•  4
    A characteristic feature of the philosophy of mathematical practice is to attend to what people do when they do mathematics. But what does it mean to do mathematics? This question raises several issues regarding the nature of action, activity, and agency in mathematical practice. The present chapter reviews contributions in the field that have attempted to theorize about these notions. It begins with some motivations for taking agents seriously in the philosophical study of mathematical practice…Read more
  •  38
    Although understanding is the object of a growing literature in epistemology and the philosophy of science, only few studies have concerned understanding in mathematics. This essay offers an account of a fundamental form of mathematical understanding: proof understanding. The account builds on a simple idea, namely that understanding a proof amounts to rationally reconstructing its underlying plan. This characterization is fleshed out by specifying the relevant notion of plan and the associated …Read more
  •  27
    Rationality in Mathematical Proofs
    with Rebecca Lea Morris
    Australasian Journal of Philosophy 101 (4): 793-808. 2023.
    Mathematical proofs are not sequences of arbitrary deductive steps—each deductive step is, to some extent, rational. This paper aims to identify and characterize the particular form of rationality at play in mathematical proofs. The approach adopted consists in viewing mathematical proofs as reports of proof activities—that is, sequences of deductive inferences—and in characterizing the rationality of the former in terms of that of the latter. It is argued that proof activities are governed by s…Read more
  •  52
    Proofs, Reliable Processes, and Justification in Mathematics
    British Journal for the Philosophy of Science 74 (4): 1027-1045. 2021.
    Although there exist today a variety of non-deductive reliable processes able to determine the truth of certain mathematical propositions, proof remains the only form of justification accepted in mathematical practice. Some philosophers and mathematicians have contested this commonly accepted epistemic superiority of proof on the ground that mathematicians are fallible: when the deductive method is carried out by a fallible agent, then it comes with its own level of reliability, and so might hap…Read more
  •  64
    Counterexample Search in Diagram‐Based Geometric Reasoning
    with John Mumma and Marie Amalric
    Cognitive Science 45 (4). 2021.
    Topological relations such as inside, outside, or intersection are ubiquitous to our spatial thinking. Here, we examined how people reason deductively with topological relations between points, lines, and circles in geometric diagrams. We hypothesized in particular that a counterexample search generally underlies this type of reasoning. We first verified that educated adults without specific math training were able to produce correct diagrammatic representations contained in the premisses of an …Read more
  •  682
    Probabilistic Proofs, Lottery Propositions, and Mathematical Knowledge
    Philosophical Quarterly 72 (1): 77-89. 2021.
    In mathematics, any form of probabilistic proof obtained through the application of a probabilistic method is not considered as a legitimate way of gaining mathematical knowledge. In a series of papers, Don Fallis has defended the thesis that there are no epistemic reasons justifying mathematicians’ rejection of probabilistic proofs. This paper identifies such an epistemic reason. More specifically, it is argued here that if one adopts a conception of mathematical knowledge in which an epistemic…Read more
  •  59
    Universal intuitions of spatial relations in elementary geometry
    with Ineke J. M. Van der Ham and John Mumma
    Journal of Cognitive Psychology 29 (3): 269-278. 2017.
    Spatial relations are central to geometrical thinking. With respect to the classical elementary geometry of Euclid’s Elements, a distinction between co-exact, or qualitative, and exact, or metric, spatial relations has recently been advanced as fundamental. We tested the universality of intuitions of these relations in a group of Senegalese and Dutch participants. Participants performed an odd-one-out task with stimuli that in all but one case display a particular spatial relation between geomet…Read more
  •  449
    Cognitive processing of spatial relations in Euclidean diagrams
    with Milan N. A. van der Kuil, Ineke J. M. van der Ham, and John Mumma
    Acta Psychologica 205 1--10. 2020.
    The cognitive processing of spatial relations in Euclidean diagrams is central to the diagram-based geometric practice of Euclid's Elements. In this study, we investigate this processing through two dichotomies among spatial relations—metric vs topological and exact vs co-exact—introduced by Manders in his seminal epistemological analysis of Euclid's geometric practice. To this end, we carried out a two-part experiment where participants were asked to judge spatial relations in Euclidean diagram…Read more
  •  105
    In recent years, philosophical work directly concerned with the practice of mathematics has intensified, giving rise to a movement known as the philosophy of mathematical practice . In this paper we offer a survey of this movement aimed at mathematics educators. We first describe the core questions philosophers of mathematical practice investigate as well as the philosophical methods they use to tackle them. We then provide a selective overview of work in the philosophy of mathematical practice …Read more
  •  104
    Plans and planning in mathematical proofs
    with Rebecca Lea Morris
    Review of Symbolic Logic 14 (4): 1030-1065. 2020.
    In practice, mathematical proofs are most often the result of careful planning by the agents who produced them. As a consequence, each mathematical proof inherits a plan in virtue of the way it is produced, a plan which underlies its “architecture” or “unity”. This paper provides an account of plans and planning in the context of mathematical proofs. The approach adopted here consists in looking for these notions not in mathematical proofs themselves, but in the agents who produced them. The sta…Read more
  •  73
    Mathematical rigor and proof
    Review of Symbolic Logic 15 (2): 409-449. 2022.
    Mathematical proof is the primary form of justification for mathematical knowledge, but in order to count as a proper justification for a piece of mathematical knowl- edge, a mathematical proof must be rigorous. What does it mean then for a mathematical proof to be rigorous? According to what I shall call the standard view, a mathematical proof is rigorous if and only if it can be routinely translated into a formal proof. The standard view is almost an orthodoxy among contemporary mathematicians…Read more
  •  32
    Conversation is one of the main contexts in which we are conducting inquiries. Yet, little attention has been paid so far in pragmatics or epistemology to the process of inquiry in conversation. In this paper, we propose to trigger such an investigation through the development of a formal modelling based on inquisitive pragmatics—a framework offering a semantic representation of questions and answers, along with an analysis of the pragmatic principles that govern questioning and answering moves …Read more
  • Poincaré and Prawitz on mathematical induction
    In Pavel Arazim & Michal Dancak (eds.), Logica Yearbook 2014, College Publications. pp. 149-164. 2015.
    Poincaré and Prawitz have both developed an account of how one can acquire knowledge through reasoning by mathematical induction. Surprisingly, their two accounts are very close to each other: both consider that what underlies reasoning by mathematical induction is a certain chain of inferences by modus ponens ‘moving along’, so to speak, the well-ordered structure of the natural numbers. Yet, Poincaré’s central point is that such a chain of inferences is not sufficient to account for the know…Read more
  •  46
    We propose a dynamic-epistemic analysis of the different epistemic operations constitutive of the process of interrogative inquiry, as described by Hintikka’s Interrogative Model of Inquiry (IMI). We develop a dynamic logic of questions for representing interrogative steps, based on Hintikka’s treatment of questions in the IMI, along with a dynamic logic of inferences for representing deductive steps, based on the tableau method. We then merge these two systems into a dynamic logic of interrogat…Read more
  •  53
    Logics of questions
    Synthese 192 (6): 1581-1584. 2015.
    Traditional logical theories are concerned with the characterization of valid reasoning. For such logical theories, the main object of investigation is the notion of entailment, a notion that is construed as a relation between two or more declarative statements, dictating when one of them can be legitimately inferred from the others. In the course of the previous century, however, and especially since the 1970s, the scope of logical theories has become much broader. In particular, logic is no lo…Read more
  •  60
    The Interrogative Model of Inquiry and Dynamic Epistemic Logics are two central paradigms in formal epistemology. This paper is motivated by the observation of a significant complementarity between them: on the one hand, the IMI provides a framework for investigating inquiry represented as an idealized game between an Inquirer and Nature, along with an account of the interaction between questions and inferences in information-seeking processes, but is lacking a formulation in the multi-agent cas…Read more
  •  95
    Prolegomena to a cognitive investigation of Euclidean diagrammatic reasoning
    with John Mumma
    Journal of Logic, Language and Information 22 (4): 421-448. 2013.
    Euclidean diagrammatic reasoning refers to the diagrammatic inferential practice that originated in the geometrical proofs of Euclid’s Elements. A seminal philosophical analysis of this practice by Manders (‘The Euclidean diagram’, 2008) has revealed that a systematic method of reasoning underlies the use of diagrams in Euclid’s proofs, leading in turn to a logical analysis aiming to capture this method formally via proof systems. The central premise of this paper is that our understanding of Eu…Read more
  •  115
    Mathematical Inference and Logical Inference
    Review of Symbolic Logic 11 (4): 665-704. 2018.
    The deviation of mathematical proof—proof in mathematical practice—from the ideal of formal proof—proof in formal logic—has led many philosophers of mathematics to reconsider the commonly accepted view according to which the notion of formal proof provides an accurate descriptive account of mathematical proof. This, in turn, has motivated a search for alternative accounts of mathematical proof purporting to be more faithful to the reality of mathematical practice. Yet, in order to develop and ev…Read more
  •  69
    Mathematical rigor is commonly formulated by mathematicians and philosophers using the notion of proof gap: a mathematical proof is rig­orous when there is no gaps in the mathematical reasoning of the proof. Any philosophical approach to mathematical rigor along this line requires then an account of what a proof gap is. However, the notion of proof gap makes sense only relatively to a given conception of valid mathematical reasoning, i.e., to a given conception of the validity of mathematical in…Read more