Is Arithmetic Synthetic? A Defense of Kant Against Frege’s Logicistic Definition of Numbers

Edited by Nick Cicchini

Abstract

Here I will assess Kant and Frege’s views on whether arithmetic is analytic or synthetic. I begin by presenting Kant’s definitions of syntheticity and how it applies to arithmetic in the Critique of Pure Reason, and then proceed with Frege’s framework of logicism in the Foundations of Arithmetic and his definitions of numbers through sets and equal numerosity. In the third section, I argue that Frege’s definition of numbers using set theory is insufficient to prove arithmetic as analytic, for the concepts and sets under the logicistically defined subject and predicate do not fulfill the two necessary containment relations that are conditions to analyticity in reciprocity. Furthermore, I argue that Frege presupposed Kantian epistemology by invoking intuition. Overall, my essay presents a defense of Kant’s assessment of arithmetic as synthetic a priori against the attack from Frege’s logicism that holds arithmetic to be analytic.