•  8
    Experimenting with Triangles
    Axiomathes 32 (1): 55-77. 2022.
    Is there anything like an experiment in mathematics? And if this is the case, what would distinguish a mathematical experiment from a mathematical thought experiment? In the present paper, a framework for the practice of mathematics will be put forward, which will consider mathematics as an experimenting activity and as a proving activity. The relationship between these two activities will be explored and more importantly a distinction between thought-experiments, real experiments, quasi experim…Read more
  • Diagrammatic Representation and Inference. Diagrams 2022 (edited book)
    with S. Linker, S. Burns, F. Bellucci, J. M. Boucheix, and P. Viana
    Springer. 2022.
  •  6
    Introduction: From Practice to Results in Mathematics and Logic
    with Amirouche Moktefi, Sandra Mols, and Jean Paul Van Bendegem
    Philosophia Scientae 16 5-11. 2012.
  •  126
    Introduction: Varieties of Iconicity
    Review of Philosophy and Psychology 6 (1): 1-25. 2015.
    This introduction aims to familiarize readers with basic dimensions of variation among pictorial and diagrammatic representations, as we understand them, in order to serve as a backdrop to the articles in this volume. Instead of trying to canvas the vast range of representational kinds, we focus on a few important axes of difference, and a small handful of illustrative examples. We begin in Section 1 with background: the distinction between pictures and diagrams, the concept of systems of repres…Read more
  •  39
    Manipulative imagination: how to move things around in mathematics
    Theoria : An International Journal for Theory, History and Fundations of Science 33 (2): 345-360. 2018.
    In the first part of the paper, previous work about embodied mathematics and the practice of topology will be presented. According to the proposed view, in order to become experts, topologists have to learn how to use manipulative imagination: representations are cognitive tools whose functioning depends from pre-existing cognitive abilities and from specific training. In the second part of the paper, the notion of imagination as “make-believe” is discussed to give an account of cognitive tools …Read more
  •  308
    Tools for Thought: The Case of Mathematics
    Endeavour 2 (42): 172-179. 2018.
    The objective of this article is to take into account the functioning of representational cognitive tools, and in particular of notations and visualizations in mathematics. In order to explain their functioning, formulas in algebra and logic and diagrams in topology will be presented as case studies and the notion of manipulative imagination as proposed in previous work will be discussed. To better characterize the analysis, the notions of material anchor and representational affordance will be …Read more
  •  8
    Sperimentare con I triangoli
    Rivista di Estetica 42 39-54. 2009.
  •  32
    In his book Gabriele Lolli discusses the notion of proof, which is, according to him, the most important and at the same time the least studied aspect of mathematics. According to Lolli, a theorem is a conditional sentence of the form ‘if T then A’ such that A is a logical consequence of T, where A is a sentence and T is a sentence or a conjunction or set of sentences. Verifying that A is a consequence of T generally involves considering infinitely many interpretations; so it is something which …Read more
  • Introduction: From Practice to Results in Mathematics and Logic
    with Amirouche Moktefi, Sandra Mois, and Jean Van Bendegem
    Philosophia Scientiae 16 (1): 5-11. 2012.
  •  694
    The objective of this article is twofold. First, a methodological issue is addressed. It is pointed out that even if philosophers of mathematics have been recently more and more concerned with the practice of mathematics, there is still a need for a sharp definition of what the targets of a philosophy of mathematical practice should be. Three possible objects of inquiry are put forward: (1) the collective dimension of the practice of mathematics; (2) the cognitives capacities requested to the pra…Read more
  •  200
    Geometria, ragionamento e scommesse
    In University of Urbino © Isonomia – Epistemologica (ed.), Mettere a fuoco il mondo, . pp. 36-46. 2014.
    Poiché i miei interessi di ricerca si concentrano sul rapporto tra spazio e rappresentazione, nel presente articolo commenterò un lavoro di Achille C. Varzi pubblicato nel 2008 e intitolato, nella sua versione italiana, «Configurazioni, regole e inferenze». Accennerò anche a un secondo articolo scritto da Varzi e Massimo Warglien e pubblicato nel 2003, intitolato «The Geometry of Negation». Mi rivolgerò poi alla psicologia sperimentale, collegando alcuni aspetti delle osservazioni di Varzi…Read more
  •  28
    Introduction: From Practice to Results in Mathematics and Logic
    with Amirouche Moktefi, Sandra Mois, and Jean Paul Van Bendegem
    Philosophia Scientae 16 5-11. 2012.
  •  276
    In this article, I will discuss the relationship between mathematical intuition and mathematical visualization. I will argue that in order to investigate this relationship, it is necessary to consider mathematical activity as a complex phenomenon, which involves many different cognitive resources. I will focus on two kinds of danger in recurring to visualization and I will show that they are not a good reason to conclude that visualization is not reliable, if we consider its use in mathematical …Read more
  •  627
    An Inquiry into the Practice of Proving in Low-Dimensional Topology
    In Gabriele Lolli, Giorgio Venturi & Marco Panza (eds.), From Logic to Practice, Springer International Publishing. pp. 315-336. 2015.
    The aim of this article is to investigate specific aspects connected with visualization in the practice of a mathematical subfield: low-dimensional topology. Through a case study, it will be established that visualization can play an epistemic role. The background assumption is that the consideration of the actual practice of mathematics is relevant to address epistemological issues. It will be shown that in low-dimensional topology, justifications can be based on sequences of pictures. Three these…Read more
  •  6
    Introduction: From Practice to Results in Mathematics and Logic
    with Amirouche Moktefi, Sandra Mois, and Jean Paul Van Bendegem
    Philosophia Scientiae 16 5-11. 2012.
  •  950
    Forms and Roles of Diagrams in Knot Theory
    Erkenntnis 79 (4): 829-842. 2014.
    The aim of this article is to explain why knot diagrams are an effective notation in topology. Their cognitive features and epistemic roles will be assessed. First, it will be argued that different interpretations of a figure give rise to different diagrams and as a consequence various levels of representation for knots will be identified. Second, it will be shown that knot diagrams are dynamic by pointing at the moves which are commonly applied to them. For this reason, experts must develop a s…Read more