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1 — 50 / 94
  1. added 2020-04-10
    Hilbert's 10th Problem for Solutions in a Subring of Q.Agnieszka Peszek & Apoloniusz Tyszka - 2019 - Scientific Annals of Computer Science 29 (1):101-111.
    Yuri Matiyasevich's theorem states that the set of all Diophantine equations which have a solution in non-negative integers is not recursive. Craig Smoryński's theorem states that the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable. Let R be a subring of Q with or without 1. By H_{10}(R), we denote the problem of whether there exists an algorithm which for any given Diophantine equation with integer coefficients, can decide whether (...)
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  2. added 2020-04-10
    Two Conjectures on the Arithmetic in ℝ and ℂ†.Apoloniusz Tyszka - 2010 - Mathematical Logic Quarterly 56 (2):175-184.
    Let G be an additive subgroup of ℂ, let Wn = {xi = 1, xi + xj = xk: i, j, k ∈ {1, …, n }}, and define En = {xi = 1, xi + xj = xk, xi · xj = xk: i, j, k ∈ {1, …, n }}. We discuss two conjectures. If a system S ⊆ En is consistent over ℝ, then S has a real solution which consists of numbers whose absolute values belong to (...)
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  3. added 2020-02-14
    Proving Quadratic Reciprocity: Explanation, Disagreement, Transparency and Depth.William D'Alessandro - 2020 - Synthese:1-44.
    Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late 18th century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, due to Gauss (...)
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  4. added 2019-09-18
    On Certain Axiomatizations of Arithmetic of Natural and Integer Numbers.Urszula Wybraniec-Skardowska - 2019 - Axioms 2019 (Deductive Systems).
    The systems of arithmetic discussed in this work are non-elementary theories. In this paper, natural numbers are characterized axiomatically in two di erent ways. We begin by recalling the classical set P of axioms of Peano’s arithmetic of natural numbers proposed in 1889 (including such primitive notions as: set of natural numbers, zero, successor of natural number) and compare it with the set W of axioms of this arithmetic (including the primitive notions like: set of natural numbers and relation of (...)
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  5. added 2019-07-15
    The Concept of “Character” in Dirichlet’s Theorem on Primes in an Arithmetic Progression.Jeremy Avigad & Rebecca Morris - 2014 - Archive for History of Exact Sciences 68 (3):265-326.
    In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. We survey implicit and explicit uses ofDirichlet characters in presentations of Dirichlet’s proof in the nineteenth and early twentieth centuries, with an eye toward understanding some of the pragmatic pressures that shaped the evolution of modern mathematical method.
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  6. added 2019-06-06
    From Fermat to Gauss: Indefinite Descent and Methods of Reduction in Number Theory. [REVIEW]Yvon Gauthier - 2008 - Dialogue 47 (2):411-414.
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  7. added 2019-06-06
    Boethian Number Theory. [REVIEW]Ivor Bulmer-Thomas - 1985 - The Classical Review 35 (1):86-87.
  8. added 2019-06-06
    On the Interpretation of Non-Finitist Proofs–Part II.G. Kreisel - 1952 - Journal of Symbolic Logic 17 (1):43-58.
  9. added 2019-01-30
    Putnam, Peano, and the Malin Génie: Could We Possibly Bewrong About Elementary Number-Theory?Christopher Norris - 2002 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 33 (2):289-321.
    This article examines Hilary Putnam's work in the philosophy of mathematics and - more specifically - his arguments against mathematical realism or objectivism. These include a wide range of considerations, from Gödel's incompleteness-theorem and the limits of axiomatic set-theory as formalised in the Löwenheim-Skolem proof to Wittgenstein's sceptical thoughts about rule-following, Michael Dummett's anti-realist philosophy of mathematics, and certain problems – as Putnam sees them – with the conceptual foundations of Peano arithmetic. He also adopts a thought-experimental approach – a (...)
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  10. added 2018-01-12
    Character and Object.Rebecca Morris & Jeremy Avigad - 2016 - Review of Symbolic Logic 9 (3):480-510.
    In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. Modern presentations of the proof are explicitly higher-order, in that they involve quantifying over and summing over Dirichlet characters, which are certain types of functions. The notion of a character is only implicit in Dirichlet’s original proof, and the subsequent history shows a very gradual transition to the modern mode of presentation. In this essay, we (...)
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  11. added 2017-05-21
    Arithmetic, Set Theory, Reduction and Explanation.William D’Alessandro - 2018 - Synthese 195 (11):5059-5089.
    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 defense (...)
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  12. added 2017-01-12
    Purity in Arithmetic: Some Formal and Informal Issues.Andrew Arana - 2014 - In Godehard Link (ed.), Formalism and Beyond: On the Nature of Mathematical Discourse. De Gruyter. pp. 315-336.
  13. added 2017-01-12
    L'infinité des nombres premiers : une étude de cas de la pureté des méthodes.Andrew Arana - 2011 - Les Etudes Philosophiques 97 (2):193.
    Une preuve est pure si, en gros, elle ne réfère dans son développement qu’à ce qui est « proche » de, ou « intrinsèque » à l’énoncé à prouver. L’infinité des nombres premiers, un théorème classique de l’arithmétique, est un cas d’étude particulièrement riche pour les recherches philosophiques sur la pureté. Deux preuves différentes de ce résultat sont ici considérées, à savoir la preuve euclidienne classique et une preuve « topologique » plus récente proposée par Furstenberg. D’un point de vue (...)
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  14. added 2016-12-08
    Intuitionistic Remarks on Husserl’s Analysis of Finite Number in the Philosophy of Arithmetic.Mark van Atten - 2004 - Graduate Faculty Philosophy Journal 25 (2):205-225.
    Brouwer and Husserl both aimed to give a philosophical account of mathematics. They met in 1928 when Husserl visited the Netherlands to deliver his Amsterdamer Vorträge. Soon after, Husserl expressed enthusiasm about this meeting in a letter to Heidegger, and he reports that they had long conversations which, for him, had been among the most interesting events in Amsterdam. However, nothing is known about the content of these conversations; and it is not clear whether or not there were any other (...)
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  15. added 2016-12-08
    Review of J. Belna, La Notion de Nombre Chez Dedekind, Cantor, Frege. Theories, Conceptions, Et Philosophie[REVIEW]E. Carson - 1998 - Philosophia Mathematica 6 (3):345-350.
  16. added 2016-11-15
    Finding Structure in a Meditative State.Bas Rasmussen - manuscript
    I have been experimenting with meditation for a long time, but just recently I seem to have come across another being in there. It may just be me looking at me, but whatever it is, it is showing me some really interesting arrangements of colored balls. At first, I thought it was just random colors and shapes, but it became very ordered. It was like this being (me?) was trying to talk to me but couldn’t, so was showing me some (...)
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  17. added 2016-03-23
    Where Our Number Concepts Come From.Susan Carey - 2009 - Journal of Philosophy 106 (4):220-254.
  18. added 2016-02-21
    An Introduction to Basic Arithmetic.Mohammad Ardeshir & Bardyaa Hesaam - 2008 - Logic Journal of the IGPL 16 (1):1-13.
    We study Basic Arithmetic BA, which is the basic logic BQC equivalent of Heyting Arithmetic HA over intuitionistic logic IQC, and of Peano Arithmetic PA over classical logic CQC. It turns out that The Friedman translation is applicable to BA. Using this translation, we prove that BA is closed under a restricted form of the Markov rule. Moreover, it is proved that all nodes of a finite Kripke model of BA are classical models of , a fragment of PA with (...)
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  19. added 2015-12-02
    Concepts and Counting.Ian Rumfitt - 2001 - Proceedings of the Aristotelian Society 102 (1):41-68.
    Frege's analysis of Zahlangaben is expounded and evaluated.
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  20. added 2015-10-30
    On Some Historical Aspects of the Theory of Riemann Zeta Function.Giuseppe Iurato - manuscript
    This comprehensive historical account concerns that non-void intersection region between Riemann zeta function and entire function theory, with a view towards possible physical applications.
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  21. added 2015-06-29
    Peirce's Potential Continuity and Pure Geometry.Jean-Louis Hudry - 2004 - Transactions of the Charles S. Peirce Society 40 (2):229 - 243.
  22. added 2015-05-26
    On the Status of Arithmetic.W. Balzer - 1979 - Erkenntnis 14 (1):57 - 85.
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  23. added 2015-05-25
    A Nonasymptotic Lower Time Bound for a Strictly Bounded Second-Order Arithmetic.Anatoly P. Beltiukov - 2006 - 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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  24. added 2015-05-25
    Incompleteness of a Free Arithmetic.E. Bencivenga - 1988 - Logique Et Analyse 31 (21):79.
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  25. added 2015-05-23
    Partitions of Trees and $${{\Sf ACA}^\Prime_{0}}$$.Bernard A. Anderson & Jeffry L. Hirst - 2009 - 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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  26. added 2014-12-27
    Prime Number Decomposition, the Hyperbolic Function and Multi-Path Michelson Interferometers.V. Tamma, C. O. Alley, W. P. Schleich & Y. H. Shih - 2012 - Foundations of Physics 42 (1):111-121.
    The phase φ of any wave is determined by the ratio x/λ consisting of the distance x propagated by the wave and its wavelength λ. Hence, the dependence of φ on λ constitutes an analogue system for the mathematical operation of division, that is to obtain the hyperbolic function f(ξ)≡1/ξ. We take advantage of this observation to decompose integers into primes and implement this approach towards factorization of numbers in a multi-path Michelson interferometer. This work is part of a larger (...)
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  27. added 2014-05-19
    A Genetic Interpretation of Neo-Pythagorean Arithmetic.Ioannis M. Vandoulakis - 2010 - Oriens - Occidens 7:113-154.
    The style of arithmetic in the treatises the Neo-Pythagorean authors is strikingly different from that of the "Elements". Namely, it is characterised by the absence of proof in the Euclidean sense and a specific genetic approach to the construction of arithmetic that we are going to describe in our paper. Lack of mathematical sophistication has led certain historians to consider this type of mathematics as a feature of decadence of mathematics in this period [Tannery 1887; Heath 1921]. The alleged absence (...)
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  28. added 2014-05-19
    Was Euclid's Approach to Arithmetic Axiomatic?Ioannis M. Vandoulakis - 1998 - Oriens - Occidens 2:141-181.
    The lack of specific arithmetical axioms in Book VII has puzzled historians of mathematics. It is hardly possible in our view to ascribe to the Greeks a conscious undertaking to axiomatize arithmetic. The view that associates the beginnings of the axiomatization of arithmetic with the works of Grassman [1861], Dedekind [1888] and Peano [1889] seems to be more plausible. In this connection a number of interesting historical problems have been raised, for instance, why arithmetic was axiomatized so late. This question (...)
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  29. added 2014-04-02
    The Successor Function and Induction Principle in a Hegelian Philosophy of Mathematics.Alan L. T. Paterson - 2000 - Idealistic Studies 30 (1):25-60.
  30. added 2014-04-02
    Classical Arithmetic as Part of Intuitionistic Arithmetic.Michael Potter - 1998 - Grazer Philosophische Studien 55:127-41.
    Argues that classical arithmetic can be viewed as a proper part of intuitionistic arithmetic. Suggests that this largely neutralizes Dummett's argument for intuitionism in the case of arithmetic.
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  31. added 2014-03-30
    New Waves in Philosophy of Mathematics.Otávio Bueno & Øystein Linnebo (eds.) - 2009 - Palgrave-Macmillan.
    Thirteen up-and-coming researchers in the philosophy of mathematics have been invited to write on what they take to be the right philosophical account of mathematics, examining along the way where they think the philosophy of mathematics is and ought to be going. A rich and diverse picture emerges. Some broader tendencies can nevertheless be detected: there is increasing attention to the practice, language and psychology of mathematics, a move to reassess the orthodoxy, as well as inspiration from philosophical logic.
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  32. added 2014-03-29
    Husserl on Axiomatization and Arithmetic.Claire Ortiz Hill - 2010 - In Mirja Hartimo (ed.), Phenomenology and Mathematics. Springer.
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  33. added 2014-03-26
    Applications of Nonstandard Analysis in Additive Number Theory.Renling Jin - 2000 - 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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  34. added 2014-03-25
    On Representing Concepts in Finite Models.Marcin Mostowski - 2001 - Mathematical Logic Quarterly 47 (4):513-523.
    We present a method of transferring Tarski's technique of classifying finite order concepts by means of truth-definitions into finite mode theory. The other considered question is the problem of representability relations on words or natural numbers in finite models. We prove that relations representable in finite models are exactly those which are of degree ≤ o′. Finally, we consider theories of sufficiently large finite models. For a given theory T we define sl as the set of all sentences true in (...)
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  35. added 2014-03-20
    Theories of Arithmetics in Finite Models.Michał Krynicki & Konrad Zdanowski - 2005 - 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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  36. added 2014-03-16
    Introduction to Mathematical Proofs: A Transition.Charles E. Roberts - 2009 - Crc Press.
    The book includes more than 75 examples and more than 600 problems. A solutions manual is available upon qualifying course adoptions.
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  37. added 2014-03-14
    Number Theory and Elementary Arithmetic.Jeremy Avigad - 2003 - 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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  38. added 2014-03-12
    Edmund Husserl, Philosophy of Arithmetic, Translated by Dallas Willard.Carlo Ierna - 2008 - Husserl Studies 24 (1):53-58.
    This volume contains an English translation of Edmund Husserl’s first major work, the Philosophie der Arithmetik, (Husserl 1891). As a translation of Husserliana XII (Husserl 1970), it also includes the first chapter of Husserl’s Habilitationsschrift (Über den Begriff der Zahl) (Husserl 1887) and various supplementary texts written between 1887 and 1901. This translation is the crowning achievement of Dallas Willard’s monumental research into Husserl’s early philosophy (Husserl 1984) and should be seen as a companion to volume V of the Husserliana: (...)
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  39. added 2014-03-10
    Hilbert's Objectivity.Lydia Patton - 2014 - Historia Mathematica 41 (2):188-203.
    Detlefsen (1986) reads Hilbert's program as a sophisticated defense of instrumentalism, but Feferman (1998) has it that Hilbert's program leaves significant ontological questions unanswered. One such question is of the reference of individual number terms. Hilbert's use of admittedly "meaningless" signs for numbers and formulae appears to impair his ability to establish the reference of mathematical terms and the content of mathematical propositions (Weyl (1949); Kitcher (1976)). The paper traces the history and context of Hilbert's reasoning about signs, which illuminates (...)
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  40. added 2014-03-07
    How Discrete Patterns Emerge From Algorithmic Fine-Tuning: A Visual Plea for Kroneckerian Finitism.Ivahn Smadja - 2010 - Topoi 29 (1):61-75.
    This paper sets out to adduce visual evidence for Kroneckerian finitism by making perspicuous some of the insights that buttress Kronecker’s conception of arithmetization as a process aiming at disclosing the arithmetical essence enshrined in analytical formulas, by spotting discrete patterns through algorithmic fine-tuning. In the light of a fairly tractable case study, it is argued that Kronecker’s main tenet in philosophy of mathematics is not so much an ontological as a methodological one, inasmuch as highly demanding requirements regarding mathematical (...)
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  41. added 2014-03-06
    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, and the (...)
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  42. added 2014-03-06
    What Does It Take to Prove Fermat's Last Theorem? Grothendieck and the Logic of Number Theory.Colin McLarty - 2010 - Bulletin of Symbolic Logic 16 (3):359-377.
    This paper explores the set theoretic assumptions used in the current published proof of Fermat's Last Theorem, how these assumptions figure in the methods Wiles uses, and the currently known prospects for a proof using weaker assumptions.
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  43. added 2014-01-30
    Partial Collapses of the Complexity Hierarchy in Models for Fragments of Bounded Arithmetic.Zofia Adamowicz & Leszek Aleksander Kołodziejczyk - 2007 - 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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  44. added 2013-01-06
    Laws of Form.G. Spencer-Brown - 1972 - New York: Julian Press.
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  45. added 2013-01-02
    Induction, More or Less.Peter Smith - unknown
    The first main topic of this paper is a weak second-order theory that sits between firstorder Peano Arithmetic PA1 and axiomatized second-order Peano Arithmetic PA2 – namely, that much-investigated theory known in the trade as ACA0. What I’m going to argue is that ACA0, in its standard form, lacks a cogent conceptual motivation. Now, that claim – when the wraps are off – will turn out to be rather less exciting than it sounds. It isn’t that all the work that (...)
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  46. added 2012-11-14
    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.Elio Conte - 2012 - 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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  47. added 2012-07-02
    Logic and Nothing Else.Jaroslav Peregrin - unknown
    Clauses (1) and (2) guarantee the inclusion of all 'intuitive' natural numbers, and (3) guarantees the exclusion of all other objects. Thus, in particular, no nonstandard numbers, which would follow after the intuitive ones are admitted (nonstandard numbers are found in nonstandard models of Peano arithmetic, in which the standard natural numbers are followed by one or more 'copies' of integers running from minus infinity to infinity).
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  48. added 2012-04-20
    Vom Zahlen Zu den Zahlen: On the Relation Between Computation and Arithmetical Structuralism.L. Horsten - 2012 - 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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  49. added 2011-08-15
    Numbers in Presence and Absence. A Study of Husserl's Philosophy of Mathematics.J. Philip Miller - 1982 - Kluwer Academic Publishers.
    CHAPTER I THE EMERGENCE AND DEVELOPMENT OF HUSSERL'S 'PHILOSOPHY OF ARITHMETIC'. HISTORICAL BACKGROUND: WEIERSTRASS AND THE ARITHMETIZATION OF ANALYSIS In ...
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  50. added 2011-07-05
    Explaining Experience In Nature: The Foundations Of Logic And Apprehension.Steven Ericsson-Zenith - forthcoming - 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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