Gottlob Frege (1848–1925) was born in a small town in Pomerania and spent almost his entire life teaching mathematics at the University of Jena. In a quiet department, scarcely known even to his own students, he became the deepest logician of his age. The short book he published in 1879 — Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens — surpassed in a single stroke the Western logic that had stood essentially unchanged for two thousand two hundred years since Aristotle. When it appeared, it was met with brief, puzzled, sometimes mocking reviews: its two-dimensional notation, read vertically rather than left to right, convinced no one.

Frege's real innovation lies not in the shape of the symbols but in the analysis they carry. Aristotelian logic had remained, for centuries, bound to the subject–predicate form: "all men are mortal." Frege breaks this form and analyses a judgement on the model of function and argument in mathematics: "x is mortal" is a function; feed it an argument, and it yields true or false. On top of this he places his quantifiers: "for every x" (∀) and "there is some x" (∃). This apparently simple addition makes it possible, for the first time, to state multiply general claims ("every number has a successor greater than itself") fully and in a form machines can handle. The symbolic foundation of modern mathematical logic, of computer science, and of artificial intelligence runs back to this page.

Frege's real target was not logic itself but the foundations of arithmetic. His thesis, which he called logicism, was that numbers can be given a genuinely logical foundation, and that arithmetic can be derived by pure inference from logic and definitions. He carried out this programme in the two volumes of his Grundgesetze der Arithmetik (1893 and 1903). While the second volume was already at the press, on 16 June 1902, the young Bertrand Russell wrote him a now-famous letter showing that the system contained a contradiction — in the notion of "the set of all sets that do not contain themselves", Russell's paradox. Frege accepted the result in an added epilogue: "Hardly anything more unwelcome can befall a scientific writer than that one of the foundations of his edifice be shaken after the work is finished." This tragic honesty is one of the most authentic scenes in the history of modern thought.

Even with his system broken, Frege's influence had already begun to flow in another channel. His short essay of 1892, Über Sinn und Bedeutung — "On Sense and Reference" — distinguishes the sense (Sinn) of an expression from what it refers to (Bedeutung): "morning star" and "evening star" refer to the same Venus yet carry different senses. This distinction has structured a century of philosophy of language. Russell and Whitehead's Principia Mathematica (1910–13), the young Wittgenstein's Tractatus, Carnap and the Vienna Circle — all pass through Frege. Almost invisible during his lifetime, the Jena professor has been recognised since his death as the quiet founder of twentieth-century analytic philosophy.