FRIPRO funding to Sorin Bangu and Kevin Cahill
The four-year project 'Mathematics with a Human Face: Set Theory within a Naturalized Wittgensteinean Framework' has received a grant from the Norwegian Research Council.
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In December 2018 the Norwegian Research Council awarded a听FRIPRO-FRIHUMSAM research grant to Sorin Bangu (Project Leader) and听Kevin Cahill (Collaborator), for the project Mathematics with a Human听Face: Set Theory within a Naturalized Wittgensteinean Framework. The听grant spans four years (2019-2023), and has a budget of ca. 10 million听kroner (ca. 1.1 mil. euro); a PhD student and a postdoctoral fellow will听join the project.
The project investigates the reasons why concepts such as 鈥榮et鈥� and听鈥榥umber鈥� have remained philosophically obscure 鈥� nobody really knows听what these things are 鈥� despite the immense success of mathematics over听centuries. The idea is to approach this issue from a perspective never听attempted before: by building on an overlooked Wittgensteinean insight,听that "mathematics is after all an anthropological phenomenon" (RFM听VII-33). The proposal is to regard mathematics, Set Theory in听particular, as a special practice, ultimately of a social nature,听constitutive of the human form of life. The research has听interdisciplinary aspects, and involves collaborations with in Bergen (WAB) and other disciplines such as mathematics,听anthropology and psychology.
Prior to being embedded into the proverbially frightening symbolism taught in school, the operations we perform on sets 鈥� e.g., doing unions or intersections 鈥� are, and have always been, part of a social practice.听This priority is, obviously, a socio-historical fact: as humans, we听group things, and manipulate these groupings, all the time. Here,听however, we argue that this priority is more than a fact: it is also of听a conceptual nature. By highlighting how the operational component of听Set Theory plays a constitutive role in the formation and functioning of听its central concept, that of a 鈥榮et鈥�, we aim to show that the unique听characteristics of mathematical reasoning and mathematical truth听(necessity, certainty, universality) can be better accounted for, when听regarded as entwined into the human form of life.
The project draws philosophical inspiration from Wittgenstein鈥檚 writings听on language, mind, and mathematics. In particular, the notion of a听(human) 鈥榝orm of life鈥� 鈥� meant to capture the complex network of听beliefs, values, practices, habits, and natural inclinations circumscribing people鈥檚 existence 鈥� features prominently in his later philosophy (cf. Wittgenstein 1958, 搂23, 搂241). Yet, as far as the later听Wittgenstein is concerned, in particular the naturalist features of his听thought relevant here, most attention has been devoted to his听masterpiece Philosophical Investigations (PI). This is somwhat understandable given the profound impact this book has had in 20th听century philosophy. Except for the remarks about following an听arithmetical rule (see especially PI 搂185-202), the Investigations pay听relatively little attention to mathematics per se. In the light of the听significance that social practice and naturalism have in his later听thought, central to our project will be an analysis of a less-studied听work, his Lectures on the Foundations of Mathematics, complemented with听his Remarks on the Foundations of Mathematics (RFM).
Furthermore, by articulating a Wittgensteinean-inspired, naturalized听social conception of mathematics, we will also show how the new insights听we bring to light impact the standard interpretations of his thinking as听a whole. Of particular interest for us will be to examine the deep听sources of Wittgenstein鈥檚 apparent hostility to the foundational role of听Set Theory. Once this is clarified, we will investigate whether听Wittgenstein鈥檚 valuable insights about mathematical practice in general听(appearing especially in his Lectures and the Remarks), do apply to Set听Theory as well.
We aim to exploit the anthropological strain in Wittgenstein鈥檚 later听work, and thus explicating how the practice of operating with sets finds听a place within it brings out one of the project鈥檚 major challenges. For听the field of anthropology itself can be regarded as containing a听biological branch, thus placing it closer to the natural sciences. Yet听it also contains a more dominant ethnographic, cultural branch, which听explicitly accounts for human societies in terms of various conventions.听Thus, coming to understand how Wittgenstein saw the conventional in听relation to the natural will be critical for getting a better handle on听how symbolic activity can be a feature of our nature. How to bring these听together in a satisfying way that naturalizes practice while preserving听a recognizably robust conception of normativity suitable for Set Theory听will be the primary focus of one part of the project. This is a point听where we will benefit from input from other relevant disciplines 鈥撎齟specially mathematics, anthropology and psychology. The key challenge,听then, of this part involves providing a coherent account of听Wittgenstein鈥檚 concepts of practice and custom that harmonize with听regarding these as natural aspects of human life.
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