Ernst Cassirer and the sui generis character of his methodological approach on an aspect of the Faktum of Science: his reception of the logicist thesis in “Substanzbegriff und Funktionsbegriff”

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.

Exemplifying: it’s true that Frege and Russell defended this thesis. Nevertheless, it’s not correct to say that both agree or defend the same kind of logicism. Frege (see, e.g., GA §§ 90 91) affirms that such thesis worth only in the case of Arithmetic, not considering that Geometry would be also reducible to logic (and this was defended by Frege already in his doctoral dissertation, in 1873). Russell, differently, defends a kind of logicism in which all mathematics is reducible to logic (see, e.g., the first chapter of his PoM, 1903).

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 Frege or Russell. 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 Frege and Russell, on the one hand, and Cassirer, on the other, go on. 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 another important figure of those times: Richard Dedekind, and particularly in his work of 1888, Was sind und was sollen die Zahlen. In this author, the philosopher of culture would find good arguments to support his criticism upon Frege and Russell, besides the fact that the Dedekind’s position agree with the general framework of the basic thesis of Substance and Function, claiming that numbers rather then things, they must be conceived as relations.

There are two main objectives in our paper. On the one hand we aim to expose Cassirer’s critique to the both authors and, on the other hand, to explore some features of his sui generis logicism. Finally, the Chapter II, ‘The concept of number’, of Cassirer’s 1910 book, Substanzbegriff und Funktionsbegriff, will serve as the basic text for the present discussion. There the philosopher makes numerous references to Frege, Russell, the neo-Kantians and Dedekind, as well to other important names present in that debate.