Dedekind’s Structuralism: HOPOS 2016

(Joint work with Wilfried Sieg)

In his 1888 essay Was sind und was sollen die Zahlen?, Dedekind set out to offer a rigorous, purely logical, account of the natural numbers.  However, a comparison of the final version of his essay with earlier manuscripts reveals that his work underwent a number of changes.  In particular, his earlier work focused on the creation of new mathematical objects, while the final version centered around the creation of concepts.

Sieg and I argue (Sieg and Morris, 2016) that the shift in Dedekind’s position is mathematically and philosophically significant.  On the mathematical side, the final version of his essay introduced, via the concept of a simply infinite system, the Dedekind-Peano axioms, as well as new and important metamathematical work.  For example, Dedekind established, in modern terms, the categoricity of the concept of simply infinite system.    On the philosophical side, Sieg and I suggest that Dedekind’s final version and earlier work reflect different kinds of structuralism.  We claim that the structuralism reflected in the published version of his work is not the same kind of structuralism that is often attributed to him in the literature.

In this talk, I will discuss the transformation of Dedekind’s work on the foundations of arithmetic, and describe the new interpretation of his account that Sieg and I have developed.  I will address the differences between our interpretation and others given in the literature, and tackle the question of why Dedekind changed his position.

Wilfried Sieg and Rebecca Morris, “Dedekind’s structuralism: creating concepts and deriving theorems”.  Forthcoming in Logic, Philosophy of Mathematics, and their History: Essays in Honor of W.W. Tait.