This project has produced two main results that go beyond the current state of the art in the philosophy of mathematics and the philosophy of mathematical practice.
First, the plan-based account of proof understanding that we developed redefines how we conceptualize what it means to understand a mathematical proof. Traditionally, proof understanding has often been conflated with verification, focusing merely on checking the correctness of individual inferential steps. Our account goes beyond this view by treating understanding as an agent-centered, active process—specifically, the rational reconstruction of the plan underlying the proof. Grounded in Bratman’s theory of planning agency, this approach introduces a clear structure for evaluating the epistemic processes that lead to understanding, including precise diagnostics for understanding failures and the notion of degrees of understanding.
This has important implications for mathematics education, where one of the longstanding challenges is how to effectively teach students not just to reproduce or verify proofs, but to genuinely understand them. The plan-based framework provides educators with a conceptual tool to analyze the structure of proofs in a way that is sensitive to the reasoning strategies behind them. It can help teachers identify where students struggle—not merely with logic, but with grasping the high-level structure and motivations of proofs—and design targeted interventions. Moreover, it can inform the development of intelligent tutoring systems capable of assessing whether a student has understood the rationale behind a proof, not just whether they’ve followed its steps. In this way, the framework could contribute to more robust pedagogical practices and improve the epistemic quality of mathematical learning environments. Developing these applications is an exciting avenue for future work at the crossroad of mathematics education and the philosophy of mathematics.
Second, I presented a novel model of collective mathematical knowledge as a socially emergent phenomenon. Modeling the mathematical community as a decentralized multi-agent system, I proposed that collective knowledge arises from the interplay of three components: collective acceptance, collective justification, and proof correctness. This model captures the subtle mechanisms through which mathematical proofs become authoritative in the community, including the roles played by reputation, trust, and selective attention. It also accounts for how epistemic failures can emerge—even in the absence of error in reasoning—due to systemic issues in collective evaluation and uptake. This opens the door for new philosophical and empirical investigations into how knowledge circulates and stabilizes in mathematical practice, and invites interdisciplinary collaborations with sociologists, historians of mathematics, and computational modelers.
To ensure further uptake and development of these results, future work should focus on (i) formalizing the aggregation functions and convergence dynamics within epistemic communities, (ii) empirically validating the model through case studies and sociological research, and (iii) exploring its application to domains such as peer review systems, open science platforms, and collaborative theorem proving. These outcomes also point to broader policy implications, including the importance of transparency and redundancy in systems of knowledge validation, and the need for educational approaches that align with the realities of mathematical practice as a social and cognitive enterprise. While the results are primarily theoretical, they establish a strong foundation for future applied work in education, artificial intelligence, and science policy.
A unifying thread across these results is the proving agency approach, which shifts the philosophical focus from mathematical proofs as static objects to the dynamic activity of proving carried out by agents. This shift proved to be especially fruitful in advancing both main lines of investigation. By centering the reasoning, intentions, and practical decisions of individual mathematical agents, the plan-based account of proof understanding captures aspects of mathematical practice that traditional, object-centered views overlook—such as the rationale behind inferential moves or the experience of understanding as a goal-directed process. Similarly, when considering mathematics as a collective endeavor, modeling the community as a decentralized multi-agent system allowed for a more nuanced account of how knowledge emerges, stabilizes, or fails to do so, through the interactions of situated, socially embedded agents. The proving agency perspective thus offers a flexible and integrative framework for analyzing both individual and collective dimensions of mathematical epistemology. It not only enables richer philosophical accounts of understanding and knowledge, but also aligns more closely with how mathematics is actually practiced—by people, in contexts, and through time.
