Frege, the complex numbers, and the identity of indiscernibles

Christian Helmut Wenzel

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.

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