Foundations of Arithmetic

Introduction

The Foundations of Arithmetic (Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl) is Gottlob Frege's 1884 informal statement of the logicist program — the philosophical case that arithmetic can be derived from purely logical principles. Written five years after the formal Begriffsschrift (1879) had introduced modern predicate logic, the Grundlagen deliberately set aside formal apparatus to make the philosophical case accessible to a wider audience of mathematicians and philosophers.

The book has three parts. The first criticizes the prevailing accounts of number (Mill's empiricism, Kant's appeal to intuition, the formalist treatments of Hankel and Heine); the second develops Frege's positive analysis, defining number in terms of the extensions of concepts; the third addresses objections and considers the philosophical consequences. The argument is conducted in clear German prose with extensive use of examples and few technical symbols, and the result is one of the most lucid pieces of mathematical philosophy in the German tradition.

Composition and publication

Frege had completed the Begriffsschrift in 1879 to limited contemporary recognition. The reviews had been harsh or uncomprehending; the book had not produced the engagement Frege had hoped for. The Grundlagen was an attempt to make the philosophical motivations of the formal project intelligible to readers who could not work with the Begriffsschrift's symbolism.

The book was published by Wilhelm Koebner in Breslau in 1884. The contemporary reception was again modest — the book did not produce the response Frege had wanted — but the work attracted the attention of a few careful readers, most importantly Bertrand Russell, who recognized its importance in the late 1890s and wrote to Frege in 1902 with the paradox that would unravel the formal version of the logicist program.

The standard German text is the 1884 edition with the 1934 reissue by Felix Meiner Verlag. The dominant English translation is J. L. Austin's The Foundations of Arithmetic (Blackwell, 1950; revised 1953), which became the standard English text and remains in print. Michael Beaney's translation in The Frege Reader (Blackwell, 1997) provides the more recent scholarly alternative.

Central doctrines

The critique of psychologism

Frege opens with a sustained attack on psychologism in the philosophy of arithmetic — the position (defended in different forms by Mill, by certain neo-Kantians, and by the broader German tradition descending from Herbart) that the laws of arithmetic are descriptive of how human minds actually reason about quantity. The critique parallels Husserl's later attack on psychologism in the Logical Investigations (1900–1901), and the two together effectively ended psychologism as a serious option in the foundations of mathematics.

Frege's central objection: the laws of arithmetic are objective truths about numbers, not empirical generalizations about human reasoning. The number of planets is the same whether or not anyone is counting; the truth that 2 + 2 = 4 holds independently of any thinker. Psychologism confuses the conditions under which we come to know arithmetical truths with the truths themselves, and the confusion produces an account on which different psychological constitutions would warrant different arithmetics — a consequence Frege treats as a reductio of the position.

The definition of number

The positive doctrine is the analysis of cardinal number in terms of concepts. The number of Fs is the number that belongs to the concept F; the number that belongs to the concept F is the extension of the concept concept equinumerous with F. Two concepts are equinumerous if their extensions can be put in one-to-one correspondence (Frege's formulation: there is a relation that maps the Fs onto the Gs bijectively).

This allows zero to be defined as the number of the concept not identical with itself (which has no instances), one as the number of the concept identical with zero (which has exactly one instance), and the natural numbers to be defined recursively. The definitions ground arithmetic in concepts and their extensions; if extensions are logical entities, arithmetic reduces to logic.

Hume's Principle

A principle Frege treats as central, in section 63, has come to be called Hume's Principle in the contemporary literature (because Frege quotes Hume in introducing it): the number of Fs equals the number of Gs if and only if the Fs and the Gs are equinumerous. Frege uses the principle in the Grundlagen's informal argument but raises the worry that it does not uniquely determine the reference of number-terms (the Julius Caesar problem: can we tell from Hume's Principle alone whether the number 2 is identical with Julius Caesar?).

The principle has become important in contemporary philosophy of mathematics through the neo-logicist program of Crispin Wright and Bob Hale, who argue that arithmetic can be derived from Hume's Principle together with second-order logic, avoiding the inconsistency that destroyed Frege's later formal system. Wright's Frege's Conception of Numbers as Objects (1983) and the subsequent literature constitute one of the most active research programs in contemporary philosophy of mathematics.

The context principle

The Grundlagen contains the famous methodological injunction (formulated in the introduction): never ask for the meaning of a word in isolation, but only in the context of a proposition. The context principle has been continuously influential in the philosophy of language; Michael Dummett treated it as central to the analytic tradition's approach to meaning. The principle anticipates later semantic holism and has been engaged by writers as different as the later Wittgenstein and contemporary inferentialists.

Reception

The immediate reception was thin. The book attracted a few perceptive readers but did not produce the engagement Frege had wanted. Edmund Husserl's review (in his early psychologistic period) was unfriendly; most contemporary German mathematicians and philosophers ignored the work.

The recovery began with Bertrand Russell's recognition of Frege's importance in the late 1890s. Russell's Principles of Mathematics (1903) treated Frege as the major precursor of the logicist program; Russell's discovery of the paradox in Frege's formal system in 1902 simultaneously demonstrated the depth of Frege's project and forced its revision. The early-twentieth-century engagement through Wittgenstein, Carnap, and the Vienna Circle established Frege as a foundational figure.

Michael Dummett's Frege: Philosophy of Language (1973) and Frege: Philosophy of Mathematics (1991) gave the contemporary scholarly framework for engaging Frege. The neo-logicist program through Crispin Wright, Bob Hale, William Demopoulos, Richard Heck, and Marcus Rossberg has produced a contemporary research program. Joan Weiner's Frege in Perspective (1990) and Tyler Burge's Truth, Thought, Reason (2005) anchor the contemporary scholarly literature.

Place in the wiki

The Grundlagen is the foundational philosophical text of the logicist program and one of the most lucid pieces of mathematical philosophy in the German tradition. It is the principal source for Frege's critique of psychologism, his definition of number, the context principle, and the principle now called Hume's Principle that anchors the contemporary neo-logicist program.

Further reading

• Frege — the author
• Analytic Philosophy — the tradition Frege founded
• Russell — the successor who discovered the paradox and developed the logicist program
• Wittgenstein — the analytic figure whose Tractatus engages Frege extensively
• Kant — the philosophical predecessor whose account of arithmetic Frege opposes

Gottlob Frege's 1884 informal statement of the logicist program. The accessible philosophical case for the reduction of arithmetic to logic.