Philosophy of Mathematics > Mathematical Cognition > Visualization in Mathematics

Visualization in Mathematics

Edited by Silvia De Toffoli (Princeton University)
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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.
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  1. Self-Graphing Equations.Samuel Alexander - manuscript
    Can 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.
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  2. A Pre-Formal Proof of Why No Planar Map Needs More Than Four Colours.Bhupinder Singh Anand - manuscript
    Although the Four Colour Theorem is passe, we give an elementary pre-formal proof that transparently illustrates why four colours suffice to chromatically differentiate any set of contiguous, simply connected and bounded, planar spaces; by showing that there is no minimal 4-coloured planar map M. We note that such a pre-formal proof of the Four Colour Theorem highlights the significance of differentiating between: (a) Plato's knowledge as justified true belief, which seeks a formal proof in a first-order mathematical language in order (...)
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  3. Who's Afraid of Mathematical Diagrams?Silvia De Toffoli - forthcoming - Philosophers' Imprint.
    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 (...)
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  4. Visual Features as Carriers of Abstract Quantitative Information.Ronald A. Rensink - forthcoming - Journal of Experimental Psychology: General.
    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, (...)
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  5. 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 (...)
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  6. Reconciling Rigor and Intuition.Silvia De Toffoli - 2021 - 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 (...)
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  7. 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 (...)
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  8. 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.
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  9. Research on the Kerr-Newman Black Hole in M82 Confirms Black Hole and White Hole Thermonuclear Binding. Pachankis - 2021 - Academia Letters 8 (3199).
    The article summarized the quadruple weak force electrodynamics on the Kerr-Newman type supermassive compact object on NGC 3034. It used both observational astronomy and data analytical techniques in the qualitative research on cosmology revolved around black hole and white hole juxtapose with nuclear astrophysics and theoretical chemistry.
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  10. 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 (...)
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  11. “Always Mixed Together”: Notation, Language, and the Pedagogy of Frege's Begriffsschrift.David E. Dunning - 2020 - Modern Intellectual History 17 (4):1099-1131.
    Gottlob Frege is considered a founder of analytic philosophy and mathematical logic, but the traditions that claim Frege as a forebear never embraced his Begriffsschrift, or “conceptual notation”—the invention he considered his most important accomplishment. Frege believed that his notation rendered logic visually observable. Rejecting the linearity of written language, he claimed Begriffsschrift exhibited a structure endogenous to logic itself. But Frege struggled to convince others to use his notation, as his frustrated pedagogical efforts at the University of Jena illustrate. (...)
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  12. 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 (...)
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  13. 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 (...)
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  14. 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 (...)
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  15. 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 (...)
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  16. 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.
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  17. 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 (...)
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  18. 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 (...)
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  19. 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.
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  20. 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.
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  21. ‘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 (...)
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  22. 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.
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  23. 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 (...)
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  24. 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 (...)
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  25. 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 (...)
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  26. 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 (...)
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  27. 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 (...)
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  28. Envisioning Transformations – The Practice of Topology.Silvia De Toffoli & Valeria Giardino - 2016 - In Brendan Larvor (ed.), Mathematical Cultures: The London Meetings 2012--2014. Zurich, Switzerland: Birkhäuser. 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; (...)
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  29. What Diagrams Argue in Late Imperial Chinese Combinatorial Texts.Andrea Bréard - 2015 - Early Science and Medicine 20 (3):241-264.
    Attitudes towards diagrammatic reasoning and visualization in mathematics were seldom spelled out in texts from pre-modern China, although illustrations figure prominently in mathematical literature since the eleventh century. Taking the sums of finite series and their combinatorial interpretation as a case study, this article investigates the epistemological function of illustrations from the eleventh to the nineteenth century that encode either the mathematical objects themselves or represent their related algorithms. It particularly focuses on the two illustrations given in Wang Lai’s Mathematical (...)
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  30. An Inquiry Into the Practice of Proving in Low-Dimensional Topology.Silvia De Toffoli & Valeria Giardino - 2015 - In Gabriele Lolli, Giorgio Venturi & Marco Panza (eds.), From Logic to Practice. Zurich, Switzerland: 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 (...)
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  31. 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 (...)
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  32. 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 (...)
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  33. 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 (...)
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  34. “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.
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  35. 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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  36. On the Prospects for a Science of Visualization.Ronald A. Rensink - 2014 - In Handbook of Human-Centric Visualization. Springer. pp. 147-175.
    This paper explores the extent to which a scientific framework for visualization might be possible. It presents several potential parts of a framework, illustrated by application to the visualization of correlation in scatterplots. The first is an extended-vision thesis, which posits that a viewer and visualization system can be usefully considered as a single system that perceives structure in a dataset, much like "basic" vision perceives structure in the world. This characterization is then used to suggest approaches to evaluation that (...)
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  37. The Advantages of Bringing Infinity to a Finite Place: Penrose Diagrams as Objects of Intuition.Aaron Sidney Wright - 2014 - Historical Studies in the Natural Sciences 44 (2):99-139.
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  38. 'Reasoning Well From Badly Drawn Figures': The Birth of Algebraic Topology.Claudio Bartocci - 2013 - Lettera Matematica 1:13-22.
    In this paper the emergence of Poincaré’s “analysis situs” is described by means of an overview of the original memoir and its supplements. In particular, the genesis of the celebrated “Poincaré conjecture” is discussed.
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  39. Prolegomena to a Cognitive Investigation of Euclidean Diagrammatic Reasoning.Yacin Hamami & John Mumma - 2013 - Journal of Logic, Language and Information 22 (4):421-448.
    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 Euclidean diagrammatic reasoning can be fruitfully advanced (...)
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  40. What is a Logical Diagram?Catherine Legg - 2013 - In Sun-Joo Shin & Amirouche Moktefi (eds.), Visual Reasoning with Diagrams. Springer. pp. 1-18.
    Robert Brandom’s expressivism argues that not all semantic content may be made fully explicit. This view connects in interesting ways with recent movements in philosophy of mathematics and logic (e.g. Brown, Shin, Giaquinto) to take diagrams seriously - as more than a mere “heuristic aid” to proof, but either proofs themselves, or irreducible components of such. However what exactly is a diagram in logic? Does this constitute a semiotic natural kind? The paper will argue that such a natural kind does (...)
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  41. The Motion Behind the Symbols: A Vital Role for Dynamism in the Conceptualization of Limits and Continuity in Expert Mathematics.Tyler Marghetis & Rafael Núñez - 2013 - Topics in Cognitive Science 5 (2):299-316.
    The canonical history of mathematics suggests that the late 19th-century “arithmetization” of calculus marked a shift away from spatial-dynamic intuitions, grounding concepts in static, rigorous definitions. Instead, we argue that mathematicians, both historically and currently, rely on dynamic conceptualizations of mathematical concepts like continuity, limits, and functions. In this article, we present two studies of the role of dynamic conceptual systems in expert proof. The first is an analysis of co-speech gesture produced by mathematics graduate students while proving a theorem, (...)
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  42. Visual Reasoning with Diagrams.Sun-Joo Shin & Amirouche Moktefi (eds.) - 2013 - Basel: Birkhaüser.
    Logic, the discipline that explores valid reasoning, does not need to be limited to a specific form of representation but should include any form as long as it allows us to draw sound conclusions from given information. The use of diagrams has a long but unequal history in logic: The golden age of diagrammatic logic of the 19th century thanks to Euler and Venn diagrams was followed by the early 20th century's symbolization of modern logic by Frege and Russell. Recently, (...)
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  43. Figures, Formulae, and Functors.Zach Weber - 2013 - In Sun-Joo Shin & Amirouche Moktefi (eds.), Visual Reasoning with Diagrams. Springer. pp. 153--170.
    This article suggests a novel way to advance a current debate in the philosophy of mathematics. The debate concerns the role of diagrams and visual reasoning in proofs—which I take to concern the criteria of legitimate representation of mathematical thought. Drawing on the so-called ‘maverick’ approach to philosophy of mathematics, I turn to mathematical practice itself to adjudicate in this debate, and in particular to category theory, because there (a) diagrams obviously play a major role, and (b) category theory itself (...)
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  44. To Diagram, to Demonstrate: To Do, To See, and To Judge in Greek Geometry.Philip Catton & Cemency Montelle - 2012 - Philosophia Mathematica 20 (1):25-57.
    Not simply set out in accompaniment of the Greek geometrical text, the diagram also is coaxed into existence manually (using straightedge and compasses) by commands in the text. The marks that a diligent reader thus sequentially produces typically sum, however, to a figure more complex than the provided one and also not (as it is) artful for being synoptically instructive. To provide a figure artfully is to balance multiple desiderata, interlocking the timelessness of insight with the temporality of construction. Our (...)
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  45. Hume on Space, Geometry, and Diagrammatic Reasoning.Graciela De Pierris - 2012 - Synthese 186 (1):169-189.
    Hume’s discussion of space, time, and mathematics at T 1.2 appeared to many earlier commentators as one of the weakest parts of his philosophy. From the point of view of pure mathematics, for example, Hume’s assumptions about the infinite may appear as crude misunderstandings of the continuum and infinite divisibility. I shall argue, on the contrary, that Hume’s views on this topic are deeply connected with his radically empiricist reliance on phenomenologically given sensory images. He insightfully shows that, working within (...)
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  46. And so On...: Reasoning with Infinite Diagrams.Solomon Feferman - 2012 - Synthese 186 (1):371 - 386.
    This paper presents examples of infinite diagrams (as well as infinite limits of finite diagrams) whose use is more or less essential for understanding and accepting various proofs in higher mathematics. The significance of these is discussed with respect to the thesis that every proof can be formalized, and a "pre" form of this thesis that every proof can be presented in everyday statements-only form.
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  47. Review of M. Giaquinto, Visual Thinking in Mathematics: An Epistemological Study. [REVIEW]Valeria Giardino - 2012 - Review of Metaphysics 66 (1):148-150.
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  48. Pictures and Pedagogy: The Role of Diagrams in Feynman's Early Lectures.Ari Gross - 2012 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 43 (3):184-194.
    This paper aims to give a substantive account of how Feynman used diagrams in the first lectures in which he explained his new approach to quantum electrodynamics. By critically examining unpublished lecture notes, Feynman’s use and interpretation of both "Feynman diagrams" and other visual representations will be illuminated. This paper discusses how the morphology of Feynman’s early diagrams were determined by both highly contextual issues, which molded his images to local needs and particular physical characterizations, and an overarching common diagrammatic (...)
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  49. Diagrams as Sketches.Brice Halimi - 2012 - Synthese 186 (1):387-409.
    This article puts forward the notion of “evolving diagram” as an important case of mathematical diagram. An evolving diagram combines, through a dynamic graphical enrichment, the representation of an object and the representation of a piece of reasoning based on the representation of that object. Evolving diagrams can be illustrated in particular with category-theoretic diagrams (hereafter “diagrams*”) in the context of “sketch theory,” a branch of modern category theory. It is argued that sketch theory provides a diagrammatic* theory of diagrams*, (...)
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  50. The Hardness of the Iconic Must: Can Peirce’s Existential Graphs Assist Modal Epistemology.Catherine Legg - 2012 - Philosophia Mathematica 20 (1):1-24.
    Charles Peirce's diagrammatic logic — the Existential Graphs — is presented as a tool for illuminating how we know necessity, in answer to Benacerraf's famous challenge that most ‘semantics for mathematics’ do not ‘fit an acceptable epistemology’. It is suggested that necessary reasoning is in essence a recognition that a certain structure has the particular structure that it has. This means that, contra Hume and his contemporary heirs, necessity is observable. One just needs to pay attention, not merely to individual (...)
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