Ontology of Mathematics

Edited by Rafal Urbaniak (Uniwersytetu Gdanskiego, Uniwersytetu Gdanskiego)
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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  1. Objects are (not) ...Friedrich Wilhelm Grafe - manuscript
    My goal in this paper is, to tentatively sketch and try defend some observations regarding the ontological dignity of object references, as they may be used from within in a formalized language. -/- Hence I try to explore, what properties objects are presupposed to have, in order to enter the universe of discourse of an interpreted formalized language. -/- First I review Frege′s analysis of the logical structure of truth value definite sentences of scientific colloquial language, to draw suggestions from (...)
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  2. A Conventionalist Account of Distinctively Mathematical Explanation.Mark Povich - forthcoming - Philosophical Problems in Science.
    Distinctively mathematical explanations (DMEs) explain natural phenomena primarily by appeal to mathematical facts. One important question is whether there can be an ontic account of DME. An ontic account of DME would treat the explananda and explanantia of DMEs as ontic structures and the explanatory relation between them as an ontic relation (e.g., Pincock 2015, Povich 2021). Here I present a conventionalist account of DME, defend it against objections, and argue that it should be considered ontic. Notably, if indeed it (...)
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  3. Rules to Infinity: The Normative Role of Mathematics in Scientific Explanation.Mark Povich - forthcoming - Oxford University Press.
    [EDIT: This book will be published open access. Check back around April 2024 to access the entire book.] One central aim of science is to provide explanations of natural phenomena. What role(s) does mathematics play in achieving this aim? How does mathematics contribute to the explanatory power of science? Rules to Infinity defends the thesis, common though perhaps inchoate among many members of the Vienna Circle, that mathematics contributes to the explanatory power of science by expressing conceptual rules, rules which (...)
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  4. A Critique of Yablo’s If-thenism.Bradley Armour-Garb & Frederick Kroon - 2023 - Philosophia Mathematica 31 (3):360-371.
    Using ideas proposed in Aboutness and developed in ‘If-thenism’, Stephen Yablo has tried to improve on classical if-thenism in mathematics, a view initially put forward by Bertrand Russell in his Principles of Mathematics. Yablo’s stated goal is to provide a reading of a sentence like ‘The number of planets is eight’ with a sort of content on which it fails to imply ‘Numbers exist’. After presenting Yablo’s framework, our paper raises a problem with his view that has gone virtually unnoticed (...)
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  5. Metamathematik der Elementarmathematik.Erwin Engeler - 1983 - New York: Springer Verlag.
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  6. Numbers as properties.Melisa Vivanco - 2023 - Synthese 202 (4):1-23.
    Although number sentences are ostensibly simple, familiar, and applicable, the justification for our arithmetical beliefs has been considered mysterious by the philosophical tradition. In this paper, I argue that such a mystery is due to a preconception of two realities, one mathematical and one nonmathematical, which are alien to each other. My proposal shows that the theory of numbers as properties entails a homogeneous domain in which arithmetical and nonmathematical truth occur. As a result, the possibility of arithmetical knowledge is (...)
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  7. Ontologies of Common Sense, Physics and Mathematics.Jobst Landgrebe & Barry Smith - 2023 - Archiv.
    The view of nature we adopt in the natural attitude is determined by common sense, without which we could not survive. Classical physics is modelled on this common-sense view of nature, and uses mathematics to formalise our natural understanding of the causes and effects we observe in time and space when we select subsystems of nature for modelling. But in modern physics, we do not go beyond the realm of common sense by augmenting our knowledge of what is going on (...)
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  8. Towards a Computational Ontology for the Philosophy of Wittgenstein: Representing Aspects of the Tractarian Philosophy of Mathematics.Jakub Gomułka - 2023 - Analiza I Egzystencja 63:27-54.
    The present paper concerns the Wittgenstein ontology project: an attempt to create a Semantic Web representation of Ludwig Wittgenstein’s philosophy. The project has been in development since 2006, and its current state enables users to search for information about Wittgenstein-related documents and the documents themselves. However, the developers have much more ambitious goals: they attempt to provide a philosophical subject matter knowledge base that would comprise the claims and concepts formulated by the philosopher. The current knowledge representation technology is not (...)
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  9. The insubstantiality of mathematical objects as positions in structures.Bahram Assadian - 2022 - Inquiry: An Interdisciplinary Journal of Philosophy 20.
    The realist versions of mathematical structuralism are often characterized by what I call ‘the insubstantiality thesis’, according to which mathematical objects, being positions in structures, have no non-structural properties: they are purely structural objects. The thesis has been criticized for being inconsistent or descriptively inadequate. In this paper, by implementing the resources of a real-definitional account of essence in the context of Fregean abstraction principles, I offer a version of structuralism – essentialist structuralism – which validates a weaker version of (...)
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  10. A Controvérsia em Torno do Estatuto dos Entes Matemáticos.Vasco Mano - manuscript
    Neste breve ensaio, exploramos alguns caminhos de uma controvérsia milenar em torno do estatuto dos entes matemáticos e apresentamos alguns argumentos a favor de uma posição platonista, aproximadamente clássica, sobre o tema. Este trabalho foi realizado no âmbito da disciplina de Filosofia das Ciências II, parte do curso de Filosofia da Faculdade de Letras da Universidade do Porto, Portugal.
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  11. “In Nature as in Geometry”: Du Châtelet and the Post-Newtonian Debate on the Physical Significance of Mathematical Objects.Aaron Wells - 2023 - In Wolfgang Lefèvre (ed.), Between Leibniz, Newton, and Kant: Philosophy and Science in the Eighteenth Century. Springer Verlag. pp. 69-98.
    Du Châtelet holds that mathematical representations play an explanatory role in natural science. Moreover, she writes that things proceed in nature as they do in geometry. How should we square these assertions with Du Châtelet’s idealism about mathematical objects, on which they are ‘fictions’ dependent on acts of abstraction? The question is especially pressing because some of her important interlocutors (Wolff, Maupertuis, and Voltaire) denied that mathematics informs us about the properties of material things. After situating Du Châtelet in this (...)
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  12. Metamathematics: foundations & physicalization.Stephen Wolfram - 2022 - [Champaign]: Wolfram Media.
    "What is mathematics?" is a question that has been debated since antiquity. This book presents a groundbreaking and surprising answer to the question-showing through the concept of the physicalization of metamathematics how both mathematics and physics as experienced by humans can be seen to emerge from the unique underlying computational structure of the recently formulated ruliad. Written with Stephen Wolfram's characteristic expositional flair and richly illustrated with remarkable algorithmic diagrams, the book takes the reader on a unprecedented intellectual journey to (...)
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  13. Numbers and the world: essays on math and beyond.David Mumford - 2023 - Providence, Rhode Island: American Mathematical Society.
    This book is a collection of essays written by a distinguished mathematician with a very long and successful career as a researcher and educator working in many areas of pure and applied mathematics. The author writes about everything he found exciting about math, its history, and its connections with art, and about how to explain it when so many smart people (and children) are turned off by it. The three longest essays touch upon the foundations of mathematics, upon quantum mechanics (...)
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  14. Is math real?: how simple questions lead us to mathematics' deepest truths.Eugenia Cheng - 2023 - New York: Basic Books.
    Where does math come from? From a textbook? From rules? From deduction? From logic? Not really, Eugenia Cheng writes in Is Math Real?: it comes from curiosity, from instinctive human curiosity, "from people not being satisfied with answers and always wanting to understand more." And most importantly, she says, "it comes from questions": not from answering them, but from posing them. Nothing could seem more at odds from the way most of us were taught math: a rigid and autocratic model (...)
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  15. Die grundsätze und das wesen des unendlichen in der mathematik und philosophie.Friedrich Jacob Kurt Geissler - 1902 - Leipzig,: B. G. Teubner.
  16. Nothing Infinite: A Summary of Forever Finite.Kip Sewell - 2023 - Rond Media Library.
    In 'Forever Finite: The Case Against Infinity' (Rond Books, 2023), the author argues that, despite its cultural popularity, infinity is not a logical concept and consequently cannot be a property of anything that exists in the real world. This article summarizes the main points in 'Forever Finite', including its overview of what debunking infinity entails for conceptual thought in philosophy, mathematics, science, cosmology, and theology.
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  17. Matematyka a Ontologiczna Estetyka Ingardena.Barry Smith - 1976 - Studia Filozoficzne 1 (122):51-56.
    This paper applies the ontological framework developed by Roman Ingarden in his Controversy over the Existence of the World to the domain of mathematics, concluding with some remarks on parallels between the mode of existence of mathematical entities on the one hand and of values on the other.
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  18. Mario Bunge's Philosophy of Mathematics: An Appraisal.Marquis Jean-Pierre - 2012 - Science & Education 21:1567-1594.
    In this paper, I present and discuss critically the main elements of Mario Bunge’s philosophy of mathematics. In particular, I explore how mathematical knowledge is accounted for in Bunge’s systemic emergent materialism.
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  19. Geist der mathematik.Max Bense - 1939 - Berlin,: R. Oldenbourg.
    Einleitung.--Das irrationale in der mathematik.--Der verfall der anschauung.--Mathematik und ästhetik.--Das unendliche.--Intuitionismus, logizismus und formalismus.--Betrachtungen über den gegenstand der mathematik.--Anmerkungen und nachweise.--Wörterbuch der disziplinen.
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  20. The liberation argument for inconsistent mathematics.Franci Mangraviti - 2023 - Australasian Journal of Logic 29 (2):278-315.
    Val Plumwood charged classical logic not only with the invalidity of some of its laws, but also with the support of systemic oppression through naturalization of the logical structure of dualisms. In this paper I show that the latter charge - unlike the former - can be carried over to classical mathematics, and I propose a new conception of inconsistent mathematics - queer incomaths - as a liberatory activity meant to undermine said naturalization.
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  21. L'existence en mathématiques.Evert Willem Beth - 1956 - Paris,: Gauthier-Villars.
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  22. Structure et objet de l'analyse mathématique.Eloi Lefebvre - 1958 - Paris,: Gauthier-Villars.
  23. A Naturalistic Paradox: Existence and Nature in the Philosophy of Mathematics.Matteo Plebani - 2012 - In Camposampiero Favaretti & Matteo Plebani (eds.), Existence and Nature: New Perspectives. De Gruyter. pp. 9-32.
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  24. A Note on von Neumann Ordinals and Dependence.Jonas Werner - 2023 - Philosophia Mathematica 31 (2):nkad007.
    This note defends the reduction of ordinals to pure sets against an argument put forward by Beau Madison Mount. In the first part I will defend the claim that dependence simpliciter can be reduced to immediate dependence and define a notion of predecessor dependence. In the second part I will provide and defend a way to model the dependence profile of ordinals akin to Mount’s proposal in terms of immediate dependence and predecessor dependence. I furthermore show that my alternative dependence (...)
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  25. Carl Posy and Yemima Ben-Menahem, eds. Mathematical Objects, Knowledge and Applications: Essays in Memory of Mark Steiner. Jerusalem Studies in Philosophy and History of Science.Robert S. D. Thomas - forthcoming - Philosophia Mathematica.
    Menachem Butler. Bibliography: Mark Steiner’s main works, pp. 3–7.
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  26. L'Esprit des mathématiques.Simone Goyard-Fabre - 1970 - Paris,: Éditions de l'École.
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  27. Qué es la matemática.Jorge Bosch - 1971 - [Buenos Aires]: Editorial Columba.
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  28. Sur la nature des mathématiques.Claude Paul Bruter - 1973 - Paris,: Gauthier-Villars.
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  29. Fundamente de matematică.G. Sâmboan - 1974 - București: Editura didactică și pedagogică.
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  30. Grundlagen der modernen Mathematik.Herbert Meschkowski - 1972 - Darmstadt: Wissenschaftliche Buchgesellschaft, [Abt. Verl.].
  31. The Turning Point in Wittgenstein’s Philosophy of Mathematics: Another Turn.Yemima Ben-Menahem - 2023 - In Carl Posy & Yemima Ben-Menahem (eds.), Mathematical Knowledge, Objects and Applications: Essays in Memory of Mark Steiner. Springer. pp. 377-393.
    According to Mark Steiner, Wittgenstein’s intense work in the philosophy of mathematics during the early 1930s brought about a distinct turning point in his philosophy. The crux of this transition, Steiner contends, is that Wittgenstein came to see mathematical truths as originating in empirical regularities that in the course of time have been hardened into rules. This interpretation, which construes Wittgenstein’s later philosophy of mathematics as more realist than his earlier philosophy, challenges another influential interpretation which reads Wittgenstein as moving (...)
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  32. Potential Infinity and De Re Knowledge of Mathematical Objects.Øystein Linnebo & Stewart Shapiro - 2023 - In Carl Posy & Yemima Ben-Menahem (eds.), Mathematical Knowledge, Objects and Applications: Essays in Memory of Mark Steiner. Springer. pp. 79-98.
    Our first goal here is to show how one can use a modal language to explicate potentiality and incomplete or indeterminate domains in mathematics, along the lines of previous work. We then show how potentiality bears on some longstanding items of concern to Mark Steiner: the applicability of mathematics, explanation, and de re propositional attitudes toward mathematical objects.
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  33. Die Begriffswelt der Mathematik.Imre Ruzsa - 1976 - Budapest: Akadémiai Kiadó.
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  34. Sur la définition euclidienne de la droite.Michel Federspiel - 1991 - In Jules Vuillemin & Rushdī Rāshid (eds.), Mathématiques et philosophie de l'antiquité à l'age classique: hommage à Jules Vuillemin. Diffusion, Presses du CNRS.
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  35. Le problème de la mesure dans la perspective de l'Etre et du non-Etre : Zénon et Platon, Eudoxe et Dedekind.Imre Tóth - 1991 - In Jules Vuillemin & Rushdī Rāshid (eds.), Mathématiques et philosophie de l'antiquité à l'age classique: hommage à Jules Vuillemin. Diffusion, Presses du CNRS.
  36. Metafizicheskai︠a︡ matematika XVII v.V. N. Katasonov - 1993 - Moskva: Nauka.
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  37. Epistemologia della matematica: ontologia, verità, valutazioni.Nicola Grana - 2001 - Napoli: L'orientale.
  38. The Maximization of Chaos.Ilexa Yardley - 2023 - Https://Medium.Com/the-Circular-Theory/.
  39. Atheistic Platonism: A Manifesto.Eric Charles Steinhart - 2022 - Springer Verlag.
    Atheistic Platonism is an alternative to both theism and nihilistic atheism. It shows how any jobs allegedly done by God are better done by impersonal Platonic objects. Without Platonic objects, atheism degenerates into an illogical nihilism. Atheistic Platonism instead provides reality with foundations that are eternal, necessary, rational, beautiful, and utterly mindless. It argues for a plenitude of mathematical objects, and an infinite plurality of possible universes. It provides mindless rational grounds for objective values, and for objective moral laws for (...)
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  40. Gödel's conceptual realism.Donald A. Martin - 2010 - In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: essays for his centennial. Association for Symbolic Logic.
  41. The Paradigm Shift in the 19th-century Polish Philosophy of Mathematics.Paweł Polak - 2022 - Studia Historiae Scientiarum 21:217-235.
    The Polish philosophy of mathematics in the 19th century had its origins in the Romantic period under the influence of the then-predominant idealist philosophies. The decline of Romantic philosophy precipitated changes in general philosophy, but what is less well known is how it triggered changes in the philosophy of mathematics. In this paper, we discuss how the Polish philosophy of mathematics evolved from the metaphysical approach that had been formed during the Romantic era to the more modern positivistic paradigm. These (...)
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  42. Mathematical Pluralism.Edward N. Zalta - 2023 - Noûs.
    Mathematical pluralism can take one of three forms: (1) every consistent mathematical theory consists of truths about its own domain of individuals and relations; (2) every mathematical theory, consistent or inconsistent, consists of truths about its own (possibly uninteresting) domain of individuals and relations; and (3) the principal philosophies of mathematics are each based upon an insight or truth about the nature of mathematics that can be validated. (1) includes the multiverse approach to set theory. (2) helps us to understand (...)
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  43. Frédéric Patras.*The Essence of Numbers.Bonnie Gold - 2023 - Philosophia Mathematica 31 (1):132-142.
    This book approaches the issue of the essence of numbers from a very broad perspective: historical, philosophical, and mathematical. The author is a practicing.
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  44. Mathematics: Method Without Metaphysics.Elaine Landry - 2023 - Philosophia Mathematica 31 (1):56-80.
    I use my reading of Plato to develop what I call as-ifism, the view that, in mathematics, we treat our hypotheses as if they were first principles and we do this with the purpose of solving mathematical problems. I then extend this view to modern mathematics showing that when we shift our focus from the method of philosophy to the method of mathematics, we see that an as-if methodological interpretation of mathematical structuralism can be used to provide an account of (...)
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  45. TGFT condensate cosmology as an example of spacetime emergence in quantum gravity.Daniele Oriti - 2022 - In Antonio Vassallo (ed.), The Foundations of Spacetime Physics: Philosophical Perspectives. Routledge.
  46. Knowledge and the Philosophy of Number. [REVIEW]Richard Lawrence - 2022 - History and Philosophy of Logic 43 (4):404-406.
    Hossack’s project in this book is to provide a new foundation for the philosophy of number inspired by the traditional idea that numbers are magnitudes.
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  47. The digital and the real world: computational foundations of mathematics, science, technology, and philosophy.Klaus Mainzer - 2017 - [Hackensack,] New Jersey: World Scientific.
    In the 21st century, digitalization is a global challenge of mankind. Even for the public, it is obvious that our world is increasingly dominated by powerful algorithms and big data. But, how computable is our world? Some people believe that successful problem solving in science, technology, and economies only depends on fast algorithms and data mining. Chances and risks are often not understood, because the foundations of algorithms and information systems are not studied rigorously. Actually, they are deeply rooted in (...)
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  48. Predicative foundations of arithmetic.with Solomon Feferman - 2020 - In Geoffrey Hellman (ed.), Mathematics and its Logics: Philosophical Essays. Cambridge University Press.
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  49. Axiomatics: mathematical thought and high modernism.Alma Steingart - 2023 - Chicago: University of Chicago Press.
    The first history of postwar mathematics, offering a new interpretation of the rise of abstraction and axiomatics in the twentieth century. Why did abstraction dominate American art, social science, and natural science in the mid-twentieth century? Why, despite opposition, did abstraction and theoretical knowledge flourish across a diverse set of intellectual pursuits during the Cold War? In recovering the centrality of abstraction across a range of modernist projects in the United States, Alma Steingart brings mathematics back into the conversation about (...)
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  50. Penelope Rush.* Ontology and the Foundations of Mathematics: Talking Past Each Other.Geoffrey Hellman - 2022 - Philosophia Mathematica 30 (3):387-392.
    This compact volume, belonging to the Cambridge Elements series, is a useful introduction to some of the most fundamental questions of philosophy and foundations of mathematics. What really distinguishes realist and platonist views of mathematics from anti-platonist views, including fictionalist and nominalist and modal-structuralist views?1 They seem to confront similar problems of justification, presenting tradeoffs between which it is difficult to adjudicate. For example, how do we gain access to the abstract posits of platonist accounts of arithmetic, analysis, geometry, etc., (...)
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