Department of Philosophy

Philosophy and mathematics

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.

Antikk kileformet tavle med assyriske tegn mot svart bakgrunn
Illustration photo: ancient cuneiform tablet with assyrian signs

In December 2018 the Norwegian Research Council awarded a FRIPRO-FRIHUMSAM research grant to Sorin Bangu (Project Leader) and Kevin Cahill (Collaborator), for the project Mathematics with a Human Face: Set Theory within a Naturalized Wittgensteinean Framework. The grant spans four years (2019-2023), and has a budget of ca. 10 million kroner (ca. 1.1 mil. euro); a PhD student and a postdoctoral fellow will join the project.

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.

Prior to being embedded into the proverbially frightening symbolism taught in school, the operations we perform on sets – e.g., doing unions or intersections – are, and have always been, part of a social practice. This priority is, obviously, a socio-historical fact: as humans, we group things, and manipulate these groupings, all the time. Here, however, we argue that this priority is more than a fact: it is also of a conceptual nature. By highlighting how the operational component of Set Theory plays a constitutive role in the formation and functioning of its central concept, that of a ‘set’, we aim to show that the unique characteristics of mathematical reasoning and mathematical truth (necessity, certainty, universality) can be better accounted for, when regarded as entwined into the human form of life.

The project draws philosophical inspiration from Wittgenstein’s writings on language, mind, and mathematics. In particular, the notion of a (human) ‘form of life’ – meant to capture the complex network of beliefs, values, practices, habits, and natural inclinations circumscribing people’s existence – features prominently in his later philosophy (cf. Wittgenstein 1958, §23, §241). Yet, as far as the later Wittgenstein is concerned, in particular the naturalist features of his thought relevant here, most attention has been devoted to his masterpiece Philosophical Investigations (PI). This is somwhat understandable given the profound impact this book has had in 20th century philosophy. Except for the remarks about following an arithmetical rule (see especially PI §185-202), the Investigations pay relatively little attention to mathematics per se. In the light of the significance that social practice and naturalism have in his later thought, central to our project will be an analysis of a less-studied work, his Lectures on the Foundations of Mathematics, complemented with his Remarks on the Foundations of Mathematics (RFM).

Furthermore, by articulating a Wittgensteinean-inspired, naturalized social conception of mathematics, we will also show how the new insights we bring to light impact the standard interpretations of his thinking as a whole. Of particular interest for us will be to examine the deep sources of Wittgenstein’s apparent hostility to the foundational role of Set Theory. Once this is clarified, we will investigate whether Wittgenstein’s valuable insights about mathematical practice in general (appearing especially in his Lectures and the Remarks), do apply to Set Theory as well.

We aim to exploit the anthropological strain in Wittgenstein’s later work, and thus explicating how the practice of operating with sets finds a place within it brings out one of the project’s major challenges. For the field of anthropology itself can be regarded as containing a biological branch, thus placing it closer to the natural sciences. Yet it also contains a more dominant ethnographic, cultural branch, which explicitly accounts for human societies in terms of various conventions. Thus, coming to understand how Wittgenstein saw the conventional in relation to the natural will be critical for getting a better handle on how symbolic activity can be a feature of our nature. How to bring these together in a satisfying way that naturalizes practice while preserving a recognizably robust conception of normativity suitable for Set Theory will be the primary focus of one part of the project. This is a point where we will benefit from input from other relevant disciplines – especially mathematics, anthropology and psychology. The key challenge, then, of this part involves providing a coherent account of Wittgenstein’s concepts of practice and custom that harmonize with regarding these as natural aspects of human life.