Number Theory

Edited by William D'Alessandro (Ludwig Maximilians Universität, München)
About this topic
Summary Number theory is the branch of pure mathematics dealing with the integers, functions on the integers and related matters. As a body of mathematical knowledge, number theory is interesting for several reasons: for instance, because many of its statements are very simple but very hard to prove, often seeming to require methods of surprising depth and complexity. (Fermat's Last Theorem, the Goldbach conjecture and the Twin Primes conjecture are famous examples.) As an axiomatic system, arithmetic and its fragments and extensions are primary objects of study for logicians. As mathematical objects, the integers are focal points of various debates in metaphysics and epistemology: about the existence of numbers and their relationship to sets, the a priori vs. a posteriori status of arithmetic, and the origins and nature of numerical cognition, to name a few.
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  1. Physical Possibility and Determinate Number Theory.Sharon Berry - manuscript
    It's currently fashionable to take Putnamian model theoretic worries seriously for mathematics, but not for discussions of ordinary physical objects and the sciences. But I will argue that (under certain mild assumptions) merely securing determinate reference to physical possibility suffices to rule out nonstandard models of our talk of numbers. So anyone who accepts realist reference to physical possibility should not reject reference to the standard model of the natural numbers on Putnamian model theoretic grounds.
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  2. 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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  3. On Cut Elimination for Subsystems of Second-Order Number Theory.William Tait - manuscript
    To appear in the Proceedings of Logic Colloquium 2006. (32 pages).
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  4. Explication and the Foundation of Number Theory.Darren Mcdonald - unknown - Eidos: The Canadian Graduate Journal of Philosophy 18.
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  5. Lorenzen's Proof of Consistency for Elementary Number Theory.Thierry Coquand & Stefan Neuwirth - 2020 - History and Philosophy of Logic 41 (3):281-290.
    We present a manuscript of Paul Lorenzen that provides a proof of consistency for elementary number theory as an application of the construction of the free countably complete pseudocomplemented semilattice over a preordered set. This manuscript rests in the Oskar-Becker-Nachlass at the Philosophisches Archiv of Universität Konstanz, file OB 5-3b-5. It has probably been written between March and May 1944. We also compare this proof to Gentzen's and Novikov's, and provide a translation of the manuscript.
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  6. 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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  7. Using Figurate Numbers in Elementary Number Theory – Discussing a ‘Useful’ Heuristic From the Perspectives of Semiotics and Cognitive Psychology.Leander Kempen & Rolf Biehler - 2020 - Frontiers in Psychology 11.
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  8. Modern Physics and Number Theory.Daniel Brox - 2019 - Foundations of Physics 49 (8):837-853.
    Despite the efforts of many individuals, the disciplines of modern physics and number theory have remained largely divorced, in the sense that the experimentally verified theories of quantum physics and gravity are written in the language of linear algebra and advanced calculus, without reference to several established branches of pure mathematics. This absence raises questions as to whether or not pure mathematics has undiscovered application to physical modeling that could have far reaching implications for human scientific understanding. In this paper, (...)
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  9. Learning the Natural Numbers as a Child.Stefan Buijsman - 2019 - Noûs 53 (1):3-22.
    How do we get out knowledge of the natural numbers? Various philosophical accounts exist, but there has been comparatively little attention to psychological data on how the learning process actually takes place. I work through the psychological literature on number acquisition with the aim of characterising the acquisition stages in formal terms. In doing so, I argue that we need a combination of current neologicist accounts and accounts such as that of Parsons. In particular, I argue that we learn the (...)
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  10. Disentangling the Mechanisms of Symbolic Number Processing in Adults’ Mathematics and Arithmetic Achievement.Josetxu Orrantia, David Muñez, Laura Matilla, Rosario Sanchez, Sara San Romualdo & Lieven Verschaffel - 2019 - Cognitive Science 43 (1).
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  11. 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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  12. 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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  13. Avicenna and Number Theory.Pascal Crozet - 2018 - In Hassan Tahiri (ed.), The Philosophers and Mathematics: Festschrift for Roshdi Rashed. Springer Verlag. pp. 67-80.
    Among the four mathematical treatises that Avicenna takes care to place within his philosophical encyclopaedia, the one he devotes to arithmetic is undoubtedly the most singular. Contrary to the treatise on geometry, which differs little from its Euclidean model, the philosopher takes as his point of departure the treatise of Nicomachus of Gerase, but modifies its spirit to incorporate results from the many disciplines which were dealing with numbers: Euclidean Theory of numbers, Nicomachean Aritmāṭīqī, Indian reckoning, Ḥisāb, Algebra, etc. We (...)
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  14. 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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  15. Counting and Number Line Trainings in Kindergarten: Effects on Arithmetic Performance and Number Sense.Ilona Friso-van den Bos, Evelyn H. Kroesbergen & Johannes E. H. Van Luit - 2018 - Frontiers in Psychology 9.
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  16. How the Abstract Becomes Concrete: Irrational Numbers Are Understood Relative to Natural Numbers and Perfect Squares.Purav Patel & Sashank Varma - 2018 - Cognitive Science 42 (5):1642-1676.
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  17. Semantic Alignment Across Whole-Number Arithmetic and Rational Numbers: Evidence From a Russian Perspective.Yulia A. Tyumeneva, Galina Larina, Ekaterina Alexandrova, Melissa DeWolf, Miriam Bassok & Keith J. Holyoak - 2018 - Thinking and Reasoning 24 (2):198-220.
    Solutions to word problems are moderated by the semantic alignment of real-world relations with mathematical operations. Categorical relations between entities are aligned with addition, whereas certain functional relations between entities are aligned with division. Similarly, discreteness vs. continuity of quantities is aligned with different formats for rational numbers. These alignments have been found both in textbooks and in the performance of college students in the USA and in South Korea. The current study examined evidence for alignments in Russia. Textbook analyses (...)
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  18. SICs and Algebraic Number Theory.Marcus Appleby, Steven Flammia, Gary McConnell & Jon Yard - 2017 - Foundations of Physics 47 (8):1042-1059.
    We give an overview of some remarkable connections between symmetric informationally complete measurements and algebraic number theory, in particular, a connection with Hilbert’s 12th problem. The paper is meant to be intelligible to a physicist who has no prior knowledge of either Galois theory or algebraic number theory.
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  19. Husserl’s Early Semiotics and Number Signs: Philosophy of Arithmetic Through the Lens of “On the Logic of Signs (Semiotic)".Thomas Byrne Iii - 2017 - Journal of the British Society of Phenomenology 48 (4):287-303.
    This paper demonstrates that Edmund Husserl’s frequently overlooked 1890 manuscript, “On the Logic of Signs ”, when closely investigated, reveals itself to be the hermeneutical touchstone for his seminal 1891 Philosophy of Arithmetic. As the former comprises Husserl’s earliest attempt to account for all of the different kinds of signitive experience, his conclusions there can be directly applied to the latter, which is focused on one particular type of sign; namely, number signs. Husserl’s 1890 descriptions of motivating and replacing signs (...)
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  20. Husserl’s Early Semiotics and Number Signs: Philosophy of Arithmetic Through the Lens of “On the Logic of Signs ”.Thomas Byrne - 2017 - Journal of the British Society for Phenomenology 48 (4):287-303.
    This paper demonstrates that Edmund Husserl’s frequently overlooked 1890 manuscript, “On the Logic of Signs,” when closely investigated, reveals itself to be the hermeneutical touchstone for his seminal 1891 Philosophy of Arithmetic. As the former comprises Husserl’s earliest attempt to account for all of the different kinds of signitive experience, his conclusions there can be directly applied to the latter, which is focused on one particular type of sign; namely, number signs. Husserl’s 1890 descriptions of motivating and replacing signs will (...)
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  21. Kant's Conception of Number.Daniel Sutherland - 2017 - Philosophical Review Current Issue 126 (2):147-190.
    Despite the importance of Kant's claims about mathematical cognition for his philosophy as a whole and for subsequent philosophy of mathematics, there is still no consensus on his philosophy of arithmetic, and in particular the role he assigns intuition in it. This inquiry sets aside the role of intuition for the nonce to investigate Kant's conception of natural number. Although Kant himself doesn't distinguish between a cardinal and an ordinal conception of number, some of the properties Kant attributes to number (...)
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  22. 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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  23. Arithmetic and Number in the Philosophy of Symbolic Forms.Jeremy Heis - 2015 - In Sebastian Luft & J. Tyler Friedman (eds.), The Philosophy of Ernst Cassirer: A Novel Assessment. De Gruyter. pp. 123-140.
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  24. 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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  25. 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.
    Over the years many mathematicians have voiced a preference for proofs that stay “close” to the statements being proved, avoiding “foreign”, “extraneous”, or “remote” considerations. Such proofs have come to be known as “pure”. Purity issues have arisen repeatedly in the practice of arithmetic; a famous instance is the question of complex-analytic considerations in the proof of the prime number theorem. This article surveys several such issues, and discusses ways in which logical considerations shed light on these issues.
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  26. 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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  27. 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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  28. Number Symbolism - Kalvesmaki the Theology of Arithmetic. Number Symbolism in Platonism and Early Christianity. Pp. X + 231, Ills. Washington, D.C.: Center for Hellenic Studies, 2013. Paper, £16.95, €20.70, Us$22.95. Isbn: 978-0-674-07330-2. [REVIEW]Svetla Slaveva-Griffin - 2014 - The Classical Review 64 (2):429-431.
  29. Study of Analytic Number Theory: Riemann’s Hypothesis and Prime Number Theorem with Addendum on Integer Partitions.Lukasz Andrzej Glinka - 2013 - Cambridge International Science Publishing.
    This monograph explores several classical issues of modern mathematics, and discusses both the historical and research aspects. The brief historical part is focused on Riemann’s zeta function and few related problems. The research part starts from direct formulation of simple proofs of both the prime number theorem and Riemann’s hypothesis, two intriguing problems of modern mathematics, which applies the concept of Mertens’s function and is based on Apostol’s and Littlewood’s criterions of equivalence. The second research problem discussed in this book (...)
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  30. Review of Graßmann, Robert, Theory of Number or Arithmetic in Strict Scientific Presentation by Strict Use of Formulas (1891). [REVIEW] Graßmann & Robert - 2013 - Philosophia Scientiæ 17 (17-1):57-70.
    The author of this book pursues the objective of treating the whole of pure mathematics [die ganze reine Mathematik] in four sections [Abtheilungen]. One half of the first of these sections is dedicated to arithmetic and is already available. The other half of the first section “A heuristic treatise on number [Zahlenlehre in freier Gedankenentwicklung]” which treats the same discipline is supposed to follow. The author may have opted for such an unusual separation [of the treatment of arithme..
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  31. On Equality and Natural Numbers in Cantor-Lukasiewicz Set Theory.P. Hajek - 2013 - Logic Journal of the IGPL 21 (1):91-100.
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  32. Review of Graßmann, Robert, Theory of Number or Arithmetic in Strict Scientific Presentation by Strict Use of Formulas (1891). [REVIEW]Otto Hölder - 2013 - Philosophia Scientiæ 17 (17-1):57-70.
    The author of this book pursues the objective of treating the whole of pure mathematics [die ganze reine Mathematik] in four sections [Abtheilungen]. One half of the first of these sections is dedicated to arithmetic and is already available. The other half of the first section “A heuristic treatise on number [Zahlenlehre in freier Gedankenentwicklung]” which treats the same discipline is supposed to follow. The author may have opted for such an unusual separation [of the treatment of arithme..
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  33. Review of Graßmann, Robert, Theory of Number or Arithmetic in Strict Scientific Presentation by Strict Use of Formulas. [REVIEW]Otto Hölder - 2013 - Philosophia Scientae 17:57-70.
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  34. Review of Graßmann, Robert, Theory of Number or Arithmetic in Strict Scientific Presentation by Strict Use of Formulas. [REVIEW]Otto Hölder - 2013 - Philosophia Scientiae 17 (1):57-70.
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  35. Reference to Numbers in Natural Language.Friederike Moltmann - 2013 - Philosophical Studies 162 (3):499 - 536.
    A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets, eight, as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this view, numbers are not primarily treated abstract objects, (...)
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  36. Review of Grassmann, Robert, Theory of Number or Arithmetic in Strict Scientific Presentation by Strict Use of Formulas (1891): Otto Hölder (1859-1937). [REVIEW]Mircea Radu - 2013 - Philosophia Scientiae 17 (1):57-70.
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  37. What Counts as a Number?Jean W. Rioux - 2013 - International Philosophical Quarterly 53 (3):229-249.
    Georg Cantor argued that pure mathematics would be better-designated “free mathematics” since mathematical inquiry need not justify its discoveries through some extra-mental standard. Even so, he spent much of his later life addressing ancient and scholastic objections to his own transfinite number theory. Some philosophers have argued that Cantor need not have bothered. Thomas Aquinas at least, and perhaps Aristotle, would have consistently embraced developments in number theory, including the transfinite numbers. The author of this paper asks whether the restriction (...)
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  38. On Infinite Number and Distance.Jeremy Gwiazda - 2012 - Constructivist Foundations 7 (2):126-130.
    Context: The infinite has long been an area of philosophical and mathematical investigation. There are many puzzles and paradoxes that involve the infinite. Problem: The goal of this paper is to answer the question: Which objects are the infinite numbers (when order is taken into account)? Though not currently considered a problem, I believe that it is of primary importance to identify properly the infinite numbers. Method: The main method that I employ is conceptual analysis. In particular, I argue that (...)
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  39. 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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  40. Frege's Changing Conception of Number.Kevin C. Klement - 2012 - Theoria 78 (2):146-167.
    I trace changes to Frege's understanding of numbers, arguing in particular that the view of arithmetic based in geometry developed at the end of his life (1924–1925) was not as radical a deviation from his views during the logicist period as some have suggested. Indeed, by looking at his earlier views regarding the connection between numbers and second-level concepts, his understanding of extensions of concepts, and the changes to his views, firstly, in between Grundlagen and Grundgesetze, and, later, after learning (...)
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  41. Indexed Natural Numbers in Mind: A Formal Model of the Basic Mature Number Competence. [REVIEW]Wojciech Krysztofiak - 2012 - 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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  42. Invited Book Review of Stanislas Debaene, The Number Sense: How the Mind Creates Mathematics (New York: Oxford University Press, 1997). [REVIEW]Stephen Palmquist - 2012 - Journal of Scientific Exploration 26 (4):928-930.
    What Stanislas Debaene dubs "the number sense" is a natural ability humans share with other animals, enabling us to "count" to four virtually instantaneously. This so-called "accumulator" provides "a direct intuition of what numbers mean". Beyond four, our ability to perceive numbers becomes approximate, though concepts enable us to move beyond approximation. Because humans typically learn number concepts in early childhood, we easily forget that our brains retain the number sense throughout life. This book examines the biological basis for this (...)
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  43. 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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  44. 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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  45. Objective Probabilities in Number Theory.J. Ellenberg & E. Sober - 2011 - Philosophia Mathematica 19 (3):308-322.
    Philosophers have explored objective interpretations of probability mainly by considering empirical probability statements. Because of this focus, it is widely believed that the logical interpretation and the actual-frequency interpretation are unsatisfactory and the hypothetical-frequency interpretation is not much better. Probabilistic assertions in pure mathematics present a new challenge. Mathematicians prove theorems in number theory that assign probabilities. The most natural interpretation of these probabilities is that they describe actual frequencies in finite sets and limits of actual frequencies in infinite sets. (...)
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  46. Logicism and the Problem of Infinity: The Number of Numbers: Articles.Gregory Landini - 2011 - Philosophia Mathematica 19 (2):167-212.
    Simple-type theory is widely regarded as inadequate to capture the metaphysics of mathematics. The problem, however, is not that some kinds of structure cannot be studied within simple-type theory. Even structures that violate simple-types are isomorphic to structures that can be studied in simple-type theory. In disputes over the logicist foundations of mathematics, the central issue concerns the problem that simple-type theory fails to assure an infinity of natural numbers as objects. This paper argues that the problem of infinity is (...)
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  47. Husserl on Axiomatization and Arithmetic.Claire Ortiz Hill - 2010 - In Mirja Hartimo (ed.), Phenomenology and Mathematics. Springer.
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  48. First-Order Reasoning and Primitive Recursive Natural Number Notations.David Isles - 2010 - Studia Logica 96 (1):49-64.
    If the collection of models for the axioms 21 of elementary number theory is enlarged to include not just the " natural numbers " or their non-standard infinitistic extensions but also what are here called "primitive recursive notations", questions arise about the reliability of first-order derivations from 21. In this enlarged set of "models" some derivations usually accepted as "reliable" may be problematic. This paper criticizes two of these derivations which claim, respectively, to establish the totality of exponentiation and to (...)
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  49. 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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  50. 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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