Ontology of Mathematics

Edited by Rafal Urbaniak (University of Ghent, University of Gdansk, University of Calgary)
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 1964, 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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  1. Kitcher's Naturalistic Epistemology and Methodology of Mathematics.Jesus Alcolea - 2012 - Poznan Studies in the Philosophy of the Sciences and the Humanities 101 (1):295-326.
    With his book The Nature of Mathematical Knowledge (1983), Ph. Kitcher, that had been doing extensive research in the history of the subject and in the contemporary debates on epistemology, saw clearly the need for a change in philosophy of mathematics. His goal was to replace the dominant, apriorist philosophy of mathematics with an empiricist philosophy. The current philosophies of mathematics all appeared, according to his analysis, not to fit well with how mathematicians actually do mathematics. A shift in orientation (...)
  2. Realism in the Philosophy of Mathematics: A Critical Discussion.T. Arrigoni - 2000 - Rivista di Filosofia Neo-Scolastica 92 (3-4):627-646.
  3. Representation of Ordinal Numbers and Derived Sets in Certain Continuous Sets.Frederick Bagemihl - 1981 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 27 (19-21):333-336.
  4. Mathematical Spandrels.Alan Baker - forthcoming - Australasian Journal of Philosophy:1-15.
    The aim of this paper is to open a new front in the debate between platonism and nominalism by arguing that the degree of explanatory entanglement of mathematics in science is much more extensive than has been hitherto acknowledged. Even standard examples, such as the prime life cycles of periodical cicadas, involve a penumbra of mathematical features whose presence can only be explained using relatively sophisticated mathematics. I introduce the term ‘mathematical spandrel’ to describe these penumbral properties, and focus on (...)
  5. 'Realism as a Philosophy of Mathematics'.F. Barker Stephen - 1969 - In Kurt Gödel, Jack J. Bulloff, Thomas C. Holyoke & Samuel Wilfred Hahn (eds.), Foundations of Mathematics. New York: Springer.
  6. Realism as a Philosophy of Mathematics.Stephen F. Barker - 1969 - In Kurt Gödel, Jack J. Bulloff, Thomas C. Holyoke & Samuel Wilfred Hahn (eds.), Journal of Symbolic Logic. New York: Springer. pp. 1--9.
  7. The Usefulness of Mathematical Learning Explained and Demonstrated Being Mathematical Lectures Read in the Public Schools of Cambridge.Isaac Barrow - 1975 - Printed for S. Austen.
  8. Mathematics, Science, and Postclassical Theory. [REVIEW]Adam Beck - 1998 - Radical Philosophy 91.
  9. Dissenting Voices.John Bell - manuscript
    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: (...)
  10. La Notion de Nombre Chez Dedekind, Cantor, Frege Th'eories, Conceptions Et Philosophie.Jean-Pierre Belna - 1996
  11. De Motu ; and, the Analyst.George Berkeley & Douglas Michael Jesseph - 1992
  12. The Construction of Logical Space, by Augustin Rayo.S. Berry - 2015 - Mind 124 (496):1375-1379.
  13. Mathematics and Metalogic.Daniel Bonevac - 1984 - The Monist 67 (1):56-71.
    In this paper I shall attempt to outline a nominalistic theory of mathematical truth. I call my theory nominalistic because it avoids a real (see [4]) ontological commitment to abstract entities. Traditionally, nominalists have found it difficult to justify any reference to infinite collections in mathematics. Even those who have tried to do so have typically restricted themselves to predicative and, thus, denumerable realms. I Indeed, many have linked impredicative definitions to platonism; nominalists have tended to agree with Weyl that (...)
  14. Logic and Arithmetic, Vol. II--Rational and Irrational Numbers.David Bostock - 1981 - Mind 90 (359):473-475.
  15. Modelle Mathematikhistorischer Entwicklung.Wolfgang Breidert - 1993 - NTM International Journal of History and Ethics of Natural Sciences, Technology and Medicine 1 (1):193-199.
    In the historiography of mathematics the concept of deviation is relative to a normal way of developing mathematics. Similarly, the concept of abbreviation is necessarily connected to some aim to which this development is directed. We should not speak of deviation or abbreviation without regard for such contects. In the history of mathematics there are some cases of treating impossible objects as possible objects of a new theory. Therefore it may be expected that there are more ways out than in (...)
  16. Science in the Looking Glass: What Do Scientists Really Know?E. Brian Davies - 2007 - Oxford University Press UK.
    How do scientific conjectures become laws? Why does proof mean different things in different sciences? Do numbers exist, or were they invented? Why do some laws turn out to be wrong? In this wide-ranging book, Brian Davies discusses the basis for scientists' claims to knowledge about the world. He looks at science historically, emphasizing not only the achievements of scientists from Galileo onwards, but also their mistakes. He rejects the claim that all scientific knowledge is provisional, by citing examples from (...)
  17. Concepts and Existence.Harold Chapman Brown - 1914 - Journal of Philosophy, Psychology and Scientific Methods 11 (13):355-357.
  18. An Anti-Realist Account of the Application of Mathematics.Otávio Bueno - 2016 - Philosophical Studies 173 (10):2591-2604.
    Mathematical concepts play at least three roles in the application of mathematics: an inferential role, a representational role, and an expressive role. In this paper, I argue that, despite what has often been alleged, platonists do not fully accommodate these features of the application of mathematics. At best, platonism provides partial ways of handling the issues. I then sketch an alternative, anti-realist account of the application of mathematics, and argue that this account manages to accommodate these features of the application (...)
  19. Why Mathematical Concepts Are Special.Bernd Buldt - unknown
  20. Constructibility and Mathematical Existence.John P. Burgess & Charles S. Chihara - 1992 - Philosophical Review 101 (4):916.
  21. Structure and Categoricity: Determinacy of Reference and Truth Value in the Philosophy of Mathematics.Tim Button & Sean Walsh - 2016 - 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.
  22. The Role of Representations in Mathematical Reasoning1.Jessica Carter - 2012 - Philosophia Scientae 16:55-70.
  23. Motivations for Realism in the Light of Mathematical Practice.Jessica Carter - 2005 - 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 (...)
  24. Meaning and Existence in Mathematics.Charles Castonguay - 1972 - New York: Springer Verlag.
  25. Meaning and Existence in Mathematics : On the Use and Abuse of the Theory of Models in the Philosophy of Mathematics.Charles Ernest Castonguay - unknown
  26. Mathematical Entities.Peter Clark - 2009 - In Robin Le Poidevin (ed.), The Routledge Companion to Metaphysics. Routledge.
  27. Aristotle's Theory of Abstraction: A Problem About the Mode of Being of Mathematical Objects.John Joseph Cleary - 1982 - Dissertation, Boston University Graduate School
    This dissertation argues that Aristotle intended his so-called theory of abstraction to serve primarily as the resolution of a special problem in the philosophy of mathematics; i.e., the ontological status of mathematical objects. My general approach is dictated by the view that Aristotle's 'theories' must be understood in terms of the particular problems that he is trying to resolve. Thus, most of my dissertation is devoted to examining his treatment of the problem which I show to be relevant to the (...)
  28. Realism, Mathematics and Modality. [REVIEW]Nino Cocchiarella - 1992 - International Studies in Philosophy 24 (3):139-141.
  29. Who’s Afraid of Inconsistent Mathematics?Mark Colyvan - 2008 - ProtoSociology 25:24-35.
    Contemporary mathematical theories are generally thought to be consistent. But it hasn’t always been this way; there have been times in the history of mathematics when the consistency of various mathematical theories has been called into question. And some theories, such as naïve set theory and the early calculus, were shown to be inconsistent. In this paper I will consider some of the philosophical issues arising from inconsistent mathematical theories.
  30. Corcoran Recommends Hambourger on the Frege-Russell Number Definition.John Corcoran - 1978 - 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) (...)
  31. Reid, Constance. Hilbert (a Biography). Reviewed by Corcoran in Philosophy of Science 39 (1972), 106–08.John Corcoran - 1972 - 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 (...)
  32. Mathematical Truth.Anthony Moncrief Coyne - 1974 - Dissertation, The University of North Carolina at Chapel Hill
  33. Some Remarks on the Physicalist Account of Mathematics.Ferenc Csatari - 2012 - Open Journal of Philosophy 2 (2):165.
    The paper comments on a rather uncommon approach to mathematics called physicalist formalism. According to this view, the formal systems mathematicians concern with are nothing more and nothing less than genuine physical systems. I give a brief review on the main theses, then I provide some arguments, concerning mostly with the practice of mathematics and the uniqueness of formal systems, aiming to show the implausibility of this radical view.
  34. Numbers and Expressions.Laura Jacobs Cunningham - 1988 - 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 (...)
  35. WRIGHT, CRISPIN.: "Frege's Conception of Numbers as Objects". [REVIEW]Gregory Currie - 1985 - British Journal for the Philosophy of Science 36:475.
  36. 'Exceeding the Age in Every Thing': Placing Sloane's Objects.James Delbourgo - 2010 - Spontaneous Generations 3 (1):41-54.
    That objects of knowledge get moved across boundaries is well known. But how they get moved often goes unexamined. Modes of movement cannot be ignored when considering objects’ historical signi?cance. Put differently, how geographies are negotiated is central to the constitution of knowledge objects. This essay offers a brief assessment of the competing agencies at work in the global collections of the Enlightenment naturalist Sir Hans Sloane (1660–1753). While discussing broadly the relationship between collecting and power in Sloane’s career, the (...)
  37. The Math Gene How Mathematical Thinking Evolved and Why Numbers Are Like Gossip.Keith J. Devlin - 2000
  38. ATZ, J. J.: "Language and Other Abstract Objects". [REVIEW]Mike Dillinger - 1984 - British Journal for the Philosophy of Science 35:301.
  39. Book Review:Language and Other Abstract Objects J. J. Katz. [REVIEW]Mike Dillinger - 1984 - Philosophy of Science 51 (1):175-.
  40. Richard Dedekind Et les Fondements des Mathématiques Avec de Nombreux Textes Inédits.Pierre Dugac - 1976 - J. Vrin.
  41. The Foundations of Arithmetic. A Logico-Mathematical Enquiry Into the Concept of Number. [REVIEW]N. E. - 1951 - Journal of Philosophy 48 (10):342-342.
  42. Richard G. Heck Jr. Reading Frege's Grundgesetze. Oxford: Oxford University Press, 2012. ISBN: 978-0-19-923370-0 ; 978-0-19-874437-5 ; 978-0-19-165535-7 . Pp. Xvii + 296. [REVIEW]Philip A. Ebert - 2015 - Philosophia Mathematica 23 (2):289-293.
  43. Mathematical Psychics.F. Y. Edgeworth - 1881 - Mind 6 (24):581-583.
  44. Real Numbers, Generalizations of the Reals and Theories of Continua (Synthese Library, Vol. 242).Philip Ehrlich & Moshe Machover - 1996 - British Journal for the Philosophy of Science 47 (2):320-324.
  45. Geometría e inteligibilidad.Miguel Espinoza - 1986 - Dialogos 21 (48):107.
  46. The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene.William G. Faris - 1998 - Complexity 4 (1):46-48.
  47. Possibility and Reality in Mathematics: A Review of Realism, Mathematics, and Modality. [REVIEW]H. Field & G. Hellman - 1992 - British Journal for the Philosophy of Science 43 (2):245-262.
  48. Science Without Numbers.Hartry Field - 2016 - Oxford University Press UK.
    Science Without Numbers caused a stir in philosophy on its original publication in 1980, with its bold nominalist approach to the ontology of mathematics and science. Hartry Field argues that we can explain the utility of mathematics without assuming it true. Part of the argument is that good mathematics has a special feature that allows it to be applied to "nominalistic" claims in a way that generates nominalistic consequences more easily without generating any new ones. There has been much debate (...)
  49. Are Our Logical and Mathematical Concepts Highly Indeterminate?Hartry Field - 1994 - Midwest Studies in Philosophy 19 (1):391-429.
  50. Chapter Three. The Status of Mathematical Expressions.Robert J. Fogelin - 2009 - In Taking Wittgenstein at His Word: A Textual Study. Princeton University Press. pp. 83-115.
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