Vincent Coumans

Understanding is a generally acknowledged aim of science, but the way in which understanding is pursued and criteria for when it is achieved differ widely across scientific disciplines. This paper offers a comparative analysis of this disciplinary variation, aimed at revealing differences as well as commonalities in methods and criteria for achieving understanding. Starting point is the contextual theory of scientific understanding (De Regt 2017), which was developed to accommodate contextual variation in criteria for understanding. While this work focused on historical variation and used mainly case studies from the history of physics, the current paper applies the contextual theory to disciplinary variation. We present brief descriptive accounts of how understanding figures in nine scientific disciplines: mathematics, theoretical physics, astrophysics, chemistry, geoscience, biology, neuroscience, psychology, and the engineering sciences. Subsequently, we compare and evaluate these accounts in light of the contextual theory of scientific understanding.

Following trends in the study of explanation in mathematics, we look at explanatory definitions. In particular, we build on the work of Lehet (2021a, 2021b), who describes explanatory definitions as those definitions that generate understanding. From this starting point, we investigate what types of understanding are associated with definitions, in order to develop a conception of explanatory definitions. Using the work of Baumberger (2014), we show that definitions can be explanatory in two ways: symbolically or factually. Furthermore, we zoom in on the impact of contextual factors on the explanatory value of definitions. © 2023 Elsevier B.V., All rights reserved.

Definitions are traditionally seen as abbreviations, as tools for notational convenience that do not increase inferential power. From a Philosophy of Mathematical Practice point of view, however, there is much more to definitions. For example, definitions can play a role in problem solving, definitions can contribute to understanding, sometimes equivalent definitions are appreciated differently, and so on. This chapter reviews the literature on definitions and (to a certain extent) concepts in mathematical practice. It is structured according to four themes through which definitions (and concepts) in mathematical practice have been studied. These themes concern (1) the nature of definitions, (2) whether and how concepts evolve, (3) definitions and concepts from a communal perspective, and (4) different values relating to definitions and concepts.

In the philosophy of mathematical practice, the aim is to understand the various aspects of this practice. Even though definitions are a central element of mathematical practice, the study of this aspect of mathematical practice is still in its infancy. In particular, there is little empirical evidence to substantiate claims about definitions in practice. In this article, we address this gap by reporting on an empirical investigation on how mathematicians create definitions and which roles and properties they attribute to them. On the basis of interviews with thirteen research mathematicians, we provide a broad range of relevant aspects of definitions. In particular, we address various roles of definitions and show that definitions are not just a product of mathematical factors, but also of social and contingent factors. Furthermore, we provide concrete examples of how mathematicians interact and think about definition. This broad empirical basis with a variety of examples provides an optimal starting point for future investigations into definitions in mathematical practice.