Georg Schiemer


University of Vienna

On the Pre-History of Mathematical Structuralism

The symposium will investigate the historical roots of mathematical structuralism, one of the most prominent positions in current philosophy of mathematics. At the position’s core lies the suggestion that mathematical theories describe abstract structures: Peano arithmetic describes the natural number structure, analysis the real number structure, geometry the structure of (Euclidian) space, and so on. Precise elaborations, or formal explications, of that suggestion have been provided in terms of a number of different versions of structuralism by now.

Carnap on model structures and invariants

Rudolf Carnap’s thinking about mathematics in the 1920s is best characterized as an early form of structuralism. This is most explicit in his work on general axiomatics in the posthumously published manuscript Untersuchungen zur allgemeinen Axiomatik (Carnap 2000). In spite of the recent increase in scholarly attention to Carnap’s philosophy of mathematics, surprisingly little has so far been said about the specific form of his structuralism. How did he conceive of abstract structures? What role did the notion play in his logical reconstruction of mathematical theories?