Search results for 'mathematical objects' (try it on Scholar)

  1.  35
    Jessica Carter (2013). Handling Mathematical Objects: Representations and Context. Synthese 190 (17):3983-3999.
    This article takes as a starting point the current popular anti realist position, Fictionalism, with the intent to compare it with actual mathematical practice. Fictionalism claims that mathematical statements do purport to be about mathematical objects, and that mathematical statements are not true. Considering these claims in the light of mathematical practice leads to questions about how mathematical objects are handled, and how we prove that certain statements hold. Based on a case (...)
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  2.  67
    A. Baker (2003). Does the Existence of Mathematical Objects Make a Difference? Australasian Journal of Philosophy 81 (2):246 – 264.
    In this paper I examine a strategy which aims to bypass the technicalities of the indispensability debate and to offer a direct route to nominalism. The starting-point for this alternative nominalist strategy is the claim that--according to the platonist picture--the existence of mathematical objects makes no difference to the concrete, physical world. My principal goal is to show that the 'Makes No Difference' (MND) Argument does not succeed in undermining platonism. The basic reason why not is that the (...)
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  3.  13
    Thomas Forster (2014). Mathematical Objects Arising From Equivalence Relations and Their Implementation in Quine's NF. Philosophia Mathematica 24 (1):nku005.
    Many mathematical objects arise from equivalence classes and invite implementation as those classes. Set-existence principles that would enable this are incompatible with ZFC's unrestricted aussonderung but there are set theories which admit more instances than does ZF. NF provides equivalence classes for stratified relations only. Church's construction provides equivalence classes for “low” sets, and thus, for example, a set of all ordinals. However, that set has an ordinal in turn which is not a member of the set constructed; (...)
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  4.  52
    Ken Akiba (2000). Indefiniteness of Mathematical Objects. Philosophia Mathematica 8 (1):26--46.
    The view that mathematical objects are indefinite in nature is presented and defended, hi the first section, Field's argument for fictionalism, given in response to Benacerraf's problem of identification, is closely examined, and it is contended that platonists can solve the problem equally well if they take the view that mathematical objects are indefinite. In the second section, two general arguments against the intelligibility of objectual indefiniteness are shown erroneous, hi the final section, the view is (...)
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  5.  1
    Thomas Forster (2016). Mathematical Objects Arising From Equivalence Relations and Their Implementation in Quine's NF. Philosophia Mathematica 24 (1):50-59.
    Many mathematical objects arise from equivalence classes and invite implementation as those classes. Set-existence principles that would enable this are incompatible with ZFC's unrestricted aussonderung but there are set theories which admit more instances than does ZF. NF provides equivalence classes for stratified relations only. Church's construction provides equivalence classes for “low” sets, and thus, for example, a set of all ordinals. However, that set has an ordinal in turn which is not a member of the set constructed; (...)
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  6.  54
    Pierre Cassou-Noguès (2005). Gödel and 'the Objective Existence' of Mathematical Objects. History and Philosophy of Logic 26 (3):211-228.
    This paper is a discussion of Gödel's arguments for a Platonistic conception of mathematical objects. I review the arguments that Gödel offers in different papers, and compare them to unpublished material (from Gödel's Nachlass). My claim is that Gödel's later arguments simply intend to establish that mathematical knowledge cannot be accounted for by a reflexive analysis of our mental acts. In other words, there is at the basis of mathematics some data whose constitution cannot be explained by (...)
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  7.  30
    Emily Katz (2013). Aristotle's Critique of Platonist Mathematical Objects: Two Test Cases From Metaphysics M 2. Apeiron 46 (1):26-47.
    Books M and N of Aristotle's Metaphysics receive relatively little careful attention. Even scholars who give detailed analyses of the arguments in M-N dismiss many of them as hopelessly flawed and biased, and find Aristotle's critique to be riddled with mistakes and question-begging. This assessment of the quality of Aristotle's critique of his predecessors (and of the Platonists in particular), is widespread. The series of arguments in M 2 (1077a14-b11) that targets separate mathematical objects is the subject of (...)
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  8.  17
    Thomas Forster (2007). Implementing Mathematical Objects in Set Theory. Logique Et Analyse 50 (197):79-86.
    In general little thought is given to the general question of how to implement mathematical objects in set theory. It is clear that—at various times in the past—people have gone to considerable lengths to devise implementations with nice properties. There is a litera- ture on the evolution of the Wiener-Kuratowski ordered pair, and a discussion by Quine of the merits of an ordered-pair implemen- tation that makes every set an ordered pair. The implementation of ordinals as Von Neumann (...)
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  9.  5
    Barry Smith (1975). Ontogenesis of Mathematical Objects. Journal of the British Society for Phenomenology 6 (2):91-101.
    Mathematical objects are divided into (1) those which are autonomous, i.e., not dependent for their existence upon mathematicians’ conscious acts, and (2) intentional objects, which are so dependent. Platonist philosophy of mathematics argues that all objects belong to group (1), Brouwer’s intuitionism argues that all belong to group (2). Here we attempt to develop a dualist ontology of mathematics (implicit in the work of, e.g., Hilbert), exploiting the theories of Meinong, Husserl and Ingarden on the relations (...)
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  10.  28
    Wolfgang Spohn, How Are Mathematical Objects Constituted? A Structuralist Answer.
    The paper proposes to amend structuralism in mathematics by saying what places in a structure and thus mathematical objects are. They are the objects of the canonical system realizing a categorical structure, where that canonical system is a minimal system in a specific essentialistic sense. It would thus be a basic ontological axiom that such a canonical system always exists. This way of conceiving mathematical objects is underscored by a defense of an essentialistic version of (...)
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  11.  79
    Charles Parsons (2008). Mathematical Thought and its Objects. Cambridge University Press.
    In Mathematical Thought and Its Objects, Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of (...)
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  12. Øystein Linnebo (2008). The Nature of Mathematical Objects. In Bonnie Gold & Roger Simons (eds.), Proof and Other Dilemmas: Mathematics and Philosophy. Mathematical Association of America 205--219.
    On the face of it, platonism seems very far removed from the scientific world view that dominates our age. Nevertheless many philosophers and mathematicians believe that modern mathematics requires some form of platonism. The defense of mathematical platonism that is both most direct and has been most influential in the analytic tradition in philosophy derives from the German logician-philosopher Gottlob Frege (1848-1925).2 I will therefore refer to it as Frege’s argument. This argument is part of the background of any (...)
     
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  13. Gian-Carlo Rota, David H. Sharp & Robert Sokolowski (1988). Syntax, Semantics, and the Problem of the Identity of Mathematical Objects. Philosophy of Science 55 (3):376-386.
    A plurality of axiomatic systems can be interpreted as referring to one and the same mathematical object. In this paper we examine the relationship between axiomatic systems and their models, the relationships among the various axiomatic systems that refer to the same model, and the role of an intelligent user of an axiomatic system. We ask whether these relationships and this role can themselves be formalized.
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  14.  89
    Irina Starikova (2010). Why Do Mathematicians Need Different Ways of Presenting Mathematical Objects? The Case of Cayley Graphs. Topoi 29 (1):41-51.
    This paper investigates the role of pictures in mathematics in the particular case of Cayley graphs—the graphic representations of groups. I shall argue that their principal function in that theory—to provide insight into the abstract structure of groups—is performed employing their visual aspect. I suggest that the application of a visual graph theory in the purely non-visual theory of groups resulted in a new effective approach in which pictures have an essential role. Cayley graphs were initially developed as exact (...) constructions. Therefore, they are legitimate components of the theory (combinatorial and geometric group theory) and the pictures of Cayley graphs are a part of practical mathematical procedures. (shrink)
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  15.  23
    Keith J. Devlin (1992). Infons as Mathematical Objects. Minds and Machines 2 (2):185-201.
    I argue that the role played by infons in the kind of mathematical theory of information being developed by several workers affiliated to CSLI is analogous to that of the various number systems in mathematics. In particular, I present a mathematical construction of infons in terms of representations and informational equivalences between them. The main theme of the paper arose from an electronic mail exchange with Pat Hayes of Xeroxparc. The exposition derives from a talk I gave at (...)
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  16. Fraser MacBride (2006). What Constitutes the Numerical Diversity of Mathematical Objects? Analysis 66 (289):63–69.
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  17. Charles Parsons (1990). The Structuralist View of Mathematical Objects. Synthese 84 (3):303 - 346.
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  18. Kit Fine (2005). Our Knowledge of Mathematical Objects. In T. Z. Gendler & J. Hawthorne (eds.), Oxford Studies in Epistemology. Clarendon Press 89-109.
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  19.  84
    Hartry Field (1993). The Conceptual Contingency of Mathematical Objects. Mind 102 (406):285-299.
  20.  53
    Charles S. Chihara (2006). Burgess's `Scientific' Arguments for the Existence of Mathematical Objects. Philosophia Mathematica 14 (3):318-337.
    This paper addresses John Burgess's answer to the ‘Benacerraf Problem’: How could we come justifiably to believe anything implying that there are numbers, given that it does not make sense to ascribe location or causal powers to numbers? Burgess responds that we should look at how mathematicians come to accept: There are prime numbers greater than 1010That, according to Burgess, is how one can come justifiably to believe something implying that there are numbers. This paper investigates what lies behind Burgess's (...)
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  21.  83
    Bob Hale & Crispin Wright (1994). A Reductio Ad Surdum? Field on the Contingency of Mathematical Objects. Mind 103 (410):169-184.
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  22. 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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  23.  5
    F. MacBride (2006). What Constitutes the Numerical Diversity of Mathematical Objects? Analysis 66 (1):63-69.
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  24.  50
    Edward Hussey (1991). Aristotle on Mathematical Objects. Apeiron 24 (4):105 - 133.
  25.  23
    Jan Woleński (1998). Michael D. Resnik (Ed.), Mathematical Objects and Mathematical Knowledge. Erkenntnis 48 (1):129-131.
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  26.  20
    David G. Robertson (2004). Chrysippus on Mathematical Objects. Ancient Philosophy 24 (1):169-191.
  27.  52
    M. A. Rozov (1989). The Mode of Existence of Mathematical Objects. Philosophia Mathematica (2):105-111.
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  28.  59
    Charles S. Chihara (1982). A Gödelian Thesis Regarding Mathematical Objects: Do They Exist? And Can We Perceive Them? Philosophical Review 91 (2):211-227.
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  29.  23
    László E. Szabó (2003). Formal Systems as Physical Objects: A Physicalist Account of Mathematical Truth. International Studies in the Philosophy of Science 17 (2):117 – 125.
    This article is a brief formulation of a radical thesis. We start with the formalist doctrine that mathematical objects have no meanings; we have marks and rules governing how these marks can be combined. That's all. Then I go further by arguing that the signs of a formal system of mathematics should be considered as physical objects, and the formal operations as physical processes. The rules of the formal operations are or can be expressed in terms of (...)
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  30.  14
    La´Szlo´ E. Szabo´ (2003). Formal Systems as Physical Objects: A Physicalist Account of Mathematical Truth. International Studies in the Philosophy of Science 17 (2):117-125.
    This article is a brief formulation of a radical thesis. We start with the formalist doctrine that mathematical objects have no meanings; we have marks and rules governing how these marks can be combined. That's all. Then I go further by arguing that the signs of a formal system of mathematics should be considered as physical objects, and the formal operations as physical processes. The rules of the formal operations are or can be expressed in terms of (...)
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  31.  15
    David G. Robertson (2004). Chrysippus on Mathematical Objects. Ancient Philosophy 24 (1):169-191.
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  32.  2
    Emily Katz (2013). Aristotle’s Critique of Platonist Mathematical Objects: Two Test Cases From Metaphysics M 2. Apeiron 46 (1).
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  33.  14
    Roman Murawski (2011). Mathematical Objects and Mathematical Knowledge. Grazer Philosophische Studien 52:257-259.
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  34.  39
    Jan Woleński (1998). Mathematical Objects and Mathematical Knowledge. Erkenntnis 48 (1).
  35.  11
    Ian Mueller (1986). On Some Academic Theories of Mathematical Objects. Journal of Hellenic Studies 106:111-120.
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  36. Michael Otte (1999). Mathematical Creativity and the Character of Mathematical Objects. Logique Et Analyse 42 (167-168):387-410.
     
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  37.  25
    Chihara Charles (2006). Burgess's ‘Scientific’ Arguments for the Existence of Mathematical Objects. Philosophia Mathematica 14 (3):318-337.
    This paper addresses John Burgess's answer to the ‘Benacerraf Problem’: How could we come justifiably to believe anything implying that there are numbers, given that it does not make sense to ascribe location or causal powers to numbers? Burgess responds that we should look at how mathematicians come to accept: There are prime numbers greater than 1010 That, according to Burgess, is how one can come justifiably to believe something implying that there are numbers. This paper investigates what lies behind (...)
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  38.  10
    Gregory Brown (1980). Vera Entia : The Nature of Mathematical Objects in Descartes. Journal of the History of Philosophy 18 (1):23-37.
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  39. Gideon Rosen (2011). The Reality of Mathematical Objects. In John Polkinghorne (ed.), Meaning in Mathematics. OUP Oxford
     
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  40.  5
    Fabrice Pataut, Mathematical Objects and Mathematical Knowledge.
  41.  1
    Alonzo Church (1975). Review: Joseph Ullian, Bernard Baumrin, Mathematical Objects; Joseph S. Ullian, Is Any Set Theory True? [REVIEW] Journal of Symbolic Logic 40 (4):593-595.
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  42.  5
    Keith Hossack (1991). Access to Mathematical Objects. Critica 23 (68):157 - 181.
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  43.  3
    Roman Murawski (1996). Review of M. D. Resnik (Ed.), Mathematical Objects and Mathematical Knowledge. [REVIEW] Grazer Philosophische Studien 52:257-259.
  44. Alan Baker, Indispensibility and the Multiple Reducibility of Mathematical Objects.
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  45. Patricia Blanchette (2007). 3 Mathematical Objects and Identity. In Michael O'Rourke Corey Washington (ed.), Situating Semantics: Essays on the Philosophy of John Perry. 73.
     
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  46. Donald A. Gillies (1992). Do We Need Mathematical Objects? [REVIEW] British Journal for the Philosophy of Science 43 (2):263-278.
     
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  47. R. Murawski (forthcoming). Michael D. Resnik (Ed.): Mathematical Objects and Mathematical Knowledge. Aldershot/Broockfield, USA/Singapore/Sydney: Dartmouth 1995. [REVIEW] Grazer Philosophische Studien.
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  48. Charles Parsons (2009). Mathematical Thought and its Objects. Cambridge University Press.
    Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a 'nature' than that confers on them. Parsons (...)
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  49. Yvonne Raley (2008). Jobless Objects: Mathematical Posits in Crisis. Protosociology 25:108-127.
    This paper focuses on an argument against the existence of mathematical objects called the “Makes No Difference Argument” . Roughly, MND claims that whether or not mathematical objects exist makes no difference, and that therefore, we have no reason to believe in them. The paper analyzes four different versions of MND for their merits. It concludes that the defender of the existence of mathematical objects does have some retorts to the first three versions of (...)
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  50.  1
    Michael D. Resnik (1995). Mathematical Objects and Mathematical Knowledge. Monograph Collection (Matt - Pseudo).
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