Explicating Dedekind: Existential Axiomatics or Logicist Abstration?

from MCMP – Mathematical Philosophy (Archive 2011/12)

Erich Reck (UCR) gives a talk at the MCMP Colloquium (18 October, 2012) titled "Explicating Dedekind: Existential Axiomatics or Logicist Abstration?". Abstract: In recent years, there has been renewed interested in Richard Dedekind as a philosopher of mathematics, especially in connection with structuralist views about the content of mathematics. In this talk, I will juxtapose two ways of interpreting Dedekind's structuralism, or better, two explications (in Carnap's sense) of his position, that seem most promising to me. One of them is Hilbertian, leading to a reading of Dedekind as a precursor of Hilbert's "existential axiomatics"; the other is neo-Fregean or neo-logicist, in the sense of being based on a distinctive kind of "abstraction principles" that can be seen as underlying Dedekind's position. I will argue that, besides being more defensible on interpretive grounds, the second explication of Dedekind points in a direction for developing mathematical structuralism that deserves further attention today.