Epistemology of Mathematics

Edited by Alan Baker (Swarthmore College)
Assistant editor: Sam Roberts (University of Sheffield)
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  1. Set-Theoretic Pluralism and the Benacerraf Problem.Justin Clarke-Doane - 2020 - Philosophical Studies 177 (7):2013-2030.
    Set-theoretic pluralism is an increasingly influential position in the philosophy of set theory (Balaguer [1998], Linksy and Zalta [1995], Hamkins [2012]). There is considerable room for debate about how best to formulate set-theoretic pluralism, and even about whether the view is coherent. But there is widespread agreement as to what there is to recommend the view (given that it can be formulated coherently). Unlike set-theoretic universalism, set-theoretic pluralism affords an answer to Benacerraf’s epistemological challenge. The purpose of this paper is (...)
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  2. Ce este matematica: Ghidul şcolar al înţelegerii conceptuale a matematicii.Catalin Barboianu - 2020 - Targu Jiu: PhilScience Press.
    Aceasta nu este o carte de matematică, ci una despre matematică, care se adresează elevului sau studentului, dar şi dascălului său, cu un scop cât se poate de practic, anume acela de a iniţia şi netezi calea către înţelegerea completă a matematicii predate în şcoală. Tradiţia predării matematicii într-o abordare preponderent procedural-formală a avut ca efect o viziune deformată a elevilor asupra matematicii, ca fiind ceva strict formal, instrumental şi calculatoriu. Pierzând contactul cu baza conceptuală a matematicii, elevii dezvoltă pe (...)
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  3. Francesca Biagioli: Space, Number, and Geometry From Helmholtz to Cassirer: Springer, Dordrecht, 2016, 239 Pp, $109.99 , ISBN: 978-3-319-31777-9. [REVIEW]Lydia Patton - 2019 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 50 (2):311-315.
    Francesca Biagioli’s Space, Number, and Geometry from Helmholtz to Cassirer is a substantial and pathbreaking contribution to the energetic and growing field of researchers delving into the physics, physiology, psychology, and mathematics of the nineteenth and twentieth centuries. The book provides a bracing and painstakingly researched re-appreciation of the work of Hermann von Helmholtz and Ernst Cassirer, and of their place in the tradition, and is worth study for that alone. The contributions of the book go far beyond that, however. (...)
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  4. Forms of Structuralism: Bourbaki and the Philosophers.Jean-Pierre Marquis - forthcoming - In Alberto Peruzzi & Silvano Zipoli Caiani (eds.), Structures Meres, Semantics, Mathematics, and Cognitive Science. New York, NY, USA:
    In this paper, we argue that, contrary to the view held by most philosophers of mathematics, Bourbaki’s technical conception of mathematical structuralism is relevant to philosophy of mathematics. In fact, we believe that Bourbaki has captured the core of any mathematical structuralism.
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  5. What is Field's Epistemological Objection to Platonism?Ylwa Sjölin Wirling - 2019 - In Robin Stenwall & Tobias Hansson Wahlberg (eds.), Maurinian Truths : Essays in Honour of Anna-Sofia Maurin on her 50th Birthday. pp. 123-133.
    This paper concerns an epistemological objection against mathematical platonism, due to Hartry Field.The argument poses an explanatory challenge – the challenge to explain the reliability of our mathematical beliefs – which the platonist, it’s argued, cannot meet. Is the objection compelling? Philosophers disagree, but they also disagree on (and are sometimes very unclear about) how the objection should be understood. Here I distinguish some options, and highlight some gaps that need to be filled in on the potentially most compelling version (...)
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  6. On the Philosophical Significance of Frege’s Constraint.Andrea Sereni - 2019 - Philosophia Mathematica 27 (2):244–275.
    Foundational projects disagree on whether pure and applied mathematics should be explained together. Proponents of unified accounts like neologicists defend Frege’s Constraint (FC), a principle demanding that an explanation of applicability be provided by mathematical definitions. I reconsider the philosophical import of FC, arguing that usual conceptions are biased by ontological assumptions. I explore more reasonable weaker variants — Moderate and Modest FC — arguing against common opinion that ante rem structuralism (and other) views can meet them. I dispel doubts (...)
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  7. Mathematical Knowledge and Naturalism.Fabio Sterpetti - 2019 - Philosophia 47 (1):225-247.
    How should one conceive of the method of mathematics, if one takes a naturalist stance? Mathematical knowledge is regarded as the paradigm of certain knowledge, since mathematics is based on the axiomatic method. Natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some naturalists try to naturalize mathematics relying on Darwinism. But several difficulties arise when one tries to naturalize (...)
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  8. Mathematician's Call for Interdisciplinary Research Effort.Catalin Barboianu - 2013 - International Gambling Studies 13 (3):430-433.
    The article addresses the necessity of increasing the role of mathematics in the psychological intervention in problem gambling, including cognitive therapies. It also calls for interdisciplinary research with the direct contribution of mathematics. The current contributions and limitations of the role of mathematics are analysed with an eye toward the professional profiles of the researchers. An enhanced collaboration between these two disciplines is suggested and predicted.
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  9. Circularities In The Contemporary Philosophical Accounts Of The Applicability Of Mathematics In The Physical Universe.Catalin Barboianu - 2015 - Revista de Filosofie 61 (5):517-542.
    Contemporary philosophical accounts of the applicability of mathematics in physical sciences and the empirical world are based on formalized relations between the mathematical structures and the physical systems they are supposed to represent within the models. Such relations were constructed both to ensure an adequate representation and to allow a justification of the validity of the mathematical models as means of scientific inference. This article puts in evidence the various circularities (logical, epistemic, and of definition) that are present in these (...)
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  10. The "Unreasonable" Effectiveness of Mathematics: The Foundational Approach of the Theoretic Alternatives.Catalin Barboianu - 2015 - Revista de Filosofie 62 (1):58-71.
    The attempts of theoretically solving the famous puzzle-dictum of physicist Eugene Wigner regarding the “unreasonable” effectiveness of mathematics as a problem of analytical philosophy, started at the end of the 19th century, are yet far from coming out with an acceptable theoretical solution. The theories developed for explaining the empirical “miracle” of applied mathematics vary in nature, foundation and solution, from denying the existence of a genuine problem to structural theories with an advanced level of mathematical formalism. Despite this variation, (...)
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  11. Filosofia Aplicabilitatii Matematicii: Intre Irational si Rational.Catalin Barboianu - 2018 - Târgu Jiu, Romania: Infarom.
    Lucrarea tratează unul dintre “misterele” filosofiei analitice şi ale raţionalităţii însăşi, anume aplicabilitatea matematicii în ştiinţe şi în investigarea matematică a realităţii înconjurătoare, a cărei filosofie este dezvoltată în jurul sintagmei – de acum paradigmatice – ‘eficacitatea iraţională a matematicii’, aparţinând fizicianului Eugene Wigner, problemă filosofică etichetată în literatură drept “puzzle-ul lui Wigner”. Odată intraţi în profunzimea acestei probleme, investigaţia nu trebuie limitată la căutarea unor răspunsuri explicative la întrebări precum “Ce este de fapt aplicabilitatea matematicii?”, “Cum explicăm prezenţa în (...)
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  12. Learning the Natural Numbers as a Child.Stefan Buijsman - 2019 - Noûs 53 (1):3-22.
    How do we get out knowledge of the natural numbers? Various philosophical accounts exist, but there has been comparatively little attention to psychological data on how the learning process actually takes place. I work through the psychological literature on number acquisition with the aim of characterising the acquisition stages in formal terms. In doing so, I argue that we need a combination of current neologicist accounts and accounts such as that of Parsons. In particular, I argue that we learn the (...)
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  13. Mathematical Models of Abstract Systems: Knowing Abstract Geometric Forms.Jean-Pierre Marquis - 2013 - Annales de la Faculté des Sciences de Toulouse 22 (5):969-1016.
    Scientists use models to know the world. It i susually assumed that mathematicians doing pure mathematics do not. Mathematicians doing pure mathematics prove theorems about mathematical entities like sets, numbers, geometric figures, spaces, etc., they compute various functions and solve equations. In this paper, I want to exhibit models build by mathematicians to study the fundamental components of spaces and, more generally, of mathematical forms. I focus on one area of mathematics where models occupy a central role, namely homotopy theory. (...)
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  14. The "Artificial Mathematician" Objection: Exploring the (Im)Possibility of Automating Mathematical Understanding.Sven Delarivière & Bart Van Kerkhove - 2017 - In B. Sriraman (ed.), Humanizing Mathematics and its Philosophy. Cham: Birkhäuser. pp. 173-198.
    Reuben Hersh confided to us that, about forty years ago, the late Paul Cohen predicted to him that at some unspecified point in the future, mathematicians would be replaced by computers. Rather than focus on computers replacing mathematicians, however, our aim is to consider the (im)possibility of human mathematicians being joined by “artificial mathematicians” in the proving practice—not just as a method of inquiry but as a fellow inquirer.
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  15. Mathematical Abstraction, Conceptual Variation and Identity.Jean-Pierre Marquis - 2014 - In Peter Schroeder-Heister, Gerhard Heinzmann, Wilfred Hodges & Pierre Edouard Bour (eds.), Logic, Methodology and Philosophy of Science, Proceedings of the 14th International Congress. London, UK: pp. 299-322.
    One of the key features of modern mathematics is the adoption of the abstract method. Our goal in this paper is to propose an explication of that method that is rooted in the history of the subject.
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  16. The Limits of Reconstructive Neologicist Epistemology.Eileen S. Nutting - 2018 - Philosophical Quarterly 68 (273):717-738.
    Wright claims that his and Hale’s abstractionist neologicist project is primarily epistemological in aim. Its epistemological aims include establishing the possibility of a priori mathematical knowledge, and establishing the possibility of reference to abstract mathematical objects. But, as Wright acknowledges, there is a question of how neologicist epistemology applies to actual, ordinary mathematical beliefs. I take up this question, focusing on arithmetic. Following a suggestion of Hale and Wright, I consider the possibility that the neologicist account provides an idealised reconstruction (...)
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  17. The Reality of Field’s Epistemological Challenge to Platonism.David Liggins - 2018 - Erkenntnis 83 (5):1027-1031.
    In the introduction to his Realism, mathematics and modality, and in earlier papers included in that collection, Hartry Field offered an epistemological challenge to platonism in the philosophy of mathematics. Justin Clarke-Doane Truth, objects, infinity: New perspectives on the philosophy of Paul Benacerraf, 2016) argues that Field’s challenge is an illusion: it does not pose a genuine problem for platonism. My aim is to show that Clarke-Doane’s argument relies on a misunderstanding of Field’s challenge.
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  18. Mathematical Knowledge, the Analytic Method, and Naturalism.Fabio Sterpetti - 2018 - In Sorin Bangu (ed.), Naturalizing Logico-Mathematical Knowledge. Approaches from Philosophy, Psychology and Cognitive Science. New York, Stati Uniti: pp. 268-293.
    This chapter tries to answer the following question: How should we conceive of the method of mathematics, if we take a naturalist stance? The problem arises since mathematical knowledge is regarded as the paradigm of certain knowledge, because mathematics is based on the axiomatic method. Moreover, natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some authors tried to naturalize (...)
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  19. (Probably) Not Companions in Guilt.Sharon Berry - 2018 - Philosophical Studies 175 (9):2285-2308.
    In this paper, I will attempt to develop and defend a common form of intuitive resistance to the companions in guilt argument. I will argue that one can reasonably believe there are promising solutions to the access problem for mathematical realism that don’t translate to moral realism. In particular, I will suggest that the structuralist project of accounting for mathematical knowledge in terms of some form of logical knowledge offers significant hope of success while no analogous approach offers such hope (...)
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  20. Stairway to Heaven: The Abstract Method and Levels of Abstraction in Mathematics.Jean Pierre Marquis & Jean-Pierre Marquis - 2016 - The Mathematical Intelligencer 38 (3):41-51.
    In this paper, following the claims made by various mathematicians, I try to construct a theory of levels of abstraction. I first try to clarify the basic components of the abstract method as it developed in the first quarter of the 20th century. I then submit an explication of the notion of levels of abstraction. In the final section, I briefly explore some of main philosophical consequences of the theory.
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  21. Emily Grosholz and Herbert Breger, Editors. The Growth of Mathematical Knowledge.Brendan P. Larvor - 2002 - Philosophia Mathematica 10 (1):93-96.
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  22. Proof and Knowledge in Mathematics.Janet Folina - 1996 - Philosophical Quarterly 46 (182):125-127.
  23. Assaying Lakatos's Philosophy of Mathematics.David Corfield - 1997 - Studies in History and Philosophy of Science Part A 28 (1):99-121.
  24. Weyl, Reichenbach and the Epistemology of Geometry.T. A. Ryckman - 1994 - Studies in History and Philosophy of Science Part A 25 (6):831-870.
  25. The Argument From Agreement and Mathematical Realism.Pieranna Garavaso - 1992 - Journal of Philosophical Research 17:173-187.
    Traditionally, in the philosophy of mathematics realists claim that mathematical objects exist independently of the human mind, whereas idealists regard them as mental constructions dependent upon human thought.It is tempting for realists to support their view by appeal to our widespread agreement on mathematical results. Roughly speaking, our agreement is explained by the fact that these results are about the same mathematical objects. It is alleged that the idealist’s appeal to mental constructions precludes any such explanation. I argue that realism (...)
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  26. The ‘Unity of Cognition’ and the Explanation of Mathematical Knowledge.Barry Stroud - 2001 - Philosophical Topics 29 (1/2):415-428.
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  27. Mathematical Epistemology: What Is the Question?Penelope Maddy - 1984 - The Monist 67 (1):46-55.
    The tenor of much recent work in the philosophy of mathematics has been dictated by the popular assumption that Platonism is defunct. Some embrace that assumption and look for alternatives, others deny it and attempt to revive Platonism, but either way it is the starting point. The fate of Platonism took center stage with the appearance of Paul Benacerraf’s “Mathematical truth”, but a decade has passed since then, and the philosophical climate has changed. Most important, the quarter from which Platonism (...)
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  28. Proof and the Virtues of Shared Enquiry.Don Berry - forthcoming - Philosophia Mathematica:nkw022.
    This paper investigates an important aspect of mathematical practice: that proof is required for a finished piece of mathematics. If follows that non-deductive arguments — however convincing — are never sufficient. I explore four aspects of mathematical research that have facilitated the impressive success of the discipline. These I call the Practical Virtues: Permanence, Reliability, Autonomy, and Consensus. I then argue that permitting results to become established on the basis of non-deductive evidence alone would lead to their deterioration. This furnishes (...)
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  29. The Nature of Mathematical Knowledge.Charles Parsons - 1986 - Philosophical Review 95 (1):129.
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  30. Mathematical Knowledge. [REVIEW]David D. Auerbach - 1977 - Philosophical Review 86 (2):247.
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  31. José Ferreirós. Mathematical Knowledge and the Interplay of Practices. [REVIEW]Dirk Schlimm - 2017 - Philosophia Mathematica 25 (1):139-143.
  32. III-Reference by Abstraction.ØYstein Linnebo - 2012 - Proceedings of the Aristotelian Society 112 (1pt1):45-71.
    Frege suggests that criteria of identity should play a central role in the explanation of reference, especially to abstract objects. This paper develops a precise model of how we can come to refer to a particular kind of abstract object, namely, abstract letter types. It is argued that the resulting abstract referents are ‘metaphysically lightweight’.
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  33. The Applicability of Mathematics to Physical Modality.Nora Berenstain - 2017 - Synthese 194 (9):3361-3377.
    This paper argues that scientific realism commits us to a metaphysical determination relation between the mathematical entities that are indispensible to scientific explanation and the modal structure of the empirical phenomena those entities explain. The argument presupposes that scientific realism commits us to the indispensability argument. The viewpresented here is that the indispensability of mathematics commits us not only to the existence of mathematical structures and entities but to a metaphysical determination relation between those entities and the modal structure of (...)
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  34. The Varieties of Indispensability Arguments.Marco Panza & Andrea Sereni - 2016 - Synthese 193 (2):469-516.
    The indispensability argument comes in many different versions that all reduce to a general valid schema. Providing a sound IA amounts to providing a full interpretation of the schema according to which all its premises are true. Hence, arguing whether IA is sound results in wondering whether the schema admits such an interpretation. We discuss in full details all the parameters on which the specification of the general schema may depend. In doing this, we consider how different versions of IA (...)
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  35. The Good, the Bad and the Ugly.Philip Ebert & Stewart Shapiro - 2009 - Synthese 170 (3):415-441.
    This paper discusses the neo-logicist approach to the foundations of mathematics by highlighting an issue that arises from looking at the Bad Company objection from an epistemological perspective. For the most part, our issue is independent of the details of any resolution of the Bad Company objection and, as we will show, it concerns other foundational approaches in the philosophy of mathematics. In the first two sections, we give a brief overview of the "Scottish" neo-logicist school, present a generic form (...)
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  36. The Role of Axioms in Mathematics.Kenny Easwaran - 2008 - 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 a wide (...)
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  37. Godel's Disjunction: The Scope and Limits of Mathematical Knowledge.Leon Horsten & Philip Welch (eds.) - 2016 - Oxford University Press UK.
    To what extent can we hope to find answers to all mathematical questions? A famous theorem from Gödel entails that if our thinking capacities do not go beyond what an electronic computer is capable of, then there are indeed absolutely unsolvable mathematical problems. Thus it is of capital importance to find out whether human mathematicians can outstrip computers. Within this context, the contributions to this book critically examine positions about the scope and limits of human mathematical knowledge.
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  38. Logicism and its Philosophical Legacy.William Demopoulos - 2012 - Cambridge University Press.
    The idea that mathematics is reducible to logic has a long history, but it was Frege who gave logicism an articulation and defense that transformed it into a distinctive philosophical thesis with a profound influence on the development of philosophy in the twentieth century. This volume of classic, revised and newly written essays by William Demopoulos examines logicism's principal legacy for philosophy: its elaboration of notions of analysis and reconstruction. The essays reflect on the deployment of these ideas by the (...)
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  39. Metaphysical Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences.Jody Azzouni - 1994 - Cambridge University Press.
    Most philosophers of mathematics try to show either that the sort of knowledge mathematicians have is similar to the sort of knowledge specialists in the empirical sciences have or that the kind of knowledge mathematicians have, although apparently about objects such as numbers, sets, and so on, isn't really about those sorts of things as well. Jody Azzouni argues that mathematical knowledge really is a special kind of knowledge with its own special means of gathering evidence. He analyses the linguistic (...)
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  40. Towards a Philosophy of Real Mathematics.David Corfield - 2003 - Cambridge University Press.
    In this ambitious study, David Corfield attacks the widely held view that it is the nature of mathematical knowledge which has shaped the way in which mathematics is treated philosophically and claims that contingent factors have brought us to the present thematically limited discipline. Illustrating his discussion with a wealth of examples, he sets out a variety of approaches to new thinking about the philosophy of mathematics, ranging from an exploration of whether computers producing mathematical proofs or conjectures are doing (...)
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  41. Mathematical Reasoning and Heuristics.Carlo Cellucci & Donald Gillies (eds.) - 2005 - College Publications.
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  42. On Principles In Sadi Carnot’s Theory (1824). Epistemological Reflections.Raffaele Pisano - 2010 - Almagest 2/1:128–179 2 (1):128-179.
    In 1824 Sadi Carnot published Réflexions sur la Puissance Motrice du Feu in which he founded almost the entire thermodynamics theory. Two years after his death, his friend Clapeyron introduced the famous diagram PV for analytically representing the famous Carnot’s cycle: one of the main and crucial ideas presented by Carnot in his booklet. Twenty-five years later, in order to achieve the modern version of the theory, Kelvin and Clausius had to reject the caloric hypothesis, which had influenced a few (...)
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  43. Proof and Knowledge in Mathematics.Michael Detlefsen (ed.) - 1992 - Routledge.
    These questions arise from any attempt to discover an epistemology for mathematics. This collection of essays considers various questions concerning the nature of justification in mathematics and possible sources of that justification. Among these are the question of whether mathematical justification is _a priori_ or _a posteriori_ in character, whether logical and mathematical differ, and if formalization plays a significant role in mathematical justification.
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  44. Of Proofs, Mathematicians, and Computers.Astrik Yepremyan - unknown
    As computers become a more prevalent commodity in mathematical research and mathematical proof, the question of whether or not a computer assisted proof can be considered a mathematical proof has become an ongoing topic of discussion in the mathematics community. The use of the computer in mathematical research leads to several implications about mathematics in the present day including the notion that mathematical proof can be based on empirical evidence, and that some mathematical conclusions can be achieved a posteriori instead (...)
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  45. Mathematical Reasoning with Diagrams From Intuition to Automation.Mateja Jamnik - 2001
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  46. Truth, Proof and Infinity a Theory of Constructions and Contructive Reasoning.Peter Fletcher - 1998
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  47. A Role for Quasi-Empiricism in Mathematics Education.Eduard Glas - 2014 - In Michael R. Matthews (ed.), International Handbook of Research in History, Philosophy and Science Teaching. Springer. pp. 731-753.
    Although there are quite a few directions in modern philosophy of mathematics that invoke some essential role for (quasi-)empirical material, this chapter will be devoted exclusively to what may be considered the seminal tradition. This enabled me to present the subject as one coherent whole and to forestall the discussion getting scattered in a diversity of directions without doing justice to any one of them. -/- Quasi-empiricism in this tradition is the view that the logic of mathematical inquiry is based, (...)
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  48. Benacerraf's Dilemma and Natural Realism for Arithmetic.Anoop K. Gupta - 2002 - Dissertation, University of Ottawa (Canada)
    A natural realist approach to the philosophy of arithmetic is defended by way of considering and arguing against contemporary attempts to solve Paul Benacerraf's dilemma . The first horn of the dilemma concerns the existence of abstract mathematical objects, which seems necessitated by a desire for a unified semantics. Benacerraf adopts an extensional semantics whereby the reference of terms for natural numbers must be abstract objects. The second horn concerns a desirable causal constraint on knowledge, according to which "for X (...)
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  49. What Is the Goal of Proof? A Reaction to "What Do Mathematicians Want?" by Don Fallis.Aaron Lercher - 2002 - Logique Et Analyse 45.
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  50. Leibniz's and Kant's Philosophical Ideas and the Development of Hilbert's Programme.Roman Murawski - 2002 - Logique Et Analyse 45.
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