Mathematical Structuralism: Weyl’s Positions

Source abstract

Abstract Weyl’s ambivalent attitude toward the foundations of mathematics is well-known; it comes out strikingly in his perspective on the intuitive character of (some) mathematical objects and the abstract conceptual organization of (parts of) mathematics. This ambivalence reflects the tension between Kronecker and Dedekind as well as that between Brouwer and Hilbert. Weyl was enmeshed in the dispute that arose from the latter and would have taken Kronecker’s side in the dispute between the two nineteenth century mathematicians. In this short note, I want to describe the position Weyl implicitly argues against when constructing the mathematical continuum in the ways of Brouwer (Weyl, 1921). It is a position he, nevertheless, implicitly agrees with, when formulating in (Weyl, 1953) the role of “structures” in the ways of Bourbaki. To recognize that it is the same position he disagrees and agrees with, one only has to observe (i) that the counterposition to Weyl (1921) is held by Dedekind and Hilbert and (ii) that Bourbaki firmly stands in their tradition. Weyl realized that the axiomatic method as practiced by Bourbaki has an important function in mathematical research. It is Weyl’s positive reevaluation of structural axiomatics I want to bring out. It is seen as complementing, not as supplanting his philosophically motivated constructive perspective.