The epistemology of mathematics has traditionally investigated the nature of mathematical knowledge. These accounts, often intertwined with a metaphysical theory of the nature of mathematics, focused on the philosophical justification of our mathematical knowledge. Flourishing as this enterprise may have been, it can no longer claim a monopoly on the epistemological study of mathematics. With the rise of the philosophy of mathematical practice, and following earlier practice-based trends in the philosophy of science, the scope of epistemological topics has been broadened significantly.

A recent topic which has garnered interest is understanding in mathematics. It is not controversial to claim that mathematicians understand the theorems they prove and teach. Yet the precise nature of understanding is itself a source of controversy. Understanding calls to mind a certain sensation, where we are suddenly struck with an insight and we feel like we have grasped the subject at hand. In attaining an understanding of a given subject, we are suddenly struck with a clarity which relieves the sense of perplexity which we may have experienced before. It is this seemingly subjective nature which in 20th century philosophy of science inspired the banishment of understanding to the field of psychology. Without denying that understanding could be an interesting topic to investigate, it was thereby considered to be well beyond of the scope of meaningful philosophical research. This preoccupation with the subjective side of understanding has not been shared by everyone though. While we may be most familiar with the feeling of understanding, this feeling is but one of the many aspects of understanding. Hence, even if we allow that there is a psychological aspect to understanding, this does not prevent us from investigating other features with a philosopher’s toolkit.

Given the developments just sketched, philosophers have fruitfully studied understanding within different fields of philosophy. The philosophy of mathematics too has seen a number of publications on the nature of mathematical understanding over the past decades. Interesting as these publications have been though, they mostly fit within a paradigm that has been dominant within epistemology since the inception of modern philosophy. Descartes famously set the agenda for the epistemological research dominant until this day. In founding the justification of all possible knowledge on the individual ratio, Descartes placed personal knowledge front and centre on the philosophical agenda. Indeed, almost all epistemological research thereafter would start from this first-person perspective. To investigate knowledge would be to investigate how I, as an actor, could attain or experience knowledge. Equally so for understanding, and, by extension, mathematical understanding.

From the paradigm introduced above, understanding of a mathematical theorem has been investigated with the aim of determining what it would mean for an individual mathematician to attain genuine understanding of said theorem. This first-person paradigm has not gone unchallenged though. During the past couple of decades, within the field of epistemology, the social nature of knowledge in general has been increasingly emphasised. Indeed, knowledge is never attained or held in a vacuum. Rather, an epistemic subject is continuously influenced in various ways by the actors surrounding it.

Beyond the recognition of social influence on individual knowledge, the possibility that knowledge itself is not exclusively held by individuals, but that groups of individuals can also be said to convey beliefs, dispositions and knowledge themselves has been considered. This new approach to epistemology, usually referred to as social epistemology, acknowledges that we assign epistemic states to groups and collectives on a daily basis, which has equally been applied to the mathematical community. According to it, individual mathematicians are not the only ones to whom we can assign beliefs and dispositions, but the mathematical community as a whole can also be said to know that this or that theorem has either or not been proven. While beliefs and even knowledge can more or less be readily assigned to groups however, the matter is not so clear with understanding. With the first forays into the idea of group understanding only being of recent date, it is far from clear what the concept would look like in mathematics. It is this topic which we shall further delve into.