Results for 'Bolzano Weierstrass theorem'

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  1.  45
    The BolzanoWeierstrass Theorem is the jump of Weak Kőnig’s Lemma.Vasco Brattka, Guido Gherardi & Alberto Marcone - 2012 - Annals of Pure and Applied Logic 163 (6):623-655.
  2.  67
    Historians and Philosophers of Logic: Are They Compatible? The Bolzano-Weierstrass Theorem as a Case Study.Gregory H. Moore - 1999 - History and Philosophy of Logic 20 (3-4):169-180.
    This paper combines personal reminiscences of the philosopher John Corcoran with a discussion of certain conflicts between historians of logic and philosophers of logic. Some mistaken claims about the history of the Bolzano-Weierstrass Theorem are analyzed in detail and corrected.
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  3.  25
    Addendum to: “The BolzanoWeierstrass theorem is the jump of weak Kőnig's lemma” [Ann. Pure Appl. Logic 163 (6) (2012) 623–655]. [REVIEW]Vasco Brattka, Andrea Cettolo, Guido Gherardi, Alberto Marcone & Matthias Schröder - 2017 - Annals of Pure and Applied Logic 168 (8):1605-1608.
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  4.  13
    On Robust Theorems Due to Bolzano, Weierstrass, Jordan, and Cantor.Dag Normann & Sam Sanders - forthcoming - Journal of Symbolic Logic:1-51.
    Reverse Mathematics (RM hereafter) is a program in the foundations of mathematics where the aim is to identify theminimalaxioms needed to prove a given theorem from ordinary, i.e., non-set theoretic, mathematics. This program has unveiled surprising regularities: the minimal axioms are very oftenequivalentto the theorem over thebase theory, a weak system of ‘computable mathematics’, while most theorems are either provable in this base theory, or equivalent to one of onlyfourlogical systems. The latter plus the base theory are called (...)
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  5.  14
    The Strength of an Axiom of Finite Choice for Branches in Trees.G. O. H. Jun Le - 2023 - Journal of Symbolic Logic 88 (4):1367-1386.
    In their logical analysis of theorems about disjoint rays in graphs, Barnes, Shore, and the author (hereafter BGS) introduced a weak choice scheme in second-order arithmetic, called the $\Sigma ^1_1$ axiom of finite choice (hereafter finite choice). This is a special case of the $\Sigma ^1_1$ axiom of choice ( $\Sigma ^1_1\text {-}\mathsf {AC}_0$ ) introduced by Kreisel. BGS showed that $\Sigma ^1_1\text {-}\mathsf {AC}_0$ suffices for proving many of the aforementioned theorems in graph theory. While it is not known (...)
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  6.  53
    Steve Russ. The mathematical works of Bernard Bolzano. Oxford: Oxford university press, 2004. Pp. XXX + 698. Isbn 0-19-853930-. [REVIEW]Ali Behboud - 2006 - Philosophia Mathematica 14 (3):352-362.
    In his book on The Mathematics of Great Amateurs Coolidge starts the chapter on Bolzano saying that he included Bolzano because it seemed interesting to him ‘that a man who was a remarkable pulpit orator, only removed from his chair for his political opinions, should have thought so far into the deepest problems of a science which he never taught in a professional capacity’ [Coolidge, 1990, p. 195]. In fact, considering Bolzano's poor health and his enormous productivity (...)
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  7.  58
    Ideal convergence of bounded sequences.Rafał Filipów, Recław Ireneusz, Mrożek Nikodem & Szuca Piotr - 2007 - Journal of Symbolic Logic 72 (2):501-512.
    We generalize the Bolzano-Weierstrass theorem on ideal convergence. We show examples of ideals with and without the Bolzano-Weierstrass property, and give characterizations of BW property in terms of submeasures and extendability to a maximal P-ideal. We show applications to Rudin-Keisler and Rudin-Blass orderings of ideals and quotient Boolean algebras. In particular we show that an ideal does not have BW property if and only if its quotient Boolean algebra has a countably splitting family.
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  8. Linguistic Functions.W. B. Vasantha Kandasamy, K. Ilanthenral & Florentin Smarandache - 2022 - Miami, FL, USA: Global Knowledge.
    In this book, for the first time, authors try to introduce the concept of linguistic variables as a continuum of linguistic terms/elements/words in par or similar to a real continuum. For instance, we have the linguistic variable, say the heights of people, then we place the heights in the linguistic continuum [shortest, tallest] unlike the real continuum (–∞, ∞) where both –∞ or +∞ is only a non-included symbols of the real continuum, but in case of the linguistic continuum we (...)
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  9.  18
    On the Foundations of Mathematical Economics.J. Barkley Rosser - unknown
    Kumaraswamy Vela Velupillai [74] presents a constructivist perspective on the foundations of mathematical economics, praising the views of Feynman in developing path integrals and Dirac in developing the delta function. He sees their approach as consistent with the Bishop constructive mathematics and considers its view on the Bolzano-Weierstrass, Hahn-Banach, and intermediate value theorems, and then the implications of these arguments for such “crown jewels” of mathematical economics as the existence of general equilibrium and the second welfare theorem. (...)
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  10.  29
    On the Mathematical Method and Correspondence with Exner: Translated by Paul Rusnock and Rolf George.Bernard Bolzano (ed.) - 2004 - BRILL.
    The Prague Philosopher Bernard Bolzano (1781-1848) has long been admired for his groundbreaking work in mathematics: his rigorous proofs of fundamental theorems in analysis, his construction of a continuous, nowhere-differentiable function, his investigations of the infinite, and his anticipations of Cantor's set theory. He made equally outstanding contributions in philosophy, most notably in logic and methodology. One of the greatest mathematician-philosophers since Leibniz, Bolzano is now widely recognised as a major figure of nineteenth-century philosophy. Praised by Husserl as (...)
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  11.  16
    Beth definability and the Stone-Weierstrass Theorem.Luca Reggio - 2021 - Annals of Pure and Applied Logic 172 (8):102990.
    The Stone-Weierstrass Theorem for compact Hausdorff spaces is a basic result of functional analysis with far-reaching consequences. We introduce an equational logic ⊨Δ associated with an infinitary variety Δ and show that the Stone-Weierstrass Theorem is a consequence of the Beth definability property of ⊨Δ, stating that every implicit definition can be made explicit. Further, we define an infinitary propositional logic ⊢Δ by means of a Hilbert-style calculus and prove a strong completeness result whereby the semantic (...)
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  12.  18
    Démonstration purement analytique du théorème : entre deux valeurs quelconques qui donnent deux résultats de signes opposés se trouve au moins une racine réelle de l'équation.Bernard Bolzano - 1964 - Revue d'Histoire des Sciences 17 (2):136-164.
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  13.  19
    Elements of Intuitionistic Analysis II the Stone‐Weierstrass Theorem and Ascoli's Theorem.H. de Swart - 1976 - Mathematical Logic Quarterly 22 (1):501-508.
  14.  29
    Elements of Intuitionistic Analysis II the Stone-Weierstrass Theorem and Ascoli's Theorem.H. de Swart - 1976 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 22 (1):501-508.
  15.  19
    Billing J.. A failure of the Bolzano-Weierstrass lemma. Arkiv för matematik, astronomi och fysik, vol. 34B , no. 11, 2 pp. [REVIEW]Alonzo Church - 1947 - Journal of Symbolic Logic 12 (3):94-94.
  16. Bolzano and the Traditions of Analysis Russell, in his History of Western Philosophy, wrote that modern analytical philosophy had its origins in the construction of modem functional analysis by Weierstrass and others.P. Rusnock - forthcoming - Grazer Philosophische Studien.
  17.  28
    Bolzano's Philosophy and the Emergence of Modern Mathematics.Paul Rusnock (ed.) - 2000 - BRILL.
    Contents: Acknowledgements. Conventions. Preface. Biographical sketch. 1 Introduction. 2 The Contributions. 3 Early work in analysis. 4 The Theory of Science . 5. Later mathematical studies. A On Kantian Intuitions. B The Bolzano-Cauchy Theorem.
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  18.  21
    Weierstrass's final theorem of arithmetic is not final.F. G. Asenjo & J. M. McKean - 1972 - Notre Dame Journal of Formal Logic 13 (1):91-94.
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  19.  15
    Bernard Bolzano et son Mémoire sur le théorème fondamental de l'Analyse.Jan Sebestik - 1964 - Revue d'Histoire des Sciences 17 (2):129-135.
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  20.  23
    On the computational content of the Bolzano-Weierstraß Principle.Pavol Safarik & Ulrich Kohlenbach - 2010 - Mathematical Logic Quarterly 56 (5):508-532.
    We will apply the methods developed in the field of ‘proof mining’ to the Bolzano-Weierstraß theorem BW and calibrate the computational contribution of using this theorem in proofs of combinatorial statements. We provide an explicit solution of the Gödel functional interpretation as well as the monotone functional interpretation of BW for the product space Πi ∈ℕ[–ki, ki] . This results in optimal program and bound extraction theorems for proofs based on fixed instances of BW, i.e. for (...)
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  21.  52
    Bolzano's Legacy.Dagfin Føllesdal - 1997 - Grazer Philosophische Studien 53 (1):1-11.
    Bernard Bolzano (1781-1848) was an original and independent thinker, who left a lasting legacy in several areas of philosophy. Four such areas are singled for special attention: political philosophy, ethics and theology, logics and semantics, and mathematics. In all these areas he was far ahead of his time. He had pioneering ideas in political philosophy and in ethics and philosophy of religion, and he argued for them in a brilliantly clear way. In logic and semantics he anticipated Frege, Carnap (...)
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  22.  8
    Bolzano's Legacy.Dagfin Føllesdal - 1997 - Grazer Philosophische Studien 53 (1):1-11.
    Bernard Bolzano (1781-1848) was an original and independent thinker, who left a lasting legacy in several areas of philosophy. Four such areas are singled for special attention: political philosophy, ethics and theology, logics and semantics, and mathematics. In all these areas he was far ahead of his time. He had pioneering ideas in political philosophy and in ethics and philosophy of religion, and he argued for them in a brilliantly clear way. In logic and semantics he anticipated Frege, Carnap (...)
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  23.  66
    Bolzano’s Concept of Consequence.Mark Siebel - 2002 - The Monist 85 (4):580-599.
    In the second volume of his Wissenschaftslehre from 1837, the Bohemian philosopher, theologian, and mathematician Bernard Bolzano introduced his concept of consequence, named derivability, together with a variety of theorems and further considerations. Derivability is an implication relation between sentences in themselves, which are not meant to be linguistic symbols but the contents of declarative sentences as well as of certain mental episodes. When Schmidt utters the sentence ‘Schnee ist weiß’, and Jones judges that snow is white, the sentence (...)
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  24.  37
    Bolzano and the Traditions of Analysis.Paul Rusnock - 1997 - Grazer Philosophische Studien 53 (1):61-85.
    Russell, in his History of Western Philosophy, wrote that modern analytical philosophy had its origins in the construction of modern functional analysis by Weierstrass and others. As it turns out, Bolzano, in the first four decades of the nineteenth century, had already made important contributions'to the creation of "Weierstrassian" analysis, some of which were well known to Weierstrass and his circle. In addition, his mathematical research was guided by a methodology which articulated many of the central principles (...)
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  25.  26
    Bolzano and the Traditions of Analysis.Paul Rusnock - 1997 - Grazer Philosophische Studien 53 (1):61-85.
    Russell, in his History of Western Philosophy, wrote that modern analytical philosophy had its origins in the construction of modern functional analysis by Weierstrass and others. As it turns out, Bolzano, in the first four decades of the nineteenth century, had already made important contributions'to the creation of "Weierstrassian" analysis, some of which were well known to Weierstrass and his circle. In addition, his mathematical research was guided by a methodology which articulated many of the central principles (...)
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  26.  15
    From BolzanoWeierstraß to Arzelà‐Ascoli.Alexander P. Kreuzer - 2014 - Mathematical Logic Quarterly 60 (3):177-183.
    We show how one can obtain solutions to the Arzelà‐Ascoli theorem using suitable applications of the BolzanoWeierstraß principle. With this, we can apply the results from and obtain a classification of the strength of instances of the Arzelà‐Ascoli theorem and a variant of it. Let be the statement that each equicontinuous sequence of functions contains a subsequence that converges uniformly with the rate and let be the statement that each such sequence contains a subsequence which converges (...)
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  27. Szemerédi’s theorem: An exploration of impurity, explanation, and content.Patrick J. Ryan - 2023 - Review of Symbolic Logic 16 (3):700-739.
    In this paper I argue for an association between impurity and explanatory power in contemporary mathematics. This proposal is defended against the ancient and influential idea that purity and explanation go hand-in-hand (Aristotle, Bolzano) and recent suggestions that purity/impurity ascriptions and explanatory power are more or less distinct (Section 1). This is done by analyzing a central and deep result of additive number theory, Szemerédi’s theorem, and various of its proofs (Section 2). In particular, I focus upon the (...)
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  28.  5
    The Completeness Theorem? So What!Göran Sundholm - 2024 - In Antonio Piccolomini D'Aragona (ed.), Perspectives on Deduction: Contemporary Studies in the Philosophy, History and Formal Theories of Deduction. Springer Verlag. pp. 39-50.
    Bolzano reduced inferential validity of the inference (from premise judgements to conclusion judgment) to the holding of logical consequence between the propositions (in themselves) that serve as contents of the respective judgements. This explicit reduction of inferential validity among judgements to logical consequence among propositions (or, alternatively, to logical truth of certain implicational propositions) has been largely taken over by current logical theory, say, by Wittgenstein’s Tractatus, by Hilbert and Ackermann, by Quine, and by Tarski also. Frege, though, stands (...)
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  29.  27
    Frege's and Bolzano's rationalist conceptions of arithmetic.Charles Chihara - 1999 - Revue d'Histoire des Sciences 52 (3):343-362.
    In this article, I compare Gottlob Frege's and Bernard Bolzano's rationalist conceptions of arithmetic. Each philosopher worked out a complicated system of propositions, all of which were set forth as true. The axioms, or basic truths, make up the foundations of the subject of arithmetic. Each member of the system which is not an axiom is related (objectively) to the axioms at the base. Even though this relation to the base may not yet be scientifically proven, the propositions of (...)
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  30.  12
    The development of the concept of uniform convergence in Karl Weierstrass’s lectures and publications between 1861 and 1886.Klaus Viertel - 2021 - Archive for History of Exact Sciences 75 (4):455-490.
    The history of uniform convergence is typically focused on the contributions of Cauchy, Seidel, Stokes, and Björling. While the mathematical contributions of these individuals to the concept of uniform convergence have been much discussed, Weierstrass is considered to be the actual inventor of today’s concept. This view is often based on his well-known article from 1841. However, Weierstrass’s works on a rigorous foundation of analytic and elliptic functions date primarily from his lecture courses at the University of Berlin (...)
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  31.  24
    Pincherle's theorem in reverse mathematics and computability theory.Dag Normann & Sam Sanders - 2020 - Annals of Pure and Applied Logic 171 (5):102788.
    We study the logical and computational properties of basic theorems of uncountable mathematics, in particular Pincherle's theorem, published in 1882. This theorem states that a locally bounded function is bounded on certain domains, i.e. one of the first ‘local-to-global’ principles. It is well-known that such principles in analysis are intimately connected to (open-cover) compactness, but we nonetheless exhibit fundamental differences between compactness and Pincherle's theorem. For instance, the main question of Reverse Mathematics, namely which set existence axioms (...)
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  32. A completeness theorem for unrestricted first- order languages.Agustin Rayo & Timothy Williamson - 2003 - In J. C. Beall (ed.), Liars and Heaps: New Essays on Paradox. Oxford, England: Oxford University Press UK. pp. 331-356.
    Here is an account of logical consequence inspired by Bolzano and Tarski. Logical validity is a property of arguments. An argument is a pair of a set of interpreted sentences (the premises) and an interpreted sentence (the conclusion). Whether an argument is logically valid depends only on its logical form. The logical form of an argument is fixed by the syntax of its constituent sentences, the meanings of their logical constituents and the syntactic differences between their non-logical constituents, treated (...)
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  33.  46
    Term extraction and Ramsey's theorem for pairs.Alexander P. Kreuzer & Ulrich Kohlenbach - 2012 - Journal of Symbolic Logic 77 (3):853-895.
    In this paper we study with proof-theoretic methods the function(al) s provably recursive relative to Ramsey's theorem for pairs and the cohesive principle (COH). Our main result on COH is that the type 2 functional provably recursive from $RCA_0 + COH + \Pi _1^0 - CP$ are primitive recursive. This also provides a uniform method to extract bounds from proofs that use these principles. As a consequence we obtain a new proof of the fact that $WKL_0 + \Pi _1^0 (...)
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  34.  84
    A Road Map of Dedekind’s Theorem 66.Ansten Klev - 2018 - Hopos: The Journal of the International Society for the History of Philosophy of Science 8 (2):241-277.
    Richard Dedekind’s theorem 66 states that there exists an infinite set. Its proof invokes such apparently nonmathematical notions as the thought-world and the self. This article discusses the content and context of Dedekind’s proof. It is suggested that Dedekind took the notion of the thought-world from Hermann Lotze. The influence of Kant and Bernard Bolzano on the proof is also discussed, and the reception of the proof in the mathematical and philosophical literature is covered in detail.
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  35.  44
    Making the hyperreal line both saturated and complete.H. Jerome Keisler & James H. Schmerl - 1991 - Journal of Symbolic Logic 56 (3):1016-1025.
    In a nonstandard universe, the κ-saturation property states that any family of fewer than κ internal sets with the finite intersection property has a nonempty intersection. An ordered field F is said to have the λ-Bolzano-Weierstrass property iff F has cofinality λ and every bounded λ-sequence in F has a convergent λ-subsequence. We show that if $\kappa < \lambda$ are uncountable regular cardinals and $\beta^\alpha < \lambda$ whenever $\alpha < \kappa$ and $\beta < \lambda$, then there is a (...)
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  36.  25
    Making the Hyperreal Line Both Saturated and Complete.H. Jerome Keisler & James H. Schmerl - 1991 - Journal of Symbolic Logic 56 (3):1016-1025.
    In a nonstandard universe, the $\kappa$-saturation property states that any family of fewer than $\kappa$ internal sets with the finite intersection property has a nonempty intersection. An ordered field $F$ is said to have the $\lambda$-Bolzano-Weierstrass property iff $F$ has cofinality $\lambda$ and every bounded $\lambda$-sequence in $F$ has a convergent $\lambda$-subsequence. We show that if $\kappa < \lambda$ are uncountable regular cardinals and $\beta^\alpha < \lambda$ whenever $\alpha < \kappa$ and $\beta < \lambda$, then there is a (...)
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  37. Ten Misconceptions from the History of Analysis and Their Debunking.Piotr Błaszczyk, Mikhail G. Katz & David Sherry - 2013 - Foundations of Science 18 (1):43-74.
    The widespread idea that infinitesimals were “eliminated” by the “great triumvirate” of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the (...)
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  38.  12
    ISCS 2013: interdisciplinary Symposium on Complex Systems.ʻAlī Ṣanāyiʻī, Ivan Zelinka & Otto E. Rössler (eds.) - 2014 - New York: Springer.
    The book you hold in your hands is the outcome of the "ISCS 2013: Interdisciplinary Symposium on Complex Systems" held at the historical capital of Bohemia as a continuation of our series of symposia in the science of complex systems. Prague, one of the most beautiful European cities, has its own beautiful genius loci. Here, a great number of important discoveries were made and many important scientists spent fruitful and creative years to leave unforgettable traces. The perhaps most significant period (...)
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  39.  5
    Husserl.Rudolf Bernet - 2017 - In Simon Critchley & William R. Schroeder (eds.), A Companion to Continental Philosophy. Oxford, UK: Blackwell. pp. 198–207.
    Edmund Husserl (1859–1938) is the founder of the phenomenological movement which has profoundly influenced twentieth‐century Continental philosophy. The historical setting in which his thought took shape was marked by the emergence of a new psychology (Herbart, von Helmholtz, James, Brentano, Stumpf, Lipps), by research into the foundation of mathematics (Gauss, Rieman, Cantor, Kronecker, Weierstrass), by a revival of logic and theory of knowledge (Bolzano, Mill, Boole, Lotze, Mach, Frege, Sigwart, Meinong, Erdmann, Schröder), as well as by the appearance (...)
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  40.  9
    Historical evolution of the concept of homotopic paths.Ria Vanden Eynde - 1992 - Archive for History of Exact Sciences 45 (2):127-188.
    The historical evolution of the homotopy concept for paths illustrates how the introduction of a concept (be it implicit or explicit) depends upon the interests of the mathematicians concerned and how it gradually acquires a more satisfactory definition. In our case the equivalence of paths first meant for certain mathematicians that they led to the same value of the integral of a given function or that they led to the same value of a multiple-valued function. (See for instance [Cau], [Pui], (...)
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  41.  52
    What is a Line?D. F. M. Strauss - 2014 - Axiomathes 24 (2):181-205.
    Since the discovery of incommensurability in ancient Greece, arithmeticism and geometricism constantly switched roles. After ninetieth century arithmeticism Frege eventually returned to the view that mathematics is really entirely geometry. Yet Poincaré, Brouwer, Weyl and Bernays are mathematicians opposed to the explication of the continuum purely in terms of the discrete. At the beginning of the twenty-first century ‘continuum theorists’ in France (Longo, Thom and others) believe that the continuum precedes the discrete. In addition the last 50 years witnessed the (...)
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  42. Philosophical method and Galileo's paradox of infinity.Matthew W. Parker - 2008 - In Bart Van Kerkhove (ed.), New Perspectives on Mathematical Practices: Essays in Philosophy and History of Mathematics : Brussels, Belgium, 26-28 March 2007. World Scientfic.
    We consider an approach to some philosophical problems that I call the Method of Conceptual Articulation: to recognize that a question may lack any determinate answer, and to re-engineer concepts so that the question acquires a definite answer in such a way as to serve the epistemic motivations behind the question. As a case study we examine “Galileo’s Paradox”, that the perfect square numbers seem to be at once as numerous as the whole numbers, by one-to-one correspondence, and yet less (...)
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  43. Dimensional theoretical properties of some affine dynamical systems.Jörg Neunhäuserer - 1999 - Dissertation,
    In this work we study dimensional theoretical properties of some a±ne dynamical systems. By dimensional theoretical properties we mean Hausdor® dimension and box- counting dimension of invariant sets and ergodic measures on theses sets. Especially we are interested in two problems. First we ask whether the Hausdor® and box- counting dimension of invariant sets coincide. Second we ask whether there exists an ergodic measure of full Hausdor® dimension on these invariant sets. If this is not the case we ask the (...)
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  44.  29
    Expansions of algebraically closed fields II: Functions of several variables.Ya'acov Peterzil & Sergei Starchenko - 2003 - Journal of Mathematical Logic 3 (01):1-35.
    Let ℛ be an o-minimal expansion of a real closed field R. We continue here the investigation we began in [11] of differentiability with respect to the algebraically closed field [Formula: see text]. We develop the basic theory of such K-differentiability for definable functions of several variables, proving theorems on removable singularities as well as analogues of the Weierstrass preparation and division theorems for definable functions. We consider also definably meromorphic functions and prove that every definable function which is (...)
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  45. New Considerations on The 'Liar' Paradox.Wojciech Żełaniec - 2004 - Filozofia Nauki 2.
    In this article the author argues that the 'Liar' Paradox sentence: "This sentence is false" is neither true nor false because it does not express any proposition or "Satz" in the sense of Bernard Bolzano. The difficulty left open is that by a similar line of reasoning also the sentence "This sentence is true" would not express any proposition, yet it is sometimes taken to be true (on the strength of a theorem by Loewe).
     
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  46.  12
    Іван слешинський - популяризатор ідей математичної логіки в україні.Marianna P. Plakhtiy - 2020 - Вісник Харківського Національного Університету Імені В. Н. Каразіна. Серія «Філософія. Філософські Перипетії» 62:99-107.
    The first half of the twentieth century was marked by the simultaneous development of logic and mathematics. Logic offered the necessary means to justify the foundations of mathematics and to solve the crisis that arose in mathematics in the early twentieth century. In European science in the late nineteenth century, the ideas of symbolic logic, based on the works of J. Bull, S. Jevons and continued by C. Pierce in the United States and E. Schroeder in Germany were getting popular. (...)
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  47.  87
    Epistemology of visual thinking in elementary real analysis.Marcus Giaquinto - 1994 - British Journal for the Philosophy of Science 45 (3):789-813.
    Can visual thinking be a means of discovery in elementary analysis, as well as a means of illustration and a stimulus to discovery? The answer to the corresponding question for geometry and arithmetic seems to be ‘yes’ (Giaquinto [1992], [1993]), and so a positive answer might be expected for elementary analysis too. But I argue here that only in a severely restricted range of cases can visual thinking be a means of discovery in analysis. Examination of persuasive visual routes to (...)
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  48.  19
    Representations and the Foundations of Mathematics.Sam Sanders - 2022 - Notre Dame Journal of Formal Logic 63 (1):1-28.
    The representation of mathematical objects in terms of (more) basic ones is part and parcel of (the foundations of) mathematics. In the usual foundations of mathematics, namely, ZFC set theory, all mathematical objects are represented by sets, while ordinary, namely, non–set theoretic, mathematics is represented in the more parsimonious language of second-order arithmetic. This paper deals with the latter representation for the rather basic case of continuous functions on the reals and Baire space. We show that the logical strength of (...)
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  49.  91
    Stefania Centrone. Logic and Philosophy of Mathematics in the Early Husserl. Synthese Library 345. Dordrecht: Springer, 2010. Pp. xxii + 232. ISBN 978-90-481-3245-4. [REVIEW]Mirja Hartimo - 2010 - Philosophia Mathematica 18 (3):344-349.
    It is beginning to be rather well known that Edmund Husserl, the founder of phenomenological philosophy, was originally a mathematician; he studied with Weierstrass and Kronecker in Berlin, wrote his doctoral dissertation on the calculus of variations, and was then a colleague of Cantor in Halle until he moved to the Göttingen of Hilbert and Klein in 1901. Much of Husserl’s writing prior to 1901 was about mathematics, and arguably the origin of phenomenology was in Husserl’s attempts to give (...)
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  50. Purity as an ideal of proof.Michael Detlefsen - 2008 - In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. Oxford University Press. pp. 179-197.
    Various ideals of purity are surveyed and discussed. These include the classical Aristotelian ideal, as well as certain neo-classical and contemporary ideals. The focus is on a type of purity ideal I call topical purity. This is purity which emphasizes a certain symmetry between the conceptual resources used to prove a theorem and those needed for the clarification of its content. The basic idea is that the resources of proof ought ideally to be restricted to those which determine its (...)
     
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