Abstract

Neo-Fregeans in the philosophy of mathematics hold that the key to a correct understanding of mathematics is the implicit definition of mathematical terms. In this paper, I discuss and advocate the rejection of abstractionism, the putative constraint (latent within the recent neo-Fregean tradition) according to which all acceptable implicit definitions take the form of abstraction principles. I argue that there is reason to think that neo-Fregean aims would be better served by construing the axioms of mathematical theories themselves as implicit definitions, and consider and respond to several lines of objection to this thought.

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