From Necessary Truths to Feelings: The Foundations of Mathematics in Leibniz and Schopenhauer

Abstract

I take into account Schopenhauer’s study of Leibniz’s work, as it emerges not only from his explicit critique in his dissertation On the Fourfold Root of the Principle of Sufficient Reason (1813) and in The World as Will and Idea (3rd edition 1859), but also from his annotations of Leibniz’s writings. Schopenhauer owned many books of Leibniz in his private library and they are full of intriguing annotations. Many of these annotations concern the discussion on logic and mathematical truths and so they are particularly relevant for the study of Schopenhauer’s philosophy of mathematics. After a comparison between Leibniz and Schopenhauer’s definition of necessary and innate truths, I put alongside what the two authors stated about system and fundamental axioms. Two questions arise from Leibniz’s interpretation of Euclid’s axioms: the role of ‘images’ in knowledge and the notion of ‘confused’ knowledge. These two questions are worth of attention, as they allow to focus on Schopenhauer’s theory of ‘feeling’ mathematical knowledge, as I show in the last section of this paper. To Schopenhauer, knowledge works with intuitive representations, intuition, perception, and, for this reason, feeling is the basis of all conceptions. Schopenhauer’s provided a new point of view regarding feeling and intuitive knowledge that involves a special meaning for his philosophy of mathematics.