Results for 'Arithmetic'

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  1. Exact and Approximate Arithmetic in an Amazonian Indigene Group.Pierre Pica, Cathy Lemer, Véronique Izard & Stanislas Dehaene - 2004 - Science 306 (5695):499-503.
    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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  2.  49
    The Foundations of Arithmetic.Gottlob Frege - 1953 - Evanston: Ill., Northwestern University Press.
    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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  3.  31
    Reason's Nearest Kin: Philosophies of Arithmetic From Kant to Carnap.Michael D. Potter - 2000 - Oxford University Press.
    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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  4. Comparing Peano Arithmetic, Basic Law V, and Hume's Principle.Sean Walsh - 2012 - Annals of Pure and Applied Logic 163 (11):1679-1709.
    This paper presents new constructions of models of Hume's Principle and Basic Law V with restricted amounts of comprehension. The techniques used in these constructions are drawn from hyperarithmetic theory and the model theory of fields, and formalizing these techniques within various subsystems of second-order Peano arithmetic allows one to put upper and lower bounds on the interpretability strength of these theories and hence to compare these theories to the canonical subsystems of second-order arithmetic. The main results of (...)
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  5.  31
    Arithmetic, Set Theory, Reduction and Explanation.William D'Alessandro - forthcoming - Synthese:1-31.
    Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In (...)
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  6. Semantic Arithmetic: A Preface.John Corcoran - 1995 - Agora 14 (1):149-156.
    SEMANTIC ARITHMETIC: A PREFACE John Corcoran Abstract Number theory, or pure arithmetic, concerns the natural numbers themselves, not the notation used, and in particular not the numerals. String theory, or pure syntax, concems the numerals as strings of «uninterpreted» characters without regard to the numbe~s they may be used to denote. Number theory is purely arithmetic; string theory is purely syntactical... in so far as the universe of discourse alone is considered. Semantic arithmetic is a broad (...)
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  7. Modal-Epistemic Arithmetic and the Problem of Quantifying In.Jan Heylen - 2013 - Synthese 190 (1):89-111.
    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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  8.  16
    On Interpretations of Arithmetic and Set Theory.Richard Kaye & Tin Lok Wong - 2007 - Notre Dame Journal of Formal Logic 48 (4):497-510.
    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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  9.  29
    A Logical Foundation of Arithmetic.Joongol Kim - 2015 - 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, (...)
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  10.  94
    Carnapian Modal and Epistemic Arithmetic.Heylen Jan - 2009 - In Carrara Massimiliano & Morato Vittorio (eds.), Language, Knowledge, and Metaphysics. Selected papers from the First SIFA Graduate Conference. College Publications. pp. 97-121.
    The subject of the first section is Carnapian modal logic. One of the things I will do there is to prove that certain description principles, viz. the ''self-predication principles'', i.e. the principles according to which a descriptive term satisfies its own descriptive condition, are theorems and that others are not. The second section will be devoted to Carnapian modal arithmetic. I will prove that, if the arithmetical theory contains the standard weak principle of induction, modal truth collapses to truth. (...)
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  11. Approximate Counting by Hashing in Bounded Arithmetic.Emil Jeřábek - 2009 - Journal of Symbolic Logic 74 (3):829-860.
    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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  12.  33
    Working Memory in Nonsymbolic Approximate Arithmetic Processing: A Dual‐Task Study With Preschoolers.Iro Xenidou‐Dervou, Ernest C. D. M. Lieshout & Menno Schoot - 2014 - Cognitive Science 38 (1):101-127.
    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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  13.  29
    Update Procedures and the 1-Consistency of Arithmetic.Jeremy Avigad - 2002 - Mathematical Logic Quarterly 48 (1):3-13.
    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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  14.  34
    Compositionality in Language and Arithmetic.Carlos Montemayor & Fuat Balci - 2007 - Journal of Theoretical and Philosophical Psychology 27 (1):53-72.
    The lack of conceptual analysis within cognitive science results in multiple models of the same phenomena. However, these models incorporate assumptions that contradict basic structural features of the domain they are describing. This is particularly true about the domain of mathematical cognition. In this paper we argue that foundational theoretic aspects of psychological models for language and arithmetic should be clarified before postulating such models. We propose a means to clarify these foundational concepts by analyzing the distinctions between metric (...)
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  15.  7
    On Interstices of Countable Arithmetically Saturated Models of Peano Arithmetic.Nicholas Bamber & Henryk Kotlarski - 1997 - Mathematical Logic Quarterly 43 (4):525-540.
    We give some information about the action of Aut on M, where M is a countable arithmetically saturated model of Peano Arithmetic. We concentrate on analogues of moving gaps and covering gaps inside M.
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  16.  2
    Herbrandizing Search Problems in Bounded Arithmetic.Jiří Hanika - 2004 - Mathematical Logic Quarterly 50 (6):577-586.
    We study search problems and reducibilities between them with known or potential relevance to bounded arithmetic theories. Our primary objective is to understand the sets of low complexity consequences of theories Si2 and Ti2 for a small i, ideally in a rather strong sense of characterization; or, at least, in the standard sense of axiomatization. We also strive for maximum combinatorial simplicity of the characterizations and axiomatizations, eventually sufficient to prove conjectured separation results. To this end two techniques based (...)
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  17.  10
    Real Closures of Models of Weak Arithmetic.Emil Jeřábek & Leszek Aleksander Kołodziejczyk - 2013 - Archive for Mathematical Logic 52 (1-2):143-157.
    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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  18.  9
    A Note on Definability in Fragments of Arithmetic with Free Unary Predicates.Stanislav O. Speranski - 2013 - Archive for Mathematical Logic 52 (5-6):507-516.
    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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  19.  17
    Complex Analysis in Subsystems of Second Order Arithmetic.Keita Yokoyama - 2007 - Archive for Mathematical Logic 46 (1):15-35.
    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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  20.  30
    Classical and Intuitionistic Models of Arithmetic.Kai F. Wehmeier - 1996 - Notre Dame Journal of Formal Logic 37 (3):452-461.
    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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  21.  14
    Fragments of Arithmetic and True Sentences.Andrés Cordón‐Franco, Alejandro Fernández‐Margarit & F. Félix Lara‐Martín - 2005 - Mathematical Logic Quarterly 51 (3):313-328.
    By a theorem of R. Kaye, J. Paris and C. Dimitracopoulos, the class of the Πn+1-sentences true in the standard model is the only consistent Πn+1-theory which extends the scheme of induction for parameter free Πn+1-formulas. Motivated by this result, we present a systematic study of extensions of bounded quantifier complexity of fragments of first-order Peano Arithmetic. Here, we improve that result and show that this property describes a general phenomenon valid for parameter free schemes. As a consequence, we (...)
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  22. The Foundations of Arithmetic in Finite Bounded Zermelo Set Theory.Richard Pettigrew - 2010 - Cahiers du Centre de Logique 17:99-118.
    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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  23.  10
    The Prehistory of the Subsystems of Second-Order Arithmetic.Walter Dean & Sean Walsh - 2017 - Review of Symbolic Logic 10 (2):357-396.
    This paper presents a systematic study of the prehistory of the traditional subsystems of second-order arithmetic that feature prominently in the reverse mathematics program of Friedman and Simpson. We look in particular at: (i) the long arc from Poincar\'e to Feferman as concerns arithmetic definability and provability, (ii) the interplay between finitism and the formalization of analysis in the lecture notes and publications of Hilbert and Bernays, (iii) the uncertainty as to the constructive status of principles equivalent to (...)
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  24.  9
    Uniform Versions of Some Axioms of Second Order Arithmetic.Nobuyuki Sakamoto & Takeshi Yamazaki - 2004 - Mathematical Logic Quarterly 50 (6):587-593.
    In this paper, we discuss uniform versions of some axioms of second order arithmetic in the context of higher order arithmetic. We prove that uniform versions of weak weak König's lemma WWKL and Σ01 separation are equivalent to over a suitable base theory of higher order arithmetic, where is the assertion that there exists Φ2 such that Φf1 = 0 if and only if ∃x0 for all f. We also prove that uniform versions of some well-known theorems (...)
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  25.  9
    A Generalization of Conservativity Theorem for Classical Versus Intuitionistic Arithmetic.S. Berardi - 2004 - Mathematical Logic Quarterly 50 (1):41.
    A basic result in intuitionism is Π02-conservativity. Take any proof p in classical arithmetic of some Π02-statement , with P decidable). Then we may effectively turn p in some intuitionistic proof of the same statement. In a previous paper [1], we generalized this result: any classical proof p of an arithmetical statement ∀x.∃y.P, with P of degree k, may be effectively turned into some proof of the same statement, using Excluded Middle only over degree k formulas. When k = (...)
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  26.  17
    William of Ockham’s Ontology of Arithmetic.Magali Roques - 2016 - Vivarium 54 (2-3):146-165.
    _ Source: _Volume 54, Issue 2-3, pp 146 - 165 Ockham’s ontology of arithmetic, specifically his position on the ontological status of natural numbers, has not yet attracted the attention of scholars. Yet it occupies a central role in his nominalism; specifically, Ockham’s position on numbers constitutes a third part of his ontological reductionism, alongside his doctrines of universals and the categories, which have long been recognized to constitute the first two parts. That is, the first part of this (...)
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  27.  5
    On the Structure of Kripke Models of Heyting Arithmetic.Zoran Marković - 1993 - Mathematical Logic Quarterly 39 (1):531-538.
    Since in Heyting Arithmetic all atomic formulas are decidable, a Kripke model for HA may be regarded classically as a collection of classical structures for the language of arithmetic, partially ordered by the submodel relation. The obvious question is then: are these classical structures models of Peano Arithmetic ? And dually: if a collection of models of PA, partially ordered by the submodel relation, is regarded as a Kripke model, is it a model of HA? Some partial (...)
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  28.  19
    Subsystems of Second-Order Arithmetic Between RCA0 and WKL0.Carl Mummert - 2008 - Archive for Mathematical Logic 47 (3):205-210.
    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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  29. On Interpretations of Bounded Arithmetic and Bounded Set Theory.Richard Pettigrew - 2009 - Notre Dame Journal of Formal Logic 50 (2):141-152.
    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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  30.  6
    Generic Cuts in Models of Arithmetic.Richard Kaye - 2008 - Mathematical Logic Quarterly 54 (2):129-144.
    We present some general results concerning the topological space of cuts of a countable model of arithmetic given by a particular indicator Y.The notion of “indicator” is de.ned in a novel way, without initially specifying what property is indicated and is used to de.ne a topological space of cuts of the model. Various familiar properties of cuts are investigated in this sense, and several results are given stating whether or not the set of cuts having the property is comeagre.A (...)
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  31.  7
    The Power of 2: How an Apparently Irregular Numeration System Facilitates Mental Arithmetic.Andrea Bender & Sieghard Beller - 2016 - Cognitive Science 40 (6):n/a-n/a.
    Mangarevan traditionally contained two numeration systems: a general one, which was highly regular, decimal, and extraordinarily extensive; and a specific one, which was restricted to specific objects, based on diverging counting units, and interspersed with binary steps. While most of these characteristics are shared by numeration systems in related languages in Oceania, the binary steps are unique. To account for these characteristics, this article draws on—and tries to integrate—insights from anthropology, archeology, linguistics, psychology, and cognitive science more generally. The analysis (...)
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  32.  13
    Four Problems Concerning Recursively Saturated Models of Arithmetic.Roman Kossak - 1995 - Notre Dame Journal of Formal Logic 36 (4):519-530.
    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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  33.  33
    Admissible Closures of Polynomial Time Computable Arithmetic.Dieter Probst & Thomas Strahm - 2011 - Archive for Mathematical Logic 50 (5-6):643-660.
    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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  34.  29
    Higher Complexity Search Problems for Bounded Arithmetic and a Formalized No-Gap Theorem.Neil Thapen - 2011 - Archive for Mathematical Logic 50 (7-8):665-680.
    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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  35.  13
    Frege, Dedekind, and the Modern Epistemology of Arithmetic.Markus Pantsar - 2016 - Acta Analytica 31 (3):297-318.
    In early analytic philosophy, one of the most central questions concerned the status of arithmetical objects. Frege argued against the popular conception that we arrive at natural numbers with a psychological process of abstraction. Instead, he wanted to show that arithmetical truths can be derived from the truths of logic, thus eliminating all psychological components. Meanwhile, Dedekind and Peano developed axiomatic systems of arithmetic. The differences between the logicist and axiomatic approaches turned out to be philosophical as well as (...)
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  36.  3
    The Strong Soundness Theorem for Real Closed Fields and Hilbert's Nullstellensatz in Second Order Arithmetic.Nobuyuki Sakamoto & Kazuyuki Tanaka - 2004 - Archive for Mathematical Logic 43 (3):337-349.
    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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  37.  61
    Kant's Theory of Arithmetic: A Constructive Approach? [REVIEW]Kristina Engelhard & Peter Mittelstaedt - 2008 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 39 (2):245 - 271.
    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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  38.  59
    Arithmetic on a Parallel Computer: Perception Versus Logic. [REVIEW]James A. Anderson - 2003 - Brain and Mind 4 (2):169-188.
    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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  39.  7
    Turing–Taylor Expansions for Arithmetic Theories.Joost J. Joosten - 2016 - Studia Logica 104 (6):1225-1243.
    Turing progressions have been often used to measure the proof-theoretic strength of mathematical theories: iterate adding consistency of some weak base theory until you “hit” the target theory. Turing progressions based on n-consistency give rise to a \ proof-theoretic ordinal \ also denoted \. As such, to each theory U we can assign the sequence of corresponding \ ordinals \. We call this sequence a Turing-Taylor expansion or spectrum of a theory. In this paper, we relate Turing-Taylor expansions of sub-theories (...)
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  40.  4
    A Remark on Independence Results for Sharply Bounded Arithmetic.Jan Johannsen - 1998 - Mathematical Logic Quarterly 44 (4):568-570.
    The purpose of this note is to show that the independence results for sharply bounded arithmetic of Takeuti [4] and Tada and Tatsuta [3] can be obtained and, in case of the latter, improved by the model-theoretic method developed by the author in [2].
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  41.  41
    Arithmetic and Logic Incompleteness: The Link.Laureano Luna & Alex Blum - 2008 - The Reasoner 2 (3):6.
    We show how second order logic incompleteness follows from incompleteness of arithmetic, as proved by Gödel.
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  42.  14
    Maximum Schemes in Arithmetic.A. Fernández‐Margarit & M. J. Pérez‐Jiménez - 1994 - Mathematical Logic Quarterly 40 (3):425-430.
    In this paper we deal with some new axiom schemes for Peano's Arithmetic that can substitute the classical induction, least-element, collection and strong collection schemes in the description of PA.
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  43. An Effective Conservation Result for Nonstandard Arithmetic.E. Palmgren - 2000 - Mathematical Logic Quarterly 46 (1):17-24.
    We prove that a nonstandard extension of arithmetic is effectively conservative over Peano arithmetic by using an internal version of a definable ultrapower. By the same method we show that a certain extension of the nonstandard theory with a saturation principle has the same proof-theoretic strength as second order arithmetic, where comprehension is restricted to arithmetical formulas.
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  44.  14
    Hyper-Slingshot. Is Fact-Arithmetic Possible?Wojciech Krysztofiak - 2015 - Foundations of Science 20 (1):59-76.
    The paper presents a new argument supporting the ontological standpoint according to which there are no mathematical facts in any set theoretic model 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 the (...)
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  45.  15
    Husserl's Psychology of Arithmetic.Carlo Ierna - 2012 - Bulletin d'Analyse Phénoménologique 8 (1):97-120.
    In 1913, in a draft for a new Preface for the second edition of the Logical Investigations, Edmund Husserl reveals to his readers that "The source of all my studies and the first source of my epistemological difficul­ties lies in my first works on the philosophy of arithmetic and mathematics in general", i.e. his Habilitationsschrift and the Philosophy of Arithmetic: "I carefully studied the consciousness constituting the amount, first the collec­tive consciousness (consciousness of quantity, of multiplicity) in its (...)
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  46.  5
    William of Ockham’s Ontology of Arithmetic.Magali Roques - 2016 - Vivarium 54 (2-3).
    _ Source: _Page Count 20 Ockham’s ontology of arithmetic, specifically his position on the ontological status of natural numbers, has not yet attracted the attention of scholars. Yet it occupies a central role in his nominalism; specifically, Ockham’s position on numbers constitutes a third part of his ontological reductionism, alongside his doctrines of universals and the categories, which have long been recognized to constitute the first two parts. That is, the first part of this program claims that the very (...)
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  47.  29
    The Complexity of Classification Problems for Models of Arithmetic.Samuel Coskey & Roman Kossak - 2010 - Bulletin of Symbolic Logic 16 (3):345-358.
    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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  48.  27
    Why Axiomatize Arithmetic?Charles Sayward - 2005 - Sorites 16:54-61.
    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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  49.  7
    Constant Regions in Models of Arithmetic.Tin Lok Wong - 2015 - Notre Dame Journal of Formal Logic 56 (4):603-624.
    This paper introduces a new theory of constant regions, which generalizes that of interstices, in nonstandard models of arithmetic. In particular, we show that two homogeneity notions introduced by Richard Kaye and the author, namely, constantness and pregenericity, are equivalent. This led to some new characterizations of generic cuts in terms of existential closedness.
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  50.  4
    William of Ockham’s Ontology of Arithmetic.Magali Roques - forthcoming - New Content is Available for Vivarium.
    _ Source: _Page Count 20 Ockham’s ontology of arithmetic, specifically his position on the ontological status of natural numbers, has not yet attracted the attention of scholars. Yet it occupies a central role in his nominalism; specifically, Ockham’s position on numbers constitutes a third part of his ontological reductionism, alongside his doctrines of universals and the categories, which have long been recognized to constitute the first two parts. That is, the first part of this program claims that the very (...)
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