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  1. The Construction of Transfinite Equivalence Algorithms.Han Geurdes - manuscript
    Context: Consistency of mathematical constructions in numerical analysis and the application of computerized proofs in the light of the occurrence of numerical chaos in simple systems. Purpose: To show that a computer in general and a numerical analysis in particular can add its own peculiarities to the subject under study. Hence the need of thorough theoretical studies on chaos in numerical simulation. Hence, a questioning of what e.g. a numerical disproof of a theorem in physics or a prediction in numerical (...)
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  2. Realism, Objectivity, and Evaluation.Justin Clarke-Doane - forthcoming - In David Kaspar (ed.), Explorations in Ethics.
    I discuss Benacerraf's epistemological challenge for realism about areas like mathematics, metalogic, and modality, and describe the pluralist response to it. I explain why normative pluralism is peculiarly unsatisfactory, and use this explanation to formulate a radicalization of Moore's Open Question Argument. According to the argument, the facts -- even the normative facts -- fail to settle the practical questions at the center of our normative lives. One lesson is that the concepts of realism and objectivity, which are widely identified, (...)
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  3. Wittgenstein Und Die Philosophie der Mathematik.Bromand Joachim & Reichert Bastian (eds.) - forthcoming - Mentis Verlag.
    Ludwig Wittgenstein selbst hielt seine Überlegungen zur Mathematik für seinen bedeutendsten Beitrag zur Philosophie. So beabsichtigte er zunächst, dem Thema einen zentralen Teil seiner Philosophischen Untersuchungen zu widmen. Tatsächlich wird kaum irgendwo sonst in Wittgensteins Werk so deutlich, wie radikal die Konsequenzen seines Denkens eigentlich sind. Vermutlich deshalb haben Wittgensteins Bemerkungen zur Mathematik unter all seinen Schriften auch den größten Widerstand provoziert: Seine Bemerkungen zu den Gödel’schen Unvollständigkeitssätzen bezeichnete Gödel selbst als Nonsens, und Alan Turing warf Wittgenstein vor, dass aufgrund (...)
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  4. New Perspectives on the Philosophy of Paul Benacerraf: Truth, Objects, Infinity (Fabrice Pataut, Editor).Fabrice Pataut Jody Azzouni, Paul Benacerraf Justin Clarke-Doane, Jacques Dubucs Sébastien Gandon, Brice Halimi Jon Perez Laraudogoitia, Mary Leng Ana Leon-Mejia, Antonio Leon-Sanchez Marco Panza, Fabrice Pataut Philippe de Rouilhan & Andrea Sereni Stuart Shapiro - forthcoming - Springer.
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  5. Wittgenstein, Peirce, and Paradoxes of Mathematical Proof.Sergiy Koshkin - forthcoming - Analytic Philosophy.
    Wittgenstein's paradoxical theses that unproved propositions are meaningless, proofs form new concepts and rules, and contradictions are of limited concern, led to a variety of interpretations, most of them centered on rule-following skepticism. We argue, with the help of C. S. Peirce's distinction between corollarial and theorematic proofs, that his intuitions are better explained by resistance to what we call conceptual omniscience, treating meaning as fixed content specified in advance. We interpret the distinction in the context of modern epistemic logic (...)
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  6. ‘Let No-One Ignorant of Geometry…’: Mathematical Parallels for Understanding the Objectivity of Ethics.James Franklin - 2021 - Journal of Value Inquiry 55:1-20.
    It may be a myth that Plato wrote over the entrance to the Academy “Let no-one ignorant of geometry enter here.” But it is a well-chosen motto for his view in the Republic that mathematical training is especially productive of understanding in abstract realms, notably ethics. That view is sound and we should return to it. Ethical theory has been bedevilled by the idea that ethics is fundamentally about actions (right and wrong, rights, duties, virtues, dilemmas and so on). That (...)
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  7. A Metasemantic Challenge for Mathematical Determinacy.Jared Warren & Daniel Waxman - 2020 - Synthese 197 (2):477-495.
    This paper investigates the determinacy of mathematics. We begin by clarifying how we are understanding the notion of determinacy before turning to the questions of whether and how famous independence results bear on issues of determinacy in mathematics. From there, we pose a metasemantic challenge for those who believe that mathematical language is determinate, motivate two important constraints on attempts to meet our challenge, and then use these constraints to develop an argument against determinacy and discuss a particularly popular approach (...)
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  8. Supertasks and Arithmetical Truth.Jared Warren & Daniel Waxman - 2020 - Philosophical Studies 177 (5):1275-1282.
    This paper discusses the relevance of supertask computation for the determinacy of arithmetic. Recent work in the philosophy of physics has made plausible the possibility of supertask computers, capable of running through infinitely many individual computations in a finite time. A natural thought is that, if supertask computers are possible, this implies that arithmetical truth is determinate. In this paper we argue, via a careful analysis of putative arguments from supertask computations to determinacy, that this natural thought is mistaken: supertasks (...)
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  9. Can Mathematical Objects Be Causally Efficacious?Seungbae Park - 2019 - Inquiry: An Interdisciplinary Journal of Philosophy 62 (3):247–255.
    Callard (2007) argues that it is metaphysically possible that a mathematical object, although abstract, causally affects the brain. I raise the following objections. First, a successful defence of mathematical realism requires not merely the metaphysical possibility but rather the actuality that a mathematical object affects the brain. Second, mathematical realists need to confront a set of three pertinent issues: why a mathematical object does not affect other concrete objects and other mathematical objects, what counts as a mathematical object, and how (...)
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  10. Infinitesimal Idealization, Easy Road Nominalism, and Fractional Quantum Statistics.Elay Shech - 2019 - Synthese 196 (5):1963-1990.
    It has been recently debated whether there exists a so-called “easy road” to nominalism. In this essay, I attempt to fill a lacuna in the debate by making a connection with the literature on infinite and infinitesimal idealization in science through an example from mathematical physics that has been largely ignored by philosophers. Specifically, by appealing to John Norton’s distinction between idealization and approximation, I argue that the phenomena of fractional quantum statistics bears negatively on Mary Leng’s proposed path to (...)
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  11. In Defense of Mathematical Inferentialism.Seungbae Park - 2017 - Analysis and Metaphysics 16:70-83.
    I defend a new position in philosophy of mathematics that I call mathematical inferentialism. It holds that a mathematical sentence can perform the function of facilitating deductive inferences from some concrete sentences to other concrete sentences, that a mathematical sentence is true if and only if all of its concrete consequences are true, that the abstract world does not exist, and that we acquire mathematical knowledge by confirming concrete sentences. Mathematical inferentialism has several advantages over mathematical realism and fictionalism.
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  12. Two Criticisms Against Mathematical Realism.Seungbae Park - 2017 - Diametros 52:96-106.
    Mathematical realism asserts that mathematical objects exist in the abstract world, and that a mathematical sentence is true or false, depending on whether the abstract world is as the mathematical sentence says it is. I raise two objections against mathematical realism. First, the abstract world is queer in that it allows for contradictory states of affairs. Second, mathematical realism does not have a theoretical resource to explain why a sentence about a tricle is true or false. A tricle is an (...)
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  13. Can the Cumulative Hierarchy Be Categorically Characterized?Luca Incurvati - 2016 - Logique Et Analyse 59 (236):367-387.
    Mathematical realists have long invoked the categoricity of axiomatizations of arithmetic and analysis to explain how we manage to fix the intended meaning of their respective vocabulary. Can this strategy be extended to set theory? Although traditional wisdom recommends a negative answer to this question, Vann McGee (1997) has offered a proof that purports to show otherwise. I argue that one of the two key assumptions on which the proof rests deprives McGee's result of the significance he and the realist (...)
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  14. Against Mathematical Convenientism.Seungbae Park - 2016 - Axiomathes 26 (2):115-122.
    Indispensablists argue that when our belief system conflicts with our experiences, we can negate a mathematical belief but we do not because if we do, we would have to make an excessive revision of our belief system. Thus, we retain a mathematical belief not because we have good evidence for it but because it is convenient to do so. I call this view ‘ mathematical convenientism.’ I argue that mathematical convenientism commits the consequential fallacy and that it demolishes the Quine-Putnam (...)
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  15. Structuring Logical Space.Alejandro Pérez Carballo - 2016 - Philosophy and Phenomenological Research 92 (2):460-491.
    I develop a non-representationalist account of mathematical thought, on which the point of mathematical theorizing is to provide us with the conceptual capacity to structure and articulate information about the physical world in an epistemically useful way. On my view, accepting a mathematical theory is not a matter of having a belief about some subject matter; it is rather a matter of structuring logical space, in a sense to be made precise. This provides an elegant account of the cognitive utility (...)
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  16. Objectivity in Ethics and Mathematics.Justin Clarke-Doane - 2015 - Proceedings of the Aristotelian Society: The Virtual Issue 3.
    How do axioms, or first principles, in ethics compare to those in mathematics? In this companion piece to G.C. Field's 1931 "On the Role of Definition in Ethics", I argue that there are similarities between the cases. However, these are premised on an assumption which can be questioned, and which highlights the peculiarity of normative inquiry.
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  17. Functions and Generality of Logic.Gabriel Sandu, Marco Panza & Hourya Benis-Sinaceur (eds.) - 2015 - Springer Verlag.
    Part I of Frege’s Grundgesetze is devoted to the “exposition [Darlegung]” of his formal system.
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  18. Gödel’s Cantorianism.Claudio Ternullo - 2015 - In Eva-Maria Engelen & Gabriella Crocco (eds.), Kurt Gödel: Philosopher-Scientist. Presses Universitaires de Provence. pp. 417-446.
    Gödel’s philosophical conceptions bear striking similarities to Cantor’s. Although there is no conclusive evidence that Gödel deliberately used or adhered to Cantor’s views, one can successfully reconstruct and see his “Cantorianism” at work in many parts of his thought. In this paper, I aim to describe the most prominent conceptual intersections between Cantor’s and Gödel’s thought, particularly on such matters as the nature and existence of mathematical entities (sets), concepts, Platonism, the Absolute Infinite, the progress and inexhaustibility of mathematics.
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  19. Relative Categoricity and Abstraction Principles.Sean Walsh & Sean Ebels-Duggan - 2015 - Review of Symbolic Logic 8 (3):572-606.
    Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory. Another great enterprise in contemporary philosophy of mathematics has been Wright's and Hale's project of founding mathematics on abstraction principles. In earlier work, it was noted that one traditional abstraction principle, namely Hume's Principle, had a certain relative categoricity property, which here we term natural relative categoricity. In this paper, we show (...)
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  20. Conventionalism, Consistency, and Consistency Sentences.Jared Warren - 2015 - Synthese 192 (5):1351-1371.
    Conventionalism about mathematics claims that mathematical truths are true by linguistic convention. This is often spelled out by appealing to facts concerning rules of inference and formal systems, but this leads to a problem: since the incompleteness theorems we’ve known that syntactic notions can be expressed using arithmetical sentences. There is serious prima facie tension here: how can mathematics be a matter of convention and syntax a matter of fact given the arithmetization of syntax? This challenge has been pressed in (...)
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  21. Moral Epistemology: The Mathematics Analogy.Justin Clarke-Doane - 2014 - Noûs 48 (2):238-255.
    There is a long tradition comparing moral knowledge to mathematical knowledge. In this paper, I discuss apparent similarities and differences between knowledge in the two areas, realistically conceived. I argue that many of these are only apparent, while others are less philosophically significant than might be thought. The picture that emerges is surprising. There are definitely differences between epistemological arguments in the two areas. However, these differences, if anything, increase the plausibility of moral realism as compared to mathematical realism. It (...)
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  22. Outscoping and Discourse Threat.Theodore Sider - 2014 - Inquiry: An Interdisciplinary Journal of Philosophy 57 (4):413-426.
    Sometimes we give truth-conditions for sentences of a discourse in other terms. According to Agustín Rayo, when doing so it is sometimes legitimate to use the terms of that very discourse, so long as the terms do not occur in the truth-conditions themselves. I argue that giving truth-conditions in this "outscoping" way prevents one from answering "discourse threat" (for example, the threat of indeterminacy).
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  23. What is Absolute Undecidability?†.Justin Clarke-Doane - 2013 - Noûs 47 (3):467-481.
    It is often alleged that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) if a mathematical (...)
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  24. Towards an Institutional Account of the Objectivity, Necessity, and Atemporality of Mathematics.Julian C. Cole - 2013 - Philosophia Mathematica 21 (1):9-36.
    I contend that mathematical domains are freestanding institutional entities that, at least typically, are introduced to serve representational functions. In this paper, I outline an account of institutional reality and a supporting metaontological perspective that clarify the content of this thesis. I also argue that a philosophy of mathematics that has this thesis as its central tenet can account for the objectivity, necessity, and atemporality of mathematics.
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  25. The Philosophical Significance of Tennenbaum’s Theorem.T. Button & P. Smith - 2012 - Philosophia Mathematica 20 (1):114-121.
    Tennenbaum's Theorem yields an elegant characterisation of the standard model of arithmetic. Several authors have recently claimed that this result has important philosophical consequences: in particular, it offers us a way of responding to model-theoretic worries about how we manage to grasp the standard model. We disagree. If there ever was such a problem about how we come to grasp the standard model, then Tennenbaum's Theorem does not help. We show this by examining a parallel argument, from a simpler model-theoretic (...)
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  26. Vom Zahlen Zu den Zahlen: On the Relation Between Computation and Arithmetical Structuralism.L. Horsten - 2012 - Philosophia Mathematica 20 (3):275-288.
    This paper sketches an answer to the question how we, in our arithmetical practice, succeed in singling out the natural-number structure as our intended interpretation. It is argued that we bring this about by a combination of what we assert about the natural-number structure on the one hand, and our computational capacities on the other hand.
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  27. Fictionalism and Mathematical Objectivity.Iulian D. Toader - 2012 - In Metaphysics and Science. University of Bucharest Press. pp. 137-158.
    This paper, written in Romanian, compares fictionalism, nominalism, and neo-Meinongianism as responses to the problem of objectivity in mathematics, and then motivates a fictionalist view of objectivity as invariance.
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  28. Social Construction in the Philosophy of Mathematics: A Critical Evaluation of Julian Cole’s Theory†: Articles.J. M. Dieterle - 2010 - Philosophia Mathematica 18 (3):311-328.
    Julian Cole argues that mathematical domains are the products of social construction. This view has an initial appeal in that it seems to salvage much that is good about traditional platonistic realism without taking on the ontological baggage. However, it also has problems. After a brief sketch of social constructivist theories and Cole’s philosophy of mathematics, I evaluate the arguments in favor of social constructivism. I also discuss two substantial problems with the theory. I argue that unless and until social (...)
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  29. Gödel and Philosophical Idealism.Charles Parsons - 2010 - Philosophia Mathematica 18 (2):166-192.
    Kurt Gödel made many affirmations of robust realism but also showed serious engagement with the idealist tradition, especially with Leibniz, Kant, and Husserl. The root of this apparently paradoxical attitude is his conviction of the power of reason. The paper explores the question of how Gödel read Kant. His argument that relativity theory supports the idea of the ideality of time is discussed critically, in particular attempting to explain the assertion that science can go beyond the appearances and ‘approach the (...)
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  30. Open-Endedness, Schemas and Ontological Commitment.Nikolaj Jang Lee Linding Pedersen & Marcus Rossberg - 2010 - Noûs 44 (2):329-339.
    Second-order axiomatizations of certain important mathematical theories—such as arithmetic and real analysis—can be shown to be categorical. Categoricity implies semantic completeness, and semantic completeness in turn implies determinacy of truth-value. Second-order axiomatizations are thus appealing to realists as they sometimes seem to offer support for the realist thesis that mathematical statements have determinate truth-values. The status of second-order logic is a controversial issue, however. Worries about ontological commitment have been influential in the debate. Recently, Vann McGee has argued that one (...)
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  31. A Mathematician Reflects on the Useful and Reliable Illusion of Reality in Mathematics.Keith Devlin - 2008 - Erkenntnis 68 (3):359-379.
    Recent years have seen a growing acknowledgement within the mathematical community that mathematics is cognitively/socially constructed. Yet to anyone doing mathematics, it seems totally objective. The sensation in pursuing mathematical research is of discovering prior (eternal) truths about an external (abstract) world. Although the community can and does decide which topics to pursue and which axioms to adopt, neither an individual mathematician nor the entire community can choose whether a particular mathematical statement is true or false, based on the given (...)
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  32. McGee on Open-Ended Schemas.Nikolaj Jang Lee Linding Pedersen & Marcus Rossberg - 2007 - In Helen Bohse & Sven Walter (eds.), Selected Contributions to GAP.6: Sixth International Conference of the German Society for Analytical Philosophy, Berlin, 11–14 September 2006. mentis.
    Vann McGee claims that open-ended schemas are more innocuous than ordinary second-order quantification, particularly in terms of ontological commitment. We argue that this is not the case.
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  33. The Objectivity of Mathematics.Stewart Shapiro - 2007 - Synthese 156 (2):337-381.
    The purpose of this paper is to apply Crispin Wright’s criteria and various axes of objectivity to mathematics. I test the criteria and the objectivity of mathematics against each other. Along the way, various issues concerning general logic and epistemology are encountered.
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  34. Gödel's Conceptual Realism.Donald A. Martin - 2005 - Bulletin of Symbolic Logic 11 (2):207-224.
  35. Review of C. O. Hill and G. E. Rosado Haddock, Husserl or Frege? Meaning, Objectivity, and Mathematics[REVIEW]Mark van Atten - 2003 - Philosophia Mathematica 11 (2):241-244.
  36. Is Cantor's Continuum Problem Inherently Vague?Kai Hauser - 2002 - Philosophia Mathematica 10 (3):257-285.
    I examine various claims to the effect that Cantor's Continuum Hypothesis and other problems of higher set theory are ill-posed questions. The analysis takes into account the viability of the underlying philosophical views and recent mathematical developments.
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  37. Is an Unpictorial Mathematical Platonism Possible?Charles W. Sayward - 2002 - Journal of Philosophical Research 27:201-214.
    In his book Wittgenstein on the Foundations of Mathematics, Crispin Wright notes that remarkably little has been done to provide an unpictorial, substantial account of what mathematical platonism comes to. Wright proposes to investigate whether there is not some more substantial doctrine than the familiar images underpinning the platonist view. He begins with the suggestion that the essential platonist claim is that mathematical truth is objective. Although he does not demarcate them as such, Wright proposes several different tests for objectivity. (...)
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  38. A Theory of Mathematical Correctness and Mathematical Truth.Mark Balaguer - 2001 - Pacific Philosophical Quarterly 82 (2):87–114.
    A theory of objective mathematical correctness is developed. The theory is consistent with both mathematical realism and mathematical anti-realism, and versions of realism and anti-realism are developed that dovetail with the theory of correctness. It is argued that these are the best versions of realism and anti-realism and that the theory of correctness behind them is true. Along the way, it is shown that, contrary to the traditional wisdom, the question of whether undecidable sentences like the continuum hypothesis have objectively (...)
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  39. Beyond the Axioms: The Question of Objectivity in Mathematics.W. W. Tait - 2001 - 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. a matter (...)
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  40. Last Bastion of Reason. [REVIEW]James Franklin - 2000 - New Criterion 18 (9):74-78.
    Attacks the irrationalism of Lakatos's Proofs and Refutations and defends mathematics as a "last bastion" of reason against postmodernist and deconstructionist currents.
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  41. Proof-Theoretic Reduction as a Philosopher's Tool.Thomas Hofweber - 2000 - Erkenntnis 53 (1-2):127-146.
    Hilbert’s program in the philosophy of mathematics comes in two parts. One part is a technical part. To carry out this part of the program one has to prove a certain technical result. The other part of the program is a philosophical part. It is concerned with philosophical questions that are the real aim of the program. To carry out this part one, basically, has to show why the technical part answers the philosophical questions one wanted to have answered. Hilbert (...)
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  42. Reality and Truth in Mathematics.M. Beeson - 1998 - Philosophia Mathematica 6 (2):131-168.
    Brouwer's positions about existence (reality) and truth are examined in the light of ninety years of scientific progress. Relevant results in proof theory, recursion theory, set theory, relativity, and quantum mechanics are used to cast light on the following philosophical questions: What is real, and how do we know it? What does it mean to say a thing exists? Can things exist that we can't know about? Can things exist that we don't know how to find? What does it mean (...)
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  43. Mathematical Objectivity and Mathematical Objects.Hartry Field - 1998 - In S. Laurence C. MacDonald (ed.), Contemporary Readings in the Foundations of Metaphysics. Blackwell.
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  44. Wittgenstein's Anti-Platonism.Sílvio Pinto - 1998 - Grazer Philosophische Studien 56 (1):109-132.
    The philosophy of mathematics of the later Wittgenstein is normally not taken very seriously. According to a popular objection, it cannot account for mathematical necessity. Other critics have dismissed Wittgenstein's approach on the grounds that his anti-platonism is unable to explain mathematical objectivity. This latter objection would be endorsed by somebody who agreed with Paul Benacerraf that any anti-platonistic view fails to describe mathematical truth. This paper focuses on the problem proposed by Benacerraf of reconciling the semantics with the epistemology (...)
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  45. Classical Arithmetic as Part of Intuitionistic Arithmetic.Michael Potter - 1998 - Grazer Philosophische Studien 55 (1):127-41.
    Argues that classical arithmetic can be viewed as a proper part of intuitionistic arithmetic. Suggests that this largely neutralizes Dummett's argument for intuitionism in the case of arithmetic.
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  46. The Metalinguistic Perspective in Mathematics.Michael Potter - 1996 - Acta Analytica 11:79-86.
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  47. On Saying What You Really Want to Say: Wittgenstein, Gödel and the Trisection of the Angle.Juliet Floyd - 1995 - In Jaakko Hintikka (ed.), From Dedekind to Gödel: The Foundations of Mathematics in the Early Twentieth Century, Synthese Library Vol. 251 (Kluwer Academic Publishers. pp. 373-426.
  48. Poincaré's Conception of the Objectivity of Mathematics.Janet Folina - 1994 - Philosophia Mathematica 2 (3):202-227.
    There is a basic division in the philosophy of mathematics between realist, ‘platonist’ theories and anti-realist ‘constructivist’ theories. Platonism explains how mathematical truth is strongly objective, but it does this at the cost of invoking mind-independent mathematical objects. In contrast, constructivism avoids mind-independent mathematical objects, but the cost tends to be a weakened conception of mathematical truth. Neither alternative seems ideal. The purpose of this paper is to show that in the philosophical writings of Henri Poincaré there is a coherent (...)
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  49. 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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  50. Mathematical Relativism.Philip Hugly & Charles Sayward - 1989 - History and Philosophy of Logic 10 (1):53-65.
    We set out a doctrine about truth for the statements of mathematics?a doctrine which we think is a worthy competitor to realist views in the philosophy of mathematics?and argue that this doctrine, which we shall call ?mathematical relativism?, withstands objections better than do other non-realist accounts.
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