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  1. Zofia Adamowicz & Paweł Zbierski (2001). On Herbrand Consistency in Weak Arithmetic. 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. Jeremy Avigad (2002). Update Procedures and the 1-Consistency of Arithmetic. Mathematical Logic Quarterly 48 (1):3-13.
    The 1-consistency of arithmetic is shown to be equiva ent 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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  3. Anthony Birch (2007). Waismann's Critique of Wittgenstein. 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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  4. Andrew Boucher, Proving Bertrand's Postulate.
    Bertand's Postulate is proved in Peano Arithmetic minus the Successor Axiom.
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  5. Andrew Boucher, Proving Quadratic Reciprocity.
    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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  6. Andrew Boucher, A A.... G.
    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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  7. Andrew Boucher, Arithmetic Without the Successor Axiom.
    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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  8. Andrew Boucher, "True" Arithmetic Can Prove its Own Consistency.
    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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  9. Andrew Boucher, Who Needs (to Assume) Hume's Principle?
    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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  10. Andrew Boucher, Equivalence of F with a Sub-Theory of Peano Arithmetic.
    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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  11. Andrew Boucher, Introduction.
    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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  12. Andrew Boucher, Systems for a Foundation of Arithmetic.
    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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  13. Andrew Boucher, Created: 9 June 2003 12 November 2003 Version 1.1 Www.Andrewboucher.Com/Papers/Quadratic_reciprocity.Pdf.
    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. Andrew Boucher, General Arithmetic.
    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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  15. Carlo Cellucci (1974). On the Role of Reducibility Principles. Synthese 27 (1-2):93 - 110.
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  16. Peter Clote & Kenneth Mcaloon (1983). Two Further Combinatorial Theorems Equivalent to the 1-Consistency of Peano Arithmetic. Journal of Symbolic Logic 48 (4):1090-1104.
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  17. Julian C. Cole (2013). Towards an Institutional Account of the Objectivity, Necessity, and Atemporality of Mathematics. 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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  18. Elio Conte (2012). On Some Considerations of Mathematical Physics: May We Identify Clifford Algebra as a Common Algebraic Structure for Classical Diffusion and Schrödinger Equations? 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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  19. E. Brian Davies (2005). A Defence of Mathematical Pluralism. 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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  20. Javier Echeverria, Andoni Ibarra & Thomas Mormann (eds.) (1992). The Space of Mathematics. de Gruyter.
    No subject index. Annotation copyright by Book News, Inc., Portland, OR.
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  21. J. Fang (1970). The Axiomatic Method in Exposition and Exploration. Philosophia Mathematica (1-2):13-24.
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  22. A. Fernández‐Margarit & M. J. Pérez‐Jiménez (1994). Maximum Schemes in Arithmetic. 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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  23. Juliet Floyd (1995). On Saying What You Really Want to Say: Wittgenstein, Gödel and the Trisection of the Angle. In Jaakko Hintikka (ed.), From Dedekind to Gödel: The Foundations of Mathematics in the Early Twentieth Century, Synthese Library Vol. 251 (Kluwer Academic Publishers. 373-426.
  24. Joseph S. Fulda, Remarks on the Argument From Design.
    Gives two pared-down versions of the argument from design, which may prove more persuasive as to a Creator, discusses briefly the mathematics underpinning disbelief and nonbelief and its misuse and some proper uses, moves to why the full argument is needed anyway, viz., to demonstrate Providence, offers a theory as to how miracles (open and hidden) occur, viz. the replacement of any particular mathematics underlying a natural law (save logic) by its most appropriate nonstandard variant. -/- Note: This is an (...)
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  25. P. Garavaso (2013). Hilary Putnam's Consistency Objection Against Wittgenstein's Conventionalism in Mathematics. Philosophia Mathematica 21 (3):279-296.
    Hilary Putnam first published the consistency objection against Ludwig Wittgenstein’s account of mathematics in 1979. In 1983, Putnam and Benacerraf raised this objection against all conventionalist accounts of mathematics. I discuss the 1979 version and the scenario argument, which supports the key premise of the objection. The wide applicability of this objection is not apparent; I thus raise it against an imaginary axiomatic theory T similar to Peano arithmetic in all relevant aspects. I argue that a conventionalist can explain the (...)
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  26. Yvon Gauthier (2001). Internal and External Consistency of Arithmetic. Logica Trianguli 5:19-41.
    What Gödel referred to as “outer” consistency is contrasted with the “inner” consistency of arithmetic from a constructivist point of view. In the settheoretic setting of Peano arithmetic, the diagonal procedure leads out of the realm of natural numbers. It is shown that Hilbert’s programme of arithmetization points rather to an “internalisation” of consistency. The programme was continued by Herbrand, Gödel and Tarski. Tarski’s method of quantifier elimination and Gödel’s Dialectica interpretation are part and parcel of Hilbert’s finitist ideal which (...)
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  27. M. Giaquinto (2002). The Search for Certainty: A Philosophical Account of Foundations of Mathematics. Oxford University Press.
    Marcus Giaquinto tells the compelling story of one of the great intellectual adventures of the modern era: the attempt to find firm foundations for mathematics. From the late nineteenth century to the present day, this project has stimulated some of the most original and influential work in logic and philosophy.
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  28. Lukasz Andrzej Glinka (2013). Study of Analytic Number Theory: Riemann’s Hypothesis and Prime Number Theorem with Addendum on Integer Partitions. 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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  29. Warren Goldfarb (2009). Carnap's Syntax Programme and the Philosophy of Mathematics. In Pierre Wagner (ed.), Carnap's Logical Syntax of Language. Palgrave Macmillan.
  30. James Hawthorne (1996). Mathematical Instrumentalism Meets the Conjunction Objection. Journal of Philosophical Logic 25 (4):363-397.
    Scientific realists often appeal to some version of the conjunction objection to argue that scientific instrumentalism fails to do justice to the full empirical import of scientific theories. Whereas the conjunction objection provides a powerful critique of scientific instrumentalism, I will show that mathematical instnrunentalism escapes the conjunction objection unscathed.
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  31. Jaakko Hintikka & Besim Karakadilar (2006). How to Prove the Consistency of Arithmetic. Acta Philosophica Fennica 78:1.
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  32. Andoni Ibarra & Thomas Mormann (1992). Structural Analogies Between Mathematical and Empirical Theories. In Javier Echeverria, Andoni Ibarra & Thomas Mormann (eds.), The Space of Mathematics. de Gruyter.
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  33. Clement F. Kent & Bernard R. Hodgson (1989). Extensions of Arithmetic for Proving Termination of Computations. Journal of Symbolic Logic 54 (3):779-794.
    Kirby and Paris have exhibited combinatorial algorithms whose computations always terminate, but for which termination is not provable in elementary arithmetic. However, termination of these computations can be proved by adding an axiom first introduced by Goodstein in 1944. Our purpose is to investigate this axiom of Goodstein, and some of its variants, and to show that these are potentially adequate to prove termination of computations of a wide class of algorithms. We prove that many variations of Goodstein's axiom are (...)
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  34. Jaegwon Kim (1981). The Role of Perception in a Priori Knowledge: Some Remarks. [REVIEW] Philosophical Studies 40 (3):339 - 354.
  35. Nicholas Maxwell (2010). Wisdom Mathematics. Friends of Wisdom Newsletter (6):1-6.
    For over thirty years I have argued that all branches of science and scholarship would have both their intellectual and humanitarian value enhanced if pursued in accordance with the edicts of wisdom-inquiry rather than knowledge-inquiry. I argue that this is true of mathematics. Viewed from the perspective of knowledge-inquiry, mathematics confronts us with two fundamental problems. (1) How can mathematics be held to be a branch of knowledge, in view of the difficulties that view engenders? What could mathematics be knowledge (...)
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  36. Storrs McCall, The Consistency of Arithmetic.
    The paper presents a proof of the consistency of Peano Arithmetic (PA) that does not lie in deducing its consistency as a theorem in an axiomatic system. PA’s consistency cannot be proved in PA, and to deduce its consistency in some stronger system PA+ is self-defeating, since the stronger system may itself be inconsistent. Instead, a semantic proof is constructed which demonstrates consistency not relative to the consistency of some other system but in an absolute sense.
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  37. N. V. Michailova (2013). Philosophical and methodological problem of consistency of mathematical theories. Liberal Arts in Russia 2 (6):552--560.
    Increased abstraction of modern mathematical theories has revived interest in traditional philosophical and methodological problem of internally consistent system of axioms where the contradicting each other statements can’t be deduced. If we are talking about axioms describing a well-known area of mathematical objects from the standpoint of local consistency this problem does not appear to be as relevant. But these problems are associated with the various attempts of formalists to explain the mathematical existence through consistency. But, for example, with regard (...)
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  38. Peter Milne (1994). Review: The Physicalization of Mathematics. [REVIEW] British Journal for the Philosophy of Science 45 (1):305 - 340.
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  39. Franco Montagna (1996). An Algebraic Treatment of Quantifier-Free Systems of Arithmetic. Archive for Mathematical Logic 35 (4):209-224.
    By algebraic means, we give an equational axiomatization of the equational fragments of various systems of arithmetic. We also introduce a faithful semantics according to which, for every reasonable system T for arithmetic, there is a model where exactly the theorems of T are true.
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  40. Thomas Mormann & Mikhail G. Katz (2013). Infinitesimals as an Issue of Neo-Kantian Philosophy of Science. HOPOS 3(2), The Journal of the International Society for the History of Phiilosophy of Science, 236 - 280 (2):236-280.
    We seek to elucidate the philosophical context in which one of the most important conceptual transformations of modern mathematics took place, namely the so-called revolution in rigor in infinitesimal calculus and mathematical analysis. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind,and Weierstrass. The dominant current of philosophy in Germany at the time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical. Our (...)
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  41. Michael Potter (1999). Intuition and Reflection in Arithmetic: Michael Potter. Aristotelian Society Supplementary Volume 73 (1):63–73.
    Classifies accounts of arithmetic into four sorts according to the resources they appeal to in constructing its subject matter.
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  42. H. E. Rose (1961). On the Consistency and Undecidability of Recursive Arithmetic. Mathematical Logic Quarterly 7 (7‐10):124-135.
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  43. Yaroslav Sergeyev (2009). Numerical Point of View on Calculus for Functions Assuming Finite, Infinite, and Infinitesimal Values Over Finite, Infinite, and Infinitesimal Domains. Nonlinear Analysis Series A 71 (12):e1688-e1707.
    The goal of this paper consists of developing a new (more physical and numerical in comparison with standard and non-standard analysis approaches) point of view on Calculus with functions assuming infinite and infinitesimal values. It uses recently introduced infinite and infinitesimal numbers being in accordance with the principle ‘The part is less than the whole’ observed in the physical world around us. These numbers have a strong practical advantage with respect to traditional approaches: they are representable at a new kind (...)
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  44. Stewart Shapiro (1991). Foundations Without Foundationalism: A Case for Second-Order Logic. Oxford University Press.
    The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages are often insufficient to codify (...)
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  45. Dirk Siefkes (1971). Undecidable Extensions of Monadic Second Order Successor Arithmetic. Mathematical Logic Quarterly 17 (1):385-394.
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  46. P. H. Stanford (1976). A Formalisation of the Integers in a Multi‐Successor Arithmetic. Mathematical Logic Quarterly 22 (1):119-121.
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  47. Eric Steinhart (1999). Nietzsche's Philosophy of Mathematics. International Studies in Philosophy 31 (3):19-27.
    Nietzsche has a surprisingly significant and strikingly positive assessment of mathematics. I discuss Nietzsche's theory of the origin of mathematical practice in the division of the continuum of force, his theory of numbers, his conception of the finite and the infinite, and the relations between Nietzschean mathematics and formalism and intuitionism. I talk about the relations between math, illusion, life, and the will to truth. I distinguish life and world affirming mathematical practice from its ascetic perversion. For Nietzsche, math is (...)
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  48. Jouko Vaananen (2001). Second-Order Logic and Foundations of Mathematics. Bulletin of Symbolic Logic 7 (4):504-520.
    We discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order set theory and second-order logic are not radically (...)
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  49. Silvio Valentini (1983). The Modal Logic of Consistency Assertions of Peano Arithmetic. Mathematical Logic Quarterly 29 (1):25-32.
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  50. Roger Wertheimer (1999). How Mathematics Isn't Logic. Ratio 12 (3):279–295.
    If logical truth is necessitated by sheer syntax, mathematics is categorially unlike logic even if all mathematics derives from definitions and logical principles. This contrast gets obscured by the plausibility of the Synonym Substitution Principle implicit in conceptions of analyticity: synonym substitution cannot alter sentence sense. The Principle obviously fails with intercepting: nonuniform term substitution in logical sentences. 'Televisions are televisions' and 'TVs are televisions' neither sound alike nor are used interchangeably. Interception synonymy gets assumed because logical sentences and their (...)
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