In this historical paper intended for non-logicians, the author focuses on the major contributions of Thoralf Skolem, who established first-order logic as the basis of mathematics. The author first outlines the history of mathematical logic, discussing Boole’s calculus, the quantifiers of Frege and Peirce (whose part is often forgotten), and Schröder’s algebra. Next, he investigates the early “collisions” between logic and mathematics, viz., the axiomatizations of geometry (by Hilbert) and of natural numbers (by Dedekind, Peano, and Russell and Whitehead), with only a cursory mention of set theory. The author gives a detailed account of the decisive influence of the emergence of Hilbertian formalism in the 1920s. The paper ends with Skolem’s work.

The author’s thesis is that logic has narrowed down to first-order with Skolem reduction. More seriously, the existence of countable models via the Löwenheim-Skolem Theorem leads to relativism, thereby going against the wishes of the pioneers of axiomatization: “the house was divided against itself.” But the history of the interface between logic and mathematics did not end in 1940. The work of Cohen and his contemporaries has significantly increased our knowledge: various new second-order axioms have been tested with no decisive achievement.

The paper is hampered by a lack of notations and symbols. Although a glossary is included, some notions of logic (e.g., axiom schemata) are used with no explanation.