Gottlob Frege

Gottlob Frege was a German mathematician, logician and philosopher who created modern logic and laid the foundations of analytic philosophy. Working almost unnoticed at the University of Jena, he invented a formal language that could express mathematical reasoning with complete precision, tried to show that arithmetic could be derived from logic alone, and developed influential ideas about meaning and language. Little read in his lifetime, he is now counted among the most important philosophers of the last two centuries.

Friedrich Ludwig Gottlob Frege was born on 8 November 1848 in Wismar, a Baltic port in the Grand Duchy of Mecklenburg-Schwerin, in northern Germany. His father, Karl Alexander Frege, ran a private school for girls, which his mother continued to run after his father's death in 1866. Frege studied mathematics, physics, chemistry and philosophy at the University of Jena from 1869, and from 1871 at Göttingen, then the leading centre of mathematics in the German-speaking world. He received his doctorate there in 1873 for a dissertation on geometry, and in 1874 he qualified to teach at Jena, where he remained for the rest of his career, becoming an associate professor in 1879 and retiring in 1918.

In 1879 Frege published a short book called Begriffsschrift, meaning concept-script. It was the most important advance in logic since Aristotle, whose system of syllogisms had dominated the subject for over two thousand years. Aristotle's logic could handle simple statements such as "all men are mortal", but not many of the complex inferences mathematicians made. Frege replaced the old analysis of sentences into subject and predicate with one based on functions and arguments, borrowed from mathematics, and introduced quantifiers, symbols for "every" and "some", that could be combined and nested. The result, now called predicate logic, could express statements such as "every number has a successor" precisely and test reasoning step by step.

His notation, laid out in two dimensions on the page, was hard to print and read, and contemporaries largely ignored or misunderstood it. Logicians later adopted a different notation developed by the Italian mathematician Giuseppe Peano and by Russell and Whitehead, but the underlying system was Frege's. In realising an old dream of Gottfried Wilhelm Leibniz for a precise language of reasoning, it also prepared the ground for the formal languages of computer science later developed by figures such as Alan Turing.

Frege wanted to show that arithmetic rests on logic alone, a view later called logicism. This contradicted Immanuel Kant, who had held that the truths of arithmetic depend on our intuition of time, and the empiricists, such as John Stuart Mill, who thought them generalisations from experience. In The Foundations of Arithmetic (1884), a brilliant and readable book, he criticised earlier theories of number and argued that numbers are objects that can be defined in terms of concepts: the number of planets, for example, is a property of the concept "planet". He also stated the context principle, that a word has meaning only in the setting of a whole sentence, and insisted that logic must be kept separate from psychology, since the laws of thought are not descriptions of how people happen to think.

He then tried to carry out the full derivation in his Basic Laws of Arithmetic, whose first volume appeared in 1893. In June 1902, as the second volume was going to press, he received a letter from the young Bertrand Russell pointing out that one of Frege's basic principles led to a contradiction, now known as Russell's paradox, concerning the set of all sets that are not members of themselves. Frege added an appendix acknowledging the problem and proposing a repair, which later proved inadequate. The discovery damaged his project at its root, and he never fully recovered from it.

In a series of papers in the early 1890s, including "Function and Concept" and "On Sense and Reference" (1892), Frege turned to the philosophy of language. He asked how a statement such as "the morning star is the evening star" could be informative, when both expressions name the same thing, the planet Venus. His answer was that expressions have both a reference, the object they stand for, and a sense, the way in which that object is presented. The two phrases share a reference but differ in sense, which is why the statement tells us something. This distinction became one of the central ideas of twentieth-century philosophy of language and logic. In later essays, such as "The Thought" (1918), he argued that thoughts exist objectively, independent of the minds that grasp them.

Frege's personal life was marked by loss. He married Margarete Lieseberg in 1887; their children died young, and after her death in 1904 he adopted a son, Alfred. After retiring he lived in Bad Kleinen, near Wismar, where he died on 26 July 1925. A diary he kept in 1924, discovered by scholars after the Second World War and published in 1994, revealed extreme right-wing and antisemitic opinions, including hostility to democracy and the wish that Jews be removed from Germany. The discovery shocked admirers who had seen him as a model of pure reason.

Frege's importance was recognised by only a few in his lifetime. Russell read him closely, and Ludwig Wittgenstein visited him in Jena in 1911 and took his advice to study with Russell in Cambridge. Rudolf Carnap attended his lectures, and his correspondence with Edmund Husserl, whose early philosophy of arithmetic Frege criticised in an 1894 review, may have helped turn the founder of phenomenology against psychologism. From the mid-twentieth century, the work of the Oxford philosopher Michael Dummett and others established Frege as the founding figure of analytic philosophy. His ideas underlie modern logic, philosophy of mathematics and the theory of meaning, and they emerged from the wider currents traced on the nineteenth-century philosophy page.