Theories of Mathematics

Edited by Roy T. Cook (University of Minnesota, University of St. Andrews, University of Minnesota)
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  1. No Easy Road to Impredicative Definabilism.Øystein Linnebo & Sam Roberts - 2024 - Philosophia Mathematica 32 (1):21-33.
    Bob Hale has defended a new conception of properties that is broadly Fregean in two key respects. First, like Frege, Hale insists that every property can be defined by an open formula. Second, like Frege, but unlike later definabilists, Hale seeks to justify full impredicative property comprehension. The most innovative part of his defense, we think, is a “definability constraint” that can serve as an implicit definition of the domain of properties. We make this constraint formally precise and prove that (...)
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  2. Solving the Mystery of Mathematics.Jared Warren - 2023 - Philosophy Now Magazine 157:16-19.
    This is a magazine article discussing the philosophy of mathematics and arguing for mathematical conventionalism, written for a non-academic audience. (As often happens with popular articles, the editors made some changes that I'm not completely happy with, e.g., the titled section headings and sub-title).
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  3. Report on some ramified-type assignment systems and their model-theoretic semantics.Harold Hodes - 2013 - In Nicholas Griffin & Bernard Linsky (eds.), The Palgrave Centenary Companion to Principia Mathematica. London and Basingstoke: Palgrave-Macmillan.
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  4. Principia mathematica, the multiple-relation theory of judgment and molecular facts.James Levine - 2013 - In Nicholas Griffin & Bernard Linsky (eds.), The Palgrave Centenary Companion to Principia Mathematica. London and Basingstoke: Palgrave-Macmillan.
  5. The logic of classes and the no-class theory.Byeong-Uk Yi - 2013 - In Nicholas Griffin & Bernard Linsky (eds.), The Palgrave Centenary Companion to Principia Mathematica. London and Basingstoke: Palgrave-Macmillan.
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  6. From logicism to metatheory.Patricia Blanchette - 2013 - In Nicholas Griffin & Bernard Linsky (eds.), The Palgrave Centenary Companion to Principia Mathematica. London and Basingstoke: Palgrave-Macmillan.
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  7. Principia mathematica in Poland.Jan Wolenski - 2013 - In Nicholas Griffin & Bernard Linsky (eds.), The Palgrave Centenary Companion to Principia Mathematica. London and Basingstoke: Palgrave-Macmillan.
  8. David Hilbert and Principia mathematica.Reinhard Kahle - 2013 - In Nicholas Griffin & Bernard Linsky (eds.), The Palgrave Centenary Companion to Principia Mathematica. London and Basingstoke: Palgrave-Macmillan.
  9. Principia mathematica : the first hundred years.Alasdair Urquhart - 2013 - In Nicholas Griffin & Bernard Linsky (eds.), The Palgrave Centenary Companion to Principia Mathematica. London and Basingstoke: Palgrave-Macmillan.
  10. Not So Simple.Colin R. Caret - 2023 - Asian Journal of Philosophy 2 (2):1-16.
    In a recent series of articles, Beall has developed the view that FDE is the formal system most deserving of the honorific “Logic”. The Simple Argument for this view is a cost-benefit analysis: the view that FDE is Logic has no drawbacks and it has some benefits when compared with any of its rivals. In this paper, I argue that both premises of the Simple Argument are mistaken. I use this as an opportunity to further reflect on how such arguments (...)
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  11. Theory of fuzzy computation.Apostolos Syropoulos - 2014 - New York: Springer.
    The book provides the first full length exploration of fuzzy computability. It describes the notion of fuzziness and present the foundation of computability theory. It then presents the various approaches to fuzzy computability. This text provides a glimpse into the different approaches in this area, which is important for researchers in order to have a clear view of the field. It contains a detailed literature review and the author includes all proofs to make the presentation accessible. Ideas for future research (...)
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  12. The origin of symbolic mathematics and the end of the science of quantity.Sören Stenlund - 2014 - Uppsala: Uppsala Universitet.
  13. Die grundsätze und das wesen des unendlichen in der mathematik und philosophie.Friedrich Jacob Kurt Geissler - 1902 - Leipzig,: B. G. Teubner.
  14. Sur la philosophie des mathématiques.Jules Richard - 1903 - Paris: Gauthier-Villars.
    La logique--La géométrie--Questions diverses--Considérations sur différentes sciences--Note I. Sur la géométrie projective--Note II. Éclaircissements divers (Notions de groupe, sur les notions premières, sur la classification des sciences).
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  15. Métamathématique.Paul Lorenzen - 1967 - Paris,: Mouton.
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  16. Fondements des mathématiques.Michel Combès - 1971 - Paris,: Presses universitaires de France.
  17. Die erkenntnistheoretischen Grundlagen der Mathematik.Gustav Kruck - 1981 - Zürich: Schulthess.
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  18. Seeing negation as always dependent frees mathematical logic from paradox, incompleteness, and undecidability-- and opens the door to its positive possibilities.Daniel A. Cowan - 2008 - San Mateo, CA: Joseph Publishing Company.
  19. Mathematical Pluralism.Edward N. Zalta - 2023 - Noûs.
    Mathematical pluralism can take one of three forms: (1) every consistent mathematical theory consists of truths about its own domain of individuals and relations; (2) every mathematical theory, consistent or inconsistent, consists of truths about its own (possibly uninteresting) domain of individuals and relations; and (3) the principal philosophies of mathematics are each based upon an insight or truth about the nature of mathematics that can be validated. (1) includes the multiverse approach to set theory. (2) helps us to understand (...)
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  20. Lower and Upper Estimates of the Quantity of Algebraic Numbers.Yaroslav Sergeyev - 2023 - Mediterranian Journal of Mathematics 20:12.
    It is well known that the set of algebraic numbers (let us call it A) is countable. In this paper, instead of the usage of the classical terminology of cardinals proposed by Cantor, a recently introduced methodology using ①-based infinite numbers is applied to measure the set A (where the number ① is called grossone). Our interest to this methodology is explained by the fact that in certain cases where cardinals allow one to say only whether a set is countable (...)
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  21. Jan von Plato.* Can Mathematics be Proved Consistent?John W. Dawson - 2023 - Philosophia Mathematica 31 (1):104-111.
    The papers of Kurt Gödel were donated to the Institute for Advanced Study by his widow Adele shortly after his death in 1978. They were catalogued by the review.
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  22. Strengthening the Russellian argument against absolutely unrestricted quantification.Laureano Luna - 2022 - Synthese 200 (3):1-13.
    The Russellian argument against the possibility of absolutely unrestricted quantification can be answered by the partisan of that quantification in an apparently easy way, namely, arguing that the objects used in the argument do not exist because they are defined in a viciously circular fashion. We show that taking this contention along as a premise and relying on an extremely intuitive Principle of Determinacy, it is possible to devise a reductio of the possibility of absolutely unrestricted quantification. Therefore, there are (...)
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  23. Set Theory INC# Based on Intuitionistic Logic with Restricted Modus Ponens Rule (Part. I).Jaykov Foukzon - 2021 - Journal of Advances in Mathematics and Computer Science 36 (2):73-88.
    In this article Russell’s paradox and Cantor’s paradox resolved successfully using intuitionistic logic with restricted modus ponens rule.
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  24. Foucault, Deleuze, and Nietzsche.Ilexa Yardley - 2021 - Https://Medium.Com/the-Circular-Theory/.
    The power of representation and the representation of power, and, the exploding NFT market. Euclid's error and the mathematics behind representation, identification, and interpretation.
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  25. Hilbert's different aims for the foundations of mathematics.Besim Karakadılar - manuscript
    The foundational ideas of David Hilbert have been generally misunderstood. In this dissertation prospectus, different aims of Hilbert are summarized and a new interpretation of Hilbert's work in the foundations of mathematics is roughly sketched out. Hilbert's view of the axiomatic method, his response to criticisms of set theory and intuitionist criticisms of the classical foundations of mathematics, and his view of the role of logical inference in mathematical reasoning are briefly outlined.
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  26. Functorial Semantics for the Advancement of the Science of Cognition.Venkata Posina, Dhanjoo N. Ghista & Sisir Roy - 2017 - Mind and Matter 15 (2):161-184.
    Cognition involves physical stimulation, neural coding, mental conception, and conscious perception. Beyond the neural coding of physical stimuli, it is not clear how exactly these component processes constitute cognition. Within mathematical sciences, category theory provides tools such as category, functor, and adjointness, which are indispensable in the explication of the mathematical calculations involved in acquiring mathematical knowledge. More speci cally, functorial semantics, in showing that theories and models can be construed as categories and functors, respectively, and in establishing the adjointness (...)
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  27. Nudging Scientific Advancement through Reviews.Venkata Rayudu Posina, Hippu Salk K. Nathan & Anshuman Behera - manuscript
    We call for a change-of-attitude towards reviews of scientific literature. We begin with an acknowledgement of reviews as pathways for the advancement of our scientific understanding of reality. The significance of the scientific struggle propelling the putting together of pieces of knowledge into parts of a cohesive body of understanding is recognized, and yet undervalued, especially in empirical sciences. Here we propose a nudge, which is prefacing the insights gained in reviewing the literature with: 'Our review reveals' (or an equivalent (...)
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  28. Foundation of paralogical nonstandard analysis and its application to some famous problems of trigonometrical and orthogonal series.Jaykov Foukzon - manuscript
    FOURTH EUROPEAN CONGRESS OF MATHEMATICS STOCKHOLM,SWEDEN JUNE27 ­ - JULY 2, 2004 Contributed papers L. Carleson’s celebrated theorem of 1965 [1] asserts the pointwise convergence of the partial Fourier sums of square integrable functions. The Fourier transform has a formulation on each of the Euclidean groups R , Z and Τ .Carleson’s original proof worked on Τ . Fefferman’s proof translates very easily to R . M´at´e [2] extended Carleson’s proof to Z . Each of the statements of the theorem (...)
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  29. 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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  30. Carnap Rudolf. On the use of Hilbert's ε-operator in scientific theories. Essays on the foundations of mathematics, dedicated to A. A. Fraenkel on his seventieth anniversary, edited by Bar-Hillel Y., Poznanski E. I. J., Rabin M. O., and A. Robinson for The Hebrew University of Jerusalem, Magnes Press, Jerusalem 1961, and North-Holland Publishing Company, Amsterdam 1962, pp. 156–164. [REVIEW]H. Bohnert - 1971 - Journal of Symbolic Logic 36 (2):320-321.
  31. Numerical infinities and infinitesimals: Methodology, applications, and repercussions on two Hilbert problems.Yaroslav Sergeyev - 2017 - EMS Surveys in Mathematical Sciences 4 (2):219–320.
    In this survey, a recent computational methodology paying a special attention to the separation of mathematical objects from numeral systems involved in their representation is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework in all the situations requiring these notions. The methodology does not contradict Cantor’s and non-standard analysis views and is based on the Euclid’s Common Notion no. 5 “The whole is greater than the (...)
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  32. A process oriented definition of number.Rolfe David - manuscript
    In this paper Russell’s definition of number is criticized. Russell’s assertion that a number is a particular kind of set implies that number has the properties of a set. It is argued that this would imply that a number contains elements and that this does not conform to our intuitive notion of number. An alternative definition is presented in which number is not seen as an object, but rather as a process and is related to the act of counting and (...)
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  33. Frank Pierobon. Kant et les mathématiques: La conception kantienne des mathématiques [Kant and mathematics: The Kantian conception of mathematics]. Bibliothèque d'Histoire de la Philosophie. Paris: J. Vrin. ISBN 2-7116-1645-2. Pp. 240. [REVIEW]Emily Carson - 2006 - Philosophia Mathematica 14 (3):370-378.
    This book is a welcome contribution to the literature on Kant's philosophy of mathematics in two particular respects. First, the author systematically traces the development of Kant's thought on mathematics from the very early pre-Critical writings through to the Critical philosophy. Secondly, it puts forward a challenge to contemporary Anglo-Saxon commentators on Kant's philosophy of mathematics which merits consideration.A central theme of the book is that an adequate understanding of Kant's pronouncements on mathematics must begin with the recognition that mathematics (...)
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  34. Genetic counseling in historical perspective: Understanding our hereditary past and forecasting our genomic future. [REVIEW]618 622 - 2013 - Studies in History and Philosophy of Science Part A 44 (4):Devon-Stillwell.
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  35. A Normative Model of Classical Reasoning in Higher Order Languages.Peter Zahn - 2006 - Synthese 148 (2):309-343.
    The present paper is concerned with a ramified type theory (cf. (Lorenzen 1955), (Russell), (Schütte), (Weyl), e.g.,) in a cumulative version. §0 deals with reasoning in first order languages. is introduced as a first order set.
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  36. Principles of Mathematics.Bertrand Russell - 1937 - New York,: Routledge.
    Published in 1903, this book was the first comprehensive treatise on the logical foundations of mathematics written in English. It sets forth, as far as possible without mathematical and logical symbolism, the grounds in favour of the view that mathematics and logic are identical. It proposes simply that what is commonly called mathematics are merely later deductions from logical premises. It provided the thesis for which _Principia Mathematica_ provided the detailed proof, and introduced the work of Frege to a wider (...)
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  37. subregular tetrahedra.John Corcoran - 2008 - Bulletin of Symbolic Logic 14 (3):411-2.
    This largely expository lecture deals with aspects of traditional solid geometry suitable for applications in logic courses. Polygons are plane or two-dimensional; the simplest are triangles. Polyhedra [or polyhedrons] are solid or three-dimensional; the simplest are tetrahedra [or triangular pyramids, made of four triangles]. -/- A regular polygon has equal sides and equal angles. A polyhedron having congruent faces and congruent [polyhedral] angles is not called regular, as some might expect; rather they are said to be subregular—a word coined for (...)
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  38. A Logic Of Sequences.Norihiro Kamide - 2011 - Reports on Mathematical Logic:29-57.
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  39. Russell ante el inicio de la Matemática.Javier de Lorenzo - 1972 - Teorema: International Journal of Philosophy 2 (4):45.
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  40. Logicism Reconsidered.Patricia A. Blanchette - 1990 - Dissertation, Stanford University
    This thesis is an examination of Frege's logicism, and of a number of objections which are widely viewed as refutations of the logicist thesis. In the view offered here, logicism is designed to provide answers to two questions: that of the nature of arithmetical truth, and that of the source of arithmetical knowledge. ;The first objection dealt with here is the view that logicism is not an epistemologically significant thesis, due to the fact that the epistemological status of logic itself (...)
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  41. Finitism: An Essay on Hilbert's Programme.David Watson Galloway - 1991 - Dissertation, Massachusetts Institute of Technology
    In this thesis, I discuss the philosophical foundations of Hilbert's Consistency Programme of the 1920's, in the light of the incompleteness theorems of Godel. ;I begin by locating the Consistency Programme within Hilbert's broader foundational project. I show that Hilbert's main aim was to establish that classical mathematics, and in particular classical analysis, is a conservative extension of finitary mathematics. Accepting the standard identification of finitary mathematics with primitive recursive arithmetic, and classical analysis with second order arithmetic, I report upon (...)
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  42. Strict Constructivism and the Philosophy of Mathematics.Feng Ye - 2000 - Dissertation, Princeton University
    The dissertation studies the mathematical strength of strict constructivism, a finitistic fragment of Bishop's constructivism, and explores its implications in the philosophy of mathematics. ;It consists of two chapters and four appendixes. Chapter 1 presents strict constructivism, shows that it is within the spirit of finitism, and explains how to represent sets, functions and elementary calculus in strict constructivism. Appendix A proves that the essentials of Bishop and Bridges' book Constructive Analysis can be developed within strict constructivism. Appendix B further (...)
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  43. Facets of Infinity: A Theory of Finitistic Truth.Zlatan Damnjanovic - 1992 - Dissertation, Princeton University
    The thesis critically examines the question of the philosophical coherence of finitism, the view which seeks to interpret mathematics without postulating an actual infinity of mathematical objects. It is argued that a widely accepted characterization of finitism, most recently expounded by Tait, is inadequate, and a new characterization based on the notion of elementary abstraction is proposed. It is further argued that the notion of elementary abstraction better explains the bearing of Godel's incompleteness theorems on the issue of the coherence (...)
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  44. The Social Problem: A Constructive Analysis. [REVIEW]Dickinson S. Miller - 1916 - Journal of Philosophy, Psychology and Scientific Methods 13 (3):81-82.
  45. Errett Bishop: Reflections on Him and His Research.Murray Rosenblatt & Errett Bishop - 1985 - Amer Mathematical Society.
    This book is the proceedings of the Memorial Meeting for Errett Bishop, held at the University of California, San Diego, 24 September 1983. During his early days as a mathematician, Errett Bishop made distinguished contributions in many branches of analysis--first in operator theory in Hilbert and Banach spaces, then in the theory of polynomial approximation in the complex plane and on Riemann surfaces, and thence to his outstanding research in function algebras. This work in turn led him to his highly (...)
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  46. What Is Mathematics, Really? by Reuben Hersh. [REVIEW]Mary Tiles - 1999 - Isis 90:344-345.
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  47. The Foundations of Intuitionistic Mathematics. [REVIEW]J. M. P. - 1965 - Review of Metaphysics 19 (1):154-155.
    The aim of the authors is to present a comprehensive study of the basis of intuitionistic mathematics by means of modern meta-mathematical devices. The first author, for whom this book is a capstone of twenty years' work on the subject, contributes three chapters on a formal system of intuitionistic analysis, notions of realizability, and order in the continuum; the second provides an analysis of the intuitionistic continuum. An extensive bibliography which includes references to almost every article on the subject makes (...)
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  48. Mystic, Geometer, and Intuitionist: The Life of L. E. J. Brouwer. Volume 1: The Dawning Revolution. [REVIEW]Diederick Raven - 2001 - British Journal for the History of Science 34 (1):97-124.
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  49. Cofinally Invariant Sequences and Revision.Edoardo Rivello - 2015 - Studia Logica 103 (3):599-622.
    Revision sequences are a kind of transfinite sequences which were introduced by Herzberger and Gupta in 1982 as the main mathematical tool for developing their respective revision theories of truth. We generalise revision sequences to the notion of cofinally invariant sequences, showing that several known facts about Herzberger’s and Gupta’s theories also hold for this more abstract kind of sequences and providing new and more informative proofs of the old results.
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  50. Brouwer versus Hilbert: 1907–1928.J. Posy Carl - 1998 - Science in Context 11 (2):291-325.
    The ArgumentL. E. J. Brouwer and David Hubert, two titans of twentieth-century mathematics, clashed dramatically in the 1920s. Though they were both Kantian constructivists, their notoriousGrundlagenstreitcentered on sharp differences about the foundations of mathematics: Brouwer was prepared to revise the content and methods of mathematics (his “Intuitionism” did just that radically), while Hilbert's Program was designed to preserve and constructively secure all of classical mathematics.Hilbert's interests and polemics at the time led to at least three misconstruals of intuitionism, misconstruals which (...)
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