This is a detailed account of Frege’s theory of real numbers from the second volume of Grundgesetze der Arithmetik, interspersed with references to Frege’s contemporaries and precursors in philosophy and in mathematics. Starting with an account of Frege’s Platonism and his take of the views of contemporaries regarding analysis, the narrative moves to Frege’s theory of quantity as presented in his Habilitationsschrift of 1874, to critical remarks made by Frege in Die Grundlagen der Arithmetik on Newton’s and on H. Hankel’s 1867 treatment of the concept of quantity, to Frege’s review of H. Cohen’s Das Prinzip der Infinitesimal-Methode und seine Geschichte, in particular regarding the Kantian distinction between extensive and intensive magnitudes, to Frege’s critique of Cantor’s theory of irrational. Next comes Russell’s theory of real numbers in The principles of mathematics of 1903, as well as in volume 3 of Principia mathematica, followed by an extensive reconstruction of Frege’s conception of the notion of quantity and his theory of real numbers, in both their informal and formal presentations, based on volume 2 of Grundgesetze der Arithmetik, together with an account of the concept of quantity in the work of Euclid, Aristotle and Euler. The study ends with a brief account of F. v. Kutschera’s “proposal of how Frege might have carried on with the logical construction of analysis in a projected third volume of Grundgesetze, had he not been shocked by Russell’s paradox” from [Arch. Math. Logik Grundlagenforsch. 9, 102–111 (1966; Zbl 0147.24609)].
For the entire collection see [Zbl 1326.03008].