Search results for 'arithmetic' (try it on Scholar)

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  1. Richard Pettigrew (2010). The Foundations of Arithmetic in Finite Bounded Zermelo Set Theory. Cahiers du Centre de Logique 17:99-118.score: 18.0
    In this paper, I pursue such a logical foundation for arithmetic in a variant of Zermelo set theory that has axioms of subset separation only for quantifier-free formulae, and according to which all sets are Dedekind finite. In section 2, I describe this variant theory, which I call ZFin0. And in section 3, I sketch foundations for arithmetic in ZFin0 and prove that certain foundational propositions that are theorems of the standard Zermelian foundation for arithmetic are independent (...)
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  2. Kristina Engelhard & Peter Mittelstaedt (2008). Kant's Theory of Arithmetic: A Constructive Approach? [REVIEW] Journal for General Philosophy of Science 39 (2):245 - 271.score: 18.0
    Kant’s theory of arithmetic is not only a central element in his theoretical philosophy but also an important contribution to the philosophy of arithmetic as such. However, modern mathematics, especially non-Euclidean geometry, has placed much pressure on Kant’s theory of mathematics. But objections against his theory of geometry do not necessarily correspond to arguments against his theory of arithmetic and algebra. The goal of this article is to show that at least some important details in Kant’s theory (...)
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  3. Richard Pettigrew (2009). On Interpretations of Bounded Arithmetic and Bounded Set Theory. Notre Dame Journal of Formal Logic 50 (2):141-152.score: 18.0
    In 'On interpretations of arithmetic and set theory', Kaye and Wong proved the following result, which they considered to belong to the folklore of mathematical logic.

    THEOREM 1 The first-order theories of Peano arithmetic and Zermelo-Fraenkel set theory with the axiom of infinity negated are bi-interpretable.

    In this note, I describe a theory of sets that is bi-interpretable with the theory of bounded arithmetic IDelta0 + exp. Because of the weakness of this theory of sets, I cannot straightforwardly adapt (...)
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  4. Gottlob Frege (1953/1968). The Foundations of Arithmetic. Evanston, Ill.,Northwestern University Press.score: 18.0
    In arithmetic, if only because many of its methods and concepts originated in India, it has been the tradition to reason less strictly than in geometry, ...
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  5. Laureano Luna & Alex Blum (2008). Arithmetic and Logic Incompleteness: The Link. The Reasoner 2 (3):6.score: 18.0
    We show how second order logic incompleteness follows from incompleteness of arithmetic, as proved by Gödel.
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  6. Michael D. Potter (2000). Reason's Nearest Kin: Philosophies of Arithmetic From Kant to Carnap. Oxford University Press.score: 18.0
    This is a critical examination of the astonishing progress made in the philosophical study of the properties of the natural numbers from the 1880s to the 1930s. Reassessing the brilliant innovations of Frege, Russell, Wittgenstein, and others, which transformed philosophy as well as our understanding of mathematics, Michael Potter places arithmetic at the interface between experience, language, thought, and the world.
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  7. Charles Sayward (2005). Why Axiomatize Arithmetic? Sorites 16:54-61.score: 18.0
    This is a dialogue in the philosophy of mathematics that focuses on these issues: Are the Peano axioms for arithmetic epistemologically irrelevant? What is the source of our knowledge of these axioms? What is the epistemological relationship between arithmetical laws and the particularities of number?
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  8. James A. Anderson (2003). Arithmetic on a Parallel Computer: Perception Versus Logic. [REVIEW] Brain and Mind 4 (2):169-188.score: 18.0
    This article discusses the properties of a controllable, flexible, hybrid parallel computing architecture that potentially merges pattern recognition and arithmetic. Humans perform integer arithmetic in a fundamentally different way than logic-based computers. Even though the human approach to arithmetic is both slow and inaccurate it can have substantial advantages when useful approximations ( intuition ) are more valuable than high precision. Such a computational strategy may be particularly useful when computers based on nanocomponents become feasible because it (...)
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  9. Jan Heylen (2013). Modal-Epistemic Arithmetic and the Problem of Quantifying In. Synthese 190 (1):89-111.score: 18.0
    The subject of this article is Modal-Epistemic Arithmetic (MEA), a theory introduced by Horsten to interpret Epistemic Arithmetic (EA), which in turn was introduced by Shapiro to interpret Heyting Arithmetic. I will show how to interpret MEA in EA such that one can prove that the interpretation of EA is MEA is faithful. Moreover, I will show that one can get rid of a particular Platonist assumption. Then I will discuss models for MEA in light of the (...)
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  10. Kai F. Wehmeier (1996). Classical and Intuitionistic Models of Arithmetic. Notre Dame Journal of Formal Logic 37 (3):452-461.score: 18.0
    Given a classical theory T, a Kripke model K for the language L of T is called T-normal or locally PA just in case the classical L-structure attached to each node of K is a classical model of T. Van Dalen, Mulder, Krabbe, and Visser showed that Kripke models of Heyting Arithmetic (HA) over finite frames are locally PA, and that Kripke models of HA over frames ordered like the natural numbers contain infinitely many PA-nodes. We show that Kripke (...)
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  11. Jessica M. Wilson (2000). Could Experience Disconfirm the Propositions of Arithmetic? Canadian Journal of Philosophy 30 (1):55-84.score: 18.0
    Alberto Casullo ("Necessity, Certainty, and the A Priori", Canadian Journal of Philosophy 18, 1988) argues that arithmetical propositions could be disconfirmed by appeal to an invented scenario, wherein our standard counting procedures indicate that 2 + 2 != 4. Our best response to such a scenario would be, Casullo suggests, to accept the results of the counting procedures, and give up standard arithmetic. While Casullo's scenario avoids arguments against previous "disconfirming" scenarios, it founders on the assumption, common to scenario (...)
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  12. Samuel Coskey & Roman Kossak (2010). The Complexity of Classification Problems for Models of Arithmetic. Bulletin of Symbolic Logic 16 (3):345-358.score: 18.0
    We observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of each of these. Finally, we show that the classification problem for pairs of recursively saturated models and for automorphisms of a fixed recursively saturated model are Borel complete.
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  13. Joongol Kim (forthcoming). A Logical Foundation of Arithmetic. Studia Logica.score: 18.0
    The aim of this paper is to shed new light on the logical roots of arithmetic by presenting a logical framework (ALA) 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 (...)
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  14. Iro Xenidou‐Dervou, Ernest C. D. M. Lieshout & Menno Schoot (2014). Working Memory in Nonsymbolic Approximate Arithmetic Processing: A Dual‐Task Study With Preschoolers. Cognitive Science 38 (1):101-127.score: 18.0
    Preschool children have been proven to possess nonsymbolic approximate arithmetic skills before learning how to manipulate symbolic math and thus before any formal math instruction. It has been assumed that nonsymbolic approximate math tasks necessitate the allocation of Working Memory (WM) resources. WM has been consistently shown to be an important predictor of children's math development and achievement. The aim of our study was to uncover the specific role of WM in nonsymbolic approximate math. For this purpose, we conducted (...)
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  15. Richard Kaye & Tin Lok Wong (2007). On Interpretations of Arithmetic and Set Theory. Notre Dame Journal of Formal Logic 48 (4):497-510.score: 18.0
    This paper starts by investigating Ackermann's interpretation of finite set theory in the natural numbers. We give a formal version of this interpretation from Peano arithmetic (PA) to Zermelo-Fraenkel set theory with the infinity axiom negated (ZF−inf) and provide an inverse interpretation going the other way. In particular, we emphasize the precise axiomatization of our set theory that is required and point out the necessity of the axiom of transitive containment or (equivalently) the axiom scheme of ∈-induction. This clarifies (...)
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  16. Jeremy Avigad, Update Procedures and the 1-Consistency of Arithmetic.score: 18.0
    The 1-consistency of arithmetic is shown to be equivalent to the existence of fixed points of a certain type of update procedure, which is implicit in the epsilon-substitution method.
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  17. António M. Fernandes (2010). Strict {Pi^1_1}-Reflection in Bounded Arithmetic. Archive for Mathematical Logic 49 (1):17-34.score: 18.0
    We prove two conservation results involving a generalization of the principle of strict ${\Pi^1_1}$ -reflection, in the context of bounded arithmetic. In this context a separation between the concepts of bounded set and binary sequence seems to emerge as fundamental.
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  18. Yoshihiro Horihata (2012). Weak Theories of Concatenation and Arithmetic. Notre Dame Journal of Formal Logic 53 (2):203-222.score: 18.0
    We define a new theory of concatenation WTC which is much weaker than Grzegorczyk's well-known theory TC. We prove that WTC is mutually interpretable with the weak theory of arithmetic R. The latter is, in a technical sense, much weaker than Robinson's arithmetic Q, but still essentially undecidable. Hence, as a corollary, WTC is also essentially undecidable.
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  19. Emil Jeřábek (2009). Approximate Counting by Hashing in Bounded Arithmetic. Journal of Symbolic Logic 74 (3):829-860.score: 18.0
    We show how to formalize approximate counting via hash functions in subsystems of bounded arithmetic, using variants of the weak pigeonhole principle. We discuss several applications, including a proof of the tournament principle, and an improvement on the known relationship of the collapse of the bounded arithmetic hierarchy to the collapse of the polynomial-time hierarchy.
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  20. Roman Kossak (1995). Four Problems Concerning Recursively Saturated Models of Arithmetic. Notre Dame Journal of Formal Logic 36 (4):519-530.score: 18.0
    The paper presents four open problems concerning recursively saturated models of Peano Arithmetic. One problems concerns a possible converse to Tarski's undefinability of truth theorem. The other concern elementary cuts in countable recursively saturated models, extending automorphisms of countable recursively saturated models, and Jonsson models of PA. Some partial answers are given.
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  21. James H. Schmerl (2012). Elementary Cuts in Saturated Models of Peano Arithmetic. Notre Dame Journal of Formal Logic 53 (1):1-13.score: 18.0
    A model $\mathscr{M} = (M,+,\times, 0,1,<)$ of Peano Arithmetic $({\sf PA})$ is boundedly saturated if and only if it has a saturated elementary end extension $\mathscr{N}$. The ordertypes of boundedly saturated models of $({\sf PA})$ are characterized and the number of models having these ordertypes is determined. Pairs $(\mathscr{N},M)$, where $\mathscr{M} \prec_{\sf end} \mathscr{N} \models({\sf PA})$ for saturated $\mathscr{N}$, and their theories are investigated. One result is: If $\mathscr{N}$ is a $\kappa$-saturated model of $({\sf PA})$ and $\mathscr{M}_0, \mathscr{M}_1 \prec_{\sf (...)
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  22. Virginie Crollen, Xavier Seron & Marie-Pascale Noël (2011). Is Finger-Counting Necessary for the Development of Arithmetic Abilities? Frontiers in Psychology 2.score: 18.0
    Is Finger-counting Necessary for the Development of Arithmetic Abilities?
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  23. Frank Domahs Elise Klein, Korbinian Moeller, Klaus Willmes, Hans-Christoph Nuerk (2011). The Influence of Implicit Hand-Based Representations on Mental Arithmetic. Frontiers in Psychology 2.score: 18.0
    Recently, a strong functional relationship between finger counting and number processing has been suggested. It has been argued that bodily experiences such as finger counting may influence the structure of the basic mental representations of numbers even in adults. However, to date it remains unclear whether the structure of finger counting systems also influences educated adults’ performance in mental arithmetic. In the present study, we pursued this question by examining finger-based sub-base-five effects in an addition production task. With the (...)
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  24. Tobias U. Hauser, Stephanie Rotzer, Roland H. Grabner, Susan Mérillat & Lutz Jäncke (2013). Enhancing Performance in Numerical Magnitude Processing and Mental Arithmetic Using Transcranial Direct Current Stimulation (tDCS). Frontiers in Human Neuroscience 7.score: 18.0
    The ability to accurately process numerical magnitudes and solve mental arithmetic is of highest importance for schooling and professional career. Although impairments in these domains in disorders such as developmental dyscalculia (DD) are highly detrimental, remediation is still sparse. In recent years, transcranial brain stimulation methods such as transcranial Direct Current Stimulation (tDCS) have been suggested as a treatment for various neurologic and neuropsychiatric disorders. The posterior parietal cortex (PPC) is known to be crucially involved in numerical magnitude processing (...)
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  25. Emil Jeřábek & Leszek Aleksander Kołodziejczyk (2013). Real Closures of Models of Weak Arithmetic. Archive for Mathematical Logic 52 (1-2):143-157.score: 18.0
    D’Aquino et al. (J Symb Log 75(1):1–11, 2010) have recently shown that every real-closed field with an integer part satisfying the arithmetic theory IΣ4 is recursively saturated, and that this theorem fails if IΣ4 is replaced by IΔ0. We prove that the theorem holds if IΣ4 is replaced by weak subtheories of Buss’ bounded arithmetic: PV or ${\Sigma^b_1-IND^{|x|_k}}$ . It also holds for IΔ0 (and even its subtheory IE 2) under a rather mild assumption on cofinality. On the (...)
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  26. Wojciech Krysztofiak (forthcoming). Hyper-Slingshot. Is Fact-Arithmetic Possible? Foundations of Science:1-18.score: 18.0
    The paper presents a new argument supporting the ontological standpoint according to which there are no mathematical facts in any set theoretic model (world) of arithmetical theories. It may be interpreted as showing that it is impossible to construct fact-arithmetic. The importance of this conclusion arises in the context of cognitive science. In the paper, a new type of slingshot argument is presented, which is called hyper-slingshot. The difference between meta-theoretical hyper-slingshots and conventional slingshots consists in the fact that (...)
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  27. Taishi Kurahashi (2013). On Predicate Provability Logics and Binumerations of Fragments of Peano Arithmetic. Archive for Mathematical Logic 52 (7-8):871-880.score: 18.0
    Solovay proved (Israel J Math 25(3–4):287–304, 1976) that the propositional provability logic of any ∑2-sound recursively enumerable extension of PA is characterized by the propositional modal logic GL. By contrast, Montagna proved in (Notre Dame J Form Log 25(2):179–189, 1984) that predicate provability logics of Peano arithmetic and Bernays–Gödel set theory are different. Moreover, Artemov proved in (Doklady Akademii Nauk SSSR 290(6):1289–1292, 1986) that the predicate provability logic of a theory essentially depends on the choice of a binumeration of (...)
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  28. Silke M. Göbel Liane Kaufmann, Silvia Pixner (2011). Finger Usage and Arithmetic in Adults with Math Difficulties: Evidence From a Case Report. Frontiers in Psychology 2.score: 18.0
    Finger Usage and Arithmetic in Adults with Math Difficulties: Evidence From a Case Report.
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  29. Carl Mummert (2008). Subsystems of Second-Order Arithmetic Between RCA0 and WKL0. Archive for Mathematical Logic 47 (3):205-210.score: 18.0
    We study the Lindenbaum algebra ${\fancyscript{A}}$ (WKL o, RCA o) of sentences in the language of second-order arithmetic that imply RCA o and are provable from WKL o. We explore the relationship between ${\Sigma^1_1}$ sentences in ${\fancyscript{A}}$ (WKL o, RCA o) and ${\Pi^0_1}$ classes of subsets of ω. By applying a result of Binns and Simpson (Arch. Math. Logic 43(3), 399–414, 2004) about ${\Pi^0_1}$ classes, we give a specific embedding of the free distributive lattice with countably many generators into (...)
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  30. Pierre Pica, Cathy Lemer, Véronique Izard & Stanislas Dehaene (2004). Exact and Approximate Arithmetic in an Amazonian Indigene Group. Science 306 (5695):499-503.score: 18.0
    Is calculation possible without language? Or is the human ability for arithmetic dependent on the language faculty? To clarify the relation between language and arithmetic, we studied numerical cognition in speakers of Mundurukú, an Amazonian language with a very small lexicon of number words. Although the Mundurukú lack words for numbers beyond 5, they are able to compare and add large approximate numbers that are far beyond their naming range. However, they fail in exact arithmetic with numbers (...)
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  31. Dieter Probst & Thomas Strahm (2011). Admissible Closures of Polynomial Time Computable Arithmetic. Archive for Mathematical Logic 50 (5-6):643-660.score: 18.0
    We propose two admissible closures ${\mathbb{A}({\sf PTCA})}$ and ${\mathbb{A}({\sf PHCA})}$ of Ferreira’s system PTCA of polynomial time computable arithmetic and of full bounded arithmetic (or polynomial hierarchy computable arithmetic) PHCA. The main results obtained are: (i) ${\mathbb{A}({\sf PTCA})}$ is conservative over PTCA with respect to ${\forall\exists\Sigma^b_1}$ sentences, and (ii) ${\mathbb{A}({\sf PHCA})}$ is conservative over full bounded arithmetic PHCA for ${\forall\exists\Sigma^b_{\infty}}$ sentences. This yields that (i) the ${\Sigma^b_1}$ definable functions of ${\mathbb{A}({\sf PTCA})}$ are the polytime functions, and (...)
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  32. Bert De Smedt Roland H. Grabner (2012). Oscillatory EEG Correlates of Arithmetic Strategies: A Training Study. Frontiers in Psychology 3.score: 18.0
    There has been a long tradition of research on mathematics education showing that children and adults use different strategies to solve arithmetic problems. Neurophysiological studies have recently begun to investigate the brain correlates of these strategies. The existing body of data, however, reflect static end points of the learning process and do not provide information on how brain activity changes in response to training or intervention. In this study, we explicitly address this issue by training participants in using fact (...)
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  33. Nobuyuki Sakamoto & Kazuyuki Tanaka (2004). The Strong Soundness Theorem for Real Closed Fields and Hilbert's Nullstellensatz in Second Order Arithmetic. Archive for Mathematical Logic 43 (3):337-349.score: 18.0
    By RCA 0 , we denote a subsystem of second order arithmetic based on Δ0 1 comprehension and Δ0 1 induction. We show within this system that the real number system R satisfies all the theorems (possibly with non-standard length) of the theory of real closed fields under an appropriate truth definition. This enables us to develop linear algebra and polynomial ring theory over real and complex numbers, so that we particularly obtain Hilbert’s Nullstellensatz in RCA 0.
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  34. Stanislav O. Speranski (2013). A Note on Definability in Fragments of Arithmetic with Free Unary Predicates. Archive for Mathematical Logic 52 (5-6):507-516.score: 18.0
    We carry out a study of definability issues in the standard models of Presburger and Skolem arithmetics (henceforth referred to simply as Presburger and Skolem arithmetics, for short, because we only deal with these models, not the theories, thus there is no risk of confusion) supplied with free unary predicates—which are strongly related to definability in the monadic SOA (second-order arithmetic) without × or + , respectively. As a consequence, we obtain a very direct proof for ${\Pi^1_1}$ -completeness of (...)
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  35. Neil Thapen (2011). Higher Complexity Search Problems for Bounded Arithmetic and a Formalized No-Gap Theorem. Archive for Mathematical Logic 50 (7-8):665-680.score: 18.0
    We give a new characterization of the strict $\forall {\Sigma^b_j}$ sentences provable using ${\Sigma^b_k}$ induction, for 1 ≤ j ≤ k. As a small application we show that, in a certain sense, Buss’s witnessing theorem for strict ${\Sigma^b_k}$ formulas already holds over the relatively weak theory PV. We exhibit a combinatorial principle with the property that a lower bound for it in constant-depth Frege would imply that the narrow CNFs with short depth j Frege refutations form a strict hierarchy with (...)
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  36. Masahiro Yasumoto (2005). Separations of First and Second Order Theories in Bounded Arithmetic. Archive for Mathematical Logic 44 (6):685-688.score: 18.0
    We prove that PTC N (n) (the polynomial time closure of the nonstandard natural number n in the model N of S 2.) cannot be a model of U 1 2. This implies that there exists a first order sentence of bounded arithmetic which is provable in U 1 2 but does not hold in PTC N (n).
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  37. Keita Yokoyama (2007). Complex Analysis in Subsystems of Second Order Arithmetic. Archive for Mathematical Logic 46 (1):15-35.score: 18.0
    This research is motivated by the program of Reverse Mathematics. We investigate basic part of complex analysis within some weak subsystems of second order arithmetic, in order to determine what kind of set existence axioms are needed to prove theorems of basic analysis. We are especially concerned with Cauchy’s integral theorem. We show that a weak version of Cauchy’s integral theorem is proved in RCAo. Using this, we can prove that holomorphic functions are analytic in RCAo. On the other (...)
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  38. Sean Walsh (2012). Comparing Peano Arithmetic, Basic Law V, and Hume's Principle. Annals of Pure and Applied Logic 163 (11):1679-1709.score: 15.0
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  39. Gottlob Frege (1980). The Foundations of Arithmetic: A Logico-Mathematical Enquiry Into the Concept of Number. Northwestern University Press.score: 15.0
    § i. After deserting for a time the old Euclidean standards of rigour, mathematics is now returning to them, and even making efforts to go beyond them. ...
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  40. Carlos Montemayor & Fuat Balci (2007). Compositionality in Language and Arithmetic. Journal of Theoretical and Philosophical Psychology 27 (1):53-72.score: 15.0
  41. Barry G. Allen (1989). Gruesome Arithmetic: Kripke's Sceptic Replies. Dialogue 28 (2):257-264.score: 15.0
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  42. Charles Sayward (2000). Remarks on Peano Arithmetic. Russell 20:27-32.score: 15.0
    Russell held that the theory of natural numbers could be derived from three primitive concepts: number, successor and zero. This leaves out multiplication and addition. Russell introduces these concepts by recursive definition. It is argued that this does not render addition or multiplication any less primitive than the other three. To this it might be replied that any recursive definition can be transformed into a complete or explicit definition with the help of a little set theory. But that is a (...)
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  43. Fernando Ferreira (1999). A Note on Finiteness in the Predicative Foundations of Arithmetic. Journal of Philosophical Logic 28 (2):165-174.score: 15.0
    Recently, Feferman and Hellman (and Aczel) showed how to establish the existence and categoricity of a natural number system by predicative means given the primitive notion of a finite set of individuals and given also a suitable pairing function operating on individuals. This short paper shows that this existence and categoricity result does not rely (even indirectly) on finite-set induction, thereby sustaining Feferman and Hellman's point in favor of the view that natural number induction can be derived from a very (...)
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  44. David E. Diamondstone, Damir D. Dzhafarov & Robert I. Soare (2010). $\Pi^0_1$ Classes, Peano Arithmetic, Randomness, and Computable Domination. Notre Dame Journal of Formal Logic 51 (1):127-159.score: 15.0
    We present an overview of the topics in the title and of some of the key results pertaining to them. These have historically been topics of interest in computability theory and continue to be a rich source of problems and ideas. In particular, we draw attention to the links and connections between these topics and explore their significance to modern research in the field.
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  45. Richard Zach (2005). Book Review: Michael Potter. Reason's Nearest Kin. Philosophies of Arithmetic From Kant to Carnap. [REVIEW] Notre Dame Journal of Formal Logic 46 (4):503-513.score: 15.0
  46. Zofia Adamowicz & Leszek Aleksander Kolodziejczyk (2010). A Note on the E1 Collection Scheme and Fragments of Bounded Arithmetic. Mathematical Logic Quarterly 56 (2):126-130.score: 15.0
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  47. Eleonore PhD Ardiale & Patrick Lemaire (2013). Within-Item Strategy Switching in Arithmetic: A Comparative Study in Children. Frontiers in Psychology 4:924.score: 15.0
    The present study investigated age-related differences in children’s within-item strategy switching. Third, fifth, and seventh graders performed a computational estimation task in which they had to provide the better estimates to two-digit addition problems (e.g., 34+57) while using the rounding-down (e.g., 30+50) or the rounding-up strategy (e.g., 40+60). After having executing the cued strategy (e.g., 30+50) during 1,000 ms, participants were given the opportunity to switch to another better strategy (e.g., 40+60) or to repeat the same strategy. The results showed (...)
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  48. Frederick Bagemihl & F. Bagemihl (1992). Ordinal Numbers in Arithmetic Progression. Mathematical Logic Quarterly 38 (1):525-528.score: 15.0
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  49. Nicholas Bamber & Henryk Kotlarski (1997). On Interstices of Countable Arithmetically Saturated Models of Peano Arithmetic. Mathematical Logic Quarterly 43 (4):525-540.score: 15.0
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  50. Arnold Beckmann & Jan Johannsen (2005). An Unexpected Separation Result in Linearly Bounded Arithmetic. Mathematical Logic Quarterly 51 (2):191-200.score: 15.0
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