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

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  1. P. Garavaso (2013). Hilary Putnam's Consistency Objection Against Wittgenstein's Conventionalism in Mathematics. Philosophia Mathematica 21 (3):279-296.score: 117.0
    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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  2. Pieranna Garavaso (1988). Wittgenstein's Philosophy of Mathematics: A Reply to Two Objections. Southern Journal of Philosophy 26 (2):179-191.score: 96.0
    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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  3. Barry Smith (1976). Historicity, Value and Mathematics. In. In A. T. Tymieniecka (ed.), Ingardeniana. 219--239.score: 76.0
    At the beginning of the present century, a series of paradoxes were discovered within mathematics which suggested a fundamental unclarity in traditional mathemati­cal methods. These methods rested on the assumption of a realm of mathematical idealities existing independently of our thinking activity, and in order to arrive at a firmly grounded mathematics different attempts were made to formulate a conception of mathematical objects as purely human constructions. It was, however, realised that such formulations necessarily result in a (...) which lacks the richness and power of the old ‘platonistic’ methods, and the latter are still defended, in various modified forms, as embodying truths about self-existent mathematical entities. Thus there is an idealism-realism dispute in the philosophy of mathematics in some respects parallel to the controversy over the existence of the experiential world to the settle­ment of which lngarden devoted his life. The present paper is an attempt to apply Ingarden’s methods to the sphere of mathematical existence. This exercise will reveal new modes of being applicable to non-real objects, and we shall put forward arguments to suggest that these modes of being have an importance outside mathematics, especially in the areas of value theory and the ontology of art. (shrink)
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  4. Emily Grosholz (2007). Representation and Productive Ambiguity in Mathematics and the Sciences. Oxford University Press.score: 66.0
    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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  5. E. Slowik (2003). Conventionalism in Reid's 'Geometry of Visibles'. Studies in History and Philosophy of Science Part A 34 (3):467-489.score: 64.0
    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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  6. Penelope Maddy (1997). Naturalism in Mathematics. Oxford University Press.score: 62.0
    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. M. Giaquinto (2007). Visual Thinking in Mathematics: An Epistemological Study. Oxford University Press.score: 62.0
    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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  8. Michael Detlefsen (ed.) (1992). Proof and Knowledge in Mathematics. Routledge.score: 62.0
    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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  9. Donald Gillies (ed.) (1992). Revolutions in Mathematics. Oxford University Press.score: 62.0
    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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  10. Peter Mittelstaedt (1977). Conventionalism in Special Relativity. Foundations of Physics 7 (7-8):573-583.score: 61.0
    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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  11. Penelope Maddy (1990). Realism in Mathematics. Oxford University Prress.score: 60.0
    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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  12. Tomasz Bigaj (2003). The Indispensability Argument – a New Chance for Empiricism in Mathematics? Foundations of Science 8 (2):173-200.score: 60.0
    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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  13. Larry Krasnoff (2012). Voluntarism and Conventionalism in Hobbes and Kant. Hobbes Studies 25 (1):43-65.score: 60.0
    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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  14. Jean Paul van Bendegem (forthcoming). Inconsistency in Mathematics and the Mathematics of Inconsistency. Synthese:1-16.score: 60.0
    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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  15. B. Pourciau (2000). Intuitionism as a (Failed) Kuhnian Revolution in Mathematics. Studies in History and Philosophy of Science Part A 31 (2):297-329.score: 59.0
    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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  16. Krzysztof Wójtowicz (2006). Independence and Justification in Mathematics. Poznan Studies in the Philosophy of the Sciences and the Humanities 91 (1):349-373.score: 59.0
    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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  17. Elizabeth de Freitas (2013). The Mathematical Event: Mapping the Axiomatic and the Problematic in School Mathematics. Studies in Philosophy and Education 32 (6):581-599.score: 59.0
    Traditional philosophy of mathematics has been concerned with the nature of mathematical objects rather than events. This traditional focus on reified objects is reflected in dominant theories of learning mathematics whereby the learner is meant to acquire familiarity with ideal mathematical objects, such as number, polygon, or tangent. I argue that the concept of event—rather than object—better captures the vitality of mathematics, and offers new ways of thinking about mathematics education. In this paper I draw on (...)
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  18. Dieter Lohmar (2012). Non-Language Thinking in Mathematics. Axiomathes 22 (1):109-120.score: 58.0
    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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  19. Feng Ye (2007). Indispensability Argument and Anti-Realism in Philosophy of Mathematics. Frontiers of Philosophy in China 2 (4):614-628.score: 57.7
    The indispensability argument for abstract mathematical entities has been an important issue in the philosophy of mathematics. The argument relies on several assumptions. Some objections have been made against these assumptions, but there are several serious defects in these objections. Ameliorating these defects leads to a new anti-realistic philosophy of mathematics, mainly: first, in mathematical applications, what really exist and can be used as tools are not abstract mathematical entities, but our inner representations that we create in imagining (...)
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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.score: 57.0
    In an influential book, 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? In this article, I argue that ethicists and philosophers of mathematics have misunderstood it. They have confused what I shall call the justificatory challenge for realism about an area, D (...)
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  21. Sarah Wu, Hitha Amin, Maria Barth, Vanessa Malcarne & Vinod Menon (2012). Math Anxiety in Second and Third Graders and Its Relation to Mathematics Achievement. Frontiers in Psychology 3.score: 57.0
    Although the detrimental effects of math anxiety in adults are well understood, few studies have examined how it affects younger children who are beginning to learn math in a formal academic setting. Here, we examine the relationship between math anxiety and math achievement in 2nd and 3rd graders. In response to the need for a grade-appropriate measure of assessing math anxiety in this group we first describe the development of Scale for Early Mathematics Anxiety (SEMA), a new measure for (...)
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  22. William Bragg Ewald (ed.) (1996). From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Oxford University Press.score: 57.0
    This massive two-volume reference presents a comprehensive selection of the most important works on the foundations of mathematics. While the volumes include important forerunners like Berkeley, MacLaurin, and D'Alembert, as well as such followers as Hilbert and Bourbaki, their emphasis is on the mathematical and philosophical developments of the nineteenth century. Besides reproducing reliable English translations of classics works by Bolzano, Riemann, Hamilton, Dedekind, and Poincare, William Ewald also includes selections from Gauss, Cantor, Kronecker, and Zermelo, all translated here (...)
     
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  23. Elijah Chudnoff (forthcoming). Intuition in Mathematics. In Barbara Held & Lisa Osbeck (eds.), Rational Intuition. Cambridge University Press.score: 56.7
    The literature on mathematics suggests that intuition plays a role in it as a ground of belief. This article explores the nature of intuition as it occurs in mathematical thinking. Section 1 suggests that intuitions should be understood by analogy with perceptions. Section 2 explains what fleshing out such an analogy requires. Section 3 discusses Kantian ways of fleshing it out. Section 4 discusses Platonist ways of fleshing it out. Section 5 sketches a proposal for resolving the main problem (...)
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  24. S. Awodey (1996). Structure in Mathematics and Logic: A Categorical Perspective. Philosophia Mathematica 4 (3):209-237.score: 56.0
    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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  25. Kenny Easwaran (2008). The Role of Axioms in Mathematics. Erkenntnis 68 (3):381 - 391.score: 56.0
    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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  26. Alexander Paseau (2005). Naturalism in Mathematics and the Authority of Philosophy. British Journal for the Philosophy of Science 56 (2):377-396.score: 56.0
    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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  27. Kajsa Bråting & Johanna Pejlare (2008). Visualizations in Mathematics. Erkenntnis 68 (3):345 - 358.score: 56.0
    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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  28. Markus Pantsar (2009). Truth, Proof and Gödelian Arguments: A Defence of Tarskian Truth in Mathematics. Dissertation, University of Helsinkiscore: 56.0
    One of the most fundamental questions in the philosophy of mathematics concerns the relation between truth and formal proof. The position according to which the two concepts are the same is called deflationism, and the opposing viewpoint substantialism. In an important result of mathematical logic, Kurt Gödel proved in his first incompleteness theorem that all consistent formal systems containing arithmetic include sentences that can neither be proved nor disproved within that system. However, such undecidable Gödel sentences can be established (...)
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  29. John Mumma & Marco Panza (2012). Diagrams in Mathematics: History and Philosophy. Synthese 186 (1):1-5.score: 56.0
    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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  30. Andrew Arana (2009). Review of M. Giaquinto's Visual Thinking in Mathematics. [REVIEW] Analysis 69:401-403.score: 56.0
    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” (...)
     
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  31. Charlotte Werndl (2009). Justifying Definitions in Mathematics—Going Beyond Lakatos. Philosophia Mathematica 17 (3):313-340.score: 56.0
    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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  32. Friedrich Waismann (1951/2003). Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics. Dover Publications.score: 56.0
    "With exceptional clarity, but with no evasion of essential ideas, the author outlines the fundamental structure of mathematics."--Carl B. Boyer, Brooklyn College. This enlightening survey of mathematical concept formation holds a natural appeal to philosophically minded readers, and no formal training in mathematics is necessary to appreciate its clear exposition. Contents include examinations of arithmetic and geometry; the rigorous construction of the theory of integers; the rational numbers and their foundation in arithmetic; and the rigorous construction of elementary (...)
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  33. W. W. Tait (2001). Beyond the Axioms: The Question of Objectivity in Mathematics. Philosophia Mathematica 9 (1):21-36.score: 56.0
    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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  34. John L. Bell (2004). Whole and Part in Mathematics. Axiomathes 14 (4):285-294.score: 56.0
    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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  35. Roman Murawski (2010). Philosophy of Mathematics in the Warsaw Mathematical School. Axiomathes 20 (2-3):279-293.score: 56.0
    The aim of this paper is to present and discuss the philosophical views concerning mathematics of the founders of the so called Warsaw Mathematical School, i.e., Wacław Sierpiński, Zygmunt Janiszewski and Stefan Mazurkiewicz. Their interest in the philosophy of mathematics and their philosophical papers will be considered. We shall try to answer the question whether their philosophical views influenced their proper mathematical investigations. Their views towards set theory and its rôle in mathematics will be emphasized.
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  36. Stojan Obradović & Slobodan Ninković (2009). The Heuristic Function of Mathematics in Physics and Astronomy. Foundations of Science 14 (4):351-360.score: 56.0
    This paper considers the role of mathematics in the process of acquiring new knowledge in physics and astronomy. The defining of the notions of continuum and discreteness in mathematics and the natural sciences is examined. The basic forms of representing the heuristic function of mathematics at theoretical and empirical levels of knowledge are studied: deducing consequences from the axiomatic system of theory, the method of generating mathematical hypotheses, “pure” proofs for the existence of objects and processes, mathematical (...)
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  37. Jairo José Da Silva (2000). Husserl's Two Notions of Completeness: Husserl and Hilbert on Completeness and Imaginary Elements in Mathematics. Synthese 125 (3):417 - 438.score: 56.0
    In this paper I discuss Husserl's solution of the problem of imaginary elements in mathematics as presented in the drafts for two lectures he gave in Göttingen in 1901 and other related texts of the same period, a problem that had occupied Husserl since the beginning of 1890, when he was planning a never published sequel to "Philosophie der Arithmetik" (1891). In order to solve the problem of imaginary entities Husserl introduced, independently of Hilbert, two notions of completeness (definiteness (...)
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  38. Angela Breitenbach (2013). Beauty in Proofs: Kant on Aesthetics in Mathematics. European Journal of Philosophy 21 (2):n/a-n/a.score: 56.0
    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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  39. Andrew Aberdein (2005). The Uses of Argument in Mathematics. Argumentation 19 (3):287-301.score: 56.0
    Stephen Toulmin once observed that ”it has never been customary for philosophers to pay much attention to the rhetoric of mathematical debate’ [Toulmin et al., 1979, An Introduction to Reasoning, Macmillan, London, p. 89]. Might the application of Toulmin’s layout of arguments to mathematics remedy this oversight? Toulmin’s critics fault the layout as requiring so much abstraction as to permit incompatible reconstructions. Mathematical proofs may indeed be represented by fundamentally distinct layouts. However, cases of genuine conflict characteristically reflect an (...)
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  40. John P. Burgess (1992). How Foundational Work in Mathematics Can Be Relevant to Philosophy of Science. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1992:433 - 441.score: 56.0
    Foundational work in mathematics by some of the other participants in the symposium helps towards answering the question whether a heterodox mathematics could in principle be used as successfully as is orthodox mathematics in scientific applications. This question is turn, it will be argued, is relevant to the question how far current science is the way it is because the world is the way it is, and how far because we are the way we are, which is (...)
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  41. James Franklin (1996). Proof in Mathematics. Quakers Hill Press.score: 56.0
    A textbook on proof in mathematics, inspired by an Aristotelian point of view on mathematics and proof. It explains how to prove statements in mathematics, from evident premises. It deals with the proof of "all" statements, "some" statements, multiple quantifiers and mathematical induction.
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  42. Mary Leng (2011). Creation and Discovery in Mathematics. In John Polkinghorne (ed.), Meaning in Mathematics. Oup Oxford.score: 56.0
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  43. Pierre Cassou-Nogués (2006). Signs, Figures and Time: Cavaillès on “Intuition” in Mathematics. Theoria 21 (1):89-104.score: 56.0
    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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  44. Joachim Frans & Erik Weber (2014). Mechanistic Explanation and Explanatory Proofs in Mathematics. Philosophia Mathematica 22 (2):231-248.score: 56.0
    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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  45. Adam Rieger (2003). Naturalism in Mathematics. [REVIEW] Philosophical Review 112 (3):425-427.score: 56.0
    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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  46. 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.score: 56.0
    Review of PENELOPE MADDY. Naturalism in Mathematics. Oxford: Clarendon Press, 1997.
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  47. I. Grattan-Guinness (2011). Omnipresence, Multipresence and Ubiquity: Kinds of Generality in and Around Mathematics and Logics. [REVIEW] Logica Universalis 5 (1):21-73.score: 56.0
    A prized property of theories of all kinds is that of generality, of applicability or least relevance to a wide range of circumstances and situations. The purpose of this article is to present a pair of distinctions that suggest that three kinds of generality are to be found in mathematics and logics, not only at some particular period but especially in developments that take place over time: ‘omnipresent’ and ‘multipresent’ theories, and ‘ubiquitous’ notions that form dependent parts, or moments, (...)
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  48. Ferdinando Arzarello, Valeria Andriano, Federica Olivero & Ornella Robutti (1998). Abduction and Conjecturing in Mathematics. Philosophica 61 (1):77-94.score: 56.0
    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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  49. Elizabeth de Freitas (2013). What Were You Thinking? A Deleuzian/Guattarian Analysis of Communication in the Mathematics Classroom. Educational Philosophy and Theory 45 (3):287-300.score: 56.0
    The primary aim of this article is to bring the work of Deleuze and Guattari to bear on the question ofcommunication in the classroom. I focus on the mathematics classroom, where agency and subjectivity are highly regulated by the rituals of the discipline, and where neoliberal psychological frameworks continue to dominate theories of teaching and learning. Moreover, the nature ofcommunication in mathematics classrooms remains highlyelusive and problematic, due in part to the distinct relationship the discipline has with verbal (...)
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  50. Abbas Edalat (1997). Domains for Computation in Mathematics, Physics and Exact Real Arithmetic. Bulletin of Symbolic Logic 3 (4):401-452.score: 56.0
    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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