Search results for 'conventionalism in mathematics' (try it on Scholar)

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  1. Pieranna Garavaso (1985). Objectivity and Consistency in Mathematics: A Critical Analysis of Two Objections to Wittgenstein's Pragmatic Conventionalism. Dissertation, The University of Nebraska - Lincoln
    Wittgenstein's views on mathematics are radically original. He criticizes most of the traditional philosophies of mathematics. His views have been subject to harsh criticisms. In this dissertation, I attempt to defend Wittgenstein's philosophy of mathematics from two objections: the objectivity objection and the consistency objection. The first claims that Wittgenstein's account of mathematics is not sufficient for the objectivity of mathematics; the second claims that it is only a partial account of mathematics because it (...)
     
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  2.  27
    P. Garavaso (2013). Hilary Putnam's Consistency Objection Against Wittgenstein's Conventionalism in Mathematics. Philosophia Mathematica 21 (3):279-296.
    Hilary Putnam first published the consistency objection against Ludwig Wittgenstein’s account of mathematics in 1979. In 1983, Putnam and Benacerraf raised this objection against all conventionalist accounts of mathematics. I discuss the 1979 version and the scenario argument, which supports the key premise of the objection. The wide applicability of this objection is not apparent; I thus raise it against an imaginary axiomatic theory T similar to Peano arithmetic in all relevant aspects. I argue that a conventionalist can (...)
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  3.  44
    Pieranna Garavaso (1988). Wittgenstein's Philosophy of Mathematics: A Reply to Two Objections. Southern Journal of Philosophy 26 (2):179-191.
    This paper has two main purposes: first to compare Wittgenstein's views to the more traditional views in the philosophy of mathematics; second, to provide a general outline for a Wittgensteinian reply to two objections against Wittgenstein's account of mathematics: the objectivity objection and the consistency objections, respectively. Two fundamental thesmes of Wittgenstein's account of mathematics title the first two sections: mathematical propositions are rules and not descritpions and mathematics is employed within a form of life. Under (...)
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  4.  1
    Josipa Gordana Petrunić (2009). Conceptions of Continuity: William Kingdon Clifford’s Empirical Conception of Continuity in Mathematics. Philosophia Scientiae 13 (2):45-83.
    The concept of continuity is fundamental to contemporary mathematical analysis. However, the seemingly well-founded definition currently used for this concept is just one of many versions that have been historically proffered, used and reshaped by mathematical practitioners over the centuries. In looking at the particular manner in which William Kingdon Clifford defined and negotiated the physical underpinnings to “continuity” in his works, and by comparing that account to Richard Dedekind’s conventionalist definition of the concept, it becomes clear that “continuity” has (...)
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  5.  51
    Alexander Paseau (2008). Naturalism in the Philosophy of Mathematics. In Stanford Encyclopedia of Philosophy.
    Contemporary philosophy’s three main naturalisms are methodological, ontological and epistemological. Methodological naturalism states that the only authoritative standards are those of science. Ontological and epistemological naturalism respectively state that all entities and all valid methods of inquiry are in some sense natural. In philosophy of mathematics of the past few decades methodological naturalism has received the lion’s share of the attention, so we concentrate on this. Ontological and epistemological naturalism in the philosophy of mathematics are discussed more briefly (...)
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  6.  83
    Penelope Maddy (1997). Naturalism in Mathematics. Oxford University Press.
    Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problem--realism--is examined and rejected in favor of another approach--naturalism. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both (...)
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  7.  7
    Ricardo Crespo & Fernando Tohmé (forthcoming). The Future of Mathematics in Economics: A Philosophically Grounded Proposal. Foundations of Science:1-17.
    The use of mathematics in economics has been widely discussed. The philosophical discussion on what mathematics is remains unsettled on why it can be applied to the study of the real world. We propose to get back to some philosophical conceptions that lead to a language-like role for the mathematical analysis of economic phenomena and present some problems of interest that can be better examined in this light. Category theory provides the appropriate tools for these analytical approach.
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  8.  19
    Emily Grosholz (2007). Representation and Productive Ambiguity in Mathematics and the Sciences. Oxford University Press.
    Viewed this way, the texts yield striking examples of language and notation that are irreducibly ambiguous and productive because they are ambiguous.
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  9. Penelope Maddy (1990). Realism in Mathematics. Oxford University Prress.
    Mathematicians tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Offering a scrupulously fair treatment of both mathematical and philosophical concerns, Penelope Maddy here delineates and defends a novel version (...)
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  10.  78
    M. Giaquinto (2007). Visual Thinking in Mathematics: An Epistemological Study. 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 (...)
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  11.  65
    Stefan Lukits (2014). Carnap's Conventionalism in Geometry. Grazer Philosophische Studien 88:123-138.
    Against Thomas Mormann's argument that differential topology does not support Carnap's conventionalism in geometry we show their compatibility. However, Mormann's emphasis on the entanglement that characterizes topology and its associated metrics is not misplaced. It poses questions about limits of empirical inquiry. For Carnap, to pose a question is to give a statement with the task of deciding its truth. Mormann's point forces us to introduce more clarity to what it means to specify the task that decides between competing (...)
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  12.  9
    Noson S. Yanofsky & Mark Zelcer (forthcoming). The Role of Symmetry in Mathematics. Foundations of Science:1-21.
    Over the past few decades the notion of symmetry has played a major role in physics and in the philosophy of physics. Philosophers have used symmetry to discuss the ontology and seeming objectivity of the laws of physics. We introduce several notions of symmetry in mathematics and explain how they can also be used in resolving different problems in the philosophy of mathematics. We use symmetry to discuss the objectivity of mathematics, the role of mathematical objects, the (...)
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  13.  30
    Dirk Schlimm (2013). Conceptual Metaphors and Mathematical Practice: On Cognitive Studies of Historical Developments in Mathematics. Topics in Cognitive Science 5 (2):283-298.
    This article looks at recent work in cognitive science on mathematical cognition from the perspective of history and philosophy of mathematical practice. The discussion is focused on the work of Lakoff and Núñez, because this is the first comprehensive account of mathematical cognition that also addresses advanced mathematics and its history. Building on a distinction between mathematics as it is presented in textbooks and as it presents itself to the researcher, it is argued that the focus of cognitive (...)
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  14.  50
    Donald Gillies (ed.) (1992). Revolutions in Mathematics. Oxford University Press.
    Social revolutions--that is critical periods of decisive, qualitative change--are a commonly acknowledged historical fact. But can the idea of revolutionary upheaval be extended to the world of ideas and theoretical debate? The publication of Kuhn's The Structure of Scientific Revolutions in 1962 led to an exciting discussion of revolutions in the natural sciences. A fascinating, but little known, off-shoot of this was a debate which began in the United States in the mid-1970's as to whether the concept of revolution could (...)
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  15.  68
    Michael Detlefsen (ed.) (1992). Proof and Knowledge in Mathematics. Routledge.
    Proof and Knowledge in Mathematics tackles the main problem that arises when considering an epistemology for mathematics, the nature and sources of mathematical justification. Focusing both on particular and general issues, these essays from leading philosophers of mathematics raise important issues for our current understanding of mathematics. Is mathematical justification a priori or a posteriori? What role, if any, does logic play in mathematical reasoning or inference? And how epistemologically important is the formalizability of proof? Michael (...)
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  16.  8
    Mikkel Willum Johansen & Morten Misfeldt (2015). Semiotic Scaffolding in Mathematics. Biosemiotics 8 (2):325-340.
    This paper investigates the notion of semiotic scaffolding in relation to mathematics by considering its influence on mathematical activities, and on the evolution of mathematics as a research field. We will do this by analyzing the role different representational forms play in mathematical cognition, and more broadly on mathematical activities. In the main part of the paper, we will present and analyze three different cases. For the first case, we investigate the semiotic scaffolding involved in pencil and paper (...)
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  17.  14
    Jean Paul van Bendegem (2014). Inconsistency in Mathematics and the Mathematics of Inconsistency. Synthese 191 (13):3063-3078.
    No one will dispute, looking at the history of mathematics, that there are plenty of moments where mathematics is “in trouble”, when paradoxes and inconsistencies crop up and anomalies multiply. This need not lead, however, to the view that mathematics is intrinsically inconsistent, as it is compatible with the view that these are just transient moments. Once the problems are resolved, consistency (in some sense or other) is restored. Even when one accepts this view, what remains is (...)
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  18.  20
    Larry Krasnoff (2012). Voluntarism and Conventionalism in Hobbes and Kant. Hobbes Studies 25 (1):43-65.
    Kant's relation to Hobbesian voluntarism has recently become a source of controversy for the interpretation of Kant's practical philosophy. Realist interpreters, most prominently Karl Ameriks, have attacked the genealogies of Kantian autonomy suggested by J. B. Schneewind and Christine Korsgaard as misleadingly voluntarist and unacceptably anti-realist. In this debate, however, there has been no real discussion of Kant's own views about Hobbes. By examining the relation of Hobbes' voluntarism to a kind of conventionalism, and through a reading of Kant's (...)
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  19.  34
    Tomasz Bigaj (2003). The Indispensability Argument – a New Chance for Empiricism in Mathematics? Foundations of Science 8 (2):173-200.
    In recent years, the so-calledindispensability argument has been given a lotof attention by philosophers of mathematics.This argument for the existence of mathematicalobjects makes use of the fact, neglected inclassical schools of philosophy of mathematics,that mathematics is part of our best scientifictheories, and therefore should receive similarsupport to these theories. However, thisobservation raises the question about the exactnature of the alleged connection betweenexperience and mathematics (for example: is itpossible to falsify empirically anymathematical theorems?). In my paper I (...)
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  20. Justin Clarke-Doane (forthcoming). Justification and Explanation in Mathematics and Morality. In Russ Shafer-Landau (ed.), Oxford Studies in Metaethics. Oxford University Press
    In an influential book, Gilbert Harman writes, "In explaining the observations that support a physical theory, scientists typically appeal to mathematical principles. On the other hand, one never seems to need to appeal in this way to moral principles [1977, 9 – 10]." What is the epistemological relevance of this contrast, if genuine? In this article, I argue that ethicists and philosophers of mathematics have misunderstood it. They have confused what I will call the justificatory challenge for realism about (...)
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  21.  20
    E. Slowik (2003). Conventionalism in Reid's 'Geometry of Visibles'. Studies in History and Philosophy of Science Part A 34 (3):467-489.
    The subject of this investigation is the role of conventions in the formulation of Thomas Reid's theory of the geometry of vision, which he calls the 'geometry of visibles'. In particular, we will examine the work of N. Daniels and R. Angell who have alleged that, respectively, Reid's 'geometry of visibles' and the geometry of the visual field are non-Euclidean. As will be demonstrated, however, the construction of any geometry of vision is subject to a choice of conventions regarding the (...)
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  22.  23
    B. Pourciau (2000). Intuitionism as a (Failed) Kuhnian Revolution in Mathematics. Studies in History and Philosophy of Science Part A 31 (2):297-329.
    In this paper it is argued, firstly, that Kuhnian revolutions in mathematics are logically possible, in the sense of not being inconsistent with the nature of mathematics; and, secondly, that Kuhnian revolutions are actually possible, in the sense that a Kuhnian paradigm for mathematics can be exhibited which would, if accepted by the mathematical community, produce a full Kuhnian revolution. These two arguments depend on first proving that a shift from a classical conception of mathematics to (...)
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  23.  1
    Mdutshekelwa Ndlovu (2013). Revisiting the Efficacy of Constructivism in Mathematics Education. Philosophy of Mathematics Education Journal 27 (April):1-13.
    The purpose of this paper is to critically analyse and discuss the views of constructivism, on the teaching and learning of mathematics. I provide a background to the learning of mathematics as constructing and reconstructing knowledge in the form of new conceptual networks; the nature, role and possibilities of constructivism as a learning theoretical framework in Mathematics Education. I look at the major criticisms and conclude that it passes the test of a learning theoretical framework but there (...)
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  24.  26
    David Liggins (forthcoming). Multiple Realization and Expressive Power in Mathematics and Ethics. In Uri D. Leibowitz & Neil Sinclair (eds.), Explanation in Ethics and Mathematics: Debunking and Dispensability. Oxford University Press
    According to a popular ‘explanationist’ argument for moral or mathematical realism the best explanation of some phenomena are moral or mathematical, and this implies the relevant form of realism. One popular way to resist the premiss of such arguments is to hold that any supposed explanation provided by moral or mathematical properties is in fact provided only by the non-moral or non-mathematical grounds of those properties. Many realists have responded to this objection by urging that the explanations provided by the (...)
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  25.  5
    James Franklin (2013). Non-Deductive Logic in Mathematics: The Probability of Conjectures. In Andrew Aberdein & Ian J. Dove (eds.), The Argument of Mathematics. Springer 11--29.
    Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective (...)
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  26.  12
    Krzysztof Wójtowicz (2006). Independence and Justification in Mathematics. Poznan Studies in the Philosophy of the Sciences and the Humanities 91 (1):349-373.
    In the article the problem of independence in mathematics is discussed. The status of the continuum hypothesis, large cardinal axioms and the axiom of constructablility is presented in some detail. The problem whether incompleteness is really relevant for ordinary mathematics and for empirical science is investigated. Another aim of the article is to give some arguments for the thesis that the problem of reliability and justification of new axioms is well-posed and worthy of attention. In my opinion, investigations (...)
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  27.  63
    Iulian D. Toader (2016). Why Did Weyl Think That Dedekind's Norm of Belief in Mathematics is Perverse? In Early Analytic Philosophy – New Perspectives on the Tradition. The Western Ontario Series in Philosophy of Science, Vol. 80, 445-451.
    This paper discusses an intriguing, though rather overlooked case of normative disagreement in the history of philosophy of mathematics: Weyl's criticism of Dedekind’s famous principle that "In science, what is provable ought not to be believed without proof." This criticism, as I see it, challenges not only a logicist norm of belief in mathematics, but also a realist view about whether there is a fact of the matter as to what norms of belief are correct.
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  28.  6
    Kenny Easwaran (2015). Rebutting and Undercutting in Mathematics. Philosophical Perspectives 29 (1):146-162.
    In my () I argued that a central component of mathematical practice is that published proofs must be “transferable” — that is, they must be such that the author's reasons for believing the conclusion are shared directly with the reader, rather than requiring the reader to essentially rely on testimony. The goal of this paper is to explain this requirement of transferability in terms of a more general norm on defeat in mathematical reasoning that I will call “convertibility”. I begin (...)
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  29. S. Awodey (1996). Structure in Mathematics and Logic: A Categorical Perspective. Philosophia Mathematica 4 (3):209-237.
    A precise notion of ‘mathematical structure’ other than that given by model theory may prove fruitful in the philosophy of mathematics. It is shown how the language and methods of category theory provide such a notion, having developed out of a structural approach in modern mathematical practice. As an example, it is then shown how the categorical notion of a topos provides a characterization of ‘logical structure’, and an alternative to the Pregean approach to logic which is continuous with (...)
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  30. Kenny Easwaran (2008). The Role of Axioms in Mathematics. Erkenntnis 68 (3):381 - 391.
    To answer the question of whether mathematics needs new axioms, it seems necessary to say what role axioms actually play in mathematics. A first guess is that they are inherently obvious statements that are used to guarantee the truth of theorems proved from them. However, this may neither be possible nor necessary, and it doesn’t seem to fit the historical facts. Instead, I argue that the role of axioms is to systematize uncontroversial facts that mathematicians can accept from (...)
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  31.  65
    John Mumma & Marco Panza (2012). Diagrams in Mathematics: History and Philosophy. Synthese 186 (1):1-5.
    Diagrams are ubiquitous in mathematics. From the most elementary class to the most advanced seminar, in both introductory textbooks and professional journals, diagrams are present, to introduce concepts, increase understanding, and prove results. They thus fulfill a variety of important roles in mathematical practice. Long overlooked by philosophers focused on foundational and ontological issues, these roles have come to receive attention in the past two decades, a trend in line with the growing philosophical interest in actual mathematical practice.
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  32.  54
    Charlotte Werndl (2009). Justifying Definitions in Mathematics—Going Beyond Lakatos. Philosophia Mathematica 17 (3):313-340.
    This paper addresses the actual practice of justifying definitions in mathematics. First, I introduce the main account of this issue, namely Lakatos's proof-generated definitions. Based on a case study of definitions of randomness in ergodic theory, I identify three other common ways of justifying definitions: natural-world justification, condition justification, and redundancy justification. Also, I clarify the interrelationships between the different kinds of justification. Finally, I point out how Lakatos's ideas are limited: they fail to show how various kinds of (...)
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  33.  49
    Angela Breitenbach (2015). Beauty in Proofs: Kant on Aesthetics in Mathematics. European Journal of Philosophy 23 (4):955-977.
    It is a common thought that mathematics can be not only true but also beautiful, and many of the greatest mathematicians have attached central importance to the aesthetic merit of their theorems, proofs and theories. But how, exactly, should we conceive of the character of beauty in mathematics? In this paper I suggest that Kant's philosophy provides the resources for a compelling answer to this question. Focusing on §62 of the ‘Critique of Aesthetic Judgment’, I argue against the (...)
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  34.  31
    Marc Lange (2015). Depth and Explanation in Mathematics. Philosophia Mathematica 23 (2):196-214.
    This paper argues that in at least some cases, one proof of a given theorem is deeper than another by virtue of supplying a deeper explanation of the theorem — that is, a deeper account of why the theorem holds. There are cases of scientific depth that also involve a common abstract structure explaining a similarity between two otherwise unrelated phenomena, making their similarity no coincidence and purchasing depth by answering why questions that separate, dissimilar explanations of the two phenomena (...)
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  35. Mark Balaguer (1998). Platonism and Anti-Platonism in Mathematics. Oxford University Press.
    In this book, Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He does this by establishing that both platonism and anti-platonism are defensible views. Introducing a form of platonism ("full-blooded platonism") that solves all problems traditionally associated with the view, he proceeds to defend anti-platonism (in particular, mathematical fictionalism) against various attacks, most notably the Quine-Putnam indispensability attack. He concludes by arguing that it is not simply that we do not currently have any good argument (...)
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  36. Alexander Paseau (2005). Naturalism in Mathematics and the Authority of Philosophy. British Journal for the Philosophy of Science 56 (2):377-396.
    Naturalism in the philosophy of mathematics is the view that philosophy cannot legitimately gainsay mathematics. I distinguish between reinterpretation and reconstruction naturalism: the former states that philosophy cannot legitimately sanction a reinterpretation of mathematics (i.e. an interpretation different from the standard one); the latter that philosophy cannot legitimately change standard mathematics (as opposed to its interpretation). I begin by showing that neither form of naturalism is self-refuting. I then focus on reinterpretation naturalism, which comes in two (...)
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  37.  70
    Kajsa Bråting & Johanna Pejlare (2008). Visualizations in Mathematics. Erkenntnis 68 (3):345 - 358.
    In this paper we discuss visualizations in mathematics from a historical and didactical perspective. We consider historical debates from the 17th and 19th centuries regarding the role of intuition and visualizations in mathematics. We also consider the problem of what a visualization in mathematical learning can achieve. In an empirical study we investigate what mathematical conclusions university students made on the basis of a visualization. We emphasize that a visualization in mathematics should always be considered in its (...)
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  38.  64
    John L. Bell (2004). Whole and Part in Mathematics. Axiomathes 14 (4):285-294.
    The centrality of the whole/part relation in mathematics is demonstrated through the presentation and analysis of examples from algebra, geometry, functional analysis,logic, topology and category theory.
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  39.  34
    Ferdinando Arzarello, Valeria Andriano, Federica Olivero & Ornella Robutti (1998). Abduction and Conjecturing in Mathematics. Philosophica 61 (1):77-94.
    The logic of discovering and that of justifying have been a permanent source of debate in mathematics, because of their different and apparently contradictory features within the processes of production of mathematical sentences. In fact, a fundamental unity appears as soon as one investigates deeply the phenomenology of conjecturing and proving using concrete examples. In this paper it is shown that abduction, in the sense of Peirce, is an essential unifying activity, ruling such phenomena. Abduction is the major ingredient (...)
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  40.  72
    Dieter Lohmar (2012). Non-Language Thinking in Mathematics. Axiomathes 22 (1):109-120.
    After a brief outline of the topic of non-language thinking in mathematics the central phenomenological tool in this concern is established, i.e. the eidetic method. The special form of eidetic method in mathematical proving is implicit variation and this procedure entails three rules that are established in a simple geometrical example. Then the difficulties and the merits of analogical thinking in mathematics are discussed in different aspects. On the background of a new phenomenological understanding of the performance of (...)
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  41. Andrew Arana (2009). Visual Thinking in Mathematics • by Marcus Giaquinto. 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 (...)
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  42.  44
    W. W. Tait (2001). Beyond the Axioms: The Question of Objectivity in Mathematics. Philosophia Mathematica 9 (1):21-36.
    This paper contains a defense against anti-realism in mathematics in the light both of incompleteness and of the fact that mathematics is a ‘cultural artifact.’. Anti-realism (here) is the view that theorems, say, of aritltmetic cannot be taken at face value to express true propositions about the system of numbers but must be reconstrued to be about somctliiiig else or about nothing at all. A ‘bite-the-bullet’ aspect of the defease is that, adopting new axioms, liitherto independent, is not. (...)
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  43.  5
    John von Neumann (2001). Unsolved Problems in Mathematics. Vienna Circle Institute Yearbook 8:231-246.
    The invitation of the Organizing Committee for me to speak about “Unsolved problems in mathematics” fills me as it should with considerable trepidation and a prevailing feeling of personal inadequacy. Hilbert gave a talk on this subject at the similar congress about 50 years ago and this is a very formidable precedent. He stated about a dozen unsolved problems in another widely separated areas of mathematics, and they proved to be prototypical for much of the development that followed (...)
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  44.  41
    Neil Tennant (2000). What is Naturalism in Mathematics, Really?: A Critical Study of P. Maddy, Naturalism in Mathematics. [REVIEW] Philosophia Mathematica 8 (3):316-338.
    Review of PENELOPE MADDY. Naturalism in Mathematics. Oxford: Clarendon Press, 1997.
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  45.  11
    Abbas Edalat (1997). Domains for Computation in Mathematics, Physics and Exact Real Arithmetic. Bulletin of Symbolic Logic 3 (4):401-452.
    We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on (...)
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  46.  14
    Jeremy Gray (2015). Depth — A Gaussian Tradition in Mathematics. Philosophia Mathematica 23 (2):177-195.
    Mathematicians use the word ‘deep’ to convey a high appreciation of a concept, theorem, or proof. This paper investigates the extent to which the term can be said to have an objective character by examining its first use in mathematics. It was a consequence of Gauss's work on number theory and the agreement among his successors that specific parts of Gauss's work were deep, on grounds that indicate that depth was a structural feature of mathematics for them. In (...)
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  47.  43
    Pierre Cassou-Nogués (2006). Signs, Figures and Time: Cavaillès on “Intuition” in Mathematics. Theoria 21 (1):89-104.
    This paper is concerned with Cavaillès’ account of “intuition” in mathematics. Cavaillès starts from Kant’s theory of constructions in intuition and then relies on various remarks by Hilbert to apply it to modern mathematics. In this context, “intuition” includes the drawing of geometrical figures, the use of algebraic or logical signs and the generation of numbers as, for example, described by Brouwer. Cavaillès argues that mathematical practice can indeed be described as “constructions in intuition” but that these constructions (...)
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  48.  21
    Joachim Frans & Erik Weber (2014). Mechanistic Explanation and Explanatory Proofs in Mathematics. Philosophia Mathematica 22 (2):231-248.
    Although there is a consensus among philosophers of mathematics and mathematicians that mathematical explanations exist, only a few authors have proposed accounts of explanation in mathematics. These accounts fit into the unificationist or top-down approach to explanation. We argue that these models can be complemented by a bottom-up approach to explanation in mathematics. We introduce the mechanistic model of explanation in science and discuss the possibility of using this model in mathematics, arguing that using it does (...)
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  49.  9
    Marcel Danesi (2013). A Semiotic Note on Accuracy and Precision in Mathematics. American Journal of Semiotics 28 (3/4):169 - 173.
    The concept of accuracy in mathematics is something that is rarely discussed. It is taken for granted, mainly because the various symbolic tools of the discipline, such as the digits and its equations, are meant to have a precise interpretation within the primary referential field. Yet, mathematics is full of inaccuracies and imprecise notions and techniques. The science of limits or the calculus, for example, is the science of imprecision, since it is based on the notions of “approximation”. (...)
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    Peter Mittelstaedt (1977). Conventionalism in Special Relativity. Foundations of Physics 7 (7-8):573-583.
    Reichenbach, Grünbaum, and others have argued that special relativity is based on arbitrary conventions concerning clock synchronizations. Here we present a mathematical framework which shows that this conventionality is almost equivalent to the arbitrariness in the choice of coordinates in an inertial system. Since preferred systems of coordinates can uniquely be defined by means of the Lorentz invariance of physical laws irrespective of the properties of light signals, a special clock synchronization—Einstein's standard synchrony—is selected by this principle. No further restrictions (...)
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