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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. Making Mathematics Visible: Mathematical Knowledge and How it Differs from Mathematical Understanding.Anne Newstead - manuscript
    This 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.
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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. 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 (...)
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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. 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, (...)
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  7. 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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  8. Visualization as a stimulus domain for vision science. [REVIEW]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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  9. 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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  10. 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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  11. 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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  12. 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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  13. ‘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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  14. 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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  15. 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 (...)
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  16. 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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  17. 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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  18. 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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  19. 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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  20. “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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  21. 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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  22. 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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  23. 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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  24. Four Futures and a History.Ronald A. Rensink - 2011 - The Encyclopedia of Human-Computer Interaction, 2nd Ed.; 35. Data Visualization for Human Perception.
    Stephen Few provides a nice overview of the reasons why we should design data visualizations to be effective, and why it’s important to understand human perception when doing so. In fact, he’s done this so well that I can’t add much to his arguments. But I can, however, push the basic message a bit further, out into the times before and after those he discusses. Out into areas that are not as well known, or not really developed, where new opportunities (...)
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  25. Content aggregation, visualization and emergent properties in computer simulations.Gordana Dodig-Crnkovic, Juan M. Durán & D. Slutej - 2010 - In Kai-Mikael Jää-Aro & Thomas Larsson (eds.), SIGRAD 2010 – Content aggregation and visualization. Linköping University Electronic Press. pp. 77-83.
    With the rapidly growing amounts of information, visualization is becoming increasingly important, as it allows users to easily explore and understand large amounts of information. However the field of information visualiza- tion currently lacks sufficient theoretical foundations. This article addresses foundational questions connecting information visualization with computing and philosophy studies. The idea of multiscale information granula- tion is described based on two fundamental concepts: information (structure) and computation (process). A new information processing paradigm of Granular Computing enables stepwise increase of (...)
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  26. The perception of correlation in scatterplots.Ronald A. Rensink & Gideon Baldridge - 2010 - Computer Graphics Forum 29:1203-1210.
    We present a rigorous way to evaluate the visual perception of correlation in scatterplots, based on classical psychophysical methods originally developed for simple properties such as brightness. Although scatterplots are graphically complex, the quantity they convey is relatively simple. As such, it may be possible to assess the perception of correlation in a similar way. Scatterplots were each of 5.0 extent, containing 100 points with a bivariate normal distribution. Means were 0.5 of the range of the points, and standard deviations (...)
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  27. Review of M. Giaquinto's Visual thinking in mathematics. [REVIEW]Andrew Arana - 2009 - Analysis 69 (2):401-403.
    Our visual experience seems to suggest that no continuous curve can cover every point of the unit square, yet in the late nineteenth century Giuseppe Peano proved that such a curve exists. Examples like this, particularly in analysis (in the sense of the infinitesimal calculus) received much attention in the nineteenth century. They helped instigate what Hans Hahn called a “crisis of intuition”, wherein visual reasoning in mathematics came to be thought to be epistemically problematic. Hahn described this “crisis” as (...)
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  28. Attention, Consciousness, and Data Display.Ronald A. Rensink - 2006 - In 2006 Proceedings of the American Statistical Association, Statistical Graphics Section.
    Recent advances in our understanding of visual perception have shown it to be a far more complex and counterintuitive process than previously believed. Several important consequences follow from this. First, the design of an effective statistical graphics system is unlikely to succeed based on intuition alone; instead, it must rely on a more sophisticated, systematic approach. The basic elements of such an approach are outlined here, along with several design principles. An overview is then given of recent advances in our (...)
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  29. Diagrammatic Reasoning and Modelling in the Imagination: The Secret Weapons of the Scientific Revolution.James Franklin - 2000 - In Guy Freeland & Anthony Corones (eds.), 1543 and All That: Image and Word, Change and Continuity in the Proto-Scientific Revolution. Kluwer Academic Publishers.
    Just before the Scientific Revolution, there was a "Mathematical Revolution", heavily based on geometrical and machine diagrams. The "faculty of imagination" (now called scientific visualization) was developed to allow 3D understanding of planetary motion, human anatomy and the workings of machines. 1543 saw the publication of the heavily geometrical work of Copernicus and Vesalius, as well as the first Italian translation of Euclid.
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