Parallels and distances between Cassirer and Frege on Logicism and Method

Lucas Alessandro Duarte Amaral

In its most elementary form, the logicist thesis claims that mathematics, be all of it or part of it, can be reducible to logic, and that means that the first one is part of the second. Some results of this would be such as: the fundamental concepts of mathematics (e.g., numbers) can be defined throughout logical concepts and its theorems can be proved by throughout logical axioms and inference rules. Although the veracity of this primary definition on the logicist thesis, there are, at very least, two interesting aspects for further reflections: (i) there were no unanimous opinions around the thesis, but on the contrary, the thesis were take into account very differently by several philosophers which intended to work with it; (ii) related to this, the second feature concerns the logical conception defended by each one of those philosophers.

It is currently said that Frege and Russell were two of most important figures that defended the mentioned thesis. Perhaps less known, Cassirer was another philosopher which defended the logicist thesis in this moment of history. However, he doesn’t defend it in such way as those two. Accordingly what were said previously, one of the reasons that underlie this difference is the fact his concept of logic differs largely from the other two. In this particular, his influences came directly from his predecessors in Marburg. And the differences between continues. For instance, another point to be fought by Cassirer rests on that aspect recognized later as "mathematical Platonism", understood as a theory in which it is considered, roughly speaking, that numbers exist in an abstract way, independent of us. This factor would be contested by Cassirer thanks to figure of Dedekind. In him, the Neo-Kantian would find good arguments to support his criticism upon Frege.

We intent here, thus, to discuss the parallels between Frege’s and Cassirer’s conception on Logicism, on the one hand, and, on the other, to explore some differences between then.