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Ontology of Mathematics

Edited by Rafal Urbaniak (University of Ghent, University of Gdansk)
Assistant editors: Sam Roberts, Pawel Pawlowski
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.), Journal of Symbolic Logic. New York, Springer 1--9.
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  4. Isaac Barrow (1975). The Usefulness of Mathematical Learning Explained and Demonstrated Being Mathematical Lectures Read in the Public Schools of Cambridge. Printed for S. Austen.
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  5. Adam Beck (1998). Mathematics, Science, and Postclassical Theory. [REVIEW] Radical Philosophy 91.
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  6. 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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  7. Jean-Pierre Belna (1996). La Notion de Nombre Chez Dedekind, Cantor, Frege Th'eories, Conceptions Et Philosophie. Monograph Collection (Matt - Pseudo).
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  8. George Berkeley & Douglas Michael Jesseph (1992). De Motu ; and, the Analyst.
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  9. S. Berry (2015). The Construction of Logical Space, by Augustin Rayo. Mind 124 (496):1375-1379.
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  10. David Bostock (1981). Logic and Arithmetic, Vol. II--Rational and Irrational Numbers. Mind 90 (359):473-475.
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  11. Harold Chapman Brown (1914). Concepts and Existence. Journal of Philosophy, Psychology and Scientific Methods 11 (13):355-357.
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  12. Tim Button & Sean Walsh (2016). Structure and Categoricity: Determinacy of Reference and Truth Value in the Philosophy of Mathematics. Philosophia Mathematica 24 (3):283-307.
    This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.
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  13. 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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  14. Charles Castonguay (1972). Meaning and Existence in Mathematics. New York,Springer-Verlag.
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  15. Charles Ernest Castonguay (1971). Meaning and Existence in Mathematics: On the Use and Abuse of the Theory of Models in the Philosophy of Mathematics. Dissertation, Mcgill University (Canada)
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  16. Peter Clark (2009). Mathematical Entities. In Robin Le Poidevin (ed.), The Routledge Companion to Metaphysics. Routledge
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  17. Nino Cocchiarella (1992). Realism, Mathematics and Modality. [REVIEW] International Studies in Philosophy 24 (3):139-141.
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  18. John Corcoran (1978). Corcoran Recommends Hambourger on the Frege-Russell Number Definition. MATHEMATICAL REVIEWS 56.
    It is widely agreed by philosophers that the so-called “Frege-Russell definition of natural number” is actually an assertion concerning the nature of the numbers and that it cannot be regarded as a definition in the ordinary mathematical sense. On the basis of the reasoning in this paper it is clear that the Frege-Russell definition contradicts the following three principles (taken together): (1) each number is the same entity in each possible world, (2) each number exists in each possible world, (3) (...)
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  19. John Corcoran (1972). Reid, Constance. Hilbert (a Biography). Reviewed by Corcoran in Philosophy of Science 39 (1972), 106–08. Philosophy of Science 39 (1):106-108.
    Reid, Constance. Hilbert (a Biography). Reviewed by Corcoran in Philosophy of Science 39 (1972), 106–08. -/- Constance Reid was an insider of the Berkeley-Stanford logic circle. Her San Francisco home was in Ashbury Heights near the homes of logicians such as Dana Scott and John Corcoran. Her sister Julia Robinson was one of the top mathematical logicians of her generation, as was Julia’s husband Raphael Robinson for whom Robinson Arithmetic was named. Julia was a Tarski PhD and, in recognition of (...)
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  20. Anthony Moncrief Coyne (1974). Mathematical Truth. Dissertation, The University of North Carolina at Chapel Hill
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  21. Laura Jacobs Cunningham (1988). Numbers and Expressions. Dissertation, City University of New York
    The objective of this dissertation is to determine whether a formalist interpretation of classical mathematics is tenable. We first argue that the best theories of linguistics and mathematics characterize both linguistic objects and mathematical objects as abstract. This eliminates one objection to a formalist construal of mathematics. These results are interesting in themselves, since they address and resolve a problem largely ignored by formalists: the ontological status of expressions. ;A second objection to formalism stems from Godel's work. He demonstrated that (...)
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  22. Keith J. Devlin (2000). The Math Gene How Mathematical Thinking Evolved and Why Numbers Are Like Gossip.
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  23. Pierre Dugac (1976). Richard Dedekind Et les Fondements des Mathématiques Avec de Nombreux Textes Inédits. J. Vrin.
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  24. F. Y. Edgeworth (1881). Mathematical Psychics. Mind 6 (24):581-583.
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  25. Gottlob Frege (1953). The Foundations of Arithmetic a Logico-Mathematical Enquiry Into the Concept of Number. English Translation by J.L. Austin. [REVIEW]
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  26. Gottlob Frege (1879). Begriffsschrift: Eine Der Arithmetische Nachgebildete Formelsprache des Reinen Denkens. L. Nebert.
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  27. Greg Fried (2016). A Challenge to Divine Psychologism. Theology and Science 14 (2):175-189.
    Alvin Plantinga proposes that mathematical objects and propositions are divine thoughts. This position, which I call divine psychologism, resonates with some remarks by contemporary thinkers. Plantinga claims several advantages for his position, and I add another: it helps to explain the glory of mathematics. But my main purpose is to issue a challenge to divine psychologism. I argue that it has an implausible consequence: it identifies an entity with God’s relation to that entity. I consider and rebut several ways in (...)
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  28. Haim Gaifman (2012). On Ontology and Realism in Mathematics. Review of Symbolic Logic 5 (3):480-512.
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  29. M. Giaquinto (1980). On Mathematical Realism.
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  30. Kurt Gödel (1944). Russell's Mathematical Logic. In Solomon Feferman, John Dawson & Stephen Kleene (eds.), Journal of Symbolic Logic. Northwestern University Press 119--141.
  31. 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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  32. 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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  33. Nelson Goodman (1961). About. Mind 70 (277):1-24.
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  34. Nelson Goodman (1951). The Structure of Appearance. Harvard University Press.
  35. Nelson Goodman (1946). A Query on Confirmation. Journal of Philosophy 43 (14):383-385.
  36. Nelson Goodman & Henry Leonard (1940). The Calculus of Individuals and its Uses. Journal of Symbolic Logic 5 (2):45-55.
  37. Robert Bates Graber (1989). Mathematical Naturalism. Southern Journal of Philosophy 27 (3):427-441.
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  38. Gg Granger (1988). The Concepts of Natural Mathematical Entities. Revue Internationale de Philosophie 42 (167):474-499.
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  39. Grattan-Guinness (1981). On the Development of Logic Between the Two World Wars. The American Mathematical Monthly 88 (7):495-509.
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  40. Thomas Greenwood (1954). Aristotle on Mathematical Constructibility. The Thomist 17:84.
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  41. Andrzej Grzegorczyk (1964). A Note on the Theory of Propositional Types. Fundamenta Mathematicae 54 (3):27-29.
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  42. Andrzej Grzegorczyk (1955). The Systems of Leśniewski in Relation to Contemporary Logical Research. Studia Logica 3 (1):77-95.
  43. John Joseph Guiniven (1975). Mathematical Ontology in Aristotle. Dissertation, University of Massachusetts, Amherst, Hampshire, Mount Holyoke and Smith Colleges
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  44. Anil Gupta (1982). Truth and Paradox. Journal of Philosophical Logic 11 (1):1-60.
  45. 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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  46. Richard John Hall (1963). Mathematical Truth. Dissertation, Princeton University
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  47. Brett K. Hayes & Susan P. Thompson (2007). Causal Relations and Feature Similarity in Children's Inductive Reasoning. Journal of Experimental Psychology 136:470-485.
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  48. Evan Heit (2000). Properties of Inductive Reasoning. Psychonomic Bulletin and Review 7:569-592.
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  49. Evan Heit & Joshua Rubinstein (1994). Similarity and Property Effects in Inductive Reasoning. Journal of Experimental Psychology 20:411-422.
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  50. Leon Henkin (1963). A Theory of Propositional Types. Fundamenta Mathematicae 52:323-334.
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