My contribution “Brentano’s Mathematical Foundation of Science” in the volume Brentano and the Positive Philosophy of Comte and Mill is now available for free through Open Access.

In the article, I examine Brentano’s strategic positioning between German Idealism and British Empiricism, using his own philosophy of mathematics, as expressed in various (mostly unpublished) manuscripts (such as the Megethologie), as a case study. The resulting position will be taken as an indication of a strategic philosophical positioning between what Brentano considered to be the excesses of idealism and empiricism. It is “strategic” in two ways. First of all, in Brentano’s opposition to Kant and Mill, mathematics plays such a pivotal role, that the implications of accepting or rejecting a specific view of the foundations of mathematics can make or break an entire philosophical position. For instance, if mathematics would be deprived of its status of synthetic a priori, this would undermine Kant’s entire philosophical project. Secondly, Kant and Mill are made into the exemplars of idealism and empiricism. Brentano does not engage only specifically with Kant and Mill for their own sake but in order to reject a whole range of approaches. Kant and Mill hence become figureheads in a broader discussion between Brentano’s particular brand of empiricism and more radical positions to the left and right. Such strategic positioning was necessary for Brentano and his school in order to support their claim that a scientific philosophy (and psychology) was possible without falling prey to dogmatism, reductionism, or skepticism.

The most important source in this respect, is the treatise titled “Über den Ursprung der mathematischen Erkenntnis” (“On the Origin of Mathematical Knowledge” Meg 40015–40038) contained in the Megethology. Brentano opens the treatise by stating that the question about the character of the foundations of mathematics has not yet been definitively answered, because of two severe errors in epistemology (Meg 40015: “zwei schwerwiegende Irrtümer der Erkenntnislehre”). These two are exemplified by Kant and Mill. The first error, Kant’s error, would be to base all science on “general immediate synthetic judgments a priori” (“allgemeinen unmittelbaren synthetischen Urteilen a priori”). In objecting to idealism, the empiricists have gone too far in the opposite direction, resulting in a position that is no less problematic than Kant’s. For Brentano, the exemplar of radical empiricism is John Stuart Mill. Despite all the respect and admiration Brentano professed for him and other positivists and empiricist, in the philosophy of mathematics this position is “far too paradoxical to be generally accepted” (Meg 40021: “zu paradox, um allgemeinere Annahme zu finden”). Brentano underscores two important problems with Mill’s view. On the one hand, it is incapable of acknowledging that also deduction can extend our knowledge; on the other, it would turn mathematics into an inductive science.

The picture of the philosophy of mathematics that emerges from the texts considered above, and particularly from the material of the Megethologie, is that Brentano defines his position in opposition to both Kant and Mill, claiming that mathematics is analytic, deductive, a priori, and capable of extending knowledge.

Hence mathematics is not an inductive, but a purely deductive, and in this sense, a priori science. Indeed, were it not, then there would be no science at all, neither deductive nor inductive. Because it is not induction that sanctions deduction, but deduction, and specifically mathematical deduction, that sanctions all rational scientific justified induction.

Brentano Meg 40025 f.

Through the analysis of Brentano’s mathematical foundations of science and comparative analysis of his critique of Kant and Mill, we can better understand Brentano’s project of philosophy as science.