One letter from Bertrand Russell shook Frege's life's work just before publication
Gottlob Frege spent decades trying to prove that all of arithmetic could be built from pure logic. In 1903, as the second volume of his masterwork was going to press, Bertrand Russell wrote to show that one of its basic laws produced a contradiction. Frege added a hurried appendix admitting the damage.
Frege was born in 1848 in Wismar on Germany's Baltic coast, where his father co-founded and ran a girls' school and wrote a German textbook for children that opened with the logic of language. A schoolteacher who was also a poet steered him to the University of Jena, where the physicist Ernst Abbe, later director of the Carl Zeiss optical firm, became his mentor and friend. He took his doctorate at Göttingen in 1873 and spent his career teaching mathematics at Jena, largely ignored.
His 1879 Begriffsschrift, or concept-script, changed logic more than anything since Aristotle. By inventing quantified variables he could handle nested uses of all and some, cleanly separating readings like each student read some book, where the books may differ, from one where a single book was read by all. Aristotle's syllogisms cannot even express Euclid's proof that the primes never run out; Frege's notation can. The machinery behind Russell and Whitehead's Principia Mathematica, and later Gödel's incompleteness theorems and Tarski's theory of truth, traces back to him.
His larger aim, called logicism, was to show arithmetic needs no intuition at all. He argued the case informally in The Foundations of Arithmetic in 1884, then attempted it symbolically in Basic Laws of Arithmetic, the second volume paid for out of his own pocket. The one genuinely new axiom, Basic Law V, said two functions have the same range of values exactly when they agree everywhere. Russell's letter showed it allowed a set of all sets not members of themselves, which both is and is not its own member.
Frege's proposed patch was later shown to imply there is only one object in existence, which made it worthless. Yet modern logicians found that his proofs really needed only Hume's principle, that two collections have the same number when their members can be paired off one to one, and that this is consistent if ordinary second-order arithmetic is. The result is now called Frege's theorem.
Source: Gottlob Frege
