Number Theory

Edited by William D'Alessandro (University of Oxford, University of Oxford)
 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.
Related

Contents
241 found
Order:
 1 — 50 / 241
1. Conventional wisdom dictates that proofs of mathematical propositions should be treated as necessary, and sufficient, for entailing `significant' mathematical truths only if the proofs are expressed in a---minimally, deemed consistent---formal mathematical theory in terms of: * Axioms/Axiom schemas * Rules of Deduction * Definitions * Lemmas * Theorems * Corollaries. Whilst Andrew Wiles' proof of Fermat's Last Theorem FLT, which appeals essentially to geometrical properties of real and complex numbers, can be treated as meeting this criteria, it nevertheless leaves two (...)

Export citation

Bookmark
2. 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.

Export citation

Bookmark   1 citation
3. 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.

Export citation

Bookmark
4. To appear in the Proceedings of Logic Colloquium 2006. (32 pages).
Remove from this list

Export citation

Bookmark
5. Explication and the Foundation of Number Theory.Darren Mcdonald - unknown - Eidos: The Canadian Graduate Journal of Philosophy 18.
Remove from this list

Export citation

Bookmark
6. Physical Possibility and Determinate Number Theory.Sharon Berry - forthcoming - Philosophia Mathematica:nkab013.
ABSTRACT It is currently fashionable to take Putnamian model-theoretic worries seriously for mathematics, but not for discussions of ordinary physical objects and the sciences. However, I will argue that merely securing determinate reference to physical possibility suffices to rule out the kind of nonstandard interpretations of our number talk Putnam invokes. So, anyone who accepts determinate reference to physical possibility should not reject determinate reference to the natural numbers on Putnamian model-theoretic grounds.

Export citation

Bookmark   2 citations
7. Turin, 1916, G. Fubini: An Experiment in the Patrimonialisation of Number Theory. [REVIEW]Erika Luciano - forthcoming - Philosophia Scientiae:123-144.
Une analyse historique du cours d’analyse supérieure donné par Guido Fubini à l’université de Turin en 1916 et entièrement consacré à la théorie des nombres ouvre d’intéressantes perspectives d’étude. L’article retrace une pratique de patrimonialisation collective, au sens où il analyse comment une communauté mathématique (l’École italienne de géométrie algébrique) liée à un lieu spécifique (l’université de Turin) organisait l’enseignement supérieur au cours d’une année donnée (1916), l’objectif étant d’identifier ce qu’elle jugeait important de préserver et de transmettre par rapport (...)

Export citation

Bookmark
8. Value Judgments in Mathematics: G. H. Hardy and the (Non-)seriousness of Mathematical Theorems.Simon Weisgerber - 2024 - Global Philosophy 34 (1):1-24.
One of the general criteria G. H. Hardy identifies and discusses in his famous essay A Mathematician’s Apology (Cambridge University Press, Cambridge, 1940) by which a mathematician’s patterns must be judged is seriousness. This article focuses on one of Hardy’s examples of a non-serious theorem, namely that 8712 and 9801 are the only numbers below 10000 which are integral multiples of their reversals, in the sense that 8712 = 4·2178, and 9801 = 9·1089. In the context of a discussion of (...)

Export citation

Bookmark
9. Introduction to proof through number theory.Bennett Chow - 2023 - Providence, Rhode Island: American Mathematical Society.
Lighten up about mathematics! Have fun. If you read this book, you will have to endure bad math puns and jokes and out-of-date pop culture references. You'll learn some really cool mathematics to boot. In the process, you will immerse yourself in living, thinking, and breathing logical reasoning. We like to call this proofs, which to some is a bogey word, but to us it is a boogie word. You will learn how to solve problems, real and imagined. After all, (...)

Export citation

Bookmark
10. Why Did Thomas Harriot Invent Binary?Lloyd Strickland - 2023 - Mathematical Intelligencer 46 (2).
From the early eighteenth century onward, primacy for the invention of binary numeration and arithmetic was almost universally credited to the German polymath Gottfried Wilhelm Leibniz (1646–1716). Then, in 1922, Frank Vigor Morley (1899–1980) noted that an unpublished manuscript of the English mathematician, astronomer, and alchemist Thomas Harriot (1560–1621) contained the numbers 1 to 8 in binary. Morley’s only comment was that this foray into binary was “certainly prior to the usual dates given for binary numeration”. Almost thirty years later, (...)

Export citation

Bookmark
11. F Things You (Probably) Didn't Know About Hexadecimal.Lloyd Strickland & Owain Daniel Jones - 2023 - The Mathematical Intelligencer 45:126-130.

Export citation

Bookmark
12. Mathematical Progress — On Maddy and Beyond.Simon Weisgerber - 2023 - Philosophia Mathematica 31 (1):1-28.
A key question of the ‘maverick’ tradition of the philosophy of mathematical practice is addressed, namely what is mathematical progress. The investigation is based on an article by Penelope Maddy devoted to this topic in which she considers only contributions ‘of some mathematical importance’ as progress. With the help of a case study from contemporary mathematics, more precisely from tropical geometry, a few issues with her proposal are identified. Taking these issues into consideration, an alternative account of ‘mathematical importance’, broadly (...)

Export citation

Bookmark   2 citations
13. Idéaux de preuve : explication et pureté.Andrew Arana - 2022 - In Andrew Arana & Marco Panza (eds.), Précis de philosophie de la logique et des mathématiques. Volume 2, philosophie des mathématiques. Paris, France: pp. 387-425.
Why do mathematics often give several proofs of the same theorem? This is the question raised in this article, introducing the notion of an epistemic ideal and discussing two such ideals, the explanatoriness and purity of proof.

Export citation

Bookmark
14. Unrealistic Models in Mathematics.William D'Alessandro - 2022 - Philosophers’ Imprint.
Models are indispensable tools of scientific inquiry, and one of their main uses is to improve our understanding of the phenomena they represent. How do models accomplish this? And what does this tell us about the nature of understanding? While much recent work has aimed at answering these questions, philosophers' focus has been squarely on models in empirical science. I aim to show that pure mathematics also deserves a seat at the table. I begin by presenting two cases: Cramér’s random (...)

Export citation

Bookmark
15. Two-sorted Frege Arithmetic is not Conservative.Stephen Mackereth & Jeremy Avigad - 2022 - Review of Symbolic Logic:1-34.
Neo-Fregean logicists claim that Hume's Principle (HP) may be taken as an implicit definition of cardinal number, true simply by fiat. A longstanding problem for neo-Fregean logicism is that HP is not deductively conservative over pure axiomatic second-order logic. This seems to preclude HP from being true by fiat. In this paper, we study Richard Kimberly Heck's Two-sorted Frege Arithmetic (2FA), a variation on HP which has been thought to be deductively conservative over second-order logic. We show that it isn't. (...)

Export citation

Bookmark   1 citation
16. Leibniz on Binary: The Invention of Computer Arithmetic.Lloyd Strickland & Harry R. Lewis - 2022 - Cambridge, MA, USA: The MIT Press.
The first collection of Leibniz's key writings on the binary system, newly translated, with many previously unpublished in any language. -/- The polymath Gottfried Wilhelm Leibniz (1646–1716) is known for his independent invention of the calculus in 1675. Another major—although less studied—mathematical contribution by Leibniz is his invention of binary arithmetic, the representational basis for today's digital computing. This book offers the first collection of Leibniz's most important writings on the binary system, all newly translated by the authors with many (...)

Export citation

Bookmark
17. On Not Saying What We Shouldn't Have to Say.Shay Logan & Leach-Krouse Graham - 2021 - Australasian Journal of Logic 18 (5):524-568.
In this paper we introduce a novel way of building arithmetics whose background logic is R. The purpose of doing this is to point in the direction of a novel family of systems that could be candidates for being the infamous R#1/2 that Meyer suggested we look for.

Export citation

Bookmark
18. The God-given Naturals, Induction and Recursion.Paulo Veloso & André Porto - 2021 - O Que Nos Faz Pensar 29 (49):115-156.
We discuss some basic issues underlying the natural numbers: induction and recursion. We examine recursive formulations and their use in establishing universal and particular properties.

Export citation

Bookmark
19. 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.

Export citation

Bookmark
20. 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 (...)

Export citation

Bookmark   4 citations
21. The use of figurate numbers (e. g. in the context of elementary number theory) can be considered a heuristic in the field of problem solving or proving. In this paper, we want to discuss this heuristic from the perspectives of the semiotic theory of Peirce (“diagrammatic reasoning” and “collateral knowledge”) and cognitive psychology (“schema theory” and “Gestalt psychology”). We will make use of several results taken from our research to illustrate first-year students’ problems when dealing with figurate numbers in the (...)

Export citation

Bookmark
22. 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, (...)

Export citation

Bookmark
23. A growing body of research has shown that symbolic number processing relates to individual differences in mathematics. However, it remains unclear which mechanisms of symbolic number processing are crucial—accessing underlying magnitude representation of symbols (i.e., symbol‐magnitude associations), processing relative order of symbols (i.e., symbol‐symbol associations), or processing of symbols per se. To address this question, in this study adult participants performed a dots‐number word matching task—thought to be a measure of symbol‐magnitude associations (numerical magnitude processing)—a numeral‐ordering task that focuses on (...)

Export citation

Bookmark
24. 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 (...)

Export citation

Bookmark
25. 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 (...)

Export citation

Bookmark
26. 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 (...)

Export citation

Bookmark
27. 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 (...)

Export citation

Bookmark   9 citations

Export citation

Bookmark   1 citation
29. 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.

Export citation

Bookmark   2 citations
30. 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 (...)

Export citation

Bookmark   1 citation
31. 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.

Export citation

Bookmark   3 citations
32. Learning the Natural Numbers as a Child.Stefan Buijsman - 2017 - 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 (...)

Export citation

Bookmark   13 citations
33. 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 (...)

Export citation

Bookmark   8 citations
34. 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 (...)

Export citation

Bookmark   3 citations
35. 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 (...)

Export citation

Bookmark   10 citations
36. Logic and discrete mathematics: a concise introduction.Willem Conradie - 2015 - Hoboken, NJ, USA: Wiley. Edited by Valentin Goranko.
A concise yet rigorous introduction to logic and discrete mathematics. This book features a unique combination of comprehensive coverage of logic with a solid exposition of the most important fields of discrete mathematics, presenting material that has been tested and refined by the authors in university courses taught over more than a decade. The chapters on logic - propositional and first-order - provide a robust toolkit for logical reasoning, emphasizing the conceptual understanding of the language and the semantics of classical (...)

Export citation

Bookmark   3 citations
37. 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.

Export citation

Bookmark   5 citations
38. 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 (...)

Export citation

Bookmark   8 citations
39. 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.

Export citation

Bookmark   2 citations
40. 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.

Export citation

Bookmark   7 citations
41. 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 (...)

Export citation

Bookmark   3 citations

Export citation

Bookmark
43. On equality and natural numbers in Cantor-Lukasiewicz set theory.P. Hajek - 2013 - Logic Journal of the IGPL 21 (1):91-100.

Export citation

Bookmark   1 citation
44. Review of Graßmann, Robert, Theory of Number or Arithmetic in Strict Scientific Presentation by Strict Use of Formulas (1891). [REVIEW]Otto Hölder - 2013 - Philosophia Scientiae 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..

Export citation

Bookmark
45. 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 (1):57-70.

Export citation

Bookmark
46. 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, (...)

Export citation

Bookmark   51 citations
47. Remove from this list

Export citation

Bookmark
48. 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 (...)

Export citation

Bookmark
49. 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 (...)