Synthese 191 (17):4231-4252 (2014)

Denis Buehler
Institut Jean Nicod
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
Keywords Mental content  Anti-Individualism  Conceptual Role Theory  Concepts  History of mathematics
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DOI 10.1007/s11229-014-0527-x
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Word and Object.Willard Van Orman Quine - 1960 - Cambridge, MA, USA: MIT Press.
Origins of Objectivity.Tyler Burge - 2010 - Oxford University Press.
The Meaning of 'Meaning'.Hillary Putnam - 1975 - Minnesota Studies in the Philosophy of Science 7:131-193.

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