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Numbers

Edited by Rafal Urbaniak (University of Ghent, University of Gdansk)
Assistant editors: Sam Roberts, Pawel Pawlowski
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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 1980 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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  1. J. Abbatucei, A. S. Abramson, E. H. Adelson, T. Adler, K. E. Adolph, J. Aerts, R. Agosti, T. Ahmad, G. Aimard & H. Akimotot (2006). Ltalicized Page Numbers Refer to Figures. In Günther Knoblich, Ian M. Thornton, Marc Grosjean & Maggie Shiffrar (eds.), Human Body Perception From the Inside Out. Oxford University Press.
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  2. G. Aldo Antonelli (2010). The Nature and Purpose of Numbers. Journal of Philosophy 107 (4):191-212.
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  3. Edward G. Armstrong (1986). Uniform Numbers. American Journal of Semiotics 4 (1/2):99-127.
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  4. David Auerbach (1994). Saying It With Numerals. 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.
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  5. William H. Austin (1988). David Lindberg & Ronald Numbers, Eds.: "God and Nature". [REVIEW] The Thomist 52 (3):562.
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  6. Jeremy Avigad, Philosophy of Mathematics.
    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 (...)
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  7. Jody Azzouni (2008). Empty de Re Attitudes About Numbers. 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 (...)
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  8. M. Baake & U. Grimm (2006). Averaged Coordination Numbers of Planar Aperiodic Tilings. Philosophical Magazine 86 (3-5):567-572.
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  9. Julian Baggini (2004). Numbers Up. The Philosophers' Magazine 27:30-33.
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  10. B. Banaschewski (1991). Fixpoints Without the Natural Numbers. Mathematical Logic Quarterly 37 (8):125-128.
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  11. Thomjas Bedürftig (1989). Another Characterization of the Natural Numbers. Mathematical Logic Quarterly 35 (2):185-186.
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  12. Eric Temple Bell (1946). The Magic of Numbers. 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.
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  13. Paul Benacerraf (1965). What Numbers Could Not Be. Philosophical Review 74 (1):47-73.
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  14. José A. Benardete (1991). Constructibility and Mathematical Existence. Review of Metaphysics 45 (1):114-115.
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  15. Kamila Bendová (2001). On Ordering and Multiplication of Natural Numbers. 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.
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  16. L. Berk (2013). Second-Order Arithmetic Sans Sets. 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.
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  17. Anindya Bhattacharyya (2009). Number and Numbers. [REVIEW] Radical Philosophy 156.
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  18. T. A. Bick (1971). Introduction to Abstract Mathematics. Academic Press.
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  19. J. D. Bishop (1956). Back Numbers Needed. Classical World: A Quarterly Journal on Antiquity 50:150.
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  20. Dougal Blyth (2000). Platonic Number in the Parmenides and Metaphysics XIII. 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 (...)
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  21. Petar Bojanic & Sanja Todorovic (2016). Realism’s Understanding of Negative Numbers. Filozofija I Društvo 27 (1):131-136.
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  22. Andrew Boucher, 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]
    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 (...)
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  23. Lester Brown (1999). The Numbers Don't Lie. Free Inquiry 19.
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  24. M. Bucciantini (1985). The Writings of Castelli, Benedetto on Negative Numbers (1631-1635)+ Mathematics in the Galilei Circle. Giornale Critico Della Filosofia Italiana 5 (2):215-228.
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  25. Otávio Bueno & Øystein Linnebo (eds.) (2009). New Waves in Philosophy of Mathematics. 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.
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  26. Yann Bugeaud & Jan-Hendrik Evertse (2009). Approximation of Complex Algebraic Numbers by Algebraic Numbers of Bounded Degree. Annali della Scuola Normale Superiore di Pisa 8 (2):333-368.
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  27. Tim Button & Sean Walsh (2016). Structure and Categoricity: Determinacy of Reference and Truth Value in the Philosophy of Mathematics. 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.
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  28. Y. I. Byeong-Uk, Numbers and Relations.
    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 (...)
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  29. Piotr Błaszczyk (2004). O przedmiocie matematycznym. 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 (...)
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  30. Patrick Caldon & Aleksandar Ignjatović (2005). On Mathematical Instrumentalism. 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 (...)
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  31. Susan Carey (2010). The Making of an Abstract Concept: Natural Number. In Denis Mareschal, Paul Quinn & Stephen E. G. Lea (eds.), The Making of Human Concepts. Oxford University Press. pp. 265.
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  32. Gregory Chaitin (2011). How Real Are Real Numbers? 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.
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  33. Qingliang Chen, Kaile Su & Xizhong Zheng (2007). Primitive Recursive Real Numbers. 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 (...)
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  34. Colin Cheyne & Charles R. Pigden (1996). Pythagorean Powers or a Challenge to Platonism. 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 (...)
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  35. Ulrich Christiansen (2004). What is a Number? Philosophy of Mathematics Education Journal 18.
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  36. Justin Clarke-Doane (2007). Multiple Reductions Revisited. 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 (...)
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  37. Nino Cocchiarella (1984). Science Without Numbers. [REVIEW] International Studies in Philosophy 16 (1):93-95.
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  38. Simon Conway Morris (forthcoming). It All Adds Up …. Or Does It? Numbers, Mathematics and Purpose. Studies in History and Philosophy of Science Part C: Studies in History and Philosophy of Biological and Biomedical Sciences.
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  39. David Neil Corfield (2004). Mathematical Kinds, or Being Kind to Mathematics. 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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  40. Leo Corry (2015). A Brief History of Numbers. Oxford University Press UK.
    Leo Corry tells the story behind the idea of number, from the early days of the Pythagoreans, up until the turn of the twentieth century. He presents an overview of how numbers were handled and conceived in classical Greek mathematics, in the mathematics of Islam, in European mathematics of the middle ages and the Renaissance, during the scientific revolution, all the way through to the mathematics of the 18th to the early 20th century.
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  41. Barbara F. Csima, Johanna N. Y. Franklin & Richard A. Shore (2013). Degrees of Categoricity and the Hyperarithmetic Hierarchy. Notre Dame Journal of Formal Logic 54 (2):215-231.
    We study arithmetic and hyperarithmetic degrees of categoricity. We extend a result of E. Fokina, I. Kalimullin, and R. Miller to show that for every computable ordinal $\alpha$, $\mathbf{0}^{}$ is the degree of categoricity of some computable structure $\mathcal{A}$. We show additionally that for $\alpha$ a computable successor ordinal, every degree $2$-c.e. in and above $\mathbf{0}^{}$ is a degree of categoricity. We further prove that every degree of categoricity is hyperarithmetic and show that the index set of structures with degrees (...)
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  42. Gregory Currie & Graham Oddie (1980). Changing Numbers. Theoria 46 (2-3):148-164.
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  43. O. Darrigol & A. J. Kox (1995). From C-Numbers to Q-Numbers: The Classical Analogy in the History of Quantum Theory. Annals of Science 52 (2):206-206.
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  44. Charles Darwin (1960). Can Man Control His Numbers? Perspectives in Biology and Medicine 3 (2):252-263.
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  45. Joseph Dauben (2008). Suan Shu Shu A Book on Numbers and Computations. Archive for History of Exact Sciences 62 (3):347-347.
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  46. James Davies (2015). Do We See Numbers? Metascience 24 (3):483-486.
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  47. Boudewijn de Bruin (2008). Wittgenstein on Circularity in the Frege-Russell Definition of Cardinal Number. Philosophia Mathematica 16 (3):354-373.
    Several scholars have argued that Wittgenstein held the view that the notion of number is presupposed by the notion of one-one correlation, and that therefore Hume's principle is not a sound basis for a definition of number. I offer a new interpretation of the relevant fragments on philosophy of mathematics from Wittgenstein's Nachlass, showing that if different uses of ‘presupposition’ are understood in terms of de re and de dicto knowledge, Wittgenstein's argument against the Frege-Russell definition of number turns out (...)
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  48. Helen De Cruz & Pierre Pica (2008). Knowledge of Number and Knowledge of Language: Number as a Test Case for the Role of Language in Cognition. Philosophical Psychology 21 (4):437 – 441.
    The relationship between language and conceptual thought is an unresolved problem in both philosophy and psychology. It remains unclear whether linguistic structure plays a role in our cognitive processes. This special issue brings together cognitive scientists and philosophers to focus on the role of language in numerical cognition: because of their universality and variability across languages, number words can serve as a fruitful test case to investigate claims of linguistic relativism.
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  49. R. Dedekind (1903). Essays on the Theory of Numbers. The Monist 13:314.
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  50. Stanislas Dehaene, Véronique Izard, Pierre Pica & Elizabeth Spelke (2009). Response to Comment on "Log or Linear? Distinct Intuitions on the Number Scale in Western and Amazonian Indigene Cultures". Science 323 (5910):38.
    The performance of the Mundurucu on the number-space task may exemplify a general competence for drawing analogies between space and other linear dimensions, but Mundurucu participants spontaneously chose number when other dimensions were available. Response placement may not reflect the subjective scale for numbers, but Cantlon et al.'s proposal of a linear scale with scalar variability requires additional hypotheses that are problematic.
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