There is no broad consensus among philosophers of mathematics on the success or viability of neo-logicism. The derivability of arithmetic from Hume's Principle plus second-order logic (Frege's Theorem) is not seriously disputed as a technical result, but its philosophical significance - whether it vindicates logicism, analyticity, or platonism - is actively contested. See Neo-Logicism - Philosophy Consensus if available.
1. Charles Parsons, “Frege's Theory of Number,” in Philosophy in America, ed. Max Black (Ithaca: Cornell University Press, 1965), pp. 180-203. The result was developed into a full derivation by Wright and subsequently given the name Frege's Theorem in the secondary literature. 2. Gottlob Frege, Grundgesetze der Arithmetik, vol. 1 (Jena: Pohle, 1893); vol. 2 (Jena: Pohle, 1903). 3. Crispin Wright, Frege's Conception of Numbers as Objects (Aberdeen: Aberdeen University Press, 1983). 4. George Boolos, “On the Proof of Frege's Theorem,” in Logic, Logic, and Logic (Cambridge, MA: Harvard University Press, 1998), pp. 275-290. 5. Bob Hale and Crispin Wright, The Reason's Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics (Oxford: Oxford University Press, 2001). 6. Kit Fine, The Limits of Abstraction (Oxford: Oxford University Press, 2002). 7. Stewart Shapiro, Philosophy of Mathematics: Structure and Ontology (Oxford: Oxford University Press, 1997). 8. Michael Dummett, Frege: Philosophy of Mathematics (London: Duckworth, 1991).