Principia Mathematica
Principia Mathematica is a three-volume work on the logical foundations of mathematics, authored by Alfred North Whitehead and Bertrand Russell and published between 1910 and 1913. The work attempts to derive all of classical mathematics from a small set of logical axioms and inference rules, using a formal symbolic language developed in part from Gottlob Frege's predicate logic and Giuseppe Peano's notation. It is among the most ambitious formal projects in the history of logic and philosophy of mathematics, and its influence extends into mathematical logic, analytic philosophy, computer science, and the philosophy of language.
The central project of Principia Mathematica is logicism - the thesis that mathematics is reducible to logic. To accomplish this, Whitehead and Russell constructed an elaborate type-theoretic hierarchy intended to block the paradoxes, particularly Russell's Paradox, that had destabilized naive set theory. The system employs the theory of types to prevent self-referential constructions, and introduces ramified type theory along with the controversial Axiom of Reducibility, which many logicians have regarded as an ad hoc concession that undermines the purity of the logicist program. The work also introduces the no-classes theory, treating classes as logical fictions rather than genuine objects.
In 1931, Kurt Gödel published his incompleteness theorems, demonstrating that any consistent formal system expressive enough to capture basic arithmetic cannot prove all truths expressible within it, and cannot prove its own consistency. This result is generally taken to place fundamental limits on the Principia program, though the precise implications for logicism as a philosophical thesis remain a matter of debate. Separately, Fregean logicism had already been severely undermined by Russell's discovery of the paradox in Frege's Grundgesetze; Principia represents the most sustained attempt to rebuild it. Later developments in set theory, particularly the Zermelo-Fraenkel axiom system, offered an alternative foundation that most working mathematicians adopted in preference to the type-theoretic framework of Principia.
The notation and methods of Principia Mathematica were foundational to the development of formal logic in the twentieth century, influencing Wittgenstein's early work, the Vienna Circle's program of logical empiricism, and the emergence of theoretical computer science through figures such as Alan Turing and Alonzo Church.
Consensus Status
There is broad consensus in mathematical logic that Gödel's incompleteness theorems constrain any program of the type Principia Mathematica represents; see Mathematical Logic Consensus. Whether this refutes logicism as a philosophical position, or merely requires its revision, is not settled by that consensus and is addressed on the Debate page.
Viewpoints
- Logicism vindicated (qualified): Some philosophers of mathematics, following neo-logicist programs such as that of Crispin Wright and Bob Hale, argue that a suitably revised logicism survives Gödel's results and the failure of Principia's original formulation. Neo-Logicist Viewpoint
- Logicism refuted: Others hold that the incompleteness theorems and the need for the Axiom of Reducibility together demonstrate that mathematics cannot be derived from pure logic, and that the Principia project failed on its own terms. Logicism Refuted Viewpoint
- Formalism: A competing foundational view holds that mathematics is the manipulation of formal symbols according to stipulated rules, without requiring reduction to logic or reference to abstract objects. Formalist Viewpoint
- Platonism/Mathematical realism: Mathematical realists argue that both logicism and formalism misconceive the nature of mathematical objects, which exist independently of any formal system. Mathematical Platonism Viewpoint
- Intuitionism: Intuitionists reject the classical logic on which Principia relies, particularly the law of excluded middle as applied to infinite domains, offering a constructivist alternative. Intuitionist Viewpoint
Related Pages
Footnotes
- Whitehead, Alfred North, and Bertrand Russell. Principia Mathematica. 3 vols. Cambridge: Cambridge University Press, 1910-1913.
- Gödel, Kurt. “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I.” Monatshefte für Mathematik und Physik 38 (1931): 173-198.
- Russell, Bertrand. “Letter to Frege.” 16 June 1902. In Jean van Heijenoort, ed., From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Cambridge, MA: Harvard University Press, 1967. pp. 124-125.
- Irvine, Andrew David. “Principia Mathematica.” Stanford Encyclopedia of Philosophy. First published 1 May 2003; substantive revision 2 December 2021. https://plato.stanford.edu/entries/principia-mathematica/
- Wright, Crispin. Frege's Conception of Numbers as Objects. Aberdeen: Aberdeen University Press, 1983.
- Zermelo, Ernst. “Untersuchungen über die Grundlagen der Mengenlehre I.” Mathematische Annalen 65 (1908): 261-281.