Seb Gandon

Moritz Pasch is usually seen today as a precursor of Hilbert. The Vorlesungen über neuere Geometrie is indeed one of the few works Hilbert referred to in the Grundlagen. Unfortunately, Hilbert's epoch-making book has eclipsed Pasch's achievement; so much so that Pasch's Vorlesungen has not been yet translated into English. But as Pollard emphasizes, it would be a mistake to reduce Pasch's research to his work on the foundations of geometry. Pasch published another book on the foundations of real analysis and arithmetic in 1882, and two other works, respectively, in 1909 and in 1914 . Pasch thus aimed not only to analyze the foundations of geometry but also to elaborate a complete theory of mathematical knowledge. This ambition brought him close to the logicists . But, unlike them, Pasch was a resolute empiricist: he considered that ‘geometry and number theory are branches of empirical science’ 1 and that all mathematical concepts must be derived from a set of notions abstracted from our experience.Pasch made his whole career in Giessen — that is, outside of the major German research centers. Heavily burdened with administrative duties, Professor Pasch seems to have found time for his research only in 1917, when he retired at the age of 67. As Pollard notes, nearly half of Pasch's published output appeared after his sixty-eighth birthday! This volume is a translation of fourteen articles dating from the later period of Pasch's life, ordered by the time of their publication. The papers are preceded by a substantial introduction by Stephen Pollard, the editor and translator of all the papers. Aside from his correspondence with Frege, this volume is thus the first translation of Pasch's work …

From 1905–1908 onward, Russell thought that his new ‘substitutional theory’ provided him with the right framework to resolve the set-theoretic paradoxes. Even if he did not finally retain this resolution, the substitutional strategy was instrumental in the development of his thought. The aim of this paper is not historical, however. It is to show that Russell's substitutional insight can shed new light on current issues in philosophy of mathematics. After having briefly expounded Russell's key notion of a ‘structured variable’, I connect it to two ongoing discussions: the Caesar problem, and absolute generality