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  1. On Herbrand Consistency in Weak Arithmetic.Zofia Adamowicz & Paweł Zbierski - 2001 - Archive for Mathematical Logic 40 (6):399-413.
    We prove that the Gödel incompleteness theorem holds for a weak arithmetic T = IΔ0 + Ω2 in the form where Cons H (T) is an arithmetic formula expressing the consistency of T with respect to the Herbrand notion of provability.
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  2. Semantical Mutation, Algorithms and Programs.Porto André - 2015 - Dissertatio:44-76.
    This article offers an explanation of perhaps Wittgenstein’s strangest and least intuitive thesis – the semantical mutation thesis – according to which one can never answer a mathematical conjecture because the new proof alters the very meanings of the terms involved in the original question. Instead of basing our justification on the distinction between mere calculation and proofs of isolated propositions, characteristic of Wittgenstein’s intermediary period, we generalize it to include conjectures involving effective procedures as well.
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  3. Avigad Jeremy. Update Procedures and the 1-Consistency of Arithmetic. Mathematical Logic Quarterly, Vol. 48 (2002), Pp. 3–13. [REVIEW]Toshiyasu Arai - 2003 - Bulletin of Symbolic Logic 9 (1):45-47.
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  4. 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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  5. Essays on the Foundations of Mathematics. Bar-Hillel, Yehoshua & [From Old Catalog] (eds.) - 1961 - Jerusalem, Magnes Press, Hebrew University;.
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  6. The Troubled Waters of Mathematics. [REVIEW]E. T. Bell - 1935 - Philosophy of Science 2 (1):115 - 117.
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  7. Review: The Troubled Waters of Mathematics. [REVIEW]E. T. Bell - 1935 - Philosophy of Science 2 (1):115 - 117.
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  8. Mathematics and Credulity.E. T. Bell - 1925 - Journal of Philosophy 22 (17):449-458.
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  9. On Volumes of Arithmetic Quotients SO.Mikhail Belolipetsky - 2004 - Annali della Scuola Normale Superiore di Pisa 3 (4):749-770.
    We apply G. Prasad’s volume formula for the arithmetic quotients of semi-simple groups and Bruhat-Tits theory to study the covolumes of arithmetic subgroups of $SO$. As a result we prove that for any even dimension $n$ there exists a unique compact arithmetic hyperbolic $n$-orbifold of the smallest volume. We give a formula for the Euler-Poincaré characteristic of the orbifolds and present an explicit description of their fundamental groups as the stabilizers of certain lattices in quadratic spaces. We also study hyperbolic (...)
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  10. Waismann's Critique of Wittgenstein.Anthony Birch - 2007 - Analysis and Metaphysics 6 (2007):263-272.
    Friedrich Waismann, a little-known mathematician and onetime student of Wittgenstein's, provides answers to problems that vexed Wittgenstein in his attempt to explicate the foundations of mathematics through an analysis of its practice. Waismann argues in favor of mathematical intuition and the reality of infinity with a Wittgensteinian twist. Waismann's arguments lead toward an approach to the foundation of mathematics that takes into consideration the language and practice of experts.
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  11. Moral Arithmetic: Seven Sins Into ten Commandments.John Bossy - 1988 - In Edmund Leites (ed.), Conscience and Casuistry in Early Modern Europe. Editions de la Maison des Sciences de L'homme. pp. 214--34.
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  12. Proving Bertrand's Postulate.Andrew Boucher - manuscript
    Bertand's Postulate is proved in Peano Arithmetic minus the Successor Axiom.
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  13. A A.... G.Andrew Boucher - unknown
    These notes are meant to continue from the paper on Consistency, in proving number-theoretic theorems from the second-order arithmetical system called FFFF. Its ultimate target is Quadratic Reciprocity, although it introduces and proves some facts about the least common multiple at the start.
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  14. Proving Quadratic Reciprocity.Andrew Boucher - manuscript
    The system of arithmetic considered in Consistency, which is essentially second-order Peano Arithmetic without the Successor Axiom, is used to prove more theorems of arithmetic, up to Quadratic Reciprocity.
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  15. Arithmetic Without the Successor Axiom.Andrew Boucher - manuscript
    Second-order Peano Arithmetic minus the Successor Axiom is developed from first principles through Quadratic Reciprocity and a proof of self-consistency. This paper combines 4 other papers of the author in a self-contained exposition.
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  16. "True" Arithmetic Can Prove its Own Consistency.Andrew Boucher - manuscript
    Using an axiomatization of second-order arithmetic (essentially second-order Peano Arithmetic without the Successor Axiom), arithmetic's basic operations are defined and its fundamental laws, up to unique prime factorization, are proven. Two manners of expressing a system's consistency are presented - the "Godel" consistency, where a wff is represented by a natural number, and the "real" consistency, where a wff is represented as a second-order sequence, which is a stronger notion. It is shown that the system can prove at least its (...)
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  17. Who Needs (to Assume) Hume's Principle?Andrew Boucher - manuscript
    Neo-logicism uses definitions and Hume's Principle to derive arithmetic in second-order logic. This paper investigates how much arithmetic can be derived using definitions alone, without any additional principle such as Hume's.
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  18. Equivalence of F with a Sub-Theory of Peano Arithmetic.Andrew Boucher - manuscript
    In a short, technical note, the system of arithmetic, F, introduced in Systems for a Foundation of Arithmetic and "True" Arithmetic Can Prove Its Own Consistency and Proving Quadratic Reciprocity, is demonstrated to be equivalent to a sub-theory of Peano Arithmetic; the sub-theory is missing, most notably, the Successor Axiom.
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  19. General Arithmetic.Andrew Boucher - manuscript
    General Arithmetic is the theory consisting of induction on a successor function. Normal arithmetic, say in the system called Peano Arithmetic, makes certain additional demands on the successor function. First, that it be total. Secondly, that it be one-to-one. And thirdly, that there be a first element which is not in its image. General Arithmetic abandons all of these further assumptions, yet is still able to prove many meaningful arithmetic truths, such as, most basically, Commutativity and Associativity of Addition and (...)
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  20. Introduction.Andrew Boucher - manuscript
    The Successor Axiom asserts that every number has a successor, or in other words, that the number series goes on and on ad infinitum. The present work investigates a particular subsystem of Frege Arithmetic, called F, which turns out to be equivalent to second-order Peano Arithmetic minus the Successor Axiom, and shows how this system can develop arithmetic up through Gauss' Quadratic Reciprocity Law. It then goes on to represent questions of provability in F, and shows that F can prove (...)
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  21. A Philosophical Introduction to the Foundations of Elementary Arithmetic by V1.03 Last Updated: 1 Jan 2001 Created: 1 Sept 2000 Please Send Your Comments to Abo. [REVIEW]Andrew Boucher - manuscript
    As it is currently used, "foundations of arithmetic" can be a misleading expression. It is not always, as the name might indicate, being used as a plural term meaning X = {x : x is a foundation of arithmetic}. Instead it has come to stand for a philosophico-logical domain of knowledge, concerned with axiom systems, structures, and analyses of arithmetic concepts. It is a bit as if "rock" had come to mean "geology." The conflation of subject matter and its study (...)
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  22. Systems for a Foundation of Arithmetic.Andrew Boucher - manuscript
    A new second-order axiomatization of arithmetic, with Frege's definition of successor replaced, is presented and compared to other systems in the field of Frege Arithmetic. The key in proving the Peano Axioms turns out to be a proposition about infinity, which a reduced subset of the axioms proves.
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  23. Consistency and Existence by V1.00 Last Updated: 1 Oct 2000 Please Send Your Comments to Abo.Andrew Boucher - manuscript
    On the one hand, first-order theories are able to assert the existence of objects. For instance, ZF set theory asserts the existence of objects called the power set, while Peano Arithmetic asserts the existence of zero. On the other hand, a first-order theory may or not be consistent: it is if and only if no contradiction is a theorem. Let us ask, What is the connection between consistency and existence?
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  24. Created: 9 June 2003 12 November 2003 Version 1.1 Www.Andrewboucher.Com/Papers/Quadratic_reciprocity.Pdf.Andrew Boucher - manuscript
    These notes are meant to continue from the paper on Consistency, in proving number-theoretic theorems from the second-order arithmetical system called FFFF. Its ultimate target is Quadratic Reciprocity, although it introduces and proves some facts about the least common multiple at the start.
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  25. The Consistency of Arithmetic, Based on a Logic of Meaning Containment.Ross T. Brady - 2012 - Logique Et Analyse 55 (219).
  26. Philip Hugly and Charles Sayward, Arithmetic and Ontology: A Non-Realist Philosophy of Arithmetic.M. Bremer - 2007 - Philosophy in Review 27 (3):188.
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  27. Philip Hugly and Charles Sayward, Arithmetic and Ontology: A Non-Realist Philosophy of Arithmetic Reviewed By.Manuel Bremer - 2007 - Philosophy in Review 27 (3):188-191.
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  28. The Earliest Arithmetic Published in America.Florian Cajori - 1927 - Isis 9 (3):391-401.
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  29. PAC Learning, VC Dimension, and the Arithmetic Hierarchy.Wesley Calvert - 2015 - Archive for Mathematical Logic 54 (7-8):871-883.
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  30. On the Role of Reducibility Principles.Carlo Cellucci - 1974 - Synthese 27 (1-2):93 - 110.
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  31. Two Further Combinatorial Theorems Equivalent to the 1-Consistency of Peano Arithmetic.Peter Clote & Kenneth Mcaloon - 1983 - Journal of Symbolic Logic 48 (4):1090-1104.
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  32. Towards an Institutional Account of the Objectivity, Necessity, and Atemporality of Mathematics.Julian C. Cole - 2013 - Philosophia Mathematica 21 (1):9-36.
    I contend that mathematical domains are freestanding institutional entities that, at least typically, are introduced to serve representational functions. In this paper, I outline an account of institutional reality and a supporting metaontological perspective that clarify the content of this thesis. I also argue that a philosophy of mathematics that has this thesis as its central tenet can account for the objectivity, necessity, and atemporality of mathematics.
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  33. On Some Considerations of Mathematical Physics: May We Identify Clifford Algebra as a Common Algebraic Structure for Classical Diffusion and Schrödinger Equations?Elio Conte - 2012 - Advanced Studies in Theoretical Physics 6 (26):1289-1307.
    We start from previous studies of G.N. Ord and A.S. Deakin showing that both the classical diffusion equation and Schrödinger equation of quantum mechanics have a common stump. Such result is obtained in rigorous terms since it is demonstrated that both diffusion and Schrödinger equations are manifestation of the same mathematical axiomatic set of the Clifford algebra. By using both such ( ) i A S and the i,±1 N algebra, it is evidenced, however, that possibly the two basic equations (...)
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  34. Quadratic Forms in Models of IΔ0+ Ω1. I.Paola D'Aquino & Angus Macintyre - 2007 - Annals of Pure and Applied Logic 148 (1):31-48.
    Gauss used quadratic forms in his second proof of quadratic reciprocity. In this paper we begin to develop a theory of binary quadratic forms over weak fragments of Peano Arithmetic, with a view to reproducing Gauss’ proof in this setting.
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  35. Capitalism and Arithmetic: The New Math of the Fifteenth CenturyFrank J. Swetz.Lorraine Daston - 1989 - Isis 80 (3):517-518.
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  36. A Defence of Mathematical Pluralism.E. Brian Davies - 2005 - Philosophia Mathematica 13 (3):252-276.
    We approach the philosophy of mathematics via an analysis of mathematics as it is practised. This leads us to a classification in terms of four concepts, which we define and illustrate with a variety of examples. We call these concepts background conventions, context, content, and intuition.
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  37. Formalism.Michael Detlefsen - 2005 - In Stewart Shapiro (ed.), The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press. pp. 236--317.
    A comprehensive historical overview of formalist ideas in the philosophy of mathematics.
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  38. Constructive Existence Claims.Michael Detlefsen - 1998 - In Matthias Schirn (ed.), The Philosophy of Mathematics Today. Clarendon Press. pp. 1998--307.
    It is a commonplace of constructivist thought that a claim that an object of a certain kind exists is to be backed by an explicit display or exhibition of an object that is manifestly of that kind. Let us refer to this requirement as the exhibition condition. The main objective of this essay is to examine this requirement and to arrive at a better understanding of its epistemic character and the role that it plays in the two main constructivist philosophies (...)
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  39. Hilbert's Formalism.Michael Detlefsen - 1993 - Revue Internationale de Philosophie 47 (186):285-304.
    Various parallels between Kant's critical program and Hilbert's formalistic program for the philosophy of mathematics are considered.
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  40. On an Alleged Refutation of Hilbert's Program Using Gödel's First Incompleteness Theorem.Michael Detlefsen - 1990 - Journal of Philosophical Logic 19 (4):343 - 377.
    It is argued that an instrumentalist notion of proof such as that represented in Hilbert's viewpoint is not obligated to satisfy the conservation condition that is generally regarded as a constraint on Hilbert's Program. A more reasonable soundness condition is then considered and shown not to be counter-exemplified by Godel's First Theorem. Finally, attention is given to the question of what a theory is; whether it should be seen as a "list" or corpus of beliefs, or as a method for (...)
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  41. The Arithmetic of Al-UqlīdisīAbū Al-Ḥasan Aḥmad Ibn Ibrahim Al-Uqlīdisī A. S. Saidan.Yvonne Dold-Samplonius - 1979 - Isis 70 (4):615-617.
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  42. The Foundations of Arithmetic?Thomas Donaldson - 2016 - Noûs 50 (4).
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  43. Preuves par excellence.Jacques Dubucs & Sandra Lapointe - 2003 - Philosophiques 30 (1):219-234.
    Bolzano fut le premier philosophe à établir une distinction explicite entre les procédés déductifs qui nous permettent de parvenir à la certitude d’une vérité et ceux qui fournissent son fondement objectif. La conception que Bolzano se fait du rapport entre ce que nous appelons ici, d’une part, « conséquence subjective », à savoir la relation de raison à conséquence épistémique et, d’autre part, la « conséquence objective », c’est-à-dire la fondation , suggère toutefois que Bolzano défendait une conception « explicativiste (...)
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  44. Ontologically Neutral Arithmetic.Rolf A. Eberle - 1974 - Philosophia 4 (1):67-94.
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  45. The Space of Mathematics.Javier Echeverria, Andoni Ibarra & Thomas Mormann (eds.) - 1992 - de Gruyter.
    No subject index. Annotation copyright by Book News, Inc., Portland, OR.
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  46. The Helpmekaar: Rescuing the "Volk" Through Reading, Writing and Arithmetic, C. 1916-C. 1965.Anton Ehlers - 2015 - História 60 (2):87-108.
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  47. The Axiomatic Method in Exposition and Exploration.J. Fang - 1970 - Philosophia Mathematica (1-2):13-24.
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  48. 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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  49. On Saying What You Really Want to Say: Wittgenstein, Gödel and the Trisection of the Angle.Juliet Floyd - 1995 - In Jaakko Hintikka (ed.), From Dedekind to Gödel: The Foundations of Mathematics in the Early Twentieth Century, Synthese Library Vol. 251 (Kluwer Academic Publishers. pp. 373-426.
  50. Number System of Arithmetic and Algebra. [REVIEW]A. C. Fox - 1924 - Australasian Journal of Philosophy 2:71.
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