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Ontology of Mathematics

Edited by Rafal Urbaniak (University of Ghent, University of Gdansk)
Assistant editors: Pawel Pawlowski, Sam Roberts
About this topic
Summary Ontology of mathematics is concerned with the existence and nature of objects that mathematics is about. An important phenomenon in the field is the need of balancing between epistemological and ontological challenges. For instance, prima facie, the ontologically simplest option is to postulate the existence of abstract mathematical objects (like numbers or sets) to which mathematical terms refer. Yet, explaining how we, mundane beings, can have knowledge of such aspatial and atemporal objects, turns out to be quite difficult. The ontologically parsimonious alternative is to deny the existence of such objects. But then, one has to explain what it is that makes mathematical theories true (or at least, correct) and how we can come to know mathematical facts. Various positions arise from various ways of addressing questions of these two sorts. 
Key works Many crucial papers are included in the following anthologies: Benacerraf & Putnam 1983, Hart 1996 and Shapiro 2005.
Introductions A good introductory survey is Horsten 2008. A readable introduction to philosophy of mathematics is Shapiro 2000. A nice, albeit somewhat biased survey of ontological options can be found in the first few chapters of Chihara 1990. A very nice introduction to the development of foundations of mathematics and the interaction between foundations, epistemology and ontology of mathematics is Giaquinto 2002.
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Subcategories:History/traditions: Ontology of Mathematics
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  1. T. Arrigoni (2000). Realism in the Philosophy of Mathematics: A Critical Discussion. Rivista di Filosofia Neo-Scolastica 92 (3-4):627-646.
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  2. F. Barker Stephen (1969). 'Realism as a Philosophy of Mathematics. In Kurt Gödel, Jack J. Bulloff, Thomas C. Holyoke & Samuel Wilfred Hahn (eds.), Foundations of Mathematics. New York, Springer.
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  3. Stephen F. Barker (1969). Realism as a Philosophy of Mathematics. In Kurt Gödel, Jack J. Bulloff, Thomas C. Holyoke & Samuel Wilfred Hahn (eds.), Foundations of Mathematics. New York, Springer. 1--9.
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  4. John Bell, Dissenting Voices.
    Continuous entities are accordingly distinguished by the feature that—in principle at least— they can be divided indefinitely without altering their essential nature. So, for instance, the water in a bucket may be indefinitely halved and yet remain water. Aristotle nowhere to my knowledge defines discreteness as such but we may take the notion as signifying the opposite of continuity—that is, incapable of being indefinitely divided into parts. Thus discrete entities, typically, cannot be divided without effecting a change in their nature: (...)
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  5. Harold Chapman Brown (1914). Concepts and Existence. Journal of Philosophy, Psychology and Scientific Methods 11 (13):355-357.
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  6. Jessica Carter (2005). Motivations for Realism in the Light of Mathematical Practice. Croatian Journal of Philosophy 5 (1):17-29.
    The aim of this paper is to identify some of the motivations that can be found for taking a realist position concerning mathematical entities and to examine these motivations in the light of a case study in contemporary mathematics. The motivations that are found are as follows: (some) mathematicians are realists, mathematical statements are true, and finally, mathematical statements have a special certainty. These claims are compared with a result in algebraic topology stating that a certain sequence, the so-called Mayer-Vietoris (...)
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  7. Charles Castonguay (1972). Meaning and Existence in Mathematics. New York,Springer-Verlag.
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  8. Peter Clark (2009). Mathematical Entities. In Robin Le Poidevin (ed.), The Routledge Companion to Metaphysics. Routledge.
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  9. Gottlob Frege (1879/1997). Begriffsschrift: Eine Der Arithmetische Nachgebildete Formelsprache des Reinen Denkens. L. Nebert.
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  10. Haim Gaifman (2012). On Ontology and Realism in Mathematics. Review of Symbolic Logic 5 (3):480-512.
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  11. Kurt Gödel (1944). Russell's Mathematical Logic. In Solomon Feferman, John Dawson & Stephen Kleene (eds.), The Philosophy of Bertrand Russell. Northwestern University Press. 119--141.
  12. David Gooding (1992). The Procedural Turn; or, Why Do Thought Experiments Work? In R. Giere & H. Feigl (eds.), Cognitive Models of Science. University of Minnesota Press. 45-76.
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  13. Nelson Goodman (1983). Fact, Fiction, and Forecast. Harvard University Press.
    In his new foreword to this edition, Hilary Putnam forcefully rejects these nativist claims.
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  14. Nelson Goodman (1972). Seven Strictures on Similarity. In Problems and Projects. Bobs-Merril.
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  15. Nelson Goodman (1961). About. Mind 70 (277):1-24.
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  16. Nelson Goodman (1951). The Structure of Appearance. Harvard University Press.
  17. Nelson Goodman (1946). A Query on Confirmation. Journal of Philosophy 43 (14):383-385.
  18. Nelson Goodman & Henry Leonard (1940). The Calculus of Individuals and its Uses. Journal of Symbolic Logic 5 (2):45-55.
  19. Robert Bates Graber (1989). Mathematical Naturalism. Southern Journal of Philosophy 27 (3):427-441.
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  20. Gg Granger (1988). The Concepts of Natural Mathematical Entities. Revue Internationale de Philosophie 42 (167):474-499.
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  21. Andrzej Grzegorczyk (1964). A Note on the Theory of Propositional Types. Fundamenta Mathematicae 54 (3):27-29.
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  22. Andrzej Grzegorczyk (1955). The Systems of Leśniewski in Relation to Contemporary Logical Research. Studia Logica 3 (1):77-95.
  23. Anil Gupta (1982). Truth and Paradox. Journal of Philosophical Logic 11 (1):1-60.
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  24. Lindsey Hair (2006). OntOlOgy and Appearing: dOcumentary Realism as a Mathematical thOught. Cosmos and History: The Journal of Natural and Social Philosophy 2 (1-2):241-262.
    This paper exposes the relation between the different mathematical orientations, on the one hand, and the modes of documentary film on the other. When we take, with Badiou, mathematics as ontology, and mathematical orientations as orientations to Being, we find in the structural similarity of mathematics and documentary an equivalence: between modes of documentaryand mathematical-ontological decisions, regarding the inscription of 'what is'. From here we move to consider Badiou's notion of 'in-appearing' through a reading of Alain Resnais' documentary Night and (...)
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  25. Leon Henkin (1963). A Theory of Propositional Types. Fundamenta Mathematicae 52:323-334.
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  26. Leon Henkin (1950). Completeness in the Theory of Types. Journal of Symbolic Logic 15 (2):81-91.
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  27. Desmond Henry (1969). Le'sniewski's Ontology and Some Medieval Logicians. Notre Dame Journal of Formal Logic 10 (3):324-326.
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  28. Reuben Hersh (2008). Mathematical Practice as a Scientific Problem. In Bonnie Gold & Roger Simons (eds.), Proof and Other Dilemmas: Mathematics and Philosophy. Mathematical Association of America. 95--108.
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  29. Hans G. Herzberger (1970). Paradoxes of Grounding in Semantics. Journal of Philosophy 67 (6):145-167.
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  30. Henry Hiz (1977). Descriptions in Russell's Theory and Ontology. Studia Logica 36 (4):271-283.
  31. Laurence R. Horn, Contradiction. Stanford Encyclopedia of Philosophy.
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  32. Edward Hussey (1991). Aristotle on Mathematical Objects. Apeiron 24 (4):105 - 133.
  33. Caroline Jullien (2012). From the Languages of Art to Mathematical Languages, and Back Again. Enrahonar: Quaderns de Filosofía 49:91-106.
    Mathematics stand in a privileged relationship with aesthetics: a relationship that follows two main directions. The first concerns the introduction of mathematical considerations into aesthetic discourse. For instance, it is common to mention the mathematical architecture of certain artistic productions. The second leads from aesthetics to mathematics. In this case, the question is that of the role and meaning that aesthetic considerations may assume in mathematics. It is indeed a widely held view among mathematicians, of whatever socio-historical context, not only (...)
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  34. H. P. K. (1967). Problems in the Philosophy of Mathematics. Review of Metaphysics 21 (1):172-173.
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  35. H. P. K. (1967). Set Theory and the Continuum Hypothesis. Review of Metaphysics 20 (4):716-716.
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  36. L. O. Kattsoff (1973). On the Nature of Mathematical Entities. International Logic Review 7:29-45.
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  37. Tom Keagy (1994). A Case for Realism in Mathematics. The Monist 77 (3):329-344.
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  38. Bart Van Kerkhove & Jean Paul Van Bendegem (2005). Mathematical Practice and Naturalist Epistemology: Structures with Potential for Interaction. Philosophia Scientiae 9:61-78.
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  39. Cassius J. Keyser (1906). Mathematical Emancipations. The Monist 16 (1):65-83.
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  40. Cassius Jackson Keyser (1929). The Pastures of Wonder the Realm of Mathematics and the Realm of Science. Columbia University Press.
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  41. Cassius Jackson Keyser (1924). Mathematical Philosophy, a Study of Fate and Freedom Lectures for Educated Laymen. Dutton.
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  42. Lilianne Rivka Kfia (1994). Realism in Mathematics. Review of Metaphysics 47 (3):628-629.
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  43. Stephen Cole Kleene (1952). Introduction to Metamathematics. North Holland.
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  44. Virginia Klenk (1990). What Mathematical Truth Need Not Be. In J. Dunn & A. Gupta (eds.), Truth or Consequences. Kluwer. 197--208.
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  45. Vojtech Kolman (2011). How to Name a Real Number? Organon F 18 (3):283-301.
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  46. Ladislav Kvasz (2011). Matematika a Skutočnosť. Organon F 18 (3):302-330.
    The aim of the present paper is to offer a new analysis of the multifarious relations between mathematics and reality. We believe that the relation of mathematics to reality is, just like in the case of the natural sciences, mediated by instruments . Therefore the kind of realism we aim to develop for mathematics can be called instrumental realism. It is a kind of realism, because it is based on the thesis, that mathematics describes certain patterns of reality. And it (...)
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  47. Shaughan Lavine (1992). Realism in Mathematics by Penelope Maddy. Journal of Philosophy 89 (6):321-326.
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  48. Shaughan Lavine (1992). Review of Maddy, Realism in Mathematics. [REVIEW] Journal of Philosophy 89 (6):321-326.
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  49. David Lewis (1970). Nominalistic Set Theory. Noûs 4 (3):225-240.
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  50. O. Linnebo & R. Pettigrew (forthcoming). Two Types of Abstraction for Structuralism. Philosophical Quarterly.
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