Incomplete understanding of complex numbers Girolamo Cardano: a case study in the acquisition of mathematical concepts

• Published in: Synthese (Dordrecht) Vol. 191; no. 17; pp. 4231 - 4252
• Main Author: Buehler, Denis
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer 01.11.2014; Springer Netherlands; Springer Nature B.V
• Subjects: Addition; Algebra; Cognition & reasoning; Complex numbers; Complexity; Education; Epistemology; Integers; Logic; Mathematical objects; Mathematical relations; Mathematics; Metaphysics; Misconception; Numbers; Philosophy; Philosophy of Language; Philosophy of Science; Possession; Reason; New York; Concepts; History of mathematics; Conceptual Role Theory; Mental content; Anti-Individualism
• ISSN: 0039-7857; 1573-0964
• DOI: 10.1007/s11229-014-0527-x

In this paper, I present the case of the discovery of complex numbers by Girolamo Cardano. Cardano acquires the concepts of (specific) complex numbers, complex addition, and complex multiplication. His understanding of these concepts is incomplete. I show that his acquisition of these concepts cannot be explained on the basis of Christopher Peacocke's Conceptual Role Theory of concept possession. I argue that Strong Conceptual Role Theories that are committed to specifying a set of transitions that is both necessary and sufficient for possession of mathematical concepts will always face counterexamples of the kind illustrated by Cardano. I close by suggesting that we should rely more heavily on resources of Anti-Individualism as a framework for understanding the acquisition and possession of concepts of abstract subject matters.