Frege, the complex numbers, and the identity of indiscernibles

Logique Et Analyse 53 (209):51-60 (2010)

Abstract
There are mathematical structures with elements that cannot be distinguished by the properties they have within that structure. For instance within the field of complex numbers the two square roots of −1, i and −i, have the same algebraic properties in that field. So how do we distinguish between them? Imbedding the complex numbers in a bigger structure, the quaternions, allows us to algebraically tell them apart. But a similar problem appears for this larger structure. There seems to be always a background and a context that we rely upon. Thus mathematicians naturally make use of Kantian intuition and references fixed by names and denotations. I argue that such features cannot be avoided.
Keywords Frege  complex numbers  identity of indiscernibles
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Naming and Necessity.S. Kripke - 1972 - Tijdschrift Voor Filosofie 45 (4):665-666.
Naming and Necessity.Saul A. Kripke - 1985 - Critica 17 (49):69-71.
Die Grundlagen der Arithmetik.Gottlob Frege - 1988 - Felix Meiner Verlag.
The Identity Problem for Realist Structuralism.J. Keranen - 2001 - Philosophia Mathematica 9 (3):308--330.

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