大象传媒

In December 2018 the Norwegian Research Council awarded a聽FRIPRO-FRIHUMSAM research grant to Sorin Bangu (Project Leader) and聽Kevin Cahill (Collaborator), for the project Mathematics with a Human聽Face: Set Theory within a Naturalized Wittgensteinean Framework. The聽grant spans four years (2019-2024), and has a budget of ca. 10 million聽kroner (ca. 1.1 mil. euro); a PhD student and a postdoctoral fellow will聽join the project.

The project investigates the reasons why concepts such as 鈥榮et鈥� and聽鈥榥umber鈥� have remained philosophically obscure 鈥� nobody really knows聽what these things are 鈥� despite the immense success of mathematics over聽centuries. The idea is to approach this issue from a perspective never聽attempted before: by building on an overlooked Wittgensteinean insight,聽that "mathematics is after all an anthropological phenomenon" (RFM聽VII-33). The proposal is to regard mathematics, Set Theory in聽particular, as a special practice, ultimately of a social nature,聽constitutive of the human form of life. The research has聽interdisciplinary aspects, and involves collaborations with in Bergen (WAB) and other disciplines such as mathematics,聽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.聽This priority is, obviously, a socio-historical fact: as humans, we聽group things, and manipulate these groupings, all the time. Here,聽however, we argue that this priority is more than a fact: it is also of聽a conceptual nature. By highlighting how the operational component of聽Set Theory plays a constitutive role in the formation and functioning of聽its central concept, that of a 鈥榮et鈥�, we aim to show that the unique聽characteristics of mathematical reasoning and mathematical truth聽(necessity, certainty, universality) can be better accounted for, when聽regarded as entwined into the human form of life.

The project draws philosophical inspiration from Wittgenstein鈥檚 writings聽on language, mind, and mathematics. In particular, the notion of a聽(human) 鈥榝orm of life鈥� 鈥� meant to capture the complex network of聽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聽Wittgenstein is concerned, in particular the naturalist features of his聽thought relevant here, most attention has been devoted to his聽masterpiece Philosophical Investigations (PI). This is somwhat understandable given the profound impact this book has had in 20th聽century philosophy. Except for the remarks about following an聽arithmetical rule (see especially PI 搂185-202), the Investigations pay聽relatively little attention to mathematics per se. In the light of the聽significance that social practice and naturalism have in his later聽thought, central to our project will be an analysis of a less-studied聽work, his Lectures on the Foundations of Mathematics, complemented with聽his Remarks on the Foundations of Mathematics (RFM).

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

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

As we detail below, a distinctive feature of our approach is that it聽incorporates an interpretive-exegetical component. But we insist that it聽goes further; it parlays the otherwise notoriously cryptic聽Wittgensteinean insights into a full-fledged, self-standing and, as far聽as we can tell, never articulated philosophical view. We carry out a聽systematic examination of Wittgenstein鈥檚 thoughts on the role and聽function of the mathematical concept of 鈥榮et鈥� (and, by extension,聽鈥榥umber鈥�) within the human form of life. A central claim we make is that聽it is the articulation of an original anthropological naturalism,聽inspired by our interpretation of several less-studied Wittgenstein聽texts, which provides a better framework to understand humans鈥櫬爀ngagement in the set theoretical practice.

Mathematics is not only older than natural science, but also much more聽deeply involved in human life. (As Wittgenstein remarked, 鈥渨hat we call聽鈥榗ounting鈥� is an important part of our life's activities. Counting and聽calculating are not simply a pastime. Counting is a technique that is聽employed daily in the most various operations of our lives.鈥� Remarks on聽the Foundations of Mathematics, I-IV.) It is well known that human聽populations who have not developed natural science have nevertheless had聽their life profoundly shaped by what is essentially set-theoretical聽practices. We thus find it prima facie implausible that a naturalist聽philosophical framework meant to accommodate such fundamental practices聽ultimately relies on a relatively recent (and geographically confined)聽phenomenon, the occurrence of natural science.

We use in essence two methods of investigation. The first method is聽exegetical-interpretive: we propose novel interpretations of聽Wittgenstein鈥檚 writings, focusing on his less known works on mathematics聽and logic. The other method is comparative: we show that, in light of聽these interpretations, the naturalized social conception of mathematics聽we propose yields a better understanding of what people actually do when聽they do mathematics, than the understanding we have by following the聽traditional ways of conceiving this discipline. The project鈥檚 approach聽to develop a suitable concept of naturalism, by harnessing resources聽from anthropology and social science more generally, will be key here.聽

One use we make of the interpretive method is in getting a clearer聽picture of what Wittgenstein took mathematical Platonism and Formalism聽to be. (In our estimation, this was never clarified in the literature.) Here we envisage that one of our key-collaborators, Prof. Juliet Floyd聽(Boston University), the Visiting Researcher for the Project, will聽assist us in this investigation, in light of her expertise on聽Wittgenstein鈥檚 understanding of the philosophical views of his two main聽Cambridge interlocutors, the brilliant mathematicians A. Turing and G.聽Hardy (both make substantial appearances in the Lectures.) As mentioned,聽the comparative method will also be used in tackling this聽sub-hypothesis, especially in comparing the philosophical credibility of聽our (Wittgensteinean) conception to the traditional positions. We intend聽to test the strengths (and weaknesses) of our anthropological naturalism聽against other forms of naturalism currently under discussion 鈥� in聽particular scientific naturalism, and especially in regard to their聽capacity to yield suitable answers to the relation between the聽conventional and the natural mentioned above.

Selected bibliography:

Wittgenstein, L.聽Philosophical Investigations. G.E.M. Anscombe and R.聽Rhees (eds.), G.E.M. Anscombe (trans.), Oxford: Blackwell. 1953

Wittgenstein, L..聽Remarks on the Foundations of Mathematics, eds. G. von聽Wright, R. Rhees, GEM Anscombe, transl. G. E. M. Anscombe. Oxford: B.聽Blackwell, 1956.

Wittgenstein, L.聽Lectures on the Foundations of Mathematics, Cambridge聽1939, edited by C. Diamond. Chicago: Univ. of Chicago Press, 1976