My contribution “Brentano as a Logicist” in the volume “The Philosophy of Brentano” is now available for free through Open Access.
In this chapter I present the following three claims: 1) There was an original and shared philosophy of mathematics in the School of Brentano, 2) mathematics was considered as the paradigmatic and foundational science, and more specifically as a deductive, analytic, and a priori science, 3) Brentano founds the concept of number on elementary logical operations, i.e. Brentano was a logicist. I will concentrate mainly on the third claim, using the other two as background and support.
I use “logicism” as the name for any position in the philosophy of mathematics that allows a progressive reduction of higher mathematics to elementary arithmetic and of arithmetic to logic. This reduction can be achieved through a definition of number based on set theory. We find support for such a reduction in several of Brentano’s manuscripts on Megethologie, among which “Von der Zahl und dem analytischen Charakter der algebraischen Wahrheiten” (Meg 2, “On Number and the Analytical Character of Algebraic Truths”) and “Der Begriff Zahl” (Meg 37, “The Concept of Number”).
Here we see that Brentano defines numbers as the answer to the question “How many?”. The options are “none”, “one”, or “many”. In particular, the dichotomy between “none” and “some” is ultimately based on negative and positive existential judgements, which are also the foundation for Brentano’s proposed reform of elementary logic.
We obtain 0 by rejection [Leugnung] of what is presented in the general concept. The opposite is its acceptance [Anerkennung](Meg 37, 40530)
Brentano applies these logical operations of acceptance and denial to collections of objects falling under a (general) concept and hence advocates what is essentially a logicist position.