Category theory and matters of trust

Ralf Krömer

University of Wuppertal

Abstract: 
In this talk, I intend to pursue the overall aim of the symposium by reading the 20th century debates related to category theory as a case study of how mathematics has been organized and done in order for it to be sound, and of questioning the grounds for having confidence in mathematical arguments. These debates are actually very rich in corresponding aspects, documented partly in earlier work of mine. First of all, the mathematical practice nourished by category-theoretic tools, as in particular in the work of Grothendieck, has been challenged by the difficulties to carry out the corresponding object constructions in any standard set-theoretic foundational framework; so one focus will be on the arguments (and the types of arguments) brought forward in the corresponding debate between set theorists and category theorists. Second, category theory (or more precisely, a conceptual framework based on the conceptual apparatus of category theory) has been proposed as an alternative foundational framework for mathematics by F. William Lawvere and others. The approval and criticisms this enterprise provoked display a wide range of divergent opinions current in the 20th century about what a foundation of mathematics is and can be and which purposes it should serve. Of course, soundness and the grounds for having confidence were major issues in these debates. When the Grothendieck school still tried to find a strengthening of standard axiomatic set theory covering the problematic category-theoretic constructions, the grounds for having confidence in axiomatic set theory in the absence of a consistency proof came to the fore, as well as those for having confidence in the strengthening in the absence of (proofs of) relative consistency. I will argue that while many arguments brought forward in favor of this confidence can be read as appealing to multiple and independent confirmations in the sense of William Wimsatt's theory of robustness, these arguments cannot be considered as conclusive in the present case. Another line of investigation concerns the way some basic concepts (like „object“, „set“, „operation“, „structure“, „function“, „space“, „identity“) are used and their use is learned and controlled in the discourse related to the foundational status of category theory. I will argue that typically these uses are not submitted to the criterion of being in accordance with some formal definition, and are not learned with the help of such a definition, but rather can be described in terms of language games in Wittgenstein's sense. This observation leads to a new interpretation of the role played by intuition in the problem of the trustworthiness of mathematical arguments, an interpretation focusing on the origins of such an intuition (relying on recent work by Gerhard Heinzmann). In conclusion, I will argue, relying on this case study, that the confidence professional mathematicians have in modern mathematics is not so much based on the ontology, especially the more or less intuitive nature, of the mathematical objects concerned, but rather on their individual and collective training and its effects on intuition.