Section: Research Program

Research Program

The aim of Anja is to develop mathematical models in selected areas of SHS, which include, at this time, economy/finance, law, and archaeology. These models are essentially probabilistic. This entails that our theoretical studies mainly lie in the fields of probability and statistics.

A major focus of Anja is on providing mathematical analyses of how performativity operates in economy/finance and law, where performativity is understood as the phenomenon by which applying models co-constructs a new reality by the very fact that the existing reality was not properly apprehended. We are chiefly concerned with performativity that results from mathematical models. Indeed, while performativity exists before and independently of such models, mathematics may, and already have, strengthened the performative power in a significant way. This has occurred so far in a uncontrolled fashion, and thus in a typically damaging way. The essence of our work is to shed light on the mechanisms mediating the increase of performativity brought by mathematical models, thus allowing one to manage their effects and hopefully orient them towards an improvement of the reality they transform.

We stress the important fact that this program allows us to go well beyond what is typically achieved in sociological studies. For instance, many such studies have evidenced the role of performativity in the context of financial theory and how it shapes today's markets [46]. This is certainly useful in order to exert political control on the tools proposed by, e.g., economists. However, such an exogenous control is not fully satisfactory because it does not provide explicit procedures to enhance the models. This is due to the fact that these studies have not permeated the technical literature to a point where it would have a significant impact on the definition and practical use of models. One explanation for this is that, though convincing, these analyses do not provide mathematical or otherwise applicable tools to modify practices: they explain general mechanisms through which, for instance, economics performs the reality of economies, but do not shed light on the precise mathematics that mediate these mechanisms. We believe that it is important that mathematicians tackle this issue. In other words, we think that it is extremely useful to reverse that statement made in [48]: “en souscrivant au programme de la performativité, la recherche en sociologie économique ne se contente plus de partager ses objets d'études avec les sciences économiques, elle inclut ces dernières dans ses propres objets d'étude” (“By subscribing to the performativity program, research in economical sociology goes beyond sharing its objects of study with economical sciences: it includes the latter in its own objects of study.). Our “reverse” statement is that economical sciences, and, more generally, mathematics applied in SHS, should include in their research objectives the sociological impact they create through performativity: mathematicians need to take into account in their models the fact that reality will be transformed by them, and thus model also this transformation.

The first goal of Anja is precisely to fill this gap, that is, to pinpoint which parts of a given mathematical model are responsible for performative effects and how this occurs. It is important to stress that, in our view, this means that the performativity of mathematical models will be itself assessed with mathematical tools. This endogeneisation permits to measure quantitatively the impact of models on reality. In turn, this quantification opens the way to our second and more ambitious goal: indeed, we propose paradigms allowing one to control the performativity of mathematical models. The main mechanisms we use in that view are as follows:

• systematically take into account the fact that the models will perform reality. This means that, when defining our models, we try as much as possible to foresee how applying them will transform practices, and then adjust them in such a way that these modifications are desirable and under control;

• impose that the output of our models always be probability distributions rather than hard prescriptions. In other words, recognizing that modelling in SHS, in addition to being typically extremely complex with a large number of variables and with many sources of errors in the process of calibrating the parameters, always involves addressing a moving reality that will be transformed by the very application of the model, we insist that, at all stages of the analysis, uncertainties be propagated, as is routinely done in industrial fields such as aeronautics, so that the answer of the system will be probabilistic. We use in particular Bayesian analysis in that view, in order to incorporate information on unknown parameters using prior probability distributions that are sequentially updated after the acquisition of each new observation.

As a longer term perspective, we intend to propose a general mathematical model of performativity. The current literature has already proposed general analyses of mechanisms through which theories can become performative: most notably, [35] has identified three main such channels, namely institutional design, social norms, and language, as well as the way in which culture and accountability affect their course of operation. Our aim will be more focused: we will concentrate on the sole performativity of mathematical models, but in this restricted frame, we wish to propose quantitative, mathematical analyses. In other words, a mathematical model of performativity should allow one to answer questions such as: when can one expect that a theory is likely to be performative, what exactly are the conditions favouring performativity, which indices should one look for in order to detect a performative influence, how can one predict whether performativity will be convergent or divergent, which aspects of reality a theory will affect and how, and finally what are the means to minimize its negative effects.

One important motivation of Anja is that we feel that, as researchers in mathematics, we are partly responsible for the way mathematics is used in social sciences. In particular, while we strongly believe that mathematical models have already and will continue to enhance our social lives as they have improved our understanding and control in natural sciences, extra caution is needed because of their performative power explained above. The core of our work is that such caution can be exercised (a) by recognizing that models impact reality and by taking into account their performative power in their very definition, and (b) by using systematically probability and statistics: in a nutshell, imposing that mathematical answers to questions in social sciences always take the form of a probability distribution should (1) remind users that no mathematical model is able to provide a definitive hard and fast answer when it comes, e.g., to computing the amount of a fine in competition law, and (2) allow one to tame the inherent complexity of human-related matters, thus providing useful guides for making informed decisions.

Of course, we do not address performativity issues in all social and human sciences. Rather, we focus on two domains where we already have an expertise: economy/finance, and law. Details on our studies in these fields are given in sections 4.1 and 4.2.

Our program cannot be realistically realised without strong collaborations from specialists in the SHS fields we deal with. In law, our expertise is brought by Jérôme Dupré, which holds a Ph.D. in law and is also a former judge. As far as archaeologyis concerned, we collaborate with Philippe Lanos (senior researcher at CNRS). He is an expert in archaeomagnetism and its applications to materials dating in archaeology.