Ontology of Mathematics

Edited by Rafal Urbaniak (Uniwersytetu Gdanskiego, Uniwersytetu Gdanskiego)
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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  1. Numerical Cognition and the Epistemology of Arithmetic.Markus Pantsar - 2024 - Cambridge University Press.
    Arithmetic is one of the foundations of our educational systems, but what exactly is it? Numbers are everywhere in our modern societies, but what is our knowledge of numbers really about? This book provides a philosophical account of arithmetical knowledge that is based on the state-of-the-art empirical studies of numerical cognition. It explains how humans have developed arithmetic from humble origins to its modern status as an almost universally possessed knowledge and skill. Central to the account is the realisation that, (...)
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  2. The Circular Theory.Ilexa Yardley - 2010 - Integrated Thought Concepts.
    Conservation of the Circle is the core (and, thus, the only) dynamic in Nature.
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  3. Abstract Objects.David Liggins - 2024 - Cambridge: Cambridge University Press.
    Philosophers often debate the existence of such things as numbers and propositions, and say that if these objects exist, they are abstract. But what does it mean to call something 'abstract'? And do we have good reason to believe in the existence of abstract objects? This Element addresses those questions, putting newcomers to these debates in a position to understand what they concern and what are the most influential considerations at work in this area of metaphysics. It also provides advice (...)
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  4. The Nature of Mathematical Objects.Carlo Cellucci - 2024 - In Bharath Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice. Cham: Springer. pp. 35-61.
    A traditional question in the philosophy of mathematics is to give an answer to the question: What is the nature of mathematical objects? This chapter considers the main answers that have been given to this question, specifically those according to which mathematical objects are independently existing entities, or abstractions, or logical objects, or simplifications, or mental constructions, or structures, or fictions, or idealizations of sensible things, or idealizations of operations. The chapter also shows the shortcomings of these answers, and considers (...)
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  5. Restricted nominalism about number and its problems.Stewart Shapiro, Richard Samuels & Eric Snyder - 2024 - Synthese 203 (5):1-23.
    Hofweber (Ontology and the ambitions of metaphysics, Oxford University Press, 2016) argues for a thesis he calls “internalism” with respect to natural number discourse: no expressions purporting to refer to natural numbers in fact refer, and no apparent quantification over natural numbers actually involves quantification over natural numbers as objects. He argues that while internalism leaves open the question of whether other kinds of abstracta exist, it precludes the existence of natural numbers, thus establishing what he calls “restricted nominalism” about (...)
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  6. Monism and the Ontology of Logic.Samuel Elgin - forthcoming - Milton Park, Abingdon, Oxon: Routledge.
    Monism is the claim that only one object exists. While few contemporary philosophers endorse monism, it has an illustrious history – stretching back to Bradley, Spinoza and Parmenides. In this paper, I show that plausible assumptions about the higher-order logic of property identity entail that monism is true. Given the higher-order framework I operate in, this argument generalizes: it is also possible to establish that there is a single property, proposition, relation, etc. I then show why this form of monism (...)
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  7. The Biological Framework for a Mathematical Universe.Ronald Williams - manuscript
    The mathematical universe hypothesis is a theory that the physical universe is not merely described by mathematics, but is mathematics, specifically a mathematical structure. Our research provides evidence that the mathematical structure of the universe is biological in nature and all systems, processes, and objects within the universe function in harmony with biological patterns. Living organisms are the result of the universe’s biological pattern and are embedded within their physiology the patterns of this biological universe. Therefore physiological patterns in living (...)
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  8. Exploring the Philosophy of Mathematics: Beyond Logicism and Platonism.Richard Startup - 2024 - Open Journal of Philosophy 14 (2):219-243.
    A perspective in the philosophy of mathematics is developed from a consideration of the strengths and limitations of both logicism and platonism, with an early focus on Frege’s work. Importantly, although many set-theoretic structures may be developed each of which offers limited isomorphism with the system of natural numbers, no one of them may be identified with it. Furthermore, the timeless, ever present nature of mathematical concepts and results itself offers direct access, in the face of a platonist account which (...)
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  9. Rules to Infinity: The Normative Role of Mathematics in Scientific Explanation.Mark Povich - 2024 - Oxford University Press USA.
    [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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  10. Et Dieu joua aux dés.Jean-Clet Martin - 2023 - Paris: Puf.
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  11. Mysticism in modern mathematics.Hastings Berkeley - 1910 - New York [etc.]: H. Frowde.
  12. Jean W. Rioux. Thomas Aquinas’ Mathematical Realism.Daniel Eduardo Usma Gómez - forthcoming - Philosophia Mathematica.
  13. Objects are (not) ...Friedrich Wilhelm Grafe - 2024 - Archive.Org.
    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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  14. A defense of Isaacson’s thesis, or how to make sense of the boundaries of finite mathematics.Pablo Dopico - 2024 - Synthese 203 (2):1-22.
    Daniel Isaacson has advanced an epistemic notion of arithmetical truth according to which the latter is the set of truths that we grasp on the basis of our understanding of the structure of natural numbers alone. Isaacson’s thesis is then the claim that Peano Arithmetic (PA) is the theory of finite mathematics, in the sense that it proves all and only arithmetical truths thus understood. In this paper, we raise a challenge for the thesis and show how it can be (...)
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  15. Why do numbers exist? A psychologist constructivist account.Markus Pantsar - forthcoming - Inquiry: An Interdisciplinary Journal of Philosophy.
    In this paper, I study the kind of questions we can ask about the existence of numbers. In addition to asking whether numbers exist, and how, I argue that there is also a third relevant question: why numbers exist. In platonist and nominalist accounts this question may not make sense, but in the psychologist account I develop, it is as well-placed as the other two questions. In fact, there are two such why-questions: the causal why-question asks what causes numbers to (...)
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  16. A Conventionalist Account of Distinctively Mathematical Explanation.Mark Povich - 2023 - Philosophical Problems in Science 74:171–223.
    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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  17. The orderly universe : how the calculus became an algorithm.Amir Alexander - 2022 - In Morgan G. Ames & Massimo Mazzotti (eds.), Algorithmic modernity: mechanizing thought and action, 1500-2000. New York, NY: Oxford University Press.
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  18. Carácter y trascendencia de las matemáticas en la época presente.Zoel Garcia de Galdeano - 1895 - Zaragoza: Impr. de C. Ariño.
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  19. Les mathématiques et la réalité.Ferdinand Gonseth - 1936 - Paris,: E. Alcan.
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  20. 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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  21. Metamathematik der Elementarmathematik.Erwin Engeler - 1983 - New York: Springer Verlag.
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  22. 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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  23. 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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  24. 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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  25. 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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  26. 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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  27. “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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  28. 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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  29. 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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  30. 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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  31. Die grundsätze und das wesen des unendlichen in der mathematik und philosophie.Friedrich Jacob Kurt Geissler - 1902 - Leipzig,: B. G. Teubner.
  32. 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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  33. Zum streit über die Grundlagen der Mathematik.Richard Hönigswald - 1912 - Heidelberg,: C. Winter.
  34. Die grundlagen der mathematik im lichte der anthroposophie.Ernst Bindel - 1928 - Stuttgart,: Waldorfschul-spielzeug und verlag.
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  35. 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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  36. 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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  37. 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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  38. Das hauptproblem der mathematik.Max Steck - 1942 - Berlin,: G. Lüttke.
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  39. El número y la realidad.Oscar Miró Quesada de la Guerra - 1944 - Lima, Perú,:
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  40. Das Wesen der Mathematik.Walter Lietzmann - 1949 - Braunschweig,: F. Vieweg.
    Altere Lehrbücher der Mathematik pflegten damit zu beginnen, die Frage nach dem Wesen der Mathematik und ihrer einzelnen Teilgebiete zu beantworten, ja, geradezu Definitionen dieser Be griffe zu geben. Wir sind heute davon abgekommen. Mit Recht! Denn eine solche Frage gehört nicht an den Anfang, sondern an den Schluß einer gewissen Beschäftigung mit Mathematik. Erst wenn man bereits etwas von der Mathematik kennengelernt hat, erscheint es angebracht, sich einmal über die mathematische Methode, über den Aufbau des Lehrgutes und über seine (...)
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  41. La philosophie des mathématiques.André Darbon - 1949 - Paris,: Presses Universitaires de France.
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  42. 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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  43. L'existence en mathématiques.Evert Willem Beth - 1956 - Paris,: Gauthier-Villars.
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  44. Structure et objet de l'analyse mathématique.Eloi Lefebvre - 1958 - Paris,: Gauthier-Villars.
  45. 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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  46. 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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  47. 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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  48. L'Esprit des mathématiques.Simone Goyard-Fabre - 1970 - Paris,: Éditions de l'École.
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  49. Qué es la matemática.Jorge Bosch - 1971 - [Buenos Aires]: Editorial Columba.
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  50. Sur la nature des mathématiques.Claude Paul Bruter - 1973 - Paris,: Gauthier-Villars.
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