Gottlob Frege

Key ideas

1. A new logic. Frege analysed sentences as functions applied to arguments, not as subject and predicate, and introduced quantifiers that can be nested. His Begriffsschrift of 1879 set out a formal system of quantificational logic with explicit rules of proof.
2. Logicism. Arithmetic, Frege held, is part of logic: its concepts can be defined and its truths proved from logical laws alone, so its truths are analytic, not synthetic as Kant thought.
3. Numbers belong to concepts. A statement of number is about a concept: “There are eight planets” says that the concept planet has eight instances. Numbers themselves are objects, not ideas in anyone’s mind.
4. The context principle. Frege’s rule in the Foundations of Arithmetic is never to ask for the meaning of a word in isolation, only in the context of a proposition.
5. Sense and reference. An expression has a reference, the thing it stands for, and a sense, the way that thing is presented. The distinction explains how an identity such as the morning star is the evening star can be informative.
6. Thoughts and the third realm. The sense of a sentence is a thought, which is objective and timeless: neither a physical thing nor an idea in a mind. Frege drew a sharp line between logic and psychology.
7. Basic Law V and the paradox. Russell showed in 1902 that Frege’s Basic Law V, which governs extensions of concepts, yields a contradiction. The paradox ended the logicist system of the Basic Laws of Arithmetic.

Gottlob Frege (1848–1925) was a German mathematician and philosopher who invented modern logic and, in doing so, gave analytic philosophy much of its method and many of its problems. He is the source of the distinction between sense and reference, of the attempt to derive arithmetic from logic, and of the system in which Bertrand Russell found his paradox. He was largely ignored in his lifetime.

He was born in Wismar on the Baltic coast, studied at Jena, where Ernst Abbe became his mentor, and took a doctorate in mathematics at Göttingen in 1873. He spent his career at Jena, as extraordinary professor from 1879, the year of his first book. He retired near the end of the First World War and died at Bad Kleinen, in Mecklenburg, in 1925.

What did Frege invent?

The traditional logic descended from Aristotle analysed sentences as subject and predicate, and could not display the difference between the two readings of Everyone admires someone: that each person admires somebody or other, or that one person is admired by everyone. In the Begriffsschrift (“concept-script”, 1879) Frege treated predicates as functions from objects to truth-values and introduced quantifiers that can be nested, so that the two readings receive different symbolism. The book set out a formal system with explicit rules of proof, partly to realise the idea of Leibniz of a universal formal language. Its two-dimensional notation, which contemporaries found hard to read, did not catch on. See logic and propositional logic.

What was logicism?

Kant had held that arithmetic is synthetic. Frege held that it is analytic, derivable from the laws of logic alone, and argued so in The Foundations of Arithmetic (1884). A statement of number, he said, is about a concept: There are eight planets says that the concept planet has eight instances. Numbers are objects, not ideas in anyone’s head, a Platonism he defended against psychologism. What is now called Hume’s principle says that the number of Fs equals the number of Gs if and only if the Fs and Gs can be paired one to one. The book also gives the context principle: “never ask for the meaning of a word in isolation, but only in the context of a proposition.” See analytic and synthetic and a priori and a posteriori.

What is the difference between sense and reference?

In “On Sense and Reference” (1892) Frege asked how an identity can be informative. The morning star is the evening star tells us something; The morning star is the morning star does not, yet both name Venus. His answer is that each expression has a reference, the thing it stands for, and a sense, a mode of presentation of it. A sentence’s sense is a thought and its reference a truth-value. In reports of belief a name refers to its usual sense, so swapping coreferring names can change the truth-value. See sense and reference.

What are thoughts?

In “The Thought” (1918) Frege held that thoughts exist in a timeless “third realm”, apart from both the mental and the physical. He insisted on a sharp line between logic and psychology, a line Husserl also drew, though how far Husserl owed it to him is disputed.

What did Russell’s letter show?

On 16 June 1902, as the second volume of the Basic Laws of Arithmetic was being printed, Russell wrote that a contradiction could be derived in the system of the first. The source was Basic Law V, which says that two concepts have the same extension just in case they apply to the same things. Consider the extension of the concept being an extension that is not a member of itself: it belongs to itself if and only if it does not. Frege, “thunderstruck” in his own word, added an appendix in which he wrote: “Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished.” His repairs failed, and in 1924 he gave up logicism for a Kantian account of arithmetic based on intuition of space.

Much survives, however. If Hume’s principle replaces Basic Law V, the axioms of arithmetic can still be derived. This is Frege’s theorem, which Crispin Wright, George Boolos and others developed into neo-logicism.

What was his influence?

Russell’s Principles of Mathematics (1903) brought Frege wider attention. Wittgenstein visited him at Jena in 1911 and Rudolf Carnap attended his lectures. Wittgenstein and Russell carried his logic into analytic philosophy, and Michael Dummett cited the Foundations as the place to pinpoint the linguistic turn. His ideas remain the base of philosophy of language. UGC NET Unit 6 lists his sense and reference; see the UGC NET guide.

A private diary of 1924, published only in 1994, expresses hostility to Jews and wishes that they would leave Germany or at least lose certain political rights. Dummett, who had admired Frege, was shocked. None of this enters his logic, but an honest account records it.

How to start reading Frege

No Frege text is in this site’s library, because the standard English translations are modern and not free online. They include J. L. Austin’s translation of the Foundations and Michael Beaney’s The Frege Reader (1997). Start with the introduction to the Foundations, then read “On Sense and Reference” and “The Thought”, and leave the Basic Laws until last.