Numbers

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. 
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321 found
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1 — 50 / 321
  1. Ltalicized Page Numbers Refer to Figures.J. Abbatucei, A. S. Abramson, E. H. Adelson, T. Adler, K. E. Adolph, J. Aerts, R. Agosti, T. Ahmad, G. Aimard & H. Akimotot - 2006 - In Günther Knoblich, Ian M. Thornton, Marc Grosjean & Maggie Shiffrar (eds.), Human Body Perception From the Inside Out. Oxford University Press.
  2. The Numbers in Italics Refer to the Pages on Which the Complete References Are Listed.R. P. Abeles, J. Adelson, A. Ahlgren, M. D. S. Ainsworth, G. W. Allport, R. Alpert, D. Anderson, M. Arnold, J. Aronfreed & Averill Jr - 1975 - In David J. DePalma & Jeanne M. Foley (eds.), Moral Development: Current Theory and Research. Halsted Press.
  3. The Nature and Purpose of Numbers.G. Aldo Antonelli - 2010 - Journal of Philosophy 107 (4):191-212.
  4. Uniform Numbers.Edward G. Armstrong - 1986 - American Journal of Semiotics 4 (1/2):99-127.
  5. Lowe on Locke{Textquoteright}s and Frege{Textquoteright}s Conceptions of Number.A. Arrieta-Urtizberea - 2010 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 17 (1):39-52.
    In his last book about Locke{\textquoteright}s philosophy, E. J. Lowe claims that Frege{\textquoteright}s arguments against the Lockean conception of number are not compelling, while at the same time he painstakingly defines the Lockean conception Lowe himself espouses. The aim of this paper is to show that the textual evidence considered by Lowe may be interpreted in another direction. This alternative reading of Frege{\textquoteright}s arguments throws light on Frege{\textquoteright}s and Lowe{\textquoteright}s different agendas. Moreover, in this paper, the problem of singular sentences (...)
  6. Saying It With Numerals.David Auerbach - 1994 - Notre Dame Journal of Formal Logic 35 (1):130-146.
    This article discusses the nature of numerals and the plausibility of their special semantic and epistemological status as proper names of numbers. Evidence is presented that minimizes the difference between numerals and other devices of direct reference. The availability of intensional contexts within formalised metamathematics is exploited to shed light on the relation between formal numerals and numerals.
  7. David Lindberg & Ronald Numbers, Eds.: "God and Nature". [REVIEW]William H. Austin - 1988 - The Thomist 52 (3):562.
  8. Philosophy of Mathematics.Jeremy Avigad - manuscript
    The philosophy of mathematics plays an important role in analytic philosophy, both as a subject of inquiry in its own right, and as an important landmark in the broader philosophical landscape. Mathematical knowledge has long been regarded as a paradigm of human knowledge with truths that are both necessary and certain, so giving an account of mathematical knowledge is an important part of epistemology. Mathematical objects like numbers and sets are archetypical examples of abstracta, since we treat such objects in (...)
  9. Empty de Re Attitudes About Numbers.Jody Azzouni - 2008 - Philosophia Mathematica 17 (2):163-188.
    I dub a certain central tradition in philosophy of language (and mind) the de re tradition. Compelling thought experiments show that in certain common cases the truth conditions for thoughts and public-language expressions categorically turn on external objects referred to, rather than on linguistic meanings and/or belief assumptions. However, de re phenomena in language and thought occur even when the objects in question don't exist. Call these empty de re phenomena. Empty de re thought with respect to numeration is explored (...)
  10. Averaged Coordination Numbers of Planar Aperiodic Tilings.M. Baake & U. Grimm - 2006 - Philosophical Magazine 86 (3-5):567-572.
  11. Representation of Ordinal Numbers and Derived Sets in Certain Continuous Sets.Frederick Bagemihl - 1981 - Mathematical Logic Quarterly 27 (19‐21):333-336.
  12. The Early Numerals.Lilian M. Bagge - 1906 - The Classical Review 20 (05):259-267.
  13. Numbers Up.Julian Baggini - 2004 - The Philosophers' Magazine 27:30-33.
  14. Fixpoints Without the Natural Numbers.B. Banaschewski - 1991 - Mathematical Logic Quarterly 37 (8):125-128.
  15. Another Characterization of the Natural Numbers.Thomjas Bedürftig - 1989 - Mathematical Logic Quarterly 35 (2):185-186.
  16. The Magic of Numbers.Eric Temple Bell - 1946 - London: Mcgraw-Hill Book Company.
    It probes the work of Pythagoras, Galileo, Berkeley, Einstein, and others, exploring how "number magic" has influenced religion, philosophy, science, and mathematics.
  17. The Real Numbers: Frege's Criticism of Cantor and Dedekind.Jean-Pierre Belna - 1997 - Revue d'Histoire des Sciences 50 (1).
  18. What Numbers Could Not Be.Paul Benacerraf - 1965 - Philosophical Review 74 (1):47-73.
  19. Constructibility and Mathematical Existence.José A. Benardete - 1991 - Review of Metaphysics 45 (1):114-115.
  20. On Ordering and Multiplication of Natural Numbers.Kamila Bendová - 2001 - Archive for Mathematical Logic 40 (1):19-23.
    Even if the ordering of all natural number is (known to be) not definable from multiplication of natural numbers and ordering of primes, there is a simple axiom system in the language $(\times,<,1)$ such that the multiplicative structure of positive integers has a unique expansion by a linear order coinciding with the standard order for primes and satisfying the axioms – namely the standard one.
  21. Second-Order Arithmetic Sans Sets.L. Berk - 2013 - Philosophia Mathematica 21 (3):339-350.
    This paper examines the ontological commitments of the second-order language of arithmetic and argues that they do not extend beyond the first-order language. Then, building on an argument by George Boolos, we develop a Tarski-style definition of a truth predicate for the second-order language of arithmetic that does not involve the assignment of sets to second-order variables but rather uses the same class of assignments standardly used in a definition for the first-order language.
  22. Number and Numbers. [REVIEW]Anindya Bhattacharyya - 2009 - Radical Philosophy 156.
  23. Introduction to Abstract Mathematics.T. A. Bick - 1971 - Academic Press.
  24. Back Numbers Needed.J. D. Bishop - 1956 - Classical World: A Quarterly Journal on Antiquity 50:150.
  25. Platonic Number in the Parmenides and Metaphysics XIII.Dougal Blyth - 2000 - International Journal of Philosophical Studies 8 (1):23 – 45.
    I argue here that a properly Platonic theory of the nature of number is still viable today. By properly Platonic, I mean one consistent with Plato's own theory, with appropriate extensions to take into account subsequent developments in mathematics. At Parmenides 143a-4a the existence of numbers is proven from our capacity to count, whereby I establish as Plato's the theory that numbers are originally ordinal, a sequence of forms differentiated by position. I defend and interpret Aristotle's report of a Platonic (...)
  26. Realism’s Understanding of Negative Numbers.Petar Bojanic & Sanja Todorovic - 2016 - Filozofija I Društvo 27 (1):131-136.
  27. Logic and Arithmetic, Vol. II--Rational and Irrational Numbers.David Bostock - 1981 - Mind 90 (359):473-475.
  28. The Existence of Numbers (Or: What is the Status of Arithmetic?) By V2.00 Created: 11 Oct 2001 Modified: 3 June 2002 Please Send Your Comments to Abo. [REVIEW]Andrew Boucher - manuscript
    I begin with a personal confession. Philosophical discussions of existence have always bored me. When they occur, my eyes glaze over and my attention falters. Basically ontological questions often seem best decided by banging on the table--rocks exist, fairies do not. Argument can appear long-winded and miss the point. Sometimes a quick distinction resolves any apparent difficulty. Does a falling tree in an earless forest make noise, ie does the noise exist? Well, if noise means that an ear must be (...)
  29. The Significance of Complex Numbers for Frege's Philosophy of Mathematics.Robert Brandom - 1996 - Proceedings of the Aristotelian Society 96 (1):293 - 315.
  30. The Numbers Don't Lie.Lester Brown - 1999 - Free Inquiry 19.
  31. The Writings of Castelli, Benedetto on Negative Numbers (1631-1635)+ Mathematics in the Galilei Circle.M. Bucciantini - 1985 - Giornale Critico Della Filosofia Italiana 5 (2):215-228.
  32. New Waves in Philosophy of Mathematics.Otávio Bueno & Øystein Linnebo (eds.) - 2009 - Palgrave-Macmillan.
    Thirteen up-and-coming researchers in the philosophy of mathematics have been invited to write on what they take to be the right philosophical account of mathematics, examining along the way where they think the philosophy of mathematics is and ought to be going. A rich and diverse picture emerges. Some broader tendencies can nevertheless be detected: there is increasing attention to the practice, language and psychology of mathematics, a move to reassess the orthodoxy, as well as inspiration from philosophical logic.
  33. Approximation of Complex Algebraic Numbers by Algebraic Numbers of Bounded Degree.Yann Bugeaud & Jan-Hendrik Evertse - 2009 - Annali della Scuola Normale Superiore di Pisa 8 (2):333-368.
  34. Counting Numbers.Michael Bulley - 1990 - Cogito 4 (1):41-47.
  35. Everybody Counts but Not Everybody Understands Numbers: The Unrecognised Handicap of Dyscalculia.B. Butterworth - forthcoming - Proceedings of the British Academy.
  36. 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.
  37. Numbers and Relations.Y. I. Byeong-Uk - unknown
    In this paper, I criticize John Bigelow's account of number and present my own account that results from the criticism. In doing so, I argue that proper understanding of the nature of number requires a radical departure from the standard conception of language and reality and outline the alternative conception that underlies my account of number. I argue that Bigelow's account of number rests on an incorrect analysis of the plural constructions underlying the talk of number and propound an analysis (...)
  38. O przedmiocie matematycznym.Piotr Błaszczyk - 2004 - Filozofia Nauki 2 (1):45-59.
    In this paper we show that the field of the real numbers is an intentional object in the sense specified by Roman Ingarden in his Das literarische Kunstwer and Der Streit um die Existenz der Welt. An ontological characteristics of a classic example of an intentional object, i.e. a literary character, is developed. There are three principal elements of such an object: the author, the text and the entity in which the literary character forms the content. In the case of (...)
  39. On Mathematical Instrumentalism.Patrick Caldon & Aleksandar Ignjatović - 2005 - Journal of Symbolic Logic 70 (3):778 - 794.
    In this paper we devise some technical tools for dealing with problems connected with the philosophical view usually called mathematical instrumentalism. These tools are interesting in their own right, independently of their philosophical consequences. For example, we show that even though the fragment of Peano's Arithmetic known as IΣ₁ is a conservative extension of the equational theory of Primitive Recursive Arithmetic (PRA). IΣ₁ has a super-exponential speed-up over PRA. On the other hand, theories studied in the Program of Reverse Mathematics (...)
  40. The Making of an Abstract Concept: Natural Number.Susan Carey - 2010 - In Denis Mareschal, Paul Quinn & Stephen E. G. Lea (eds.), The Making of Human Concepts. Oxford University Press. pp. 265.
  41. How Real Are Real Numbers?Gregory Chaitin - 2011 - Manuscrito 34 (1):115-141.
    We discuss mathematical and physical arguments against continuity and in favor of discreteness, with particular emphasis on the ideas of Émile Borel.
  42. Primitive Recursive Real Numbers.Qingliang Chen, Kaile Su & Xizhong Zheng - 2007 - Mathematical Logic Quarterly 53 (4‐5):365-380.
    In mathematics, various representations of real numbers have been investigated. All these representations are mathematically equivalent because they lead to the same real structure – Dedekind-complete ordered field. Even the effective versions of these representations are equivalent in the sense that they define the same notion of computable real numbers. Although the computable real numbers can be defined in various equivalent ways, if “computable” is replaced by “primitive recursive” , these definitions lead to a number of different concepts, which we (...)
  43. Pythagorean Powers or a Challenge to Platonism.Colin Cheyne & Charles R. Pigden - 1996 - Australasian Journal of Philosophy 74 (4):639 – 645.
    The Quine/Putnam indispensability argument is regarded by many as the chief argument for the existence of platonic objects. We argue that this argument cannot establish what its proponents intend. The form of our argument is simple. Suppose indispensability to science is the only good reason for believing in the existence of platonic objects. Either the dispensability of mathematical objects to science can be demonstrated and, hence, there is no good reason for believing in the existence of platonic objects, or their (...)
  44. What is a Number?Ulrich Christiansen - 2004 - Philosophy of Mathematics Education Journal 18.
  45. How to Deal with Janus'face of Natural Numbers?Demetra Christopoulou - unknown
    This paper addresses a dilemma arising from the linguistic behaviour of arithmetical expressions in two basic ways: they occur, either as singular terms in identity statements or as predicates of concepts in adjectival statements. However, those forms of syntactical behaviour give rise to opposite accounts of the ontological status of natural numbers. The substantival use of arithmetical expressions is associated with the interpretation of natural numbers as abstract particulars while the adjectival use of arithmetical expressions ordinarily supports the interpretation of (...)
  46. Multiple Reductions Revisited.Justin Clarke-Doane - 2007 - Philosophia Mathematica 16 (2):244-255.
    Paul Benacerraf's argument from multiple reductions consists of a general argument against realism about the natural numbers (the view that numbers are objects), and a limited argument against reductionism about them (the view that numbers are identical with prima facie distinct entities). There is a widely recognized and severe difficulty with the former argument, but no comparably recognized such difficulty with the latter. Even so, reductionism in mathematics continues to thrive. In this paper I develop a difficulty for Benacerraf's argument (...)
  47. Science Without Numbers. [REVIEW]Nino Cocchiarella - 1984 - International Studies in Philosophy 16 (1):93-95.
  48. On the Question 'Do Numbers Exist?'.Arthur W. Collins - 1998 - Philosophical Quarterly 48 (190):23-36.
    Since we know that there are four prime numbers less than 8 we know that there are numbers. This ‘short argument’ is correct but it is not an ontological claim or part of philosophy of mathematics. Both realists (Quine) and nominalists (Field) reject the short argument and adopt the idea that the existence of numbers might be posited to explain known mathematical truths. Philosophers operate with a negative conception of what numbers are: they are not in space and time, not (...)
  49. It All Adds Up …. Or Does It? Numbers, Mathematics and Purpose.Simon Conway Morris - forthcoming - Studies in History and Philosophy of Science Part C: Studies in History and Philosophy of Biological and Biomedical Sciences.
  50. Mathematical Kinds, or Being Kind to Mathematics.David Neil Corfield - 2004 - Philosophica 74.
    In 1908, Henri Poincar? claimed that: ...the mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a mathematical law, just as experimental facts lead us to the knowledge of a physical law. They are those which reveal to us unsuspected kinship between other facts, long known, but wrongly believed to be strangers to one another. Towards the end of the twentieth century, with many more mathematical (...)
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