Dedekind Lecture 2019

Dedekind's and Frege's views on arithmetic and logic

Dedekind's famous assertion, in Was sind und sollen die Zahlen?, that arithmetic is "a part of logic" has been taken as an expression of something close to Frege's logicism. Dedekind shared indeed with Frege the same goal, namely replacing intuitive importants from spatio-temporal experience with, ex ante, logical standards of rigour in the deductive presentation of arithmetic. But he does not intend to reduce arithmetic to logic nor does he intend to invent a formal system allowing such a reduction. Then, what did Dedekind mean? A meticulous, contextual and comparative analysis shows that Dedekind and Frege did not understand the terms 'logic', 'arithmetic' and 'number' in the same way. The meaning and the scope of the concepts denoted by those terms are indeed essentially different for the two scientists. Consequently Dedekind and Frege differ completely on the relationship between arithmetic and logic. Drawing from these observations, I will show that by "arithmetic is a part of logic" Dedekind means that arithmetic, intrinsically, constitutes a purely rational (logical in a definitely pre-Fregean sense) norm of thinking. Dedekind's "logic of the mind" is arithmetic generalized to unspecified elements.