Colloquia Theologica Ottoniana

Michael Resnik, the founder of modern structuralism in the philosophy of mathematics, changed his views and proposed a new non-ontological structuralism. Resnik is considered a prominent figure in modern structuralism within the realm of contemporary philosophy of mathematics, and his sui generis structuralism is regarded as one of the most significant and frequently discussed positions in the field. This article examines the motivations behind Resnik’s change of perspective. His new position is presented in detail, and an attempt is made to contrast it with selected views in the philosophy of mathematics. The discussion is conducted in the context of the Frege-Hilbert Controversy, which centers on the status of mathematical theory axioms and the meaning attributed to primary terms. The three-stage concept of the development of deductive sciences, proposed by Kazimierz Ajdukiewicz, is also introduced. This concept, inspired by Hilbert’s ideas, outlines three stages in the evolution of deductive theories: (1) pre-axiomatic deductive, (2) axiomatic deductive, (3) abstract axiomatic. Each of these stages possesses unique characteristics that illuminate the nature of deductive theories, particularly in relation to the approach to sets of axioms and primary terms within these theories. Additionally, two styles of practicing deductive theories (assertive and hypothetical) are discussed. Ultimately, an exploration of Resnik’s non-ontological structuralism provides insight into how this novel structuralist concept should be understood and clarifies the author’s actual claims. Alongside these distinctions, a new conception of hypothetical structuralism, distinct from non-ontological structuralism, is formulated. This novel structuralism is rooted in the hypothetical approach to practicing deductive theories.