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  1. Thinking with maps.Elisabeth Camp - 2007 - Philosophical Perspectives 21 (1):145–182.
    Most of us create and use a panoply of non-sentential representations throughout our ordinary lives: we regularly use maps to navigate, charts to keep track of complex patterns of data, and diagrams to visualize logical and causal relations among states of affairs. But philosophers typically pay little attention to such representations, focusing almost exclusively on language instead. In particular, when theorizing about the mind, many philosophers assume that there is a very tight mapping between language and thought. Some analyze utterances (...)
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  • Image structure.John Kulvicki - 2003 - Journal of Aesthetics and Art Criticism 61 (4):323–340.
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  • Why do mathematicians re-prove theorems?John W. Dawson Jr - 2006 - Philosophia Mathematica 14 (3):269-286.
    From ancient times to the present, the discovery and presentation of new proofs of previously established theorems has been a salient feature of mathematical practice. Why? What purposes are served by such endeavors? And how do mathematicians judge whether two proofs of the same theorem are essentially different? Consideration of such questions illuminates the roles that proofs play in the validation and communication of mathematical knowledge and raises issues that have yet to be resolved by mathematical logicians. The Appendix, in (...)
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  • (1 other version)Languages of Art.Nelson Goodman - 1968 - Indianapolis,: Hackett Publishing Company.
    "Like Dewey, he has revolted against the empiricist dogma and the Kantian dualisms which have compartmentalized philosophical thought.... Unlike Dewey, he has provided detailed incisive argumentation, and has shown just where the dogmas and dualisms break down." --Richard Rorty, _The Yale Review_.
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  • 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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  • 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 (...)
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  • Philosophy of Mathematical Practice — Motivations, Themes and Prospects†.Jessica Carter - 2019 - Philosophia Mathematica 27 (1):1-32.
    A number of examples of studies from the field ‘The Philosophy of Mathematical Practice’ (PMP) are given. To characterise this new field, three different strands are identified: an agent-based, a historical, and an epistemological PMP. These differ in how they understand ‘practice’ and which assumptions lie at the core of their investigations. In the last part a general framework, capturing some overall structure of the field, is proposed.
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  • Exploring the fruitfulness of diagrams in mathematics.Jessica Carter - 2019 - Synthese 196 (10):4011-4032.
    The paper asks whether diagrams in mathematics are particularly fruitful compared to other types of representations. In order to respond to this question a number of examples of propositions and their proofs are considered. In addition I use part of Peirce’s semiotics to characterise different types of signs used in mathematical reasoning, distinguishing between symbolic expressions and 2-dimensional diagrams. As a starting point I examine a proposal by Macbeth. Macbeth explains how it can be that objects “pop up”, e.g., as (...)
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  • ‘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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  • (2 other versions)Languages of Art.Nelson Goodman - 1970 - Philosophy and Rhetoric 3 (1):62-63.
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  • (2 other versions)What Metaphors Mean.Donald Davidson - 1978 - Critical Inquiry 5 (1):31-47.
    The concept of metaphor as primarily a vehicle for conveying ideas, even if unusual ones, seems to me as wrong as the parent idea that a metaphor has a special meaning. I agree with the view that metaphors cannot be paraphrased, but I think this is not because metaphors say something too novel for literal expression but because there is nothing there to paraphrase. Paraphrase, whether possible or not, inappropriate to what is said: we try, in paraphrase, to say it (...)
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  • (2 other versions)What Metaphors Mean.Donald Davidson - 2013 - In Maite Ezcurdia & Robert J. Stainton (eds.), The Semantics-Pragmatics Boundary in Philosophy. Peterborough, CA: Broadview Press. pp. 453-465.
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  • The Euclidean Diagram.Kenneth Manders - 2008 - In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oxford, England: Oxford University Press. pp. 80--133.
    This chapter gives a detailed study of diagram-based reasoning in Euclidean plane geometry (Books I, III), as well as an exploration how to characterise a geometric practice. First, an account is given of diagram attribution: basic geometrical claims are classified as exact (equalities, proportionalities) or co-exact (containments, contiguities); exact claims may only be inferred from prior entries in the demonstration text, but co-exact claims may be asserted based on what is seen in the diagram. Diagram control by constructions is necessary (...)
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  • Rigor and Structure.John P. Burgess - 2015 - Oxford, England: Oxford University Press UK.
    While we are commonly told that the distinctive method of mathematics is rigorous proof, and that the special topic of mathematics is abstract structure, there has been no agreement among mathematicians, logicians, or philosophers as to just what either of these assertions means. John P. Burgess clarifies the nature of mathematical rigor and of mathematical structure, and above all of the relation between the two, taking into account some of the latest developments in mathematics, including the rise of experimental mathematics (...)
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  • Bolzano's ideal of algebraic analysis.Philip Kitcher - 1975 - Studies in History and Philosophy of Science Part A 6 (3):229-269.
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  • 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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  • A formal system for euclid’s elements.Jeremy Avigad, Edward Dean & John Mumma - 2009 - Review of Symbolic Logic 2 (4):700--768.
    We present a formal system, E, which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning.
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  • (2 other versions)What metaphors mean.Donald Davidson - 2010 - In Darragh Byrne & Max Kölbel (eds.), Arguing about language. New York: Routledge. pp. 31.
    The concept of metaphor as primarily a vehicle for conveying ideas, even if unusual ones, seems to me as wrong as the parent idea that a metaphor has a special meaning. I agree with the view that metaphors cannot be paraphrased, but I think this is not because metaphors say something too novel for literal expression but because there is nothing there to paraphrase. Paraphrase, whether possible or not, inappropriate to what is said: we try, in paraphrase, to say it (...)
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  • (1 other version)Languages of Art: An Approach to a Theory of Symbols.Nelson Goodman - 1968 - Indianapolis,: Bobbs-Merrill.
    . . . Unlike Dewey, he has provided detailed incisive argumentation, and has shown just where the dogmas and dualisms break down." -- Richard Rorty, The Yale Review.
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  • The imaginary museum of musical works: an essay in the philosophy of music.Lydia Goehr - 1992 - New York: Oxford University Press.
    What is the difference between a performance of Beethoven's Fifth Symphony and the symphony itself? What does it mean for musicians to be faithful to the works they perform? To answer this question, Goehr combines philosophical and historical methods of enquiry. She describes how the concept of a musical work emerged as late as 1800, and how it subsequently defined the norms, expectations, and behavior characteristic of classical musical practice. Out of the historical thesis, Goehr draws philosophical conclusions about the (...)
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  • (1 other version)From Euclidean geometry to knots and nets.Brendan Larvor - 2019 - Synthese 196 (7):2715-2736.
    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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  • The Philosophy of Mathematical Practice.Paolo Mancosu - 2009 - Studia Logica 92 (1):137-141.
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  • 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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  • The Philosophy of Mathematical Practice.Paolo Mancosu (ed.) - 2008 - Oxford, England: Oxford University Press.
    There is an urgent need in philosophy of mathematics for new approaches which pay closer attention to mathematical practice. This book will blaze the trail: it offers philosophical analyses of important characteristics of contemporary mathematics and of many aspects of mathematical activity which escape purely formal logical treatment.
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  • Mathematical Knowledge and the Interplay of Practices.Jose Ferreiros - 2009 - In Mauricio Suárez, Mauro Dorato & Miklós Rédei (eds.), EPSA Philosophical Issues in the Sciences · Launch of the European Philosophy of Science Association. Dordrecht, Netherland: Springer. pp. 55--64.
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  • Mathematical Knowledge and the Interplay of Practices.José Ferreirós - 2015 - Princeton, USA: Princeton University Press.
    On knowledge and practices: a manifesto -- The web of practices -- Agents and frameworks -- Complementarity in mathematics -- Ancient Greek mathematics: a role for diagrams -- Advanced math: the hypothetical conception -- Arithmetic certainty -- Mathematics developed: the case of the reals -- Objectivity in mathematical knowledge -- The problem of conceptual understanding.
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  • (2 other versions)Languages of Art: An Approach to a Theory of Symbols.Nelson Goodman - 1971 - British Journal for the Philosophy of Science 22 (2):187-198.
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  • (1 other version)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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  • Review of Sun-Joo Shin: The Logical Status of Diagrams[REVIEW]Sun-joo Shin - 1997 - British Journal for the Philosophy of Science 48 (2):290-291.
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  • (1 other version)Visual thinking in mathematics: an epistemological study.Marcus Giaquinto - 2007 - New York: Oxford University Press.
    Visual thinking -- visual imagination or perception of diagrams and symbol arrays, and mental operations on them -- is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? By examining the many kinds of visual representation in mathematics and the diverse ways in which they are used, Marcus Giaquinto argues that visual thinking in (...)
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  • The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History.Reviel Netz - 1999 - Cambridge and New York: Cambridge University Press.
    An examination of the emergence of the phenomenon of deductive argument in classical Greek mathematics.
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  • An introduction to the philosophy of mathematics.Mark Colyvan - 2012 - Cambridge: Cambridge University Press.
    This introduction to the philosophy of mathematics focuses on contemporary debates in an important and central area of philosophy. The reader is taken on a fascinating and entertaining journey through some intriguing mathematical and philosophical territory, including such topics as the realism/anti-realism debate in mathematics, mathematical explanation, the limits of mathematics, the significance of mathematical notation, inconsistent mathematics and the applications of mathematics. Each chapter has a number of discussion questions and recommended further reading from both the contemporary literature and (...)
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  • Visualizing in Mathematics.Marcus Giaquinto - 2008 - In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oxford, England: Oxford University Press. pp. 22-42.
    Visual thinking in mathematics is widespread; it also has diverse kinds and uses. Which of these uses is legitimate? What epistemic roles, if any, can visualization play in mathematics? These are the central philosophical questions in this area. In this introduction I aim to show that visual thinking does have epistemically significant uses. The discussion focuses mainly on visual thinking in proof and discovery and touches lightly on its role in understanding.
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  • Greek Mathematical Diagrams: Their Use and Their Meaning’.R. Netz - 1998 - For the Learning of Mathematics 18:33-39.
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  • (1 other version)Visual Thinking in Mathematics: An Epistemological Study.Marcus Giaquinto - 2007 - Oxford, England: Oxford University Press.
    Marcus Giaquinto presents an investigation into the different kinds of visual thinking involved in mathematical thought, drawing on work in cognitive psychology, philosophy, and mathematics. He argues that mental images and physical diagrams are rarely just superfluous aids: they are often a means of discovery, understanding, and even proof.
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  • (1 other version)Proof: Its nature and significance.Michael Detlefsen - 2008 - In Bonnie Gold & Roger A. Simons (eds.), Proof and Other Dilemmas: Mathematics and Philosophy. Mathematical Association of America. pp. 1.
    I focus on three preoccupations of recent writings on proof. -/- I. The role and possible effects of empirical reasoning in mathematics. Do recent developments (specifically, the computer-assisted proof of the 4CT) point to something essentially new as regards the need for and/or effects of using broadly empirical and inductive reasoning in mathematics? In particular, should we see such things as the computer-assisted proof of the 4CT as pointing to the existence of mathematical truths of which we cannot have a (...)
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  • That We See That Some Diagrammatic Proofs Are Perfectly Rigorous.Jody Azzouni - 2013 - Philosophia Mathematica 21 (3):323-338.
    Mistaken reasons for thinking diagrammatic proofs aren't rigorous are explored. The main result is that a confusion between the contents of a proof procedure (what's expressed by the referential elements in a proof procedure) and the unarticulated mathematical aspects of a proof procedure (how that proof procedure is enabled) gives the impression that diagrammatic proofs are less rigorous than language proofs. An additional (and independent) factor is treating the impossibility of naturally generalizing a diagrammatic proof procedure as an indication of (...)
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  • On Frege’s Begriffsschrift Notation for Propositional Logic: Design Principles and Trade-Offs.Dirk Schlimm - 2017 - History and Philosophy of Logic 39 (1):53-79.
    Well over a century after its introduction, Frege's two-dimensional Begriffsschrift notation is still considered mainly a curiosity that stands out more for its clumsiness than anything else. This paper focuses mainly on the propositional fragment of the Begriffsschrift, because it embodies the characteristic features that distinguish it from other expressively equivalent notations. In the first part, I argue for the perspicuity and readability of the Begriffsschrift by discussing several idiosyncrasies of the notation, which allow an easy conversion of logically equivalent (...)
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  • Visual Thinking in Mathematics. [REVIEW]Marcus Giaquinto - 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 19th century Giuseppe Peano proved that such a curve exists. Examples like this, particularly in analysis received much attention in the 19th century. They helped to 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 follows : " Mathematicians had for (...)
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  • And so on... : reasoning with infinite diagrams.Solomon Feferman - 2012 - Synthese 186 (1):371-386.
    This paper presents examples of infinite 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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  • (1 other version)Purity as an ideal of proof.Michael Detlefsen - 2008 - In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oxford, England: Oxford University Press. pp. 179-197.
    Various ideals of purity are surveyed and discussed. These include the classical Aristotelian ideal, as well as certain neo-classical and contemporary ideals. The focus is on a type of purity ideal I call topical purity. This is purity which emphasizes a certain symmetry between the conceptual resources used to prove a theorem and those needed for the clarification of its content. The basic idea is that the resources of proof ought ideally to be restricted to those which determine its content.
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  • Empiricism, Logic, and Mathematics.H. Hahn - 1980
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