#### Abstract / Synopsis

The narrative about the nineteenth century favored by many philosophers of mathematics strongly influenced by either logic or algebra, is that geometric intuition led real and complex analysis astray until Cauchy and Kronecker in one sense and Dedekind in another guided mathematicians out of the labyrinth through the arithmetization of analysis. Yet the use of geometry in most cases in nineteenth century mathematics was *not* misleading and was often key to important developments. Thus the geometrization of complex numbers was essential to their acceptance and to the development of complex analysis; geometry provided the canonical examples that led to the formulation of group theory; and geometry, transformed by Riemann, lay at the heart of topology, which in turn transformed much of modern mathematics. Using complex numbers as my case study, I argue that the best way to teach students mathematics is through a repertoire of modes of representation, which is also the best way to make mathematical discoveries.

#### DOI

10.5642/jhummath.201301.06

#### Rights Information

© Emily R. Grosholz

#### Recommended Citation

Grosholz, E.
R.
"Teaching the Complex Numbers: What History and Philosophy of Mathematics Suggest," *Journal of Humanistic Mathematics*, Volume 3
Issue 1
(January 2013), pages 62-73.
DOI: 10.5642/jhummath.201301.06
. Available at: https://scholarship.claremont.edu/jhm/vol3/iss1/6

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History of Science, Technology, and Medicine Commons, Logic and Foundations of Mathematics Commons, Mathematics Commons, Philosophy of Science Commons, Science and Mathematics Education Commons