The Axiom of Choice as Paradigm Shift: The Case for the Distinction Between the Ontological and the Methodological Crisis in the Foundations of Mathematics

Abstract

Seldom has a mathematical axiom engendered the kind of criticism and controversy as did Zermelo’s (1904) Axiom of Choice (henceforth, AC). In this paper, we intend to place the development of the Axiom of Choice in its proper historical context relative to the period often called “the crisis in the foundations of mathematics.” To this end, we propose that the nature of the controversy surrounding AC warrants a division of the Grundlagenkrise der Mathematik into two separate horns: (1) an ontological crisis related to the nature and status of mathematics itself (viz., the nature of its foundation and the logical paradoxes that surrounded early attempts to logically formalize mathematics); and (2) a methodological branch concerned rather with the nature of mathematical practice (viz., the nature of mathematical proofs). These two strands are inexorably intertwined and, though it is not new to suggest that the controversy surrounding AC was related either to the foundational crisis or to a polemic about the nature of mathematical demonstration, it is perhaps new to state that the question of the validity of AC not only was a central question of this period but also, furthermore, was one of its primary drivers—one which led to a profound paradigm shift in the way we construe mathematical reasoning, whether it has led us down a path of embracing realism/Platonism or intuitionism/constructivism.