Edited by Rafal Urbaniak (University of Ghent, University of Gdansk)
Assistant editors: Pawel Pawlowski, Sam Roberts
About this topic
Summary Various theories concerned with numbers (arithmetic, real number theory, ...) are among the most often taught and applied mathematical theories. Accordingly, philosophers paid a significant amount of attention to considerations pertaining the status of such theories and the nature of numbers and number-theoretic discourse. Because of their relative simplicity, philosophical discussion surrounding such theories provide a neat proving ground for various wider philosophical accounts of mathematics, which makes this category fairly closely intertwined with other categories falling under Ontology of Mathematics.
Key works Frege 1950 is a seminal work on the philosophy of numbers (his approached has been further developed byWright 1983). A very good anthology of classic papers is Heijenoort 1967.
Introductions Potter 2000 is a nice book to start with. 
Related categories

344 found
1 — 50 / 344
  1. added 2019-03-01
    On the Reality of the Continuum Discussion Note: A Reply to Ormell, ‘Russell's Moment of Candour’, Philosophy: Anne Newstead and James Franklin.Anne Newstead - 2008 - Philosophy 83 (1):117-127.
    In a recent article, Christopher Ormell argues against the traditional mathematical view that the real numbers form an uncountably infinite set. He rejects the conclusion of Cantor’s diagonal argument for the higher, non-denumerable infinity of the real numbers. He does so on the basis that the classical conception of a real number is mys- terious, ineffable, and epistemically suspect. Instead, he urges that mathematics should admit only ‘well-defined’ real numbers as proper objects of study. In practice, this means excluding as (...)
  2. added 2019-02-05
    On the Varieties of Abstract Objects.James E. Davies - forthcoming - Australasian Journal of Philosophy:1-15.
    I reconcile the spatiotemporal location of repeatable artworks and impure sets with the non-location of natural numbers despite all three being varieties of abstract objects. This is possible because, while the identity conditions for all three can be given by abstraction principles, in the former two cases spatiotemporal location is a congruence for the equivalence relation featuring in the relevant principle, whereas in the latter it is not. I then generalize this to other ‘physical’ properties like shape, mass, and causal (...)
  3. added 2018-12-22
    Plato’s Philosophy of Cognition by Mathematical Modelling.Roman S. Kljujkov & Sergey Kljujkov - 2014 - Dialogue and Universalism 24 (3):110-115.
    By the end of his life Plato had rearranged the theory of ideas into his teaching about ideal numbers, but no written records have been left. The Ideal mathematics of Plato is present in all his dialogues. It can be clearly grasped in relation to the effective use of mathematical modelling. Many problems of mathematical modelling were laid in the foundation of the method by cutting the three-level idealism of Plato to the single-level “ideism” of Aristotle. For a long time, (...)
  4. added 2018-12-08
    Approaching Infinity.Michael Huemer - 2016 - New York: Palgrave Macmillan.
  5. added 2018-12-08
    Aristotelian Finitism.Tamer Nawar - 2015 - Synthese 192 (8):2345-2360.
    It is widely known that Aristotle rules out the existence of actual infinities but allows for potential infinities. However, precisely why Aristotle should deny the existence of actual infinities remains somewhat obscure and has received relatively little attention in the secondary literature. In this paper I investigate the motivations of Aristotle’s finitism and offer a careful examination of some of the arguments considered by Aristotle both in favour of and against the existence of actual infinities. I argue that Aristotle has (...)
  6. added 2018-11-23
    How Abstract Objects Strike Us.Michael Liston - 1994 - Dialectica 48 (1):3-27.
    SummaryBenacerraf challenges us to account for the reliability of our mathematical beliefs given that there appear to be no natural connections between mathematical believers and mathematical ontology. In this paper I try to do two things. I argue that the interactionist view underlying this challenge renders inexplicable not only the reliability of our mathematical beliefs, construed either platonistically or naturalistically , but also the reliability of most of our beliefs in physics. I attempt to counter Benacerraf's challenge by sketching an (...)
  7. added 2018-11-22
    Arbitrary Reference, Numbers, and Propositions.Michele Palmira - 2018 - European Journal of Philosophy 26 (3):1069-1085.
    Reductionist realist accounts of certain entities, such as the natural numbers and propositions, have been taken to be fatally undermined by what we may call the problem of arbitrary identification. The problem is that there are multiple and equally adequate reductions of the natural numbers to sets (see Benacerraf, 1965), as well as of propositions to unstructured or structured entities (see, e.g., Bealer, 1998; King, Soames, & Speaks, 2014; Melia, 1992). This paper sets out to solve the problem by canvassing (...)
  8. added 2018-10-09
    Ontology and the Ambitions of Metaphysics.Thomas Hofweber - 2016 - Oxford University Press UK.
    Many significant problems in metaphysics are tied to ontological questions, but ontology and its relation to larger questions in metaphysics give rise to a series of puzzles that suggest that we don't fully understand what ontology is supposed to do, nor what ambitions metaphysics can have for finding out about what the world is like. Thomas Hofweber aims to solve these puzzles about ontology and consequently to make progress on four metaphysical debates tied to ontology: the philosophy of arithmetic, the (...)
  9. added 2018-10-01
    Deflationary Nominalism and Puzzle Avoidance†.David Mark Kovacs - 2019 - Philosophia Mathematica 27 (1):88-104.
    In a series of works, Jody Azzouni has defended deflationary nominalism, the view that certain sentences quantifying over mathematical objects are literally true, although such objects do not exist. One alleged attraction of this view is that it avoids various philosophical puzzles about mathematical objects. I argue that this thought is misguided. I first develop an ontologically neutral counterpart of Field’s reliability challenge and argue that deflationary nominalism offers no distinctive answer to it. I then show how this reasoning generalizes (...)
  10. added 2018-09-21
    The Adverbial Theory of Numbers: Some Clarifications.Joongol Kim - forthcoming - Synthese:1-20.
    In a forthcoming paper in this journal, entitled “Bad company objection to Joongol Kim’s adverbial theory of numbers”, Namjoong Kim presents an ingenious Russell-style paradox based on an analogue of Kim’s definition of the number 1, and argues that Kim’s theory needs to provide a criterion of demarcation between acceptable and unacceptable definitions of adverbial entities. This paper addresses this ‘bad company’ objection and some other related issues concerning Kim’s adverbial theory by clarifying the purposes and uses of the formal (...)
  11. added 2018-09-20
    Bad Company Objection to Joongol Kim’s Adverbial Theory of Numbers.Namjoong Kim - forthcoming - Synthese.
  12. added 2018-09-06
    Frege Numbers and the Relativity Argument.Christopher Menzel - 1988 - Canadian Journal of Philosophy 18 (1):87-98.
  13. added 2018-08-17
    Numbers and Propositions: Reply to Melia.Tim Crane - 1992 - Analysis 52 (4):253-256.
    Is the way we use propositions to individuate beliefs and other intentional states analogous to the way we use numbers to measure weights and other physical magnitudes? In an earlier paper [2], I argued that there is an important disanalogy. One and the same weight can be 'related to' different numbers under different units of measurement. Moreover, the choice of a unit of measurement is arbitrary,in the sense that which way we choose doesn't affect the weight attributed to the object. (...)
  14. added 2018-07-16
    Maths, Logic and Language.Tetsuaki Iwamoto - 2018 - Geneva: Logic Forum.
    A work on the philosophy of mathematics (2017) -/- ‘Number’, such a simple idea, and yet it fascinated and absorbed the greatest proportion of human geniuses over centuries, not to mention the likes of Pythagoras, Euclid, Newton, Leibniz, Descartes and countless maths giants like Euler, Gauss and Hilbert, etc.. Einstein thought of pure maths as the poetry of logical ideas, the exactitude of which, although independent of experience, strangely seems to benefit the study of the objects of reality. And, interestingly (...)
  15. added 2018-04-02
    Beyond Witches, Angels and Unicorns. The Possibility of Expanding Russell´s Existential Analysis.Olga Ramirez - 2018 - E-Logos Electronic Journal for Philosophy 25 (1):4-15.
    This paper attempts to be a contribution to the epistemological project of explaining complex conceptual structures departing from more basic ones. The central thesis of the paper is that there are what I call “functionally structured concepts”, these are non-harmonic concepts in Dummett’s sense that might be legitimized if there is a function that justifies the tie between the inferential connection the concept allows us to trace. Proving this requires enhancing the russellian existential analysis of definite descriptions to apply to (...)
  16. added 2018-03-31
    Our Incorrigible Ontological Relations and Categories of Being.Julian M. Galvez Bunge (ed.) - 2017 - USA: Amazon.
    The object of this book is to present a radical novel conception of the ontological categories, their nature and epistemic importance. A conception that constitutes a challenge to the prevailing tenets, if not paradigms, of ontology today. The arguments and observations are given without addressing nor directly contesting the current theories on the subject. However, its author emphasises some of the main conclusions that entail from the new perspective, in particular regarding the role of philosophy among the sciences. Departing from (...)
  17. added 2018-02-19
    Time and the Russell Definition of Number.Charles Byron Cross - 1979 - Southwestern Journal of Philosophy 10 (2):177-180.
  18. added 2018-02-17
    Where Do the Natural Numbers Come From?Harold T. Hodes - 1990 - Synthese 84 (3):347-407.
  19. added 2018-02-16
    The Indefinability of €œOne”.Laurence Goldstein - 2002 - Journal of Philosophical Logic 31 (1):29-42.
    Logicism is one of the great reductionist projects. Numbers and the relationships in which they stand may seem to possess suspect ontological credentials – to be entia non grata – and, further, to be beyond the reach of knowledge. In seeking to reduce mathematics to a small set of principles that form the logical basis of all reasoning, logicism holds out the prospect of ontological economy and epistemological security. This paper attempts to show that a fundamental logicist project, that of (...)
  20. added 2018-02-16
    Why Numbers Are Sets.Eric Steinhart - 2002 - Synthese 133 (3):343-361.
    I follow standard mathematical practice and theory to argue that the natural numbers are the finite von Neumann ordinals. I present the reasons standardly given for identifying the natural numbers with the finite von Neumann's. I give a detailed mathematical demonstration that 0 is {} and for every natural number n, n is the set of all natural numbers less than n. Natural numbers are sets. They are the finite von Neumann ordinals.
  21. added 2017-12-31
    To Reduce Nothingness Into a Reference by Falsity.Hazhir Roshangar - manuscript
    Assuming the absolute nothingness as the most basic object of thought, I present a way to refer to this object, by reducing it onto a primitive object that supersedes and comes right after the absolute nothingness. The new primitive object that is constructed can be regarded as a formal system that can generate some infinite variety of symbols. [The PDF here is outdated, for a recent draft please contact me.].
  22. added 2017-11-28
    What We Talk About When We Talk About Numbers.Richard Pettigrew - manuscript
    In this paper, I describe and motivate a new species of mathematical structuralism, which I call Instrumental Nominalism about Set-Theoretic Structuralism. As the name suggests, this approach takes standard Set-Theoretic Structuralism of the sort championed by Bourbaki and removes its ontological commitments by taking an instrumental nominalist approach to that ontology of the sort described by Joseph Melia and Gideon Rosen. I argue that this avoids all of the problems that plague other versions of structuralism.
  23. added 2017-11-09
    Fourteen Arguments in Favour of a Formalist Philosophy of Real Mathematics.Karlis Podnieks - 2015 - Baltic Journal of Modern Computing 3 (1):1-15.
    The formalist philosophy of mathematics (in its purest, most extreme version) is widely regarded as a “discredited position”. This pure and extreme version of formalism is called by some authors “game formalism”, because it is alleged to represent mathematics as a meaningless game with strings of symbols. Nevertheless, I would like to draw attention to some arguments in favour of game formalism as an appropriate philosophy of real mathematics. For the most part, these arguments have not yet been used or (...)
  24. added 2017-11-06
    Chapter 6: Reifying Terms.Friederike Moltmann - 2013 - In Abstract Objects and the Semantics of Natural Language. Oxford: Oxford University Press.
    This chapter develops a semantics for 'reifying terms' of the sort 'the proposition that S', 'the fact that S', 'the property of being P', 'the number eight', 'the concept horse', 'the truth value true', 'the kind humane being'. This semantics is developed within the broader perspective of the ontology of natural language involving abstract objects only at its periphery, not its core.
  25. added 2017-09-28
    Das Unendliche und die Zahl.Hugo Bergmann - 1914 - Revue de Métaphysique et de Morale 22 (2):16-16.
  26. added 2017-09-23
    Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege's Grundgesetze in Object Theory.Edward N. Zalta - 1999 - Journal of Philosophical Logic 28 (6):619-660.
    In this paper, the author derives the Dedekind-Peano axioms for number theory from a consistent and general metaphysical theory of abstract objects. The derivation makes no appeal to primitive mathematical notions, implicit definitions, or a principle of infinity. The theorems proved constitute an important subset of the numbered propositions found in Frege's *Grundgesetze*. The proofs of the theorems reconstruct Frege's derivations, with the exception of the claim that every number has a successor, which is derived from a modal axiom that (...)
  27. added 2017-09-07
    Learning the Natural Numbers as a Child.Stefan Buijsman - 2019 - Noûs 53 (1):3-22.
    How do we get out knowledge of the natural numbers? Various philosophical accounts exist, but there has been comparatively little attention to psychological data on how the learning process actually takes place. I work through the psychological literature on number acquisition with the aim of characterising the acquisition stages in formal terms. In doing so, I argue that we need a combination of current neologicist accounts and accounts such as that of Parsons. In particular, I argue that we learn the (...)
  28. added 2017-09-06
    Does Ontology Rest on a Mistake?Stephen Yablo & Andre Gallois - 1998 - Proceedings of the Aristotelian Society, Supplementary Volumes( 72:229-283.
    [Stephen Yablo] The usual charge against Carnap's internal/external distinction is one of 'guilt by association with analytic/synthetic'. But it can be freed of this association, to become the distinction between statements made within make-believe games and those made outside them-or, rather, a special case of it with some claim to be called the metaphorical/literal distinction. Not even Quine considers figurative speech committal, so this turns the tables somewhat. To determine our ontological commitments, we have to ferret out all traces of (...)
  29. added 2017-06-30
    Talking About Numbers: Easy Arguments for Mathematical Realism. [REVIEW]Richard Lawrence - 2017 - History and Philosophy of Logic 38 (4):390-394.
  30. added 2017-06-21
    Numbers and Everything.Gonçalo Santos - 2013 - Philosophia Mathematica 21 (3):297-308.
    I begin by drawing a parallel between the intuitionistic understanding of quantification over all natural numbers and the generality relativist understanding of quantification over absolutely everything. I then argue that adoption of an intuitionistic reading of relativism not only provides an immediate reply to the absolutist's charge of incoherence but it also throws a new light on the debates surrounding absolute generality.
  31. added 2017-06-01
    Number Words as Number Names.Friederike Moltmann - 2017 - Linguistics and Philosophy 40 (4):331-345.
    This paper critically evaluates the view according to which number words in argument position retain the meaning they have when occurring as determiners or adjectives. In particular, it argues against syntactic evidence from German given by myself in support of that view.
  32. added 2017-03-25
    The Psychology and Philosophy of Natural Numbers.Oliver R. Marshall - 2017 - Philosophia Mathematica (1):nkx002.
    ABSTRACT I argue against both neuropsychological and cognitive accounts of our grasp of numbers. I show that despite the points of divergence between these two accounts, they face analogous problems. Both presuppose too much about what they purport to explain to be informative, and also characterize our grasp of numbers in a way that is absurd in the light of what we already know from the point of view of mathematical practice. Then I offer a positive methodological proposal about the (...)
  33. added 2017-02-18
    Abstract Entities.Cowling Sam - 2017 - Routledge.
    Think of a number, any number, or properties like _fragility_ and _humanity_. These and other abstract entities are radically different from concrete entities like electrons and elbows. While concrete entities are located in space and time, have causes and effects, and are known through empirical means, abstract entities like meanings and possibilities are remarkably different. They seem to be immutable and imperceptible and to exist "outside" of space and time. This book provides a comprehensive critical assessment of the problems raised (...)
  34. added 2017-02-15
    Review of Gabriele Lolli, Numeri. La creazione continua della matematica. [REVIEW]Longa Gianluca - 2016 - Lo Sguardo. Rivista di Filosofia 21 (II):377-380.
  35. added 2017-02-15
    In Measure, Number, and Weight: Studies in Mathematics and Culture. [REVIEW]George Molland - 1996 - British Journal for the History of Science 29 (2):229-230.
  36. added 2017-02-15
    Counting on Number: Plato on the Goodness of Arithmos.David Roochnik - 1994 - American Journal of Philology 115 (4):543-563.
  37. added 2017-02-15
    Where Do the Cardinal Numbers Come From?Harold T. Hodes - 1983 - Journal of Philosophy 80 (9999):655-656.
  38. added 2017-02-14
    Numbers and Narratives of Modernity.U. Kalpagam - 2000 - In A. K. Raina, B. N. Patnaik & Monima Chadha (eds.), Science and Tradition. Inter-University Centre for Humanities and Social Sciences, Indian Institute of Advanced Study. pp. 168.
  39. added 2017-02-14
    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.
  40. added 2017-02-14
    In Measure, Number, and Weight: Studies in Mathematics and Culture.J. Hoyrup & I. Grattan-Guinness - 1995 - Annals of Science 52 (6):623.
  41. added 2017-02-13
    Institutional Critique-by-Numbers-a Reply to Esther Leslie.D. Willsdon - 2002 - Radical Philosophy 111:55-56.
  42. added 2017-02-13
    Governing by Numbers: Why Ranking Systems Matter.Peter Miller - 2001 - Social Research 68 (2):379-96.
  43. added 2017-02-13
    Why the Largest Number Imaginable is Still a Finite Number.Jean Paul Van Bendegem - 1999 - Logique Et Analyse 42 (165-166).
  44. added 2017-02-13
    X2. The Arithmetic of Cardinal Numbers. Cardinal Numbers Were Intro.Thomas Jech - 1995 - Bulletin of Symbolic Logic 1 (4).
  45. added 2017-02-13
    Some Numerological Features of Beethoven's Output.I. Grattan-Guinness - 1994 - Annals of Science 51 (2):103-135.
    It is argued that Beethoven used a system of numbers to guide aspects of many of his works, especially major ones. The numbers manifest themselves in the number of notes in a melody and/or of bars in a work or part of it, in groupings and numberings of works of a given kind, and in his deliberate choice of Opus numbers. They are not only small ones such as 3 , which of course turn up frequently anyway; larger ones are (...)
  46. added 2017-02-12
    How to Name a Real Number?Vojtech Kolman - 2011 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 18 (3):283-301.
  47. added 2017-02-12
    The Reality of Numbers: A Physicalist's Philosophy of Mathematics.A. J. Dale - 1990 - Philosophical Books 31 (1):61-62.
  48. added 2017-02-12
    Representation of Ordinal Numbers and Derived Sets in Certain Continuous Sets.Frederick Bagemihl - 1981 - Mathematical Logic Quarterly 27 (19‐21):333-336.
  49. added 2017-02-11
    Whitehead's Early Philosophy of Mathematics and the Development of Formalism.Rosen Lutskanov - 2011 - Logique Et Analyse 54 (214).
  50. added 2017-02-11
    Gestures Expressing Numbers—or—Numbers Expressed by Gestures.Vilmos Voigt - 2010 - American Journal of Semiotics 26 (1/4):111 - 127.
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