1. Bootstraps and Scaffolds: What a Cognitive-Historical Analysis of the Complex Number System Reveals About Numerical Cognition.
- Author
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Card, Charles R. and Miller, Gary G.
- Subjects
- *
COMPLEX numbers , *MODEL-based reasoning , *NUMBER systems , *MENTAL rotation , *SQUARE root - Abstract
The following investigation is a cognitive-historical analysis of the conceptual development of complex numbers. The history we explore spans nearly two millennia, from the earliest appearance of the square root of a negative quantity in the calculations of Heron of Alexandra (first century CE) to the full flowering of complex numbers in the first half of the 19th century. The approach we use is Nersessian's, including her formulations of model-based reasoning and mental models. Additional aspects 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 con- ceptual 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. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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