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Visualization in Mathematics
Edited by Silvia De Toffoli (University School of Advanced Studies IUSS Pavia)
About this topic
| Summary | Visualization in mathematics comes in many different varieties. It is often connected with 1) the use of spatiotemporal intuition and 2) the use of diagrams and illustrations in mathematics. Traditionally, visualization has been associated with geometry. Euclid’s Elements includes diagrams of figures and geometrical constructions. Understanding what role these diagrams played in Euclid’s proofs has been the focus of extensive researches. Visualization is, however, not limited to the realm of geometry and nowadays enters different mathematical domains, such as abstract algebra, logic, and category theory. Philosophical issues relating to visualization range from traditional debates about the a priori nature of mathematical knowledge to questions about the reliability of proofs involving diagrams. While according to the received view in philosophy of mathematics, diagrams are merely heuristic devices, recent literature challenges such view. Other questions concern the cognitive abilities at play when engaging in mathematical visualization and the relation between visualization and mathematical understanding. |
| Key works | A monograph on visualization in mathematics is Giaquinto 2007. A relevant edited collection is Mancosu et al 2005. The relation between visualization and intuition is explored in Bråting & Pejlare 2008, and Giardino 2010. Friedman 2000 focuses on intuition and geometry in the Kantian tradition. For an analysis of diagrams in Euclidean geometry, see Netz 1999, Manders 2008, Macbeth 2010, Mumma 2010, and Panza 2012. For a monograph on Peirce's logical diagrams see Shin 2002. Article focusing on diagrams in contemporary mathematics are, for example, Carter 2010, Feferman 2012, and de Toffoli 2017. The relationship between visualization in mathematics and cognitive science is investigated in Giardino 2018. |
| Introductions | Giaquinto 2008 and Mancosu 2005. For a discussion focused on mathematical diagrams see Mumma & Panza 2012. |
Show all references
Related
Siblings
- Mathematical Intuition (171)
- Mathematical Cognition, Misc (38)
- Phenomenology of Mathematics (97)
- Numerical Cognition (202)
See also
- Mathematical Practice (347)
- Mathematical Intuition (171)
- Epistemology of Mathematics (1,639 | 366)
Contents
111 found
Order:
| 1 — 50 / 111 |
- Self-graphing equations.Samuel Alexander - manuscriptCan you find an xy-equation that, when graphed, writes itself on the plane? This idea became internet-famous when a Wikipedia article on Tupper’s self-referential formula went viral in 2012. Under scrutiny, the question has two flaws: it is meaningless (it depends on fonts) and it is trivial. We fix these flaws by formalizing the problem.
- Making Mathematics Visible: Mathematical Knowledge and How it Differs from Mathematical Understanding.Anne Newstead - manuscriptThis is a grant proposal for a research project conceived and written as a Research Associate at UNSW in 2011. I have plans to spin it into an article.
- Kant’s Crucial Contribution to Euler Diagrams.Jens Lemanski - 2024 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 55 (1):59–78.Logic diagrams have been increasingly studied and applied for a few decades, not only in logic, but also in many other fields of science. The history of logic diagrams is an important subject, as many current systems and applications of logic diagrams are based on historical predecessors. While traditional histories of logic diagrams cite pioneers such as Leibniz, Euler, Venn, and Peirce, it is not widely known that Kant and the early Kantians in Germany and England played a crucial role (...)
- Who's Afraid of Mathematical Diagrams?Silvia De Toffoli - 2023 - Philosophers' Imprint 23 (1).Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for non-diagrammatic expressions. Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs. In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic proofs, one from topology and one from algebra, I show that (...)
- Diagrams, Visual Imagination, and Continuity in Peirce's Philosophy of Mathematics.Vitaly Kiryushchenko - 2023 - New York, NY, USA: Springer.This book is about the relationship between necessary reasoning and visual experience in Charles S. Peirce’s mathematical philosophy. It presents mathematics as a science that presupposes a special imaginative connection between our responsiveness to reasons and our most fundamental perceptual intuitions about space and time. Central to this view on the nature of mathematics is Peirce’s idea of diagrammatic reasoning. In practicing this kind of reasoning, one treats diagrams not simply as external auxiliary tools, but rather as immediate visualizations of (...)
- Analyzing the philosophy of travel with Schopenhauerian argument maps.Jens Lemanski - 2023 - Southern Journal of Philosophy 61 (4):588-606.Emily Thomas's seminal book The Meaning of Travel has brought the philosophy of travel back into the public eye in recent years. Thomas has shown that the topic of travel can be approached from numerous different perspectives, ranging from the historical to the conceptual‐analytical, to the political or even social‐philosophical perspectives. This article introduces another perspective, which Thomas only indirectly addresses, namely the argumentation‐theoretical perspective. It is notable that contemporary philosophy of travel lacks the nineteenth‐century approach of using diagrams and (...)
- Naive cubical type theory.Bruno Bentzen - 2022 - Mathematical Structures in Computer Science:1-27.This article proposes a way of doing type theory informally, assuming a cubical style of reasoning. It can thus be viewed as a first step toward a cubical alternative to the program of informalization of type theory carried out in the homotopy type theory book for dependent type theory augmented with axioms for univalence and higher inductive types. We adopt a cartesian cubical type theory proposed by Angiuli, Brunerie, Coquand, Favonia, Harper, and Licata as the implicit foundation, confining our presentation (...)
- What are mathematical diagrams?Silvia De Toffoli - 2022 - Synthese 200 (2):1-29.Although traditionally neglected, mathematical diagrams have recently begun to attract attention from philosophers of mathematics. By now, the literature includes several case studies investigating the role of diagrams both in discovery and justification. Certain preliminary questions have, however, been mostly bypassed. What are diagrams exactly? Are there different types of diagrams? In the scholarly literature, the term “mathematical diagram” is used in diverse ways. I propose a working definition that carves out the phenomena that are of most importance for a (...)
- Objectivity and Rigor in Classical Italian Algebraic Geometry.Silvia De Toffoli & Claudio Fontanari - 2022 - Noesis 38:195-212.The classification of algebraic surfaces by the Italian School of algebraic geometry is universally recognized as a breakthrough in 20th-century mathematics. The methods by which it was achieved do not, however, meet the modern standard of rigor and therefore appear dubious from a contemporary viewpoint. In this article, we offer a glimpse into the mathematical practice of the three leading exponents of the Italian School of algebraic geometry: Castelnuovo, Enriques, and Severi. We then bring into focus their distinctive conception of (...)
- Schopenhauers Logikdiagramme in den Mathematiklehrbüchern Adolph Diesterwegs.Jens Lemanski - 2022 - Siegener Beiträge Zur Geschichte Und Philosophie der Mathematik 16:97-127.Ein Beispiel für die Rezeption und Fortführung der schopenhauerschen Logik findet man in den Mathematiklehrbüchern Friedrich Adolph Wilhelm Diesterwegs (1790–1866), In diesem Aufsatz werden die historische und systematische Dimension dieser Anwendung von Logikdiagramme auf die Mathematik skizziert. In Kapitel 2 wird zunächst die frühe Rezeption der schopenhauerschen Logik und Philosophie der Mathematik vorgestellt. Dabei werden einige oftmals tradierte Vorurteile, die das Werk Schopenhauers betreffen, in Frage gestellt oder sogar ausgeräumt. In Kapitel 3 wird dann die Philosophie der Mathematik und der (...)
- Five dogmas of logic diagrams and how to escape them.Jens Lemanski, Andrea Anna Reichenberger, Theodor Berwe, Alfred Olszok & Claudia Anger - 2022 - Language & Communication 87 (1):258-270.In the vein of a renewed interest in diagrammatic reasoning, this paper challenges an opposition between logic diagrams and formal languages that has traditionally been the common view in philosophy of logic and linguistics. We examine, from a philosophical point of view, what we call five dogmas of logic diagrams. These are as follows: (1) diagrams are non-linguistic; (2) diagrams are visual representations; (3) diagrams are iconic, and not symbolic; (4) diagrams are non-linear; (5) diagrams are heterogenous, and not homogenous. (...)
- Visual features as carriers of abstract quantitative information.Ronald A. Rensink - 2022 - Journal of Experimental Psychology: General 8 (151):1793-1820.Four experiments investigated the extent to which abstract quantitative information can be conveyed by basic visual features. This was done by asking observers to estimate and discriminate Pearson correlation in graphical representations where the first data dimension of each element was encoded by its horizontal position, and the second by the value of one of its visual features; perceiving correlation then requires combining the information in the two encodings via a common abstract representation. Four visual features were examined: luminance, color, (...)
- Tables as powerful representational tools.Dirk Schlimm - 2022 - In Valeria Giardino, Sven Linker, Tony Burns, Francesco Bellucci, J. M. Boucheix & Diego Viana (eds.), Diagrammatic Representation and Inference. 13th International Conference, Diagrams 2022, Rome, Italy, September 14–16, 2022, Proceedings. Springer. pp. 185-201.Tables are widely used for storing, retrieving, communicating, and processing information, but in the literature on the study of representations they are still somewhat neglected. The strong structural constraints on tables allow for a clear identification of their characteristic features and the roles these play in the use of tables as representational and cognitive tools. After introducing syntactic, spatial, and semantic features of tables, we give an account of how these affect our perception and cognition on the basis of fundamental (...)
- Visual Proofs as Counterexamples to the Standard View of Informal Mathematical Proofs?Simon Weisgerber - 2022 - In Giardino V., Linker S., Burns R., Bellucci F., Boucheix J.-M. & Viana P. (eds.), Diagrammatic Representation and Inference. 13th International Conference, Diagrams 2022, Rome, Italy, September 14–16, 2022, Proceedings. Springer, Cham. pp. 37-53.A passage from Jody Azzouni’s article “The Algorithmic-Device View of Informal Rigorous Mathematical Proof” in which he argues against Hamami and Avigad’s standard view of informal mathematical proof with the help of a specific visual proof of 1/2+1/4+1/8+1/16+⋯=1 is critically examined. By reference to mathematicians’ judgments about visual proofs in general, it is argued that Azzouni’s critique of Hamami and Avigad’s account is not valid. Nevertheless, by identifying a necessary condition for the visual proof to be considered a proper proof (...)
- Counterexample Search in Diagram‐Based Geometric Reasoning.Yacin Hamami, John Mumma & Marie Amalric - 2021 - Cognitive Science 45 (4):e12959.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 inference. Our first experiment then revealed that subjects who correctly judged an (...)
- A Diagrammatic Representation of Hegel’s Science of Logic.Jens Lemanski & Valentin Pluder - 2021 - In Stapleton G. Basu A. (ed.), Diagrams 2021: Diagrammatic Representation and Inference. 93413 Cham, Deutschland: Springer. pp. 255-259.In this paper, we interpret a 19th century diagram, which is meant to visualise G.W.F. Hegel’s entire method of the `Science of Logic' on the basis of bitwise operations. For the interpretation of the diagram we use a binary numeral system, and discuss whether the anti-Hegelian argument associated with it is valid or not. The reinterpretation is intended to make more precise rules of construction, a stricter binary code and a review of strengths and weaknesses of the critique.
- Visualization as a stimulus domain for vision science.Ronald A. Rensink - 2021 - Journal of Vision 21 (3):1–18.Traditionally, vision science and information/data visualization have interacted by using knowledge of human vision to help design effective displays. It is argued here, however, that this interaction can also go in the opposite direction: the investigation of successful visualizations can lead to the discovery of interesting new issues and phenomena in visual perception. Various studies are reviewed showing how this has been done for two areas of visualization, namely, graphical representations and interaction, which lend themselves to work on visual processing (...)
- Signs as a Theme in the Philosophy of Mathematical Practice.David Waszek - 2021 - In Bharath Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice. Springer.Why study notations, diagrams, or more broadly the variety of nonverbal “representations” or “signs” that are used in mathematical practice? This chapter maps out recent work on the topic by distinguishing three main philosophical motivations for doing so. First, some work (like that on diagrammatic reasoning) studies signs to recover norms of informal or historical mathematical practices that would get lost if the particular signs that these practices rely on were translated away; work in this vein has the potential to (...)$1399.99 new (collection) View on Amazon.comRemove from this list Direct download (2 more)Export citation
Bookmark - Reconciling Rigor and Intuition.Silvia De Toffoli - 2020 - Erkenntnis 86 (6):1783-1802.Criteria of acceptability for mathematical proofs are field-dependent. In topology, though not in most other domains, it is sometimes acceptable to appeal to visual intuition to support inferential steps. In previous work :829–842, 2014; Lolli, Panza, Venturi From logic to practice, Springer, Berlin, 2015; Larvor Mathematical cultures, Springer, Berlin, 2016) my co-author and I aimed at spelling out how topological proofs work on their own terms, without appealing to formal proofs which might be associated with them. In this article, I (...)
- Stress-free math: a visual guide to acing math in grades 4-9.Theresa Fitzgerald - 2020 - Waco, TX: Prufrock Press Inc. ;.Quick reference guide includes illustrated explanations of the most common terms used in general math classes. Discusses how students can use manipulatives and basic math tools to improve their understanding. With measurement conversion tables, guides to geometric shapes, and more.
- Cognitive processing of spatial relations in Euclidean diagrams.Yacin Hamami, Milan N. A. van der Kuil, Ineke J. M. van der Ham & John Mumma - 2020 - Acta Psychologica 205:1--10.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 diagrams in a visual half field task design. In the first part, we (...)
- Multiple readability in principle and practice: Existential Graphs and complex symbols.Dirk Schlimm & David Waszek - 2020 - Logique Et Analyse 251:231-260.Since Sun-Joo Shin's groundbreaking study (2002), Peirce's existential graphs have attracted much attention as a way of writing logic that seems profoundly different from our usual logical calculi. In particular, Shin argued that existential graphs enjoy a distinctive property that marks them out as "diagrammatic": they are "multiply readable," in the sense that there are several di erent, equally legitimate ways to translate one and the same graph into a standard logical language. Stenning (2000) and Bellucci and Pietarinen (2016) have (...)
- Iconicity in mathematical notation: commutativity and symmetry.Theresa Wege, Sophie Batchelor, Matthew Inglis, Honali Mistry & Dirk Schlimm - 2020 - Journal of Numerical Cognition 3 (6):378-392.Mathematical notation includes a vast array of signs. Most mathematical signs appear to be symbolic, in the sense that their meaning is arbitrarily related to their visual appearance. We explored the hypothesis that mathematical signs with iconic aspects—those which visually resemble in some way the concepts they represent—offer a cognitive advantage over those which are purely symbolic. An early formulation of this hypothesis was made by Christine Ladd in 1883 who suggested that symmetrical signs should be used to convey commutative (...)
- A fresh look at research strategies in computational cognitive science: The case of enculturated mathematical problem solving.Regina E. Fabry & Markus Pantsar - 2019 - Synthese 198 (4):3221-3263.Marr’s seminal distinction between computational, algorithmic, and implementational levels of analysis has inspired research in cognitive science for more than 30 years. According to a widely-used paradigm, the modelling of cognitive processes should mainly operate on the computational level and be targeted at the idealised competence, rather than the actual performance of cognisers in a specific domain. In this paper, we explore how this paradigm can be adopted and revised to understand mathematical problem solving. The computational-level approach applies methods from (...)
- Foundations of geometric cognition.Mateusz Hohol - 2019 - London-New York: Routledge.The cognitive foundations of geometry have puzzled academics for a long time, and even today are mostly unknown to many scholars, including mathematical cognition researchers. -/- Foundations of Geometric Cognition shows that basic geometric skills are deeply hardwired in the visuospatial cognitive capacities of our brains, namely spatial navigation and object recognition. These capacities, shared with non-human animals and appearing in early stages of the human ontogeny, cannot, however, fully explain a uniquely human form of geometric cognition. In the book, (...)
- Naturalizing Logico-Mathematical Knowledge: Approaches from Philosophy, Psychology and Cognitive Science.Markus Pantsar - 2019 - Philosophical Quarterly 69 (275):432-435.Naturalizing Logico-Mathematical Knowledge: Approaches from Philosophy, Psychology and Cognitive Science. Edited by Bangu Sorin.
- A Priori Concepts in Euclidean Proof.Peter Fisher Epstein - 2018 - Proceedings of the Aristotelian Society 118 (3):407-417.With the discovery of consistent non-Euclidean geometries, the a priori status of Euclidean proof was radically undermined. In response, philosophers proposed two revisionary interpretations of the practice: some argued that Euclidean proof is a purely formal system of deductive logic; others suggested that Euclidean reasoning is empirical, employing concepts derived from experience. I argue that both interpretations fail to capture the true nature of our geometrical thought. Euclidean proof is not a system of pure logic, but one in which our (...)
- The Epistemology of Mathematical Necessity.Catherine Legg - 2018 - In Peter Chapman, Gem Stapleton, Amirouche Moktefi, Sarah Perez-Kriz & Francesco Bellucci (eds.), Diagrammatic Representation and Inference10th International Conference, Diagrams 2018, Edinburgh, UK, June 18-22, 2018, Proceedings. Berlin: Springer-Verlag. pp. 810-813.It seems possible to know that a mathematical claim is necessarily true by inspecting a diagrammatic proof. Yet how does this work, given that human perception seems to just (as Hume assumed) ‘show us particular objects in front of us’? I draw on Peirce’s account of perception to answer this question. Peirce considered mathematics as experimental a science as physics. Drawing on an example, I highlight the existence of a primitive constraint or blocking function in our thinking which we might (...)
- Geometrie.Jens Lemanski - 2018 - In Daniel Schubbe & Matthias Koßler (eds.), Schopenhauer-Handbuch: Leben – Werk – Wirkung. Springer. pp. 329-333.In Mathematiklehrbüchern und mathematischen Spezialabhandlungen tauchen bis heute immer wieder Themen und Thesen der Schopenhauerschen Elementargeometrie auf. Da Schopenhauers Geometrie bzw. Philosophie der Geometrie in ihrer Figuren- und damit Anschauungsbezogenheit im 19. und frühen 20. Jahrhundert exemplarisch galt, folgt die hier skizzenhaft dargestellte zweihundertjährige Rezeptionsgeschichte auch der von den mathematischen Paradigmen abhängenden Bewertung anschauungsbezogener Geometrien.
- Tools of Reason: The Practice of Scientific Diagramming from Antiquity to the Present.Greg Priest, Silvia De Toffoli & Paula Findlen - 2018 - Endeavour 42 (2-3):49-59.
- Rigor and the Context-Dependence of Diagrams: The Case of Euler Diagrams.David Waszek - 2018 - In Peter Chapman, Gem Stapleton, Amirouche Moktefi, Sarah Perez-Kriz & Francesco Bellucci (eds.), Diagrammatic Representation and Inference. Cham: Springer. pp. 382-389.Euler famously used diagrams to illustrate syllogisms in his Lettres à une princesse d’Allemagne [1]. His diagrams are usually seen as suffering from a fatal “ambiguity problem” [11]: as soon as they involve intersecting circles, which are required for the representation of existential statements, it becomes unclear what exactly may be read off from them, and as Hammer & Shin conclusively showed, any set of reading conventions can lead to erroneous conclusions. I claim that Euler diagrams can, however, be used (...)
- A diagrammatic representation for entities and mereotopological relations in ontologies.José M. Parente de Oliveira & Barry Smith - 2017 - In CEUR, vol. 1908.In the graphical representation of ontologies, it is customary to use graph theory as the representational background. We claim here that the standard graph-based approach has a number of limitations. We focus here on a problem in the graph-based representation of ontologies in complex domains such as biomedical, engineering and manufacturing: lack of mereotopological representation. Based on such limitation, we proposed a diagrammatic way to represent an entity’s structure and various forms of mereotopological relationships between the entities.
- ‘Chasing’ the diagram—the use of visualizations in algebraic reasoning.Silvia de Toffoli - 2017 - Review of Symbolic Logic 10 (1):158-186.The aim of this article is to investigate the roles of commutative diagrams (CDs) in a specific mathematical domain, and to unveil the reasons underlying their effectiveness as a mathematical notation; this will be done through a case study. It will be shown that CDs do not depict spatial relations, but represent mathematical structures. CDs will be interpreted as a hybrid notation that goes beyond the traditional bipartition of mathematical representations into diagrammatic and linguistic. It will be argued that one (...)
- On the Norms of Visual Argument: A Case for Normative Non-revisionism.David Godden - 2017 - Argumentation 31 (2):395-431.Visual arguments can seem to require unique, autonomous evaluative norms, since their content seems irreducible to, and incommensurable with, that of verbal arguments. Yet, assertions of the ineffability of the visual, or of visual-verbal incommensurability, seem to preclude counting putatively irreducible visual content as functioning argumentatively. By distinguishing two notions of content, informational and argumentative, I contend that arguments differing in informational content can have equivalent argumentative content, allowing the same argumentative norms to be rightly applied in their evaluation.
- Abstraction and Diagrammatic Reasoning in Aristotle’s Philosophy of Geometry.Justin Humphreys - 2017 - Apeiron 50 (2):197-224.Aristotle’s philosophy of geometry is widely interpreted as a reaction against a Platonic realist conception of mathematics. Here I argue to the contrary that Aristotle is concerned primarily with the methodological question of how universal inferences are warranted by particular geometrical constructions. His answer hinges on the concept of abstraction, an operation of “taking away” certain features of material particulars that makes perspicuous universal relations among magnitudes. On my reading, abstraction is a diagrammatic procedure for Aristotle, and it is through (...)
- From Euclidean geometry to knots and nets.Brendan Larvor - 2017 - Synthese:1-22.This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or (...)
- Perceiving Necessity.Catherine Legg & James Franklin - 2017 - Pacific Philosophical Quarterly 98 (3).In many diagrams one seems to perceive necessity – one sees not only that something is so, but that it must be so. That conflicts with a certain empiricism largely taken for granted in contemporary philosophy, which believes perception is not capable of such feats. The reason for this belief is often thought well-summarized in Hume's maxim: ‘there are no necessary connections between distinct existences’. It is also thought that even if there were such necessities, perception is too passive or (...)
- The Psychology and Philosophy of Natural Numbers.Oliver R. Marshall - 2017 - Philosophia Mathematica (1):nkx002.ABSTRACT I argue against both neuropsychological and cognitive accounts of our grasp of numbers. I show that despite the points of divergence between these two accounts, they face analogous problems. Both presuppose too much about what they purport to explain to be informative, and also characterize our grasp of numbers in a way that is absurd in the light of what we already know from the point of view of mathematical practice. Then I offer a positive methodological proposal about the (...)
- The nature of correlation perception in scatterplots.Ronald A. Rensink - 2017 - Psychonomic Bulletin & Review 24 (3):776-797.For scatterplots with gaussian distributions of dots, the perception of Pearson correlation r can be described by two simple laws: a linear one for discrimination, and a logarithmic one for perceived magnitude (Rensink & Baldridge, 2010). The underlying perceptual mechanisms, however, remain poorly understood. To cast light on these, four different distributions of datapoints were examined. The first had 100 points with equal variance in both dimensions. Consistent with earlier results, just noticeable difference (JND) was a linear function of the (...)
- Universal intuitions of spatial relations in elementary geometry.Ineke J. M. Van der Ham, Yacin Hamami & John Mumma - 2017 - Journal of Cognitive Psychology 29 (3):269-278.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 geometric objects. As the exact/co-exact distinction is closely related to (...)
- Diagrams of the past: How timelines can aid the growth of historical knowledge.Marc Champagne - 2016 - Cognitive Semiotics 9 (1):11-44.Historians occasionally use timelines, but many seem to regard such signs merely as ways of visually summarizing results that are presumably better expressed in prose. Challenging this language-centered view, I suggest that timelines might assist the generation of novel historical insights. To show this, I begin by looking at studies confirming the cognitive benefits of diagrams like timelines. I then try to survey the remarkable diversity of timelines by analyzing actual examples. Finally, having conveyed this (mostly untapped) potential, I argue (...)
- Envisioning Transformations – The Practice of Topology.Silvia De Toffoli & Valeria Giardino - 2016 - In Brendan Larvor (ed.), Mathematical Cultures: The London Meetings 2012-2014. Springer International Publishing. pp. 25-50.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; (...)
- Basic mathematical cognition.David Gaber & Dirk Schlimm - 2015 - WIREs Cognitive Science 4 (6):355-369.Mathematics is a powerful tool for describing and developing our knowledge of the physical world. It informs our understanding of subjects as diverse as music, games, science, economics, communications protocols, and visual arts. Mathematical thinking has its roots in the adaptive behavior of living creatures: animals must employ judgments about quantities and magnitudes in the assessment of both threats (how many foes) and opportunities (how much food) in order to make effective decisions, and use geometric information in the environment for (...)
- Maps, Diagrams, and Signs: Visual Experience in Peirce's Semiotics.Vitaly Kiryushchenko - 2015 - In Peter Pericles Trifonas (ed.), International Handbook of Semiotics. Dordrecht: Springer. pp. 115-124.
- Semantic properties of diagrams and their cognitive potentials.Atsushi Shimojima - 2015 - Stanford, California: CSLI Publications.Why are diagrams sometimes so useful, while other times unhelpful and even misguiding? There are systematic reasons for this. Drawing on modern research in logic, Artificial Intelligence, cognitive psychology, and graphic design, "Semantic Properties of Diagrams and their Cognitive Potentials" shows that diagrams' cognitive functions are rooted in the characteristic ways they carry information about their targets. The analysis leads to an answer for the deeper question of What makes a diagram a diagram?, which is of crucial importance to the (...)
- An Inquiry into the Practice of Proving in Low-Dimensional Topology.Silvia De Toffoli & Valeria Giardino - 2014 - In Giorgio Venturi, Marco Panza & Gabriele Lolli (eds.), From Logic to Practice: Italian Studies in the Philosophy of Mathematics. Cham: Springer International Publishing. pp. 315-336.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 (...)$42.77 new (collection) View on Amazon.comRemove from this list Direct downloadExport citation Bookmark 17 citations
- Forms and Roles of Diagrams in Knot Theory.Silvia De Toffoli & Valeria Giardino - 2014 - Erkenntnis 79 (4):829-842.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 (...)
- A Perceptual Account of Symbolic Reasoning.David Landy, Colin Allen & Carlos Zednik - 2014 - Frontiers in Psychology 5.People can be taught to manipulate symbols according to formal mathematical and logical rules. Cognitive scientists have traditionally viewed this capacity—the capacity for symbolic reasoning—as grounded in the ability to internally represent numbers, logical relationships, and mathematical rules in an abstract, amodal fashion. We present an alternative view, portraying symbolic reasoning as a special kind of embodied reasoning in which arithmetic and logical formulae, externally represented as notations, serve as targets for powerful perceptual and sensorimotor systems. Although symbolic reasoning often (...)
- “Things Unreasonably Compulsory”: A Peircean Challenge to a Humean Theory of Perception, Particularly With Respect to Perceiving Necessary Truths.Catherine Legg - 2014 - Cognitio 15 (1):89-112.Much mainstream analytic epistemology is built around a sceptical treatment of modality which descends from Hume. The roots of this scepticism are argued to lie in Hume’s (nominalist) theory of perception, which is excavated, studied and compared with the very different (realist) theory of perception developed by Peirce. It is argued that Peirce’s theory not only enables a considerably more nuanced and effective epistemology, it also (unlike Hume’s theory) does justice to what happens when we appreciate a proof in mathematics.
- On the status and role of instrumental images in contemporary science: some epistemological issues.Hermínio Martins - 2014 - Scientiae Studia 12 (SPE):11-36.The controversy over imageless thought versus picture thinking , with the recent reconsideration of model-based reasoning in the physical sciences is briefly examined. The main focus of the article is on the role of instrumentally elicited images in the sciences, especially in the physical sciences, with special reference to optics, experimental particle physics and observational astronomy, against the background of the civilization of digital images, though to some degree every scientific discipline is implicated. Imaging, today chiefly in the mode of (...)
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