Periodic Reporting for period 1 - ProvingAgency (Proving Agency in Mathematical Practice: Practical, Social, and Mental Aspects)
Reporting period: 2023-06-01 to 2025-05-31
Mathematical reasoning is essential to the development of mathematical and scientific knowledge, and is a crucial skill in our science- and technology-based European societies. Yet, despite the revolutionary advances in formal logic of the past century, the nature of reasoning and proofs in ordinary mathematical practice remains unclear.
The ProvingAgency project aims to move forwards on this general question by shifting the focus from mathematical proofs themselves to the activity of proving that gives rise to them. The approach to be developed proposes to structure the inquiry around the different dimensions of proving agency—the capacities of proving agents necessary to the realization of the activity of proving. The general idea is then to attend to what mathematicians do rather than the end products of their activity. This immediately raises a number of issues: What are the necessary characteristics that agents need to possess to prove theorems? What are the conceptual, metaphysical, and normative resources required to account for the activity of proving? In which sense should proving be considered as a mental activity? Such a perspective would structure the inquiry around the different aspects of proving agency, that is, around the different dimensions of the proving agent and of the proving activity itself. Furthermore, it may help identify the most appropriate theoretical and empirical methods to model the different aspects of proving agency.
The aim of the ProvingAgency project is to implement this approach concretely, within a restricted yet substantial perimeter, in order to investigate three central aspects of proving agency: (i) practical aspects which concern the different kinds of practical knowledge or know-how required by mathematical agents to prove theorems; (ii) social aspects which concern the issues that arise when mathematical agents are interacting in proving; (iii) mental aspects which concern the nature of proving taken as a mental activity. The main objective of the ProvingAgency project can be stated as follows: to implement a systematic, multi-disciplinary approach to the study of proofs and proving in mathematical practice, centered around the notion of proving agency, in order to account for some of the key practical, social, and mental aspects of proving agents and the activity of proving.
Practical aspects are concerned with the practical knowledge required by mathematical agents to prove theorems. The objective here is to theorize the kind of mathematical understanding required to engage in the activity of proving, with a specific focus on practical knowledge and practical reasoning.
Social aspects are concerned with the issues that arise when mathematicians are interacting in the activity of proving. This happens whenever several mathematical agents are proving a theorem together or when a mathematical proof is submitted by one or more agents for examination by the mathematical community. Using resources from social epistemology, we aim here to identify the social mechanisms at work when the activity of proving is conceived as a social endeavor.
Mental aspects are concerned with the mental dimension of the activity of proving. We will aim here to provide a conception of proving as a mental activity that does justice to the fact that proving often requires to rely on mathematical artifacts, which we will do by building on recent developments in the emerging field of extended epistemology.
The ProvingAgency project aims to move forwards on this general question by shifting the focus from mathematical proofs themselves to the activity of proving that gives rise to them. The approach to be developed proposes to structure the inquiry around the different dimensions of proving agency—the capacities of proving agents necessary to the realization of the activity of proving. The general idea is then to attend to what mathematicians do rather than the end products of their activity. This immediately raises a number of issues: What are the necessary characteristics that agents need to possess to prove theorems? What are the conceptual, metaphysical, and normative resources required to account for the activity of proving? In which sense should proving be considered as a mental activity? Such a perspective would structure the inquiry around the different aspects of proving agency, that is, around the different dimensions of the proving agent and of the proving activity itself. Furthermore, it may help identify the most appropriate theoretical and empirical methods to model the different aspects of proving agency.
The aim of the ProvingAgency project is to implement this approach concretely, within a restricted yet substantial perimeter, in order to investigate three central aspects of proving agency: (i) practical aspects which concern the different kinds of practical knowledge or know-how required by mathematical agents to prove theorems; (ii) social aspects which concern the issues that arise when mathematical agents are interacting in proving; (iii) mental aspects which concern the nature of proving taken as a mental activity. The main objective of the ProvingAgency project can be stated as follows: to implement a systematic, multi-disciplinary approach to the study of proofs and proving in mathematical practice, centered around the notion of proving agency, in order to account for some of the key practical, social, and mental aspects of proving agents and the activity of proving.
Practical aspects are concerned with the practical knowledge required by mathematical agents to prove theorems. The objective here is to theorize the kind of mathematical understanding required to engage in the activity of proving, with a specific focus on practical knowledge and practical reasoning.
Social aspects are concerned with the issues that arise when mathematicians are interacting in the activity of proving. This happens whenever several mathematical agents are proving a theorem together or when a mathematical proof is submitted by one or more agents for examination by the mathematical community. Using resources from social epistemology, we aim here to identify the social mechanisms at work when the activity of proving is conceived as a social endeavor.
Mental aspects are concerned with the mental dimension of the activity of proving. We will aim here to provide a conception of proving as a mental activity that does justice to the fact that proving often requires to rely on mathematical artifacts, which we will do by building on recent developments in the emerging field of extended epistemology.
This MSCA project was interrupted in the second year of the outgoing phase at the ETH Zürich. I am thus reporting here on the work performed and the main achievements reached during this period which focused on the practical and social aspects of proving agency.
To achieve the objective of theorizing the kind of mathematical understanding required to engage in the activity of proving, I conducted a comprehensive philosophical and conceptual investigation into the nature of proof understanding. This work builds on Bratman’s theory of planning agency and provides a detailed characterization of the practical knowledge and reasoning involved in proving mathematical theorems. Through rigorous conceptual analysis and case studies, this work demonstrates how understanding a proof goes beyond mere verification and instead involves reconstructing the rational plan underlying the proof. This contribution directly addresses the practical aspects of mathematical reasoning by elucidating the kinds of practical decisions and inferential structures that constitute mathematical understanding. The main achievement is the development of a robust framework that captures the agent-centered and goal-directed nature of proving, offering both explanatory power for understanding failures and a methodology for analyzing and improving proof comprehension. The results are reported in an article co-authored with Rebecca Morris and published in Noûs entitled “Understanding in mathematics: The case of mathematical proofs”.
To address the objective of examining the social dimensions involved in the activity of proving, I developed a conceptual model of collective mathematical knowledge. Drawing on resources from social epistemology and the social philosophy of science, the work explored how proofs circulate within the mathematical community and become accepted or contested over time. A central activity was the articulation of a framework that treats the mathematical community as a decentralized multi-agent system, in which individual agents engage in public and private epistemic actions such as publishing, accepting, or verifying proofs. The main achievement was the proposal of a tripartite characterization of collective mathematical knowledge based on collective acceptance, collective justification, and proof correctness. This model accounts for both the dynamics of reputational influence and the challenges posed by errors or unexamined results within the community. A central idea is that collective knowledge in mathematics is not merely the aggregation of individual beliefs, but emerges from structured interactions and social mechanisms, thus offering a new lens through which to understand the epistemic life of mathematical proofs. This ongoing work is conducted in cooperation with Line Andersen from the Vrije Universiteit Brussel. I have presented this work in two workshops in Nice (November, 2024) and Brussels (February, 2025). In the Spring 2025, we will write an article reporting our main results.
To achieve the objective of theorizing the kind of mathematical understanding required to engage in the activity of proving, I conducted a comprehensive philosophical and conceptual investigation into the nature of proof understanding. This work builds on Bratman’s theory of planning agency and provides a detailed characterization of the practical knowledge and reasoning involved in proving mathematical theorems. Through rigorous conceptual analysis and case studies, this work demonstrates how understanding a proof goes beyond mere verification and instead involves reconstructing the rational plan underlying the proof. This contribution directly addresses the practical aspects of mathematical reasoning by elucidating the kinds of practical decisions and inferential structures that constitute mathematical understanding. The main achievement is the development of a robust framework that captures the agent-centered and goal-directed nature of proving, offering both explanatory power for understanding failures and a methodology for analyzing and improving proof comprehension. The results are reported in an article co-authored with Rebecca Morris and published in Noûs entitled “Understanding in mathematics: The case of mathematical proofs”.
To address the objective of examining the social dimensions involved in the activity of proving, I developed a conceptual model of collective mathematical knowledge. Drawing on resources from social epistemology and the social philosophy of science, the work explored how proofs circulate within the mathematical community and become accepted or contested over time. A central activity was the articulation of a framework that treats the mathematical community as a decentralized multi-agent system, in which individual agents engage in public and private epistemic actions such as publishing, accepting, or verifying proofs. The main achievement was the proposal of a tripartite characterization of collective mathematical knowledge based on collective acceptance, collective justification, and proof correctness. This model accounts for both the dynamics of reputational influence and the challenges posed by errors or unexamined results within the community. A central idea is that collective knowledge in mathematics is not merely the aggregation of individual beliefs, but emerges from structured interactions and social mechanisms, thus offering a new lens through which to understand the epistemic life of mathematical proofs. This ongoing work is conducted in cooperation with Line Andersen from the Vrije Universiteit Brussel. I have presented this work in two workshops in Nice (November, 2024) and Brussels (February, 2025). In the Spring 2025, we will write an article reporting our main results.
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.
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.