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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. Alison Adam (2010). Femininity, Mathematics and Science, 1880–1914. [REVIEW] British Journal for the History of Science 43 (3):494-496.
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  2. I. A. Akchurin (1967). The Place of Mathematics in the System of the Sciences. Russian Studies in Philosophy 6 (3):3-13.
    The deep and many-sided penetration of mathematical methods into virtually all branches of scientific knowledge is a characteristic feature of the present period of development of human culture. Even fields so remote from mathematics as the theory of versification, jurisprudence, archeology, and medical diagnostics have now proved to be associated with the accelerating process of application of disciplines such as probability theory, information theory, algorithm theory, etc. Mathematical methods are rapidly penetrating the sphere of the social sciences. One can no (...)
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  3. 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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  4. Jeremy Avigad (2008). Philosophical Relevance of Computers in Mathematics. In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oup Oxford.
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  5. Frederick Bagemihl (1981). Representation of Ordinal Numbers and Derived Sets in Certain Continuous Sets. Mathematical Logic Quarterly 27 (19‐21):333-336.
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  6. 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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  7. 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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  8. Jeffrey A. Barrett (2013). On the Coevolution of Basic Arithmetic Language and Knowledge. Erkenntnis 78 (5):1025-1036.
    Skyrms-Lewis sender-receiver games with invention allow one to model how a simple mathematical language might be invented and become meaningful as its use coevolves with the basic arithmetic competence of primitive mathematical inquirers. Such models provide sufficient conditions for the invention and evolution of a very basic sort of arithmetic language and practice, and, in doing so, provide insight into the nature of a correspondingly basic sort of mathematical knowledge in an evolutionary context. Given traditional philosophical reflections concerning the nature (...)
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  9. 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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  10. Paul Benacerraf (2003). What Mathematical Truth Could Not Be--1. In Matthias Schirn (ed.), The Philosophy of Mathematics Today. Clarendon Press.
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  11. Boran Berčić (2005). Zašto 2+2=4? Filozofska Istrazivanja 25 (4):945-961.
    The starting point of this article is the ontological question: What makes it true that2+2=4?, that is, what are the truth makers of mathematical propositions? Of course,the satisfactory theory in the philosophy of mathematics has to answer semantical question: What are mathematical propositions about? Also, epistemological question:How do we know them?, as well. Author compares five theories in the philosophy of mathematics, that is, five accounts of the nature of truth makers in mathematical discourse: fictionalism ; nominalism ; physicalism ; (...)
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  12. Enrico Bombieri (2011). The Mathematical Infinity. In Michał Heller & W. H. Woodin (eds.), Infinity: New Research Frontiers. Cambridge University Press. 55.
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  13. Harold Chapman Brown (1914). Concepts and Existence. Journal of Philosophy, Psychology and Scientific Methods 11 (13):355-357.
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  14. Charles Burnett (2010). Expounding the Mathematical Seed: A Translation of Bhaskara I on the Mathematical Chapter of the Aryabhatiya. Annals of Science 67 (1):132-134.
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  15. Jessica Carter (2008). Categories for the Working Mathematician: Making the Impossible Possible. Synthese 162 (1):1 - 13.
    This paper discusses the notion of necessity in the light of results from contemporary mathematical practice. Two descriptions of necessity are considered. According to the first, necessarily true statements are true because they describe ‘unchangeable properties of unchangeable objects’. The result that I present is argued to provide a counterexample to this description, as it concerns a case where objects are moved from one category to another in order to change the properties of these objects. The second description concerns necessary (...)
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  16. 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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  17. Charles Castonguay (1972). Meaning and Existence in Mathematics. New York,Springer-Verlag.
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  18. Peter Clark (2009). Mathematical Entities. In Robin Le Poidevin (ed.), The Routledge Companion to Metaphysics. Routledge.
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  19. C. A. J. Coady (1981). Mathematical Knowledge and Reliable Authority. Mind 90 (360):542-556.
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  20. Randall Collins & Sal Restivo (2010). Robber Barons and Politicians in Mathematics: A Conflict Model of Science. Philosophy of Mathematics Education Journal 25.
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  21. Paul Ernest (1997). Texts And The Objects Of Mathematics. Philosophy of Mathematics Education Journal 10.
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  22. K. P. F. (1964). The Mathematical Principles of Natural Philosophy. Review of Metaphysics 18 (1):181-181.
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  23. Fernando Ferreira (1999). A Note on Finiteness in the Predicative Foundations of Arithmetic. Journal of Philosophical Logic 28 (2):165-174.
    Recently, Feferman and Hellman (and Aczel) showed how to establish the existence and categoricity of a natural number system by predicative means given the primitive notion of a finite set of individuals and given also a suitable pairing function operating on individuals. This short paper shows that this existence and categoricity result does not rely (even indirectly) on finite-set induction, thereby sustaining Feferman and Hellman's point in favor of the view that natural number induction can be derived from a very (...)
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  24. H. Field (1990). Mathematics Without Truth (a Reply to Maddy). Pacific Philosophical Quarterly 71 (3):206-222.
    This paper elaborates on the fictionalist conception of mathematics, and on how it accommodates the obvious fact that mathematical claims are important in application to the physical world. It also replies to Maddy's argument that fictionalism does not have the epistemological advantage over Platonism that it appears to have; the reply involves a discussion of whether mathematics should be regarded as conservative over second order physical theories as well as first order ones.
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  25. Kevin L. Flannery (1998). Frege's Philosophy of Mathematics. Review of Metaphysics 51 (3):670-672.
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  26. Karen François & Laurent De Sutter (2004). Where Mathematics Becomes Political. Representing Humans. Philosophica 74.
  27. Gottlob Frege (1879/1997). Begriffsschrift: Eine Der Arithmetische Nachgebildete Formelsprache des Reinen Denkens. L. Nebert.
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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. José García García (1994). A. Bowie: estética y subjetividad. Logos 28 (2):337-348.
    This paper shows an analysis of some found solutions for the famous aporia of the race between Achilles and the Tortoise. As an introduction, we present the mechanical solution, to establish that it is not in the field of matters of fact where you can resolve a purely rational problem like the one raised by Zeno of Elea. And so, the main part of the article is dedicated to the mathematical solutions, which face the problem under the point of view (...)
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  30. 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.
  31. Martin Goldstern (1997). Strongly Amorphous Sets and Dual Dedekind Infinity. Mathematical Logic Quarterly 43 (1):39-44.
    1. If A is strongly amorphous , then its power set P is dually Dedekind infinite, i. e., every function from P onto P is injective. 2. The class of “inexhaustible” sets is not closed under supersets unless AC holds.
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  32. 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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  33. 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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  34. Nelson Goodman (1972). Seven Strictures on Similarity. In Problems and Projects. Bobs-Merril.
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  35. Nelson Goodman (1961). About. Mind 70 (277):1-24.
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  36. Nelson Goodman (1951). The Structure of Appearance. Harvard University Press.
  37. Nelson Goodman (1946). A Query on Confirmation. Journal of Philosophy 43 (14):383-385.
  38. Nelson Goodman & Henry Leonard (1940). The Calculus of Individuals and its Uses. Journal of Symbolic Logic 5 (2):45-55.
  39. Robert Bates Graber (1989). Mathematical Naturalism. Southern Journal of Philosophy 27 (3):427-441.
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  40. Gg Granger (1988). The Concepts of Natural Mathematical Entities. Revue Internationale de Philosophie 42 (167):474-499.
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  41. Aleksandr A. Grigorian (2012). Sociocultural and Metaphysical Circles and Their Overcoming in the Development of Mathematics. Russian Studies in Philosophy 50 (4):73-93.
    The author shows how mathematics in ancient and early medieval Europe was constrained by deeply rooted metaphysical conceptions and how these constraints have been overcome since the late medieval period. As examples, he focuses on changing conceptions of chance, motion, and infinity.
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  42. Andrzej Grzegorczyk (1964). A Note on the Theory of Propositional Types. Fundamenta Mathematicae 54 (3):27-29.
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  43. Andrzej Grzegorczyk (1955). The Systems of Leśniewski in Relation to Contemporary Logical Research. Studia Logica 3 (1):77-95.
  44. Anil Gupta (1982). Truth and Paradox. Journal of Philosophical Logic 11 (1):1-60.
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  45. Guillermo E. Rosado Haddock (1992). Interderivability of Seemingly Unrelated Mathematical Statements and the Philosophy of Mathematics. Dialogos 27 (59):121-134.
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  46. 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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  47. Leon Henkin (1963). A Theory of Propositional Types. Fundamenta Mathematicae 52:323-334.
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  48. Leon Henkin (1950). Completeness in the Theory of Types. Journal of Symbolic Logic 15 (2):81-91.
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  49. 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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  50. 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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