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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. 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. Harold Chapman Brown (1914). Concepts and Existence. Journal of Philosophy, Psychology and Scientific Methods 11 (13):355-357.
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  3. Charles Castonguay (1972). Meaning and Existence in Mathematics. New York,Springer-Verlag.
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  4. 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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  5. Gottlob Frege (1879/1997). Begriffsschrift: Eine Der Arithmetische Nachgebildete Formelsprache des Reinen Denkens. L. Nebert.
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  6. 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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  7. 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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  8. 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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  9. Nelson Goodman (1972). Seven Strictures on Similarity. In Problems and Projects. Bobs-Merril.
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  10. Nelson Goodman (1961). About. Mind 70 (277):1-24.
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  11. Nelson Goodman (1951). The Structure of Appearance. Harvard University Press.
  12. Nelson Goodman (1946). A Query on Confirmation. Journal of Philosophy 43 (14):383-385.
  13. Nelson Goodman & Henry Leonard (1940). The Calculus of Individuals and its Uses. Journal of Symbolic Logic 5 (2):45-55.
  14. Andrzej Grzegorczyk (1964). A Note on the Theory of Propositional Types. Fundamenta Mathematicae 54:27-29.
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  15. Andrzej Grzegorczyk (1955). The Systems of Leśniewski in Relation to Contemporary Logical Research. Studia Logica 3 (1):77-95.
  16. Anil Gupta (1982). Truth and Paradox. Journal of Philosophical Logic 11 (1):1-60.
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  17. Leon Henkin (1963). A Theory of Propositional Types. Fundamenta Mathematicae 52:323-334.
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  18. Leon Henkin (1950). Completeness in the Theory of Types. Journal of Symbolic Logic 15 (2):81-91.
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  19. 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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  20. Hans G. Herzberger (1970). Paradoxes of Grounding in Semantics. Journal of Philosophy 67 (6):145-167.
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  21. Henry Hiz (1977). Descriptions in Russell's Theory and Ontology. Studia Logica 36 (4):271-283.
  22. Laurence R. Horn, Contradiction. Stanford Encyclopedia of Philosophy.
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  23. Edward Hussey (1991). Aristotle on Mathematical Objects. Apeiron 24 (4):105 - 133.
  24. Stephen Cole Kleene (1952). Introduction to Metamathematics. North Holland.
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  25. Penelope Maddy (1980). Perception and Mathematical Intuition. Philosophical Review 89 (2):163-196.
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  26. Roman Murawski (2011). Mathematical Objects and Mathematical Knowledge. Grazer Philosophische Studien 52:257-259.
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  27. John E. Nolt (1983). Sets and Possible Worlds. Philosophical Studies 44 (1):21-35.
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  28. Alejandro Pérez Carballo (2014). Structuring Logical Space. Philosophy and Phenomenological Research 89 (2).
  29. 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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  30. Marcin Poręba (2012). Poglądy Kanta na matematykę a konstruktywizm. Filozofia Nauki 1.
    The author rejects the opinion that Kant’s views on mathematics lend in any interesting sense support to constructivism, understood as the thesis that the truth conditions of mathematical propositions consist in the existence of their constructive proofs or in the possibility of proving them constructively. Kant’s insistence on the role of intuitive construction in mathematics is here interpreted as a thesis concerning mathematical concepts, not mathematical objects, and therefore not in any sense implying that the objects of mathematical cognition cannot (...)
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  31. W. V. Quine (1986). Philosophy of Logic. Harvard University Press.
    With his customary incisiveness, W. V. Quine presents logic as the product of two factors, truth and grammar--but argues against the doctrine that the logical ...
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  32. W. V. Quine (1968). Ontological Relativity. Journal of Philosophy 65 (7):185-212.
  33. W. V. Quine (1955). On Frege's Way Out. Mind 64 (254):145-159.
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  34. Paula Quinon (2011). La Métalangue d'Une Syntaxe Inscriptionnelle. History and Philosophy of Logic 32 (2):191 - 193.
    History and Philosophy of Logic, Volume 32, Issue 2, Page 191-193, May 2011.
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  35. Greg Restall, Anti-Realist Classical Logic and Realist Mathematics.
    I sketch an application of a semantically anti-realist understanding of the classical sequent calculus to the topic of mathematics. The result is a semantically anti-realist defence of a kind of mathematical realism. In the paper, I begin the development of the view and compare it to orthodox positions in the philosophy of mathematics.
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  36. V. Frederick Rickey (1977). A Survey of Leśniewski's Logic. Studia Logica 36 (4):407 - 426.
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  37. Lance J. Rips (2001). Necessity and Natural Categories. Psychological Bulletin 127:827-852.
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  38. Gideon Rosen, Abstract Objects. Stanford Encyclopedia of Philosophy.
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  39. Peter Simons (1997). Higher-Order Quantification and Ontological Commitment. Dialectica 51 (4):255–271.
  40. Peter Simons (1995). Lesniewski and Ontological Commitment. In Denis Miéville & D. Vernant (eds.), Stanislaw Lesniewski Aujourd'hui. Université De Grenoble. 103-119.
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  41. Peter Simons (1994). Discovering Lesniewski: Collected Works. History and Philosophy of Logic 15 (2):227-235.
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  42. Peter M. Simons (1987). Frege's Theory of Real Numbers. History and Philosophy of Logic 8 (1):25--44.
    Frege's theory of real numbers has undeservedly received almost no attention, in part because what we have is only a fragment. Yet his theory is interesting for the light it throws on logicism, and it is quite different from standard modern approaches. Frege polemicizes vigorously against his contemporaries, sketches the main features of his own radical alternative, and begins the formal development. This paper summarizes and expounds what he has to say, and goes on to reconstruct the most important steps (...)
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  43. Peter M. Simons (1987/2000). Parts: A Study in Ontology. Oxford University Press.
    Although the relationship of part to whole is one of the most fundamental there is, this is the first full-length study of this key concept. Showing that mereology, or the formal theory of part and whole, is essential to ontology, Simons surveys and critiques previous theories--especially the standard extensional view--and proposes a new account that encompasses both temporal and modal considerations. Simons's revised theory not only allows him to offer fresh solutions to long-standing problems, but also has far-reaching consequences for (...)
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  44. Richard Tieszen (2006). After Gödel: Mechanism, Reason, and Realism in the Philosophy of Mathematics. Philosophia Mathematica 14 (2):229-254.
    In his 1951 Gibbs Lecture Gödel formulates the central implication of the incompleteness theorems as a disjunction: either the human mind infinitely surpasses the powers of any finite machine or there exist absolutely unsolvable diophantine problems (of a certain type). In his later writings in particular Gödel favors the view that the human mind does infinitely surpass the powers of any finite machine and there are no absolutely unsolvable diophantine problems. I consider how one might defend such a view in (...)
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  45. Richard Tieszen (1994). Mathematical Realism and Gödel's Incompleteness Theorems. Philosophia Mathematica 2 (3):177-201.
    In this paper I argue that it is more difficult to see how Godel's incompleteness theorems and related consistency proofs for formal systems are consistent with the views of formalists, mechanists and traditional intuitionists than it is to see how they are consistent with a particular form of mathematical realism. If the incompleteness theorems and consistency proofs are better explained by this form of realism then we can also see how there is room for skepticism about Church's Thesis and the (...)
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  46. Eric P. Tsui-James (1998). Dummett, Brouwer and the Metaphysics of Mathematics. Grazer Philosophische Studien 55:143-168.
    Although Brouwer is well known for his Intuitionistic philosophy of mathematics, a constructivist philosophy which calls for restricted use of certain logical principles, there is much less awareness of the well-developed metaphysical basis which underlies those restrictions. In the first half of this paper I outline a basic interpretation of Brouwer's metaphysics, and then in the second half consider the compatibility of that metaphysics with Dummett's argument for a principled non-metaphysical approach to intuitionism. I conclude that once the variously misleading (...)
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  47. Jan Woleński (1998). Mathematical Objects and Mathematical Knowledge. Erkenntnis 48 (1).
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  48. Jan Woleński (1998). Michael D. Resnik (Ed.), Mathematical Objects and Mathematical Knowledge. Erkenntnis 48 (1):129-131.
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Mathematical Fictionalism
  1. B. Armour-Garb (2011). The Implausibility of Hermeneutic Non-Assertivism. Philosophia Mathematica 19 (3):349-353.
    In a recent paper, Mark Balaguer has responded to the argument that I launched against Hermeneutic Non-Assertivism, claiming that, as a matter of empirical fact, ‘when typical mathematicians utter mathematical sentences, they are doing something that differs from asserting in a pretty subtle way, so that the difference between [asserting] and this other kind of speech act is not obvious’. In this paper, I show the implausibility of this empirical hypothesis.
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  2. B. Armour-Garb (2011). Understanding and Mathematical Fictionalism. Philosophia Mathematica 19 (3):335-344.
    In a recent paper in this journal, Mark Balaguer develops and defends a new version of mathematical fictionalism, what he calls ‘Hermeneutic non-assertivism’, and responds to some recent objections to mathematical fictionalism that were launched by John Burgess and others. In this paper I provide some fairly compelling reasons for rejecting Hermeneutic non-assertivism — ones that highlight an important feature of what understanding mathematics involves (or, as we shall see, does not involve).
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