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  1. Zofia Adamowicz & Leszek Aleksander Kołodziejczyk (2007). Partial Collapses of the Complexity Hierarchy in Models for Fragments of Bounded Arithmetic. Annals of Pure and Applied Logic 145 (1):91-95.
    For any n, we construct a model of in which each formula is equivalent to an formula.
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  2. Bernard A. Anderson & Jeffry L. Hirst (2009). Partitions of Trees and {{Sf ACA}^Prime_{0}}. Archive for Mathematical Logic 48 (3-4):227-230.
    We show that a version of Ramsey’s theorem for trees for arbitrary exponents is equivalent to the subsystem ${{\sf ACA}^\prime_{0}}$ of reverse mathematics.
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  3. F. G. Asenjo (1965). The Arithmetic of the Term-Relation Number Theory. Notre Dame Journal of Formal Logic 6 (3):223-228.
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  4. Jeremy Avigad (2003). Number Theory and Elementary Arithmetic. Philosophia Mathematica 11 (3):257-284.
    is a fragment of first-order aritlimetic so weak that it cannot prove the totality of an iterated exponential fimction. Surprisingly, however, the theory is remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context.
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  5. Jeremy Avigad, Kevin Donnelly, David Gray & Adam Kramer, Number Theory.
    1.1 Some examples of rule induction on permutations . . . . . . . 6 1.2 Ways of making new permutations . . . . . . . . . . . . . . . 7 1.3 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Removing elements . . . . . . . . . . (...)
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  6. W. Balzer (1979). On the Status of Arithmetic. Erkenntnis 14 (1):57 - 85.
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  7. Anatoly P. Beltiukov (2006). A Nonasymptotic Lower Time Bound for a Strictly Bounded Second-Order Arithmetic. Annals of Pure and Applied Logic 141 (3):320-324.
    We obtain a nonasymptotic lower time bound for deciding sentences of bounded second-order arithmetic with respect to a form of the random access machine with stored programs. More precisely, let P be an arbitrary program for the model under consideration which recognized true formulas with a given range of parameters. Let p be the length of P and let N be an arbitrary natural number. We show how to construct a formula G with one free variable with length not more (...)
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  8. E. Bencivenga (1988). Incompleteness of a Free Arithmetic. Logique Et Analyse 31 (21):79.
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  9. D. Bollman & M. Laplaza (1973). A Set-Theoretic Model for Nonassociative Number Theory. Notre Dame Journal of Formal Logic 14 (1):107-110.
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  10. Dorothy Bollman (1967). Formal Nonassociative Number Theory. Notre Dame Journal of Formal Logic 8 (1-2):9-16.
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  11. Andrew Boucher, Proving Quadratic Reciprocity.
    The system of arithmetic considered in Consistency, which is essentially second-order Peano Arithmetic without the Successor Axiom, is used to prove more theorems of arithmetic, up to Quadratic Reciprocity.
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  12. Andrew Boucher, General Arithmetic.
    General Arithmetic is the theory consisting of induction on a successor function. Normal arithmetic, say in the system called Peano Arithmetic, makes certain additional demands on the successor function. First, that it be total. Secondly, that it be one-to-one. And thirdly, that there be a first element which is not in its image. General Arithmetic abandons all of these further assumptions, yet is still able to prove many meaningful arithmetic truths, such as, most basically, Commutativity and Associativity of Addition and (...)
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  13. Alan C. Bowen (1989). Boethian Number Theory: A Translation of the de Institutione Arithmetica with Introduction and Notes. Ancient Philosophy 9 (1):137-143.
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  14. 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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  15. Ivor Bulmer-Thomas (1985). Boethian Number Theory Michael Masi: Boethian Number Theory: A Translation of the De Institutione Arithmetica (with Introduction and Notes). (Studies in Classical Antiquity, 6.) Pp. 198; 8 Figures with Mathematical Diagrams and Musical Notation in Text. Amsterdam: Editions Rodopi, 1983. Paper, Fl. 60. [REVIEW] The Classical Review 35 (01):86-87.
  16. Susan Carey (2009). Where Our Number Concepts Come From. Journal of Philosophy 106 (4):220-254.
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  17. Emily Carson (1998). Review of J. Belna, La Notion de Nombre Chez Dedekind, Cantor, Frege. Theories, Conceptions, Et Philosophie. [REVIEW] Philosophia Mathematica 6 (3):345-350.
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  18. Nino Cocchiarella (1984). Formal Number Theory and Compatibility. [REVIEW] Teaching Philosophy 7 (4):361-362.
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  19. Mark Colyvan, The Pursuit of the Riemann Hypothesis.
    With Fermat’s Last Theorem finally disposed of by Andrew Wiles in 1994, it’s only natural that popular attention should turn to arguably the most outstanding unsolved problem in mathematics: the Riemann Hypothesis. Unlike Fermat’s Last Theorem, however, the Riemann Hypothesis requires quite a bit of mathematical background to even understand what it says. And of course both require a great deal of background in order to understand their significance. The Riemann Hypothesis was first articulated by Bernhard Riemann in an address (...)
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  20. Elio Conte (2012). What is The Reason to Use Clifford Algebra in Quantum Cognition? Part I: “It From Qubit” On The Possibility That the Amino Acids Can Discern Between Two Quantum Spin States. Neuroquantology 10 (3):561-565.
    Starting with 1985, we discovered the possible existence of electrons with net helicity in biomolecules as amino acids and their possibility to discern between the two quantum spin states. It is well known that the question of a possible fundamental role of quantum mechanics in biological matter constitutes still a long debate. In the last ten years we have given a rather complete quantum mechanical elaboration entirely based on Clifford algebra whose basic entities are isomorphic to the well known spin (...)
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  21. Tobias Dantzig (1954/1967). Number, the Language of Science. New York, Free Press.
    A new edition of the classic introduction to mathematics, first published in 1930 and revised in the 1950s, explains the history and tenets of mathematics, ...
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  22. J. Michael Dunn (1979). A Theorem in 3-Valued Model Theory with Connections to Number Theory, Type Theory, and Relevant Logic. Studia Logica 38 (2):149 - 169.
    Given classical (2 valued) structures and and a homomorphism h of onto , it is shown how to construct a (non-degenerate) 3-valued counterpart of . Classical sentences that are true in are non-false in . Applications to number theory and type theory (with axiom of infinity) produce finite 3-valued models in which all classically true sentences of these theories are non-false. Connections to relevant logic give absolute consistency proofs for versions of these theories formulated in relevant logic (the proof for (...)
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  23. Steven Ericsson-Zenith (forthcoming). Explaining Experience In Nature: The Foundations Of Logic And Apprehension. Institute for Advanced Science & Engineering.
    At its core this book is concerned with logic and computation with respect to the mathematical characterization of sentient biophysical structure and its behavior. -/- Three related theories are presented: The first of these provides an explanation of how sentient individuals come to be in the world. The second describes how these individuals operate. And the third proposes a method for reasoning about the behavior of individuals in groups. -/- These theories are based upon a new explanation of experience in (...)
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  24. Howard F. Fehr (1940). A Study of the Number Concept of Secondary School Mathematics. [New York]Teachers College, Columbia University.
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  25. Harvey Friedman, Unprovable Theorems in Discrete Mathematics.
    An unprovable theorem is a mathematical result that can-not be proved using the com-monly accepted axioms for mathematics (Zermelo-Frankel plus the axiom of choice), but can be proved by using the higher infinities known as large cardinals. Large car-dinal axioms have been the main proposal for new axioms originating with Gödel.
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  26. Yvon Gauthier (2008). From Fermat to Gauss: Indefinite Descent and Methods of Reduction in Number Theory Paolo Bussotti Augsburg, Erwin Rauner Verlag, 2006, 574 p. [REVIEW] Dialogue 47 (02):411-.
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  27. Yvon Gauthier (1978). Foundational Problems of Number Theory. Notre Dame Journal of Formal Logic 19 (1):92-100.
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  28. George Goe (1965). Concerning Professor Sawyer's Reflections on Irrational Numbers. Philosophia Mathematica (1):38-43.
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  29. W. D. Goldfarb & T. M. Scanlon (1974). The Ω-Consistency of Number Theory Via Herbrand's Theorem. Journal of Symbolic Logic 39 (4):678-692.
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  30. Harry Gonshor (1980). Number Theory for the Ordinals with a New Definition for Multiplication. Notre Dame Journal of Formal Logic 21 (4):708-710.
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  31. R. L. Goodstein (1947). Transfinite Ordinals in Recursive Number Theory. Journal of Symbolic Logic 12 (4):123-129.
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  32. Claire Ortiz Hill (2010). Husserl on Axiomatization and Arithmetic. In Mirja Hartimo (ed.), Phenomenology and Mathematics. Springer.
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  33. Harold T. Hodes (2008). On Some Concepts Associated with Finite Cardinal Numbers. Behavioral and Brain Sciences 31 (6):657-658.
    I catalog several concepts associated with finite cardinals, and then invoke them to interpret and evaluate several passages in Rips et al.'s target article. Like the literature it discusses, the article seems overly quick to ascribe the possession of certain concepts to children (and of set-theoretic concepts to non-mathematicians).
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  34. L. Horsten (2012). Vom Zahlen Zu den Zahlen: On the Relation Between Computation and Arithmetical Structuralism. Philosophia Mathematica 20 (3):275-288.
    This paper sketches an answer to the question how we, in our arithmetical practice, succeed in singling out the natural-number structure as our intended interpretation. It is argued that we bring this about by a combination of what we assert about the natural-number structure on the one hand, and our computational capacities on the other hand.
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  35. Jean-Louis Hudry (2004). Peirce's Potential Continuity and Pure Geometry. Transactions of the Charles S. Peirce Society 40 (2):229 - 243.
  36. Edmund Husserl (1972). On the Concept of Number: Psychological Analysis. Philosophia Mathematica (1):44-52.
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  37. Carlo Ierna (2008). Edmund Husserl, Philosophy of Arithmetic, Translated by Dallas Willard. Husserl Studies 24 (1):53-58.
  38. Renling Jin (2000). Applications of Nonstandard Analysis in Additive Number Theory. Bulletin of Symbolic Logic 6 (3):331-341.
    This paper reports recent progress in applying nonstandard analysis to additive number theory, especially to problems involving upper Banach density.
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  39. John Kadvany (2008). Review of Alain Badiou, Number and Numbers. [REVIEW] Notre Dame Philosophical Reviews 2008 (10).
    This review takes seriously Badiou's use of set theory and mathematics, explaining the book's subtle technical content while maintaining a critical distance on Badiou's interpretative views.
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  40. Emelie Kenney (1989). Cardano: “Arithmetic Subtlety” and Impossible Solutions. Philosophia Mathematica (2):195-216.
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  41. Joongol Kim (2015). A Logical Foundation of Arithmetic. Studia Logica 103 (1):113-144.
    The aim of this paper is to shed new light on the logical roots of arithmetic by presenting a logical framework that takes seriously ordinary locutions like ‘at least n Fs’, ‘n more Fs than Gs’ and ‘n times as many Fs as Gs’, instead of paraphrasing them away in terms of expressions of the form ‘the number of Fs’. It will be shown that the basic concepts of arithmetic can be intuitively defined in the language of ALA, and the (...)
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  42. S. C. Kleene (1945). On the Interpretation of Intuitionistic Number Theory. Journal of Symbolic Logic 10 (4):109-124.
  43. W. Knorr (1976). Problems in the Interpretation of Greek Number Theory: Euclid and the 'Fundamental Theorem of Arithmetic'. Studies in History and Philosophy of Science Part A 7 (4):353-368.
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  44. G. Kreisel (1952). On the Interpretation of Non-Finitist Proofs: Part II. Interpretation of Number Theory. Applications. Journal of Symbolic Logic 17 (1):43-58.
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  45. M. Krynicki & K. Zdanowski (2005). Theories of Arithmetics in Finite Models. Journal of Symbolic Logic 70 (1):1-28.
    We investigate theories of initial segments of the standard models for arithmetics. It is easy to see that if the ordering relation is definable in the standard model then the decidability results can be transferred from the infinite model into the finite models. On the contrary we show that the Σ₂—theory of multiplication is undecidable in finite models. We show that this result is optimal by proving that the Σ₁—theory of multiplication and order is decidable in finite models as well (...)
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  46. Wojciech Krysztofiak (2012). Indexed Natural Numbers in Mind: A Formal Model of the Basic Mature Number Competence. [REVIEW] Axiomathes 22 (4):433-456.
    The paper undertakes three interdisciplinary tasks. The first one consists in constructing a formal model of the basic arithmetic competence, that is, the competence sufficient for solving simple arithmetic story-tasks which do not require any mathematical mastery knowledge about laws, definitions and theorems. The second task is to present a generalized arithmetic theory, called the arithmetic of indexed numbers (INA). All models of the development of counting abilities presuppose the common assumption that our simple, folk arithmetic encoded linguistically in the (...)
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  47. Mathieu Le Corre (2008). Why Cardinalities Are the “Natural” Natural Numbers. Behavioral and Brain Sciences 31 (6):659-659.
    According to Rips et al., numerical cognition develops out of two independent sets of cognitive primitives – one that supports enumeration, and one that supports arithmetic and the concepts of natural numbers. I argue against this proposal because it incorrectly predicts that natural number concepts could develop without prior knowledge of enumeration.
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  48. Steven C. Leth (1988). Some Nonstandard Methods in Combinatorial Number Theory. Studia Logica 47 (3):265 - 278.
    A combinatorial result about internal subsets of *N is proved using the Lebesgue Density Theorem. This result is then used to prove a standard theorem about difference sets of natural numbers which provides a partial answer to a question posed by Erdös and Graham.
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  49. David Lewis (1989). Review of John Bigelow, The Reality of Numbers. [REVIEW] Australasian Journal of Philosophy 67 (4):487-489.
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  50. Øystein Linnebo (2009). The Individuation of the Natural Numbers. In Otavio Bueno & Øystein Linnebo (eds.), New Waves in Philosophy of Mathematics. Palgrave.
    It is sometimes suggested that criteria of identity should play a central role in an account of our most fundamental ways of referring to objects. The view is nicely illustrated by an example due to (Quine, 1950). Suppose you are standing at the bank of a river, watching the water that floats by. What is required for you to refer to the river, as opposed to a particular segment of it, or the totality of its water, or the current temporal (...)
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