Results for 'mathematical objects'

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  1.  5
    Avicenna on the Nature of Mathematical Objects.Mohammad Saleh Zarepour - 2016 - Dialogue: Canadian Philosophical Review / Revue canadienne de philosophie 55 (3):511-536.
    Some authors have proposed that Avicenna considers mathematical objects, i.e., geometric shapes and numbers, to be mental existents completely separated from matter. In this paper, I will show that this description, though not completely wrong, is misleading. Avicenna endorses, I will argue, some sort of literalism, potentialism, and finitism.
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  2.  13
    Mathematical Objects and the Object of Theology.Victoria S. Harrison - forthcoming - Religious Studies:1-18.
    In what ways might God be like an abstract mathematical object, such as a number or a geometrical shape? The objects of mathematics are often regarded as having three key negative characteristics. They are unknowable by the senses, not located at any point in space-time, and are not involved in physical causal chains. God can also be thought of as possessing these three characteristics. Exploring this convergence between mathematical objects and the God of classical western theism (...)
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  3.  53
    Handling Mathematical Objects: Representations and Context.Jessica Carter - 2013 - 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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  4.  89
    Does the Existence of Mathematical Objects Make a Difference?A. Baker - 2003 - 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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  5.  2
    Mathematical Objects Arising From Equivalence Relations and Their Implementation in Quine's NF.Thomas Forster - 2016 - 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.  14
    Mathematical Objects Arising From Equivalence Relations and Their Implementation in Quine's NF.Thomas Forster - 2014 - 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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  7.  71
    Indefiniteness of Mathematical Objects.Ken Akiba - 2000 - 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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  8.  4
    Referring to Mathematical Objects Via Definite Descriptions.Stefan Buijsman - 2017 - Philosophia Mathematica 25 (1):128-138.
    Linsky and Zalta try to explain how we can refer to mathematical objects by saying that this happens through definite descriptions which may appeal to mathematical theories. I present two issues for their account. First, there is a problem of finding appropriate pre-conditions to reference, which are currently difficult to satisfy. Second, there is a problem of ensuring the stability of the resulting reference. Slight changes in the properties ascribed to a mathematical object can result in (...)
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  9.  58
    Gödel and 'the Objective Existence' of Mathematical Objects.Pierre Cassou-Noguès - 2005 - 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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  10.  36
    Aristotle's Critique of Platonist Mathematical Objects: Two Test Cases From Metaphysics M 2.Emily Katz - 2013 - 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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  11.  25
    Implementing Mathematical Objects in Set Theory.Thomas Forster - 2007 - 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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  12.  30
    How Are Mathematical Objects Constituted? A Structuralist Answer.Wolfgang Spohn - unknown
    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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  13.  9
    The Ontogenesis of Mathematical Objects.Barry Smith - 1975 - 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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  14. The Nature of Mathematical Objects.Øystein Linnebo - 2008 - In Bonnie Gold & Roger Simons (eds.), Proof and Other Dilemmas: Mathematics and Philosophy. Mathematical Association of America. pp. 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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  15. The Structuralist View of Mathematical Objects.Charles Parsons - 1990 - Synthese 84 (3):303 - 346.
  16. Syntax, Semantics, and the Problem of the Identity of Mathematical Objects.Gian-Carlo Rota, David H. Sharp & Robert Sokolowski - 1988 - 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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  17. What Constitutes the Numerical Diversity of Mathematical Objects?Fraser MacBride - 2006 - Analysis 66 (289):63–69.
  18. Why Do Mathematicians Need Different Ways of Presenting Mathematical Objects? The Case of Cayley Graphs.Irina Starikova - 2010 - 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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  19.  26
    Infons as Mathematical Objects.Keith J. Devlin - 1992 - 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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  20. Our Knowledge of Mathematical Objects.Kit Fine - 2005 - In T. Z. Gendler & J. Hawthorne (eds.), Oxford Studies in Epistemology. Clarendon Press. pp. 89-109.
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  21. The Conceptual Contingency of Mathematical Objects.Hartry Field - 1993 - Mind 102 (406):285-299.
  22.  85
    A Reductio Ad Surdum? Field on the Contingency of Mathematical Objects.Bob Hale & Crispin Wright - 1994 - Mind 103 (410):169-184.
  23. 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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  24.  58
    Burgess's `Scientific' Arguments for the Existence of Mathematical Objects.Charles S. Chihara - 2006 - 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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  25.  78
    Formal Systems as Physical Objects: A Physicalist Account of Mathematical Truth.E. Szabo´ La´Szlo´ - 2003 - 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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  26.  69
    A Gödelian Thesis Regarding Mathematical Objects: Do They Exist? And Can We Perceive Them?Charles S. Chihara - 1982 - Philosophical Review 91 (2):211-227.
  27.  13
    What is the Nature of Mathematical–Logical Objects?Stathis Livadas - 2017 - Axiomathes 27 (1):79-112.
    This article deals with a question of a most general, comprehensive and profound content as it is the nature of mathematical–logical objects insofar as these are considered objects of knowledge and more specifically objects of formal mathematical theories. As objects of formal theories they are dealt with in the sense they have acquired primarily from the beginnings of the systematic study of mathematical foundations in connection with logic dating from the works of G. (...)
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  28.  64
    Aristotle on Mathematical Objects.Edward Hussey - 1991 - Apeiron 24 (4):105 - 133.
  29.  36
    Chrysippus on Mathematical Objects.David G. Robertson - 2004 - Ancient Philosophy 24 (1):169-191.
  30.  58
    The Mode of Existence of Mathematical Objects.M. A. Rozov - 1989 - Philosophia Mathematica (2):105-111.
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  31.  24
    Michael D. Resnik (Ed.), Mathematical Objects and Mathematical Knowledge.Jan Woleński - 1998 - Erkenntnis 48 (1):129-131.
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  32.  9
    What Constitutes the Numerical Diversity of Mathematical Objects?F. MacBride - 2006 - Analysis 66 (1):63-69.
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  33. Mathematical Creativity and the Character of Mathematical Objects.Michael Otte - 1999 - Logique Et Analyse 42 (167-168):387-410.
     
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  34.  39
    Mathematical Objects and Mathematical Knowledge.Jan Woleński - 1998 - Erkenntnis 48 (1).
  35.  14
    On Some Academic Theories of Mathematical Objects.Ian Mueller - 1986 - Journal of Hellenic Studies 106:111-120.
  36.  15
    Mathematical Objects and Mathematical Knowledge.Roman Murawski - 1996 - Grazer Philosophische Studien 52:257-259.
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  37.  29
    Burgess's ‘Scientific’ Arguments for the Existence of Mathematical Objects.Chihara Charles - 2006 - 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.  12
    Vera Entia : The Nature of Mathematical Objects in Descartes.Gregory Brown - 1980 - Journal of the History of Philosophy 18 (1):23-37.
  39. The Reality of Mathematical Objects.Gideon Rosen - 2011 - In John Polkinghorne (ed.), Meaning in Mathematics. Oxford University Press.
     
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  40.  2
    Aristotle’s Critique of Platonist Mathematical Objects: Two Test Cases From Metaphysics M 2.Emily Katz - 2013 - Apeiron 46 (1).
  41.  5
    Mathematical Objects and Mathematical Knowledge.Fabrice Pataut - unknown
  42.  6
    Access to Mathematical Objects.Keith Hossack - 1991 - Critica 23 (68):157 - 181.
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  43.  4
    Review of M. D. Resnik (Ed.), Mathematical Objects and Mathematical Knowledge[REVIEW]Roman Murawski - 1996 - Grazer Philosophische Studien 52:257-259.
  44.  1
    Review: Joseph Ullian, Bernard Baumrin, Mathematical Objects; Joseph S. Ullian, Is Any Set Theory True? [REVIEW]Alonzo Church - 1975 - Journal of Symbolic Logic 40 (4):593-595.
  45.  1
    Mathematical Objects.Joseph Ullian, Bernard Baumrin & Joseph S. Ullian - 1975 - Journal of Symbolic Logic 40 (4):593-595.
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  46. Indispensibility and the Multiple Reducibility of Mathematical Objects.Alan Baker - manuscript
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  47. 3 Mathematical Objects and Identity.Patricia Blanchette - 2007 - In Michael O'Rourke Corey Washington (ed.), Situating Semantics: Essays on the Philosophy of John Perry. pp. 73.
     
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  48. Ullian Joseph. Mathematical Objects. Philosophy of Science, The Delaware Seminar, Volume 1, 1961-1962, Edited by Baumrin Bernard, Interscience Publishers, New York and London 1963, Pp. 187–205. [REVIEW]Alonzo Church - 1975 - Journal of Symbolic Logic 40 (4):593-595.
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  49. Do We Need Mathematical Objects[REVIEW]Donald A. Gillies - 1992 - British Journal for the Philosophy of Science 43 (2):263-278.
  50. Access to Mathematical Objects.Keith Hossack - 1991 - Crítica: Revista Hispanoamericana de Filosofía 23 (68):157-181.
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