Weyl on Generality, Circularity, and the Foundations of Mathematics

Source abstract

Abstract This contribution argues that Weyl’s motives in The Continuum were primarily epistemological. Though the paradoxes were an important catalyst for the work, consistency was not his only concern. For example, Weyl regarded Zermelo’s axiomatic solution as “unsatisfactory” because it merely blocked the main contradictions. In contrast, Weyl’s predicativism aimed to purify analysis by eliminating the methods that permitted the contradictions in the first place. He argues for this positive goal of purification, as well as his negative critiques of set-theoretic approaches to arithmetic, in broadly epistemic ways, focusing on issues like insight, intuition, cognitive resources, and epistemic significance. For comparison and support of Weyl, some views from Bernays are discussed; though his argument is somewhat different, Bernays’ conclusions are similar to Weyl’s. Also, like Weyl, his emphasis is on epistemological issues in mathematics (e.g., his critiques of logicism). Seeing Weyl’s philosophical motives as generally epistemological facilitates a more unified view of several strands in his philosophy of mathematics, including his rejection of contemporary alternatives in the foundations of mathematics, his concerns about circularity, and of course his positive program of predicative analysis.