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 representation, and that (...) of the properties of usage associated with signs. In Section 2 we illustrate these ideas with a case study of diagrammatic representation: the evolution from Euler diagrams to Venn diagrams. Section 3 is correspondingly devoted to pictorial representation, illustrated by the comparison between parallel and linear perspective drawing. We conclude with open questions, and then briefly summarize the articles to follow. (shrink)

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 introduced.

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 in mathematics as props; to (...) better specify the claim, the notion of “affordance” is explored in its possible extension from concrete objects to representations. (shrink)

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 practice. Then, I (...) will give an example of mathematical reasoning with a figure, and show that both visualization and intuition are involved. I claim that mathematical intuition depends on background knowledge and expertise, and that it allows to see the generality of the conclusions obtained by means of visualization. (shrink)

In the past 10 years, contemporary philosophy of mathematics has seen the development of a trend that conceives mathematics as first and foremost a human activity and in particular as a kind of practice. However, only recently the need for a general framework to account for the target of the so-called philosophy of mathematical practice has emerged. The purpose of the present article is to make progress towards the definition of a more precise general framework for the philosophy of mathematical (...) practice by exploring two strategies. A first strategy is to turn to philosophy of mind and Edwin Hutchins' view of distributed cognition in order to better understand the cognitive issues at play when considering a community of mathematicians; a second strategy is to refer to philosophy of language and focus on Robert Brandom's inferentialism and mathematical conceptual content. A possible combination of these two views, called enhanced material inferentialism, is then put forward as a promising framework to account for the philosophy of mathematical practice. (shrink)

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 specific form of enhanced manipulative imagination, in order to draw inferences from knot diagrams by performing epistemic actions. Moreover, it will be argued that knot diagrams not only can promote discovery, but also provide evidence. This case study is an experimentation ground to evaluate the role of space and action in making inferences by reasoning diagrammatically. (shrink)

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 a un articolo di Johnson- Laird del 2005 intitolato «The (...) Shape of Problems». Nelle conclusioni, farò un ultimo riferimento a un articolo di Varzi pubblicato nel 2000, scritto con Philip Kitcher e intitolato «Some Pictures Are Worth 2 אo Sentences». Le parole configurazioni, geometry, shape e picture nei titoli degli articoli di cui mi occuperò sono dei buoni indizi per capire quale sia il filo rosso che lega a mio avviso tutti questi lavori. (shrink)

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 theses will be defended. First, the representations used (...) in the practice are an integral part of the mathematical reasoning. As a matter of fact, they convey in a material form the relevant transitions and thus allow experts to draw inferential connections. Second, in low-dimensional topology experts exploit a particular type of manipulative imagination which is connected to intuition of two- and three-dimensional space and motor agency. This imagination allows recognizing the transformations which connect different pictures in an argument. Third, the epistemic—and inferential—actions performed are permissible only within a specific practice: this form of reasoning is subject-matter dependent. Local criteria of validity are established to assure the soundness of representationally heterogeneous arguments in low-dimensional topology. (shrink)

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 experiments and proofs in pure mathematics will be provided. (...) To do so, three examples of experimenting with triangles will be introduced and discussed. (shrink)

1 Mathematical practice: a short overview This volume is a collection of essays that discuss the relationships between the practices deployed by logicians and mathematicians, either as individuals or as members of research communities, and the results from their research. We are interested in exploring the concept of 'practices' in the formal sciences. Though common in the history, philosophy and sociology of science, this concept has surprisingly thus far been little reflected upon in logic...

1 Mathematical practice: a short overview This volume is a collection of essays that discuss the relationships between the practices deployed by logicians and mathematicians, either as individuals or as members of research communities, and the results from their research. We are interested in exploring the concept of 'practices' in the formal sciences. Though common in the history, philosophy and sociology of science, this concept has surprisingly thus far been little reflected upon in logic...

1 Mathematical practice: a short overview This volume is a collection of essays that discuss the relationships between the practices deployed by logicians and mathematicians, either as individuals or as members of research communities, and the results from their research. We are interested in exploring the concept of 'practices' in the formal sciences. Though common in the history, philosophy and sociology of science, this concept has surprisingly thus far been little reflected upon in logic...

8 chapters are available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.

1 Mathematical practice: a short overview This volume is a collection of essays that discuss the relationships between the practices deployed by logicians and mathematicians, either as individuals or as members of research communities, and the results from their research. We are interested in exploring the concept of 'practices' in the formal sciences. Though common in the history, philosophy and sociology of science, this concept has surprisingly thus far been little reflected upon in logic...

It is possible to argue that the only ‘genuine’ experiments in mathematics are in fact thought experiments. Nevertheless, one preliminary question is to ask what genuine mathematica experiments are, and whether the activity of experimentation is a proper mathematical activity. In this article, I will consider which kind of conception of mathematics allows for the existence of experiments in mathematics and I will present the constraints these experiments are subject to. Then, I will discuss three examples to arrive at a (...) general conclusion: experimentation is indeed a genuine mathematical activity, but its mechanisms are particular with respect to experimentation in the natural sciences. (shrink)

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 practitioners; (...) and (3) the specific forms of representation and notation shared and selected by the practitioners. Moreover, it is claimed that a broadening of the notion of ‘permissible action’ as introduced by Larvor (2012) with respect to mathematical arguments, allows for a consideration of all these three elements simultaneously. Second, a case from topology – the proof of Alexander’s theorem – is presented to illustrate a concrete analysis of a mathematical practice and to exemplify the proposed method. It is discussed that the attention to the three elements of the practice identified above brings to the emergence of philosophically relevant features in the practice of topology: the need for a revision in the definition of criteria of validity, the interest in tracking the operations that are performed on the notation, and the constant and fruitful back-and-forth from one representation to another in dealing with mathematical content. Finally, some suggestions for further research are given in the conclusions. (shrink)

Achille Varzi è uno dei maggiori metafisici viventi. Nel corso degli anni ha scritto testi fondamentali di logica, metafisica, mereologia, filosofia del linguaggio. Ha sconfinato nella topologia, nella geografia, nella matematica, ha ragionato di mostri e confini, percezione e buchi, viaggi nel tempo, nicchie, eventi e ciambelle; e non ha disdegnato di dialogare con gli abitanti di Flatlandia, con Neo e con Terminator. Tra le sue opere principali: Holes and Other Superficialities e Parts and Places. The Structures of Spatial Representation, (...) entrambi scritti insieme a R. Casati per MIT Press; Il mondo messo a fuoco, Laterza; e il suo libro più recente: Le tribolazioni del filosofare, con C. Calosi, per Laterza. -/- Da una giornata all’Università di Urbino nasce questa conversazione a molte voci sulla e con la filosofia di Achille C. Varzi. In un dialogo critico al quale l’Autore si presta con generosità e onestà intellettuale, Andrea Borghini, Francesco Calemi, Claudio Calosi, Elena Casetta, Valeria Giardino, Pierluigi Graziani, Patrizia Pedrini, Daniele Santoro e Giuliano Torrengo lo interrogano e mettono alla prova sui temi affrontati, nel corso degli anni, in campi diversi. Il risultato è un percorso che si snoda attraverso molti mondi, dalla logica alla metafisica, dalla filosofia del linguaggio alla filosofia della matematica, dalla mereologia alla filosofia del tempo, spingendosi in qualche caso oltre i confini del saggio filosofico. (shrink)

Public deliberation has been defended as a rational and noncoercive way to overcome paradoxical results from democratic voting, by promoting consensus on the available alternatives on the political agenda. Some critics have argued that full consensus is too demanding and inimical to pluralism and have pointed out that single-peakedness, a much less stringent condition, is sufficient to overcome voting paradoxes. According to these accounts, deliberation can induce single-peakedness through the creation of a ‘meta-agreement’, that is, agreement on the dimension according (...) to which the issues at stake are ‘conceptualized’. We argue here that once all the conditions needed for deliberation to bring about single-peakedness through meta-agreement are unpacked and made explicit, meta-agreement turns out to be a highly demanding condition, and one that is very inhospitable to pluralism. (shrink)

En 2014 et en 2015, deux campagnes se sont déroulées pour la continuation de l’étude des vases en métal et des instruments apparentés de Delphes, de 600 av. J.‑C. à l’époque romaine. La recherche, qui a pour but la publication systématique de ce corpus, porte sur des centaines d’objets, dont la plupart inédits. La majorité provient évidemment des sanctuaires d’Apollon et d’Athéna à Marmaria, mais des attestations relèvent aussi des contextes funéraires et domestiques du site. L’activité au mu...

The following paper explores the categorisation of groups. Indeed, there are different ways to distinguish human groups from one another: on the one hand, sociological analyses focus their attention on the distinction between being inside and outside of groups; on the other hand, collective action theories mainly focus on the distinction between collectives and aggregates, based on the kind of action that groups can perform, i.e., joint or not. In this paper, we offer an alternative view by adopting the agent’s (...) perspective and creating a scale based on the importance the agent attributes to his or her membership to different groups. While this perspective seems to lose objectivity as it does not allow the formulation of an unambiguous scale that applies to all individuals, it does allow us to understand the process of identification and the consequent phenomenon whereby very often we act in accordance with other people’s actions, in a collective way, without prior coordination. A practical example based on the so-called “acting white” will be explored in order to test the paper’s proposal. The analysis intends to offer a common ground on the basis of which it would be possible to have further inquiries on the social dimension of self-experience. (shrink)

O advento da sociedade em rede promoveu mudanças significativas na experiência-com-o-mundo. Observa-se nos últimos anos a crescente toxidade no ambiente informacional, marcado tanto pela vivência tóxica da informação que circula legalmente nas redes sociais como por informações que intoxicam os indivíduos e grupos. O objetivo é discutir, em tempos de pós-verdade e da sociedade non-stop, o aspecto tóxico da informação, que ao tornar o ambiente informacional tóxico, adoece indivíduos e também as sociedades democráticas, ao esmaecer a empatia para com os (...) demais e de pôr em risco a própria ideia de democracia e de vida partilhada de modo democrático. A proposta é discutir a intoxicação pela informação não pelo excesso (infoxicação), mas pelo que nomeio informação tóxica, i.e., o aspecto tóxico da informação e de ambiente informacional tóxico tendo como referência os discursos de ódio. Para tanto serão consideradas a desconstrução de experiências de estados democráticos mediante o crescimento da circulação de discursos de ódio e de discursos de ódio fascistas; o crescimento da desinformação; o crescimento da circulação de notícias fraudulentas (Fake news) e de notícias falsas (por exemplo, as não endossadas pelos saberes científicos, como o terraplanismo, a ideia da supremacia branca); os robots e os algoritmos. (shrink)

Le laboratoire scientifique est ici appréhendé dans sa spécificité propre. Il est bien situé en tant qu'objet anthropologique. Le lecteur trouvera dans ce livre la problématique du pouvoir dans le laboratoire, il découvrira une manière spécifique de traiter la dialectique entre production identitaire individuelle et collective.

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 is impossible to do in finite terms. Proofs may serve as ‘shortcuts’ in this respect. A proof is defined by Lolli as any finite argument certifying that A is a consequence of T. A proof is a shortcut in the sense that it spares us considering infinitely many interpretations.The reason for such a very general definition of proof is Lolli's strong belief that mathematics is not a rigid system of explicit rules, but rather a set of tools; as a consequence, there is no prescription as to what a proof should or should not be. Actually, mathematics is historically situated and not timeless, and the history of mathematics is the …. (shrink)

This article describes the methods involved in building a diachronic multilingual corpus devoted to Fine Arts, beginning with G. Vasari's Lives of the most excellent Italian architects, sculptors and painters (1568) as the fundamental source text in the field of Art History. Attention is given to automatic pre-alignment, the special proofreading protocol and segmentation rules developed to allow multilingual and/or diachronic alignment of multiple texts, and the difficulties inherent in annotating a multilingual database. A case study is offered, comparing the (...) term “tondo” and its variants (e.g. “mezzo tondo”, “di tondo rilievo”) in Italian with their translations in English (De Vere 1912-15) and French (Leclanché 1839, Chastel 1981, Luciani 2002, Powell 2007). These later translations contain a greater variety of both lexicalized counterparts such as “medallion” or “lunette” as well as paraphrases, used to disambiguate and distinguish the various meanings of “tondo”, thus demonstrating how artistic vocabulary, which was still developing in 16th c. Italian, through its transmission to other languages, became more diversified and stable over time. (shrink)

In the age of Anthropocene, can aesthetics propose a not prevaricating behavioural model towards nature, leading to a new conception of humanity based on our being moved by nature? In this article, we will try to answer this question, exploring how some exponents of contemporary Environmental Aesthetics have examined the significance of human emotional responses to nature. Starting from the assumption that Kant defines the sublime as a motion of the soul, we wonder how aesthetics can reinterpret this concept to (...) affirm not human superiority over nature but his being part of it. To do so, we will turn our attention not only to some exponents of Environmental Aesthetics but also to Schopenhauer's concept of the sublime and the thought of Arne Næss, the father of the Deep Ecology movement. (shrink)

En este artículo realizamos un relevamiento exhaustivo de las publicaciones de Médicos por la Verdad, uno de los principales grupos antivacunas de Latinoamérica, en la red social Twitter durante la pandemia de COVID-19. Clasificamos sus tipos de razonamiento y el contenido de sus mensajes y mostramos que las propuestas existentes de análisis de discursos anticientíficos no pueden aplicarse a este caso particular. Proponemos, en consecuencia, una nueva categorización y su aplicación focalizada en las herramientas disponibles para comunicadores de la ciencia, (...) por un lado, y público no especializado, por el otro. (shrink)

En este trabajo, se propone una clasificación epistemológica teórica para el discurso astrológico a partir de evidencia empírica a fin de abordar el problema de cómo determinar específicamente qué es lo que lo hace ilegítimo. A partir de esta clasificación, se diseñó una intervención didáctica, enmarcada en el enfoque de Enseñanza de las Ciencias Naturales en Contexto (ECNC), con el objetivo de fomentar, en estudiantes de nivel secundario, la capacidad de distinguir enunciados cognoscitivamente ilegítimos -por infalsables o por falsos- respecto (...) de aquellos que pueden constituir genuino conocimiento científico. La secuencia didáctica se implementó con un grupo de 51 estudiantes de 14 a 17 años. Se evaluó su impacto por comparación entre un pretest y un postest. Los resultados mostraron que su capacidad de discernimiento entre enunciados falsables y no falsables se incrementó apreciablemente. (shrink)