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  1. Alex A. B. Aspeitia, Internalism and Externalism in the Foundations of Mathematics.
    Without a doubt, one of the main reasons Platonsim remains such a strong contender in the Foundations of Mathematics debate is because of the prima facie plausibility of the claim that objectivity needs objects. It seems like nothing else but the existence of external referents for the terms of our mathematical theories and calculations can guarantee the objectivity of our mathematical knowledge. The reason why Frege – and most Platonists ever since – could not adhere to the idea that mathematical (...)
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  2. Mark Balaguer (2001). A Theory of Mathematical Correctness and Mathematical Truth. Pacific Philosophical Quarterly 82 (2):87–114.
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  3. M. Beeson (1998). Reality and Truth in Mathematics. 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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  4. T. Button & P. Smith (2012). The Philosophical Significance of Tennenbaum's Theorem. 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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  5. Justin Clarke-Doane, Moral Realism and Mathematical Realism.
    Ethics and mathematics are normally treated independently in philosophical discussions. When comparisons are drawn between problems in the two areas, those comparisons tend to be highly local, concerning just one or two issues. Nevertheless, certain metaethicists have made bold claims to the effect that moral realism is on “no worse footing” than mathematical realism -- i.e. that one cannot reasonably reject moral realism without also rejecting mathematical realism. -/- In the absence of any remotely systematic survey of the relevant arguments, (...)
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  6. Justin Clarke-Doane (2014). Moral Epistemology: The Mathematics Analogy. 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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  7. Justin Clarke-Doane (2013). What is Absolute Undecidability?†. 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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  8. Justin Clarke-Doane, Flawless Disagreement in Mathematics.
    A disagrees with B with respect to a proposition, p, flawlessly just in case A believes p and B believes not-p, or vice versa, though neither A nor B is guilty of a cognitive shortcoming – i.e. roughly, neither A nor B is being irrational, lacking evidence relevant to p, conceptually incompetent, insufficiently imaginative, etc.
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  9. Julian C. Cole (2013). Towards an Institutional Account of the Objectivity, Necessity, and Atemporality of Mathematics. 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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  10. Keith Devlin (2008). A Mathematician Reflects on the Useful and Reliable Illusion of Reality in Mathematics. 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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  11. J. M. Dieterle (2010). Social Construction in the Philosophy of Mathematics: A Critical Evaluation of Julian Cole's Theory. 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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  12. Hartry Field (1998). Mathematical Objectivity and Mathematical Objects. In S. Laurence C. MacDonald (ed.), Contemporary Readings in the Foundations of Metaphysics. Basil Blackwell.
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  13. Juliet Floyd (1995). On Saying What You Really Want to Say: Wittgenstein, Gödel and the Trisection of the Angle. In Jaakko Hintikka (ed.), From Dedekind to Gödel: The Foundations of Mathematics in the Early Twentieth Century, Synthese Library Vol. 251 (Kluwer Academic Publishers. 373-426.
  14. Janet Folina (1994). Poincaré's Conception of the Objectivity of Mathematics. 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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  15. Pieranna Garavaso (1992). The Argument From Agreement and Mathematical Realism. 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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  16. Han Geurdes, The Construction of Transfinite Equivalence Algorithms.
    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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  17. Kai Hauser (2002). Is Cantor's Continuum Problem Inherently Vague? 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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  18. Thomas Hofweber (2000). Proof-Theoretic Reduction as a Philosopher's Tool. 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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  19. L. Horsten (2012). Vom Zahlen Zu den Zahlen: On the Relation Between Computation and Arithmetical Structuralism. 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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  20. Philip Hugly & Charles Sayward (1989). Mathematical Relativism. History and Philosophy of Logic 10:53-65.
    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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  21. Philip Hugly & Charles Sayward (1987). Relativism and Ontology. Philosophical Quarterly 37 (148):278-290.
    This paper deals with the question of whether there is objectivist truth about set-theoretic matters. The dogmatist and skeptic agree that there is such truth. They disagree about whether this truth is knowable. In contrast, the relativist says there is no objective truth to be known. Two versions of relativism are distinguished in the paper. One of these versions is defended.
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  22. 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). New Perspectives on the Philosophy of Paul Benacerraf: Truth, Objects, Infinity (Fabrice Pataut, Editor). Springer.
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  23. Donald A. Martin (2005). Gödel's Conceptual Realism. Bulletin of Symbolic Logic 11 (2):207-224.
  24. Charles Parsons (2010). Gödel and Philosophical Idealism. 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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  25. Nikolaj Jang Lee Linding Pedersen & Marcus Rossberg (2010). Open-Endedness, Schemas and Ontological Commitment. 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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  26. Nikolaj Jang Lee Linding Pedersen & Marcus Rossberg (2007). McGee on Open-Ended Schemas. 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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  27. Alejandro Pérez Carballo (2014). Structuring Logical Space. Philosophy and Phenomenological Research 89 (2).
  28. Sílvio Pinto (1998). Wittgenstein's Anti-Platonism. Grazer Philosophische Studien 56:109-132.
    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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  29. Michael Potter (1998). Classical Arithmetic as Part of Intuitionistic Arithmetic. Grazer Philosophische Studien 55:127-41.
    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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  30. Michael Potter (1993). The Metalinguistic Perspective in Mathematics. Acta Analytica 11:79-86.
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  31. Hilary Putnam (1975). What is Mathematical Truth? In Mathematics, Matter and Method. Cambridge University Press. 60--78.
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  32. Charles Sayward (2002). Is an Unpictorial Mathematical Platonism Possible? Journal of Philosophical Research 27:199-212.
    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 platoninism 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 element in the platonist claim is that mathematical truth is objective. Although he does not demarcate them as such, Wright proposes several different (...)
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  33. Stewart Shapiro (2007). The Objectivity of Mathematics. 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. W. W. Tait (2001). Beyond the Axioms: The Question of Objectivity in Mathematics. Philosophia Mathematica 9 (1):21-36.
    This paper contains a defense against anti-realism in mathematics in the light both of incompleteness and of the fact that mathematics is a ‘cultural artifact.’. Anti-realism (here) is the view that theorems, say, of aritltmetic cannot be taken at face value to express true propositions about the system of numbers but must be reconstrued to be about somctliiiig else or about nothing at all. A ‘bite-the-bullet’ aspect of the defease is that, adopting new axioms, liitherto independent, is not. a matter (...)
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  35. Timm Triplett (1986). Relativism and the Sociology of Mathematics: Remarks on Bloor, Flew, and Frege. Inquiry 29 (1-4):439-450.
    Antony Flew's ?A Strong Programme for the Sociology of Belief (Inquiry 25 {1982], 365?78) critically assesses the strong programme in the sociology of knowledge defended in David Bloor's Knowledge and Social Imagery. I argue that Flew's rejection of the epistemological relativism evident in Bloor's work begs the question against the relativist and ignores Bloor's focus on the social relativity of mathematical knowledge. Bloor attempts to establish such relativity via a sociological analysis of Frege's theory of number. But this analysis only (...)
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  36. Mark van Atten (2003). Review of C. O. Hill and G. E. Rosado Haddock, Husserl or Frege? Meaning, Objectivity, and Mathematics. [REVIEW] Philosophia Mathematica 11 (2):241-244.
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