Abstract / Synopsis
The following investigation is a cognitive-historical analysis of the conceptual development of complex numbers. The history of this development spans nearly two millennia, from the earliest appearance of the square root of a negative quantity in the calculations of Heron of Alexandra (1st Century CE) to the full flowering of complex numbers in the first half of the 19th Century. The approach used for this analysis is Nersessian's, including her formulations of model-based reasoning and mental models. Additional aspects of the analysis feature the prominent roles played by process representations, including object-process complementarities, and by core numerical systems. Our analysis provides support for mental rotation as a fourth core system and for bootstrapping scenarios in both the senses of Nersessian and of Carey. We use the results of our analysis and discussion to update previous conceptual metaphor mapping approaches to complex numbers by Fauconnier and Turner and by Lakoff and Núñez. Finally, we review recent empirical studies of complex number learning and find them to be consistent with the present work.
DOI
10.5642/jhummath.VTJD5740
Recommended Citation
Charles R. Card & Gary G. Miller, "Bootstraps and Scaffolds: What a Cognitive-Historical Analysis of the Complex Number System Reveals About Numerical Cognition," Journal of Humanistic Mathematics, Volume 14 Issue 2 (July 2024), pages 451-514. DOI: 10.5642/jhummath.VTJD5740. Available at: https://scholarship.claremont.edu/jhm/vol14/iss2/14
