Frege

Gottlob Frege

Working in near-obscurity as a professor of mathematics at Jena, Gottlob Frege invented modern logic and founded analytic philosophy almost single-handedly. His new logic was the greatest advance in the subject since Aristotle; his distinction between the sense and the reference of an expression reshaped the philosophy of language; his program for grounding arithmetic in logic set the agenda for a century of work on the foundations of mathematics. Almost no one read him while he lived. Then Bertrand Russell found him, and the young Wittgenstein, and Carnap, and through them Frege became the source from which the whole analytic tradition flows.

The corpus and how the chat answers. Frege wrote in German, and his major works reach English only through twentieth-century translations that remain under copyright (J. L. Austin's Foundations of Arithmetic of 1950, Max Black's "On Sense and Reference," the Geach and Black readings). The German originals are public domain, the Begriffsschrift (1879), Die Grundlagen der Arithmetik (1884), and the great essays of 1891 and 1892, as are Philip Jourdain's 1915 to 1917 partial translations in The Monist, and a later pass could add the German originals in the site's original-language practice. For now this room answers from training knowledge, says so plainly, and cites Frege's works by title rather than by a retrieved passage. Expect citations to The Foundations of Arithmetic, "On Sense and Reference," and the other works by title.

Life. Frege was born in 1848 in Wismar, on the Baltic coast, studied at Jena and at Göttingen, and then spent his entire working life as a mathematician at the University of Jena, producing in quiet isolation a body of work whose importance the philosophical world of his day very nearly missed entirely. His writings found their first true readers in Bertrand Russell and the rising generation of logicians, and the recognition that eluded him in his own time arrived in full afterward, when it became clear that he had reinvented the discipline.

Where to begin. Start with The Foundations of Arithmetic (1884), his most readable book, which lays out the philosophy of logicism in graceful prose with almost no symbolism. Move then to "On Sense and Reference" (1892) for the philosophy of language, to the Begriffsschrift (1879) for the birth of modern logic, and to the late essay "The Thought" (1918) for the metaphysics behind it all.

The invention of modern logic. The Begriffsschrift, the "concept-script" of 1879, created in a single work the first complete system of quantificational logic. Aristotle's syllogistic had been helpless before statements of multiple generality, sentences with nested every and some, like "every boy loves some girl"; Frege's analysis of a sentence into function and argument, together with his invention of the quantifier and the bound variable, handled them with ease. It was the largest advance in logic in two thousand years, and it remains the foundation of formal logic, the foundations of mathematics, and computer science.

Function and argument. Frege swept away the old analysis of a sentence into grammatical subject and predicate and put in its place the mathematician's analysis into function and argument. A concept, for Frege, is a function that maps objects to truth-values: the concept of a horse maps each object to the True or the False according to whether it is a horse. This sharp distinction between concept and object, and the idea that a concept-expression is "unsaturated" until completed by an argument, underlies his entire logic and much of the philosophy of language that followed.

Logicism. The thesis of The Foundations of Arithmetic is that arithmetic is reducible to logic. Against Kant, who held that arithmetic is synthetic a priori, against Mill, who held it an empirical generalization, and against the psychologistic logicians, who reduced numbers to ideas in the mind, Frege argued that the truths of arithmetic are analytic, derivable from logic alone, and that numbers are objective things, neither mental nor physical. A number, he proposed, belongs to a concept, and two concepts have the same number when their objects can be paired off one to one. Two principles guide the book and the tradition after it: a fierce anti-psychologism that separates the logical from the merely psychological, and the context principle, that one should never ask for the meaning of a word in isolation but only in the setting of a whole sentence.

Sense and reference. Frege's most influential gift to the philosophy of language came in 1892. Every meaningful expression, he argued, has both a reference, the object it stands for, and a sense, the mode under which that object is presented. "The morning star" and "the evening star" share a single reference, the planet Venus, but differ in sense, which is why the identity "the morning star is the evening star" carries real information while "the morning star is the morning star" carries none. Sense is objective and shareable, not a private mental picture. The reference of a whole declarative sentence, Frege held, is its truth-value, the True or the False, and its sense is the thought that it expresses.

The thought, and Russell's paradox. In "The Thought" (1918) Frege argued that thoughts, the senses of sentences, are objective and independent of any mind, inhabiting a third realm distinct from outer things and inner ideas, the timeless bearers of truth. It is a bold Platonism about logic and the final blow in his long war on psychologism. And here falls the great reversal of his life: as Frege was completing the formal system of his Basic Laws of Arithmetic to carry logicism through, Bertrand Russell wrote to him in 1902 with a paradox, the set of all sets that are not members of themselves, that Frege's own basic law permitted, and a contradiction opened at the foundation of the whole structure. Frege's published reply, acknowledging that the ground had shifted beneath his life's work, is one of the most admired acts of intellectual honesty in the history of philosophy.

Reception, and adjacent reading. Almost unread in his own day, Frege became the fountainhead of analytic philosophy and of modern logic. Read now, he is the thinker who taught philosophy to analyze language and thought with mathematical precision. He set out to realize the old dream of Gottfried Leibniz of a universal logical calculus and very nearly did; he wrote against the synthetic a priori arithmetic of Immanuel Kant; Bertrand Russell inherited his logicism, nearly shattered it with the paradox that bears his name, and carried the project forward; the young Ludwig Wittgenstein took Frege's logic and his notion of sense as the starting point of the Tractatus; his campaign against psychologism reshaped the phenomenology of Edmund Husserl; and the later analytic tradition down to W. V. O. Quine is unthinkable without him. Every time anyone writes "for all x" or "there exists an x," they write in Frege's hand.