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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. 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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  2. Paul Benacerraf (1973). Mathematical Truth. Journal of Philosophy 70 (19):661-679.
  3. Giovanni Boniolo & Silvio Valentini (2012). Objects: A Study in Kantian Formal Epistemology. Notre Dame Journal of Formal Logic 53 (4):457-478.
    We propose a formal representation of objects , those being mathematical or empirical objects. The powerful framework inside which we represent them in a unique and coherent way is grounded, on the formal side, in a logical approach with a direct mathematical semantics in the well-established field of constructive topology, and, on the philosophical side, in a neo-Kantian perspective emphasizing the knowing subject’s role, which is constructive for the mathematical objects and constitutive for the empirical ones.
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  4. 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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  5. Harold Chapman Brown (1914). Concepts and Existence. Journal of Philosophy, Psychology and Scientific Methods 11 (13):355-357.
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  6. Charles Castonguay (1972). Meaning and Existence in Mathematics. New York,Springer-Verlag.
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  7. Charles Castonguay, Meaning and Existence in Mathematics : On the Use and Abuse of the Theory of Models in the Philosophy of Mathematics.
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  8. Hartry Field (1980). Science Without Numbers. Princeton University Press.
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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. P. T. Geach (1956). On Frege's Way Out. Mind 65 (259):408-409.
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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.
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  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. Alison Gopnik (1997). Words, Thoughts, and Theories. Mit Press.
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  20. Adam Grove (1988). Two Modellings for Theory Change. Journal of Philosophical Logic 17 (2):157-170.
  21. Andrzej Grzegorczyk (1964). A Note on the Theory of Propositional Types. Fundamenta Mathematicae 54:27-29.
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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. Sven Ove Hansson, Logic of Belief Revision. Stanford Encyclopedia of Philosophy.
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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. Hans G. Herzberger (1970). Paradoxes of Grounding in Semantics. Journal of Philosophy 67 (6):145-167.
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  29. Henry Hiz (1977). Descriptions in Russell's Theory and Ontology. Studia Logica 36 (4):271-283.
  30. Laurence R. Horn, Contradiction. Stanford Encyclopedia of Philosophy.
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  31. Colin Howson (2000). Induction and the Justification of Belief. Oxford University Press, Inc..
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  32. Paul Hoyningen-Huene (1993). Reconstructing Scientific Revolutions: Thomas S. Kuhn's Philosophy of Science. University of Chicago Press.
    Few philosophers of science have influenced as many readers as Thomas S. Kuhn. Yet no comprehensive study of his ideas has existed--until now. In this volume, Paul Hoyningen-Huene examines Kuhn's work over four decades, from the days before The Structure of Scientific Revolutions to the present, and puts Kuhn's philosophical development in a historical framework. Scholars from disciplines as diverse as political science and art history have offered widely differing interpretations of Kuhn's ideas, appropriating his notions of paradigm shifts and (...)
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  33. Franz Huber, Confirmation and Induction. Internet Encyclopedia of Philosophy.
  34. Edward Hussey (1991). Aristotle on Mathematical Objects. Apeiron 24 (4):105 - 133.
  35. Arata Ishimoto (1977). A Propositional Fragment of Leśniewski's Ontology. Studia Logica 36 (4):285-299.
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  36. Bogusław Iwanuś (1973). On Leśniewski's Elementary Ontology. Studia Logica 31 (1):73 - 125.
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  37. R. J. (1994). [Z Nowości Zagranicznych] Zagadnienia Filozoficzne W Matematyce J.M. Folina, Poincaré and the Philosophy of Mathematics, 1992. K. Jacobs, Invitation to Mathematics, 1992. D. M. Davis, The Nature and Power of Mathematics, 1993. G. Hellman, Mathemati. [REVIEW] Zagadnienia Filozoficzne W Nauce 16.
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  38. Stanisław Jaśkowski (1969). Propositional Calculus for Contradictory Deductive Systems. Studia Logica 24 (1):143 - 160.
  39. Richard Jeffrey (1983). The Logic of Decision. University of Chicago Press.
    "Frederic Schick, Journal of Philosophy This book uses elementary logical and mathematical means to philosophical end: elucidation of the notions of subjective ...
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  40. W. E. Johnson (1932). Probability: The Deductive and Inductive Problems. Mind 41 (164):409-423.
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  41. Cory F. Juhl (1994). The Speed-Optimality of Reichenbach's Straight Rule of Induction. British Journal for the Philosophy of Science 45 (3):857-863.
    , Hans Reichenbach made a bold and original attempt to ‘vindicate’ induction. He proposed a rule, the ‘straight rule’ of induction, which would guarantee inductive success if any rule of induction would. A central problem facing his attempt to vindicate the straight rule is that too many other rules are just as good as the straight rule if our only constraint on what counts as ‘success’ for an inductive rule is that it is ‘asymptotic’, i.e. that it converges in the (...)
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  42. John Kearns (1973). Review: The Logical Systems of Le'sniewski} by E. Luschei. [REVIEW] Journal of Symbolic Logic 38 (1):147-148.
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  43. John Kearns (1969). Two Views of Variables. Notre Dame Journal of Formal Logic 10 (2):163-180.
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  44. John Kearns (1967). The Contribution of Le'sniewski. Notre Dame Journal of Formal Logic 8 (1-2):61-93.
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  45. John Kearns (1962). Lesniewski, Language, and Logic. Dissertation, Yale University
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  46. Stephen Cole Kleene (1952). Introduction to Metamathematics. North Holland.
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  47. Penelope Maddy (1980). Perception and Mathematical Intuition. Philosophical Review 89 (2):163-196.
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  48. Roman Murawski (2011). Mathematical Objects and Mathematical Knowledge. Grazer Philosophische Studien 52:257-259.
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  49. John E. Nolt (1983). Sets and Possible Worlds. Philosophical Studies 44 (1):21-35.
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  50. Roberto Poli & Massimo Libardi (1999). Logic, Theory of Science, and Metaphysics According to Stanislaw Lesniewski. Grazer Philosophische Studien 57:183-219.
    Due to the current availability of the English translation of almost all of Lesniewski's works it is now possible to give a clear and detailed picture of his ideas. Lesniewski's system of the foundation of mathematics is discussed. In abrief ouüine of his three systems Mereology, Ontology and Protothetics his positions conceming the problems of the forms of expression, proper names, synonymity, analytic and synthetic propositions, existential propositions, the concept of logic, and his views of theory of science and metaphysics (...)
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