The project investigates the reasons why concepts such as 'set' and 'number' have remained philosophically obscure - nobody really knows what these things are - despite the immense success of mathematics over centuries. The idea is to approach this issue from a perspective never attempted before: by building on an overlooked Wittgensteinean insight, that "mathematics is after all an anthropological phenomenon" (RFM VII-33). The proposal is to regard mathematics, Set Theory in particular, as a special practice, ultimately of a social nature, constitutive of the human form of life. The research has interdisciplinary aspects, and involves collaborations with Wittgenstein Archives in Bergen (WAB) and other disciplines such as mathematics, anthropology and psychology.
Set Theory is widely regarded as the foundation of mathematics, and yet there is consensus among both mathematicians and philosophers that no one really knows what a set is. There are profound philosophical reasons why this notion has remained obscure. It is the main aim of this project to investigate them. We approach this issue from a perspective never attempted before: we build on an overlooked Wittgensteinean insight, that "mathematics is after all an anthropological phenomenon" (1956; VII-33) - i.e., the idea to regard mathematics, set theory in particular, as a special practice, constitutive of the human form of life.
Philosophers and mathematicians alike need an account of how it is possible on the one hand to develop a foundational theory for mathematics while on the other hand lacking a clear understanding of the foundation itself. Here we aim to offer such an account, developed within a naturalist framework. Given the foundational role of Set Theory within mathematics, this account would mark significant philosophical progress in understanding what we actually do when we do mathematics. Furthermore, this account offers a new approach to the three perennial puzzles about mathematics, for which there is no agreed-upon solution: why are the axioms of Set Theory (i) necessary, (ii) certain, and (iii) universally applicable?
We aim to draw on Wittgenstein's thoughts on the role of mathematical concepts - especially 'set' (and 'number') - within the human form of life, in order to support our working hypothesis: mathematics, set theory in particular, is best understood as a special type of practice, a codification of the rules governing archetypal human activities (grouping, segregating, etc.). This codification, and the subsequent development of a theory, is possible without explicitly answering the question 'what is a set?', since the notion of a 'set' is not a theoretical one, but inextricably embedded into the human practices and our form of life.