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.

This talk argues that hinge epistemology adds new insights which help us understand the role of certainties in mathematical practices. Subsequently, we employ these insights to account for disagreements on a local level coexisting with a high level of consensus in modern mathematics globally. We characterize mathematics as a set of partially overlapping practices instead of simply a tree or a building with secure foundations at the bottom. Explicit disagreements are often casted out of the forefront of mathematics and qualified as mere philosophical background of different practices, thus conventionally contributing to the prevalent image of mathematics as objective and uber-consensual. In a short excurse we also talk why this might have consequences for the idea of inner-mathematical explanations pushing a context-sensitive or even relativistic approach to the issue at hand.

This is jww: José Antonio Perez-Escobar and Jordi Fairhurst.

In my talk I will discuss the meaning of ‘mathematical explanation’ from an enactivist perspective. Enactivism was so named by Varela, Thompson and Rosch (1991) and has its origins in Maturana’s Biology of Cognition (Maturana, 1970/1980; Maturana & Varela, 1992; Maturana, & Poerksen, 2004). A central principle is the observation, derived from the biological study of perception, that living beings do not directly perceive features of an external world. This means in the current context that I must abandon the question, “What is a ‘mathematical explanation’?” and ask instead, “What happens in me when I say that I see something as a ‘mathematical explanation’?”. Hence it is this latter question that I will explore. I will make reference to the feelings of certainty and ‘knowing why’ that I experience in connection with proving and explaining. Along the way, I will also make connections with the ways ‘mathematical explanation’ has previously been characterised in the mathematics education literature, and discuss some implication of this perspective for teaching.

Sources:

Maturana, H. & Varela, F. (1992). The Tree of Knowledge: The biological roots of human understanding. Shambhala.

Maturana, H. R., & Poerksen, B. (2004). From being to doing: The origins of the biology of cognition. Carl-Auer Verlag.

Maturana, H. (1980). Biology of cognition. In Maturana, H. & Varela, F. (Eds.) Autopoiesis and Cognition: The Realization of the Living (pp. 5–58). Reidel. Originally published in 1970 by the Biological Computer Laboratory, Department of Electrical Engineering, University of Illinois.

Varela, F., Thompson, E. & Rosch, E. (1991). The Embodied Mind: Cognitive Science and Human Experience. MIT Press.

This presentation investigates the pivotal role of mathematics in providing explanation and scientific understanding, grounded in Immanuel Kant’s anti-metaphysical philosophy of mathematics and my own research on abduction, diagrams, and cognitive affordances. Kant’s framework casts mathematics as a transformative tool that organizes empirical phenomena into a coherent system of relations, enabling advanced cognitive processes aimed at making the universe intelligible. I extend this perspective to explore three key aspects:

1) the generalization of Kant’s “Aesthetics” and “Logic” to support a naturalized and historicized view of mathematics, emphasizing the interplay between mathematical concepts and their historical development;

2) the critical role of mathematical modeling as a powerful method for generating explanatory scientific understanding, serving as a counterbalance to the overemphasis on big data; and

3) the importance of mathematically sound cognitive schemata to prevent superficial modeling mischaracterized as scientific.

Incorporating my own focus on abductive reasoning and diagrammatic thinking, I argue that mathematics functions as a dynamic explanatory framework that structures scientific knowledge and enhances our capacity to comprehend the world. This analysis offers a fresh perspective on the cognitive and epistemic contributions of mathematics to scientific inquiry and intelligibility.

This presentation explores the “practical turn” in contemporary philosophy of mathematics, focusing on the use of Interactive Theorem Provers (ITPs) and their intersection with artificial intelligence. In mathematical practice, the distinction between knowing-that a proof is correct and knowing-why it is correct is central. Knowing-that refers to the recognition of correctness grounded in evidential reasons (rationes cognoscendi), whereas knowing-why emphasizes conceptual understanding and explanatory reasons (rationes essendi). This distinction is especially relevant in mathematics education, where explanation—not mere verification—is key to fostering deep comprehension and intellectual autonomy.

The rise of ITPs, particularly in an era shaped by AI, raises important questions about the nature of mathematical explanation. While ITPs are highly effective at verifying proofs, their ability to clarify mathematical ideas and support explanation remains contested. This presentation critically examines the extent to which ITPs—alone or with AI tools—can contribute to the explanatory dimension of mathematical practice. Drawing on insights from philosophy, mathematics, and computer science, I assess the potential and limits of these technologies not only for discovering and verifying proofs but also for cultivating understanding and explanation. The discussion aims to clarify the evolving role of ITPs in the explanatory and practical landscape of mathematics.