Search results for 'mathematical definitions' (try it on Scholar)

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  1. Christopher Belanger (2013). On Two Mathematical Definitions of Observational Equivalence: Manifest Isomorphism and Epsilon-Congruence Reconsidered. Studies in History and Philosophy of Science Part B 44 (2):69-76.
    In this article I examine two mathematical definitions of observational equivalence, one proposed by Charlotte Werndl and based on manifest isomorphism, and the other based on Ornstein and Weiss’s ε-congruence. I argue, for two related reasons, that neither can function as a purely mathematical definition of observational equivalence. First, each definition permits of counterexamples; second, overcoming these counterexamples will introduce non-mathematical premises about the systems in question. Accordingly, the prospects for a broadly applicable and purely (...) definition of observational equivalence are unpromising. Despite this critique, I suggest that Werndl’s proposals are valuable because they clarify the distinction between provable and unprovable elements in arguments for observational equivalence. (shrink)
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  2.  4
    Christopher Belanger (2013). On Two Mathematical Definitions of Observational Equivalence: Manifest Isomorphism and Reconsidered. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 44 (2):69-76.
    In this paper I examine two mathematical definitions of observational equivalence, one proposed by Charlotte Werndl and based on manifest isomorphism, and the other based on Ornstein and Weiss's ε-congruenceε-congruence. I argue, for two related reasons, that neither can function as a purely mathematical definition of observational equivalence. First, each definition permits of counterexamples; second, overcoming these counterexamples will introduce non-mathematical premises about the systems in question. Accordingly, the prospects for a broadly applicable and purely (...) definition of observational equivalence are unpromising. Despite this critique, I suggest that Werndl's proposals are valuable because they clarify the distinction between provable and unprovable elements in arguments for observational equivalence. (shrink)
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  3.  2
    Charlotte Werndl, The Formulation and Justification of Mathematical Definitions Illustrated By Deterministic Chaos.
    The general theme of this article is the actual practice of how definitions are justified and formulated in mathematics. The theoretical insights of this article are based on a case study of topological definitions of chaos. After introducing this case study, I identify the three kinds of justification which are important for topological definitions of chaos: natural-world-justification, condition-justification and redundancy-justification. To my knowledge, the latter two have not been identified before. I argue that these three kinds of (...)
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  4.  8
    John-Michael Kuczynski (2016). Mathematical Logic and Formal Arithmetic: Key Definitions and Principles. Amazon Digital Services LLC.
    This books states, as clearly and concisely as possible, the most fundamental principles of set-theory and mathematical logic. Included is an original proof of the incompleteness of formal logic. Also included are clear and rigorous definitions of the primary arithmetical operations, as well as clear expositions of the arithmetic of transfinite cardinals.
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  5.  75
    Jamie Tappenden (2008). Mathematical Concepts and Definitions. In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice. OUP Oxford 256--275.
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  6.  8
    S. C. Kleene (1949). Review: K. R. Popper, On the Theory of Deduction, Part I. Derivation and its Generalizations; K. R. Popper, On the Theory of Deduction, Part II. The Definitions of Classical and Intuitionist Negation; K. R. Popper, The Trivialization of Mathematical Logic. [REVIEW] Journal of Symbolic Logic 14 (1):62-63.
  7.  2
    Robert McNaughton (1997). Robinson Raphael M.. Restricted Set-Theoretical Definitions in Arithmetic. Proceedings of the American Mathematical Society, Vol. 9 (1958), Pp. 238–242. Robinson Raphael M.. Restricted Set-Theoretical Definitions in Arithmetic. Summaries of Talks Presented at the Summer Institute for Symbolic Logic, Cornell University, 1957, 2nd Edn., Communications Research Division, Institute for Defense Analyses, Princeton, NJ, 1960, Pp. 139–140. [REVIEW] Journal of Symbolic Logic 31 (4):659-660.
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  8.  4
    Marwan Rashed (1996). Ex Aequali Ratios in the Greek and Arabic Euclidean Traditions Gregg De Young Euclid Discusses the Ex Aequali Relationship Twice in the Elements. The First is in Book V (Based on Definitions 17 and 18, Propositions 22 and 23), During His Discussion of Arithmetical Relations Between Mathematical Magnitudes In. [REVIEW] Arabic Sciences and Philosophy 6.
  9. Gregory W. Jones (1973). Grilliot Thomas J.. Inductive Definitions and Computability. Transactions of the American Mathematical Society, Vol. 158 , Pp. 309–317. [REVIEW] Journal of Symbolic Logic 38 (4):654.
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  10. S. C. Kleene (1949). Popper K. R.. On the Theory of Deduction, Part I. Derivation and its Generalizations. Koninklijke Nederlandsche Akademie van Wetenschappen, Proceedings of the Section of Sciences, Vol. 51 , Pp. 173–183; Also Indagationes Mathematicae, Vol. 10 , Pp. 44–54.Popper K. R.. On the Theory of Deduction, Part II. The Definitions of Classical and Intuitionist Negation. Koninklijke Nederlandsche Akademie van Wetenschappen, Proceedings of the Section of Sciences, Vol. 51 , Pp. 322–331; Also Ibid., Pp. 111–120.Popper K. R.. The Trivialization of Mathematical Logic. Library of the Xlh International Congress of Philosophy . Vol. I. Proceedings of the Congress. Preprint 1948, Pp. 510–515. [REVIEW] Journal of Symbolic Logic 14 (1):62-63.
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  11. Richard Montague (1957). Wang Hao. Truth Definitions and Consistency Proofs. Transactions of the American Mathematical Society, Vol. 73 , Pp. 243–275. [REVIEW] Journal of Symbolic Logic 22 (4):365-367.
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  12. Joan Rand Moschovakis (1970). Brouwer Luitzen Egbertus Jan. On the Significance of the Principle of Excluded Middle in Mathematics, Especially in Function Theory, English Translation of 15516 by Bauer-Mengelberg Stefan and Heijenoort Jean Van. From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931, Edited by Heijenoort Jean van, Harvard University Press, Cambridge, Massachusetts, 1967, Pp. 334–341. Addenda and Corrigenda, English Translation of XXIV 189 by Stefan Bauer-Mengelberg, Claske M. Berndes Franck, Dirk van Dalen, and Jean van Heijenoort. Ibid., Pp. 341–342. Further Addenda and Corrigenda. English Translation of XXIV 189 by Stefan Bauer-Mengelberg, Dirk van Dalen, and Jean van Heijenoort. Ibid., Pp. 342–345.Brouwer Luitzen Egbertus Jan. On the Domains of Definition of Functions. From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931, Edited by Heijenoort Jean van, Harvard University Press, Cambridge, Massachusetts, 1967, Pp. 446–463. English Translation of §§1–3 of Über Definitions. [REVIEW] Journal of Symbolic Logic 35 (2):332-333.
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  13. Steven Orey (1965). Wang Hao. Truth Definitions and Consistency Proofs. A Survey of Mathematical Logic, Studies in Logic and the Foundations of Mathematics, Science Press, Peking, and North-Holland Publishing Company, Amsterdam, 1963, Pp. 443–477. [REVIEW] Journal of Symbolic Logic 30 (1):100.
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  14. Rózsa Péter (1952). Robinson Raphael M.. Arithmetical Definitions in the Ring of Integers. Proceedings of the American Mathematical Society, Bd. 2 , S.279–284. [REVIEW] Journal of Symbolic Logic 17 (4):269-270.
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  15.  90
    Federico Zalamea (2015). The Mathematical Description of a Generic Physical System. Topoi 34 (2):339-348.
    When dealing with a certain class of physical systems, the mathematical characterization of a generic system aims to describe the phase portrait of all its possible states. Because they are defined only up to isomorphism, the mathematical objects involved are “schematic structures”. If one imposes the condition that these mathematical definitions completely capture the physical information of a given system, one is led to a strong requirement of individuation for physical states. However, we show there are (...)
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  16.  73
    John Corcoran (2014). Review of Macbeth, D. Diagrammatic Reasoning in Frege's Begriffsschrift. Synthese 186 (2012), No. 1, 289–314. Mathematical Reviews MR 2935338. MATHEMATICAL REVIEWS 2014:2935338.
    A Mathematical Review by John Corcoran, SUNY/Buffalo -/- Macbeth, Danielle Diagrammatic reasoning in Frege's Begriffsschrift. Synthese 186 (2012), no. 1, 289–314. ABSTRACT This review begins with two quotations from the paper: its abstract and the first paragraph of the conclusion. The point of the quotations is to make clear by the “give-them-enough-rope” strategy how murky, incompetent, and badly written the paper is. I know I am asking a lot, but I have to ask you to read the quoted passages—aloud (...)
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  17.  57
    Sanford Shieh (2009). Teaching & Learning Guide For: Frege on Definitions. Philosophy Compass 4 (5):885-888.
    Three clusters of philosophically significant issues arise from Frege's discussions of definitions. First, Frege criticizes the definitions of mathematicians of his day, especially those of Weierstrass and Hilbert. Second, central to Frege's philosophical discussion and technical execution of logicism is the so-called Hume's Principle, considered in The Foundations of Arithmetic . Some varieties of neo-Fregean logicism are based on taking this principle as a contextual definition of the operator 'the number of …', and criticisms of such neo-Fregean programs (...)
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  18.  85
    R. Eugene Collins (2005). The Mathematical Basis for Physical Laws. Foundations of Physics 35 (5):743-785.
    Laws of mechanics, quantum mechanics, electromagnetism, gravitation and relativity are derived as “related mathematical identities” based solely on the existence of a joint probability distribution for the position and velocity of a particle moving on a Riemannian manifold. This probability formalism is necessary because continuous variables are not precisely observable. These demonstrations explain why these laws must have the forms previously discovered through experiment and empirical deduction. Indeed, the very existence of electric, magnetic and gravitational fields is predicted by (...)
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  19. Uri Nodelman & Edward N. Zalta (2014). Foundations for Mathematical Structuralism. Mind 123 (489):39-78.
    We investigate the form of mathematical structuralism that acknowledges the existence of structures and their distinctive structural elements. This form of structuralism has been subject to criticisms recently, and our view is that the problems raised are resolved by proper, mathematics-free theoretical foundations. Starting with an axiomatic theory of abstract objects, we identify a mathematical structure as an abstract object encoding the truths of a mathematical theory. From such foundations, we derive consequences that address the main questions (...)
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  20.  53
    S. Friederich (2011). Motivating Wittgenstein's Perspective on Mathematical Sentences as Norms. Philosophia Mathematica 19 (1):1-19.
    The later Wittgenstein’s perspective on mathematical sentences as norms is motivated for sentences belonging to Hilbertian axiomatic systems where the axioms are treated as implicit definitions. It is shown that in this approach the axioms are employed as norms in that they function as standards of what counts as using the concepts involved. This normative dimension of their mode of use, it is argued, is inherited by the theorems derived from them. Having been motivated along these lines, Wittgenstein’s (...)
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  21.  23
    Matthew Donald (1995). A Mathematical Characterization of the Physical Structure of Observers. Foundations of Physics 25 (4):529-571.
    It is proposed that the physical structure of an observer in quantum mechanics is constituted by a pattern of elementary localized switching events. A key preliminary step in giving mathematical expression to this proposal is the introduction of an equivalence relation on sequences of spacetime sets which relates a sequence to any other sequence to which it can be deformed without change of causal arrangement. This allows an individual observer to be associated with a finite structure. The identification of (...)
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  22.  12
    Vladimir Drekalović (2015). Some Aspects of Understanding Mathematical Reality: Existence, Platonism, Discovery. Axiomathes 25 (3):313-333.
    The sum of all objects of a science, the objects’ features and their mutual relations compose the reality described by that sense. The reality described by mathematics consists of objects such as sets, functions, algebraic structures, etc. Generally speaking, the use of terms reality and existence, in relation to describing various objects’ characteristics, usually implies an employment of physical and perceptible attributes. This is not the case in mathematics. Its reality and the existence of its objects, leaving aside its application, (...)
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  23.  32
    A. T. Balaban (2005). Reflections About Mathematical Chemistry. Foundations of Chemistry 7 (3):289-306.
    A personal account is presented for the present status of mathematical chemistry, with emphasis on non-numerical applications. These use mainly graph-theoretical concepts. Most computational chemical applications involve quantum chemistry and are therefore largely reducible to physics, while discrete mathematical applications often do not. A survey is provided for opinions and definitions of mathematical chemistry, and then for journals, books and book series, as well as symposia of mathematical chemistry.
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  24.  46
    Paul Cortois (1996). The Structure of Mathematical Experience According to Jean Cavaillèst. Philosophia Mathematica 4 (1):18-41.
    In this expository article one of the contributions of Jean Cavailles to the philosophy of mathematics is presented: the analysis of ‘mathematical experience’. The place of Cavailles on the logico-philosophical scene of the 30s and 40s is sketched. I propose a partial interpretation of Cavailles's epistemological program of so-called ‘conceptual dialectics’: mathematical holism, duality principles, the notion of formal contents, and the specific temporal structure of conceptual dynamics. The structure of mathematical abstraction is analysed in terms of (...)
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  25.  17
    Thomas Forster (2007). Implementing Mathematical Objects in Set Theory. Logique Et Analyse 50 (197):79-86.
    In general little thought is given to the general question of how to implement mathematical objects in set theory. It is clear that—at various times in the past—people have gone to considerable lengths to devise implementations with nice properties. There is a litera- ture on the evolution of the Wiener-Kuratowski ordered pair, and a discussion by Quine of the merits of an ordered-pair implemen- tation that makes every set an ordered pair. The implementation of ordinals as Von Neumann ordinals (...)
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  26.  21
    Ali Moussa (2011). Mathematical Methods in Abū Al-Wafāʾ's Almagest and the Qibla Determinations. Arabic Sciences and Philosophy 21 (1):1-56.
    The problem of the Qibla was one of the central issues in the scientific culture of Medieval Islam, and to solve it properly, one needed mathematics and observation. The mathematics consisted of two parts: plane trigonometry and spherical trigonometry . Observation and its instruments were needed to find the geographical coordinates of Mecca and the given location; these coordinates will be the input data in the formulas of the Qibla . In his Almagest , Ab?? al-Waf???? produced a brilliant work (...)
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  27.  10
    Jamie Tappenden (2011). Définitions mathématiques pour philosophes. Les Etudes Philosophiques 97 (2):179.
    Le choix de définitions « naturelles » ou « correctes » est un aspect fondamental de la recherche mathématique qui a été négligé dans l’étude de la connaissance mathématique. L’une des raisons qui expliquent cet abandon tient au sentiment qu’ont eu de nombreux auteurs que la préférence pour une définition au détriment d’une autre ne pouvait être que « simplement psychologique » ou « subjective » en sorte que de tels jugements ne pouvaient pas être philosophiquement intéressants. Je discute ici (...)
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  28.  26
    Reinhard Kahle (2002). Mathematical Proof Theory in the Light of Ordinal Analysis. Synthese 133 (1/2):237 - 255.
    We give an overview of recent results in ordinal analysis. Therefore, we discuss the different frameworks used in mathematical proof-theory, namely "subsystem of analysis" including "reverse mathematics", "Kripke-Platek set theory", "explicit mathematics", "theories of inductive definitions", "constructive set theory", and "Martin-Löf's type theory".
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  29.  6
    Demetra Christopoulou (2014). Weyl on Fregean Implicit Definitions: Between Phenomenology and Symbolic Construction. Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 45 (1):35-47.
    This paper aims to investigate certain aspects of Weyl’s account of implicit definitions. The paper takes under consideration Weyl’s approach to a certain kind of implicit definitions i.e. abstraction principles introduced by Frege.ion principles are bi-conditionals that transform certain equivalence relations into identity statements, defining thereby mathematical terms in an implicit way. The paper compares the analytic reading of implicit definitions offered by the Neo-Fregean program with Weyl’s account which has phenomenological leanings. The paper suggests that (...)
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  30.  5
    Steven Kieffer, Jeremy Avigad & Harvey Friedman, A Language for Mathematical Knowledge Management.
    We argue that the language of Zermelo Fraenkel set theory with definitions and partial functions provides the most promising bedrock semantics for communicating and sharing mathematical knowledge. We then describe a syntactic sugaring of that language that provides a way of writing remarkably readable assertions without straying far from the set-theoretic semantics. We illustrate with some examples of formalized textbook definitions from elementary set theory and point-set topology. We also present statistics concerning the complexity of these (...), under various complexity measures. (shrink)
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  31. Ruth Berger (1997). The Dynamics of Explanation: Mathematical Modeling and Scientific Understanding. Dissertation, Indiana University
    This dissertation challenges two prevalent views on the topic of scientific explanation: that science explains by revealing causal mechanisms, and that science explains by unifying our knowledge of the world. ;My methodological strategy is to compare our best current philosophical accounts of scientific explanation with evidence from contemporary scientific research. In particular, I focus on evidence from dynamical explanations, that is, explanations which appeal to nonlinear dynamical modeling for their force. Nonlinear dynamical modeling is a type of mathematical modeling (...)
     
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  32. Edmund Harriss & Wilfrid Hodges (2007). Logic for Mathematical Writing. Logic Journal of the IGPL 15 (4):313-320.
    In the School of Mathematical Sciences at Queen Mary in the University of London we have been running a module that teaches the students to write good mathematical English. The module is for second-year undergraduates and has been running for three years. It is based on logic, but the logic—though mathematically precise—is informal and doesn't use logical symbols. Some theory of definitions is taught in order to give a structure for mathematical descriptions, and some natural deduction (...)
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  33.  16
    Kai-Uwe Kühnberger, Benedikt Löwe, Michael Möllerfeld & Philip Welch (2005). Comparing Inductive and Circular Definitions: Parameters, Complexity and Games. Studia Logica 81 (1):79 - 98.
    Gupta-Belnap-style circular definitions use all real numbers as possible starting points of revision sequences. In that sense they are boldface definitions. We discuss lightface versions of circular definitions and boldface versions of inductive definitions.
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  34.  14
    Maricarmen Martinez (2001). Some Closure Properties of Finite Definitions. Studia Logica 68 (1):43-68.
    There is no known syntactic characterization of the class of finite definitions in terms of a set of basic definitions and a set of basic operators under which the class is closed. Furthermore, it is known that the basic propositional operators do not preserve finiteness. In this paper I survey these problems and explore operators that do preserve finiteness. I also show that every definition that uses only unary predicate symbols and equality is bound to be finite.
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  35.  21
    Witold A. Pogorzelski & Piotr Wojtylak (2001). Cn-Definitions of Propositional Connectives. Studia Logica 67 (1):1-26.
    We attempt to define the classical propositional logic by use of appropriate derivability conditions called Cn-definitions. The conditions characterize basic properties of propositional connectives.
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  36.  12
    Samuel R. Buss & Alan S. Johnson (2010). The Quantifier Complexity of Polynomial‐Size Iterated Definitions in First‐Order Logic. Mathematical Logic Quarterly 56 (6):573-590.
    We refine the constructions of Ferrante-Rackoff and Solovay on iterated definitions in first-order logic and their expressibility with polynomial size formulas. These constructions introduce additional quantifiers; however, we show that these extra quantifiers range over only finite sets and can be eliminated. We prove optimal upper and lower bounds on the quantifier complexity of polynomial size formulas obtained from the iterated definitions. In the quantifier-free case and in the case of purely existential or universal quantifiers, we show that (...)
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  37.  4
    Erik Palmgren (1992). Type-Theoretic Interpretation of Iterated, Strictly Positive Inductive Definitions. Archive for Mathematical Logic 32 (2):75-99.
    We interpret intuitionistic theories of (iterated) strictly positive inductive definitions (s.p.-ID i′ s) into Martin-Löf's type theory. The main purpose being to obtain lower bounds of the proof-theoretic strength of type theories furnished with means for transfinite induction (W-type, Aczel's set of iterative sets or recursion on (type) universes). Thes.p.-ID i′ s are essentially the wellknownID i -theories, studied in ordinal analysis of fragments of second order arithmetic, but the set variable in the operator form is restricted to occur (...)
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  38.  6
    Benno van den Berg (2013). Non-Deterministic Inductive Definitions. Archive for Mathematical Logic 52 (1-2):113-135.
    We study a new proof principle in the context of constructive Zermelo-Fraenkel set theory based on what we will call “non-deterministic inductive definitions”. We give applications to formal topology as well as a predicative justification of this principle.
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  39.  33
    John Mandel (1964). The Statistical Analysis of Experimental Data. Dover.
    First half of book presents fundamental mathematical definitions, concepts and facts while remaining half deals with statistics primarily as an interpretive tool. Well-written text, numerous worked examples with step-by-step presentation. 116 tables.
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  40. Paolo Mancosu (2014). The Adventure of Reason: Interplay Between Philosophy of Mathematics and Mathematical Logic, 1900-1940. Oxford University Press Uk.
    Paolo Mancosu presents a series of innovative studies in the history and the philosophy of logic and mathematics in the first half of the twentieth century. The Adventure of Reason is divided into five main sections: history of logic ; foundational issues ; mathematics and phenomenology ; nominalism ; semantics. Mancosu exploits extensive untapped archival sources to make available a wealth of new material that deepens in significant ways our understanding of these fascinating areas of modern intellectual history. At the (...)
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  41.  31
    Tyler Marghetis & Rafael Núñez (2013). The Motion Behind the Symbols: A Vital Role for Dynamism in the Conceptualization of Limits and Continuity in Expert Mathematics. Topics in Cognitive Science 5 (2):299-316.
    The canonical history of mathematics suggests that the late 19th-century “arithmetization” of calculus marked a shift away from spatial-dynamic intuitions, grounding concepts in static, rigorous definitions. Instead, we argue that mathematicians, both historically and currently, rely on dynamic conceptualizations of mathematical concepts like continuity, limits, and functions. In this article, we present two studies of the role of dynamic conceptual systems in expert proof. The first is an analysis of co-speech gesture produced by mathematics graduate students while proving (...)
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  42. Jamie Tappenden (2005). The Caesar Problem in its Historical Context: Mathematical Background. Dialectica 59 (2):237–264.
    The issues surrounding the Caesar problem are assumed to be inert as far as ongoing mathematics is concerned. This paper aims to correct this impression by spelling out the ways that, in their historical context, Frege's remarks would have had considerable resonance with work that other mathematicians such as Riemann and Dedekind were doing. The search for presentation‐independent characterizations of objects and global definitions was seen as bound up with fundamental methodological questions in complex analysis and number theory.
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  43.  44
    Andrea Cantini (1996). Logical Frameworks for Truth and Abstraction: An Axiomatic Study. Elsevier Science B.V..
    This English translation of the author's original work has been thoroughly revised, expanded and updated. The book covers logical systems known as type-free or self-referential . These traditionally arise from any discussion on logical and semantical paradoxes. This particular volume, however, is not concerned with paradoxes but with the investigation of type-free sytems to show that: (i) there are rich theories of self-application, involving both operations and truth which can serve as foundations for property theory and formal semantics; (ii) these (...)
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  44.  51
    Volker Peckhaus (1999). 19th Century Logic Between Philosophy and Mathematics. Bulletin of Symbolic Logic 5 (4):433-450.
    The history of modern logic is usually written as the history of mathematical or, more general, symbolic logic. As such it was created by mathematicians. Not regarding its anticipations in Scholastic logic and in the rationalistic era, its continuous development began with George Boole's The Mathematical Analysis of Logic of 1847, and it became a mathematical subdiscipline in the early 20th century. This style of presentation cuts off one eminent line of development, the philosophical development of logic, (...)
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  45.  3
    Richard A. Shore (2007). Direct and Local Definitions of the Turing Jump. Journal of Mathematical Logic 7 (2):229-262.
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  46.  13
    Alexander Abian (1968). On Definitions of Cuts and Completion of Partially Ordered Sets. Mathematical Logic Quarterly 14 (19):299-302.
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  47.  5
    Evangelos Kranakis (1982). Invisible Ordinals and Inductive Definitions. Mathematical Logic Quarterly 28 (8‐12):137-148.
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  48.  17
    Francis Heylighen (1999). Advantages and Limitations of Formal Expression. Foundations of Science 4 (1):25-56.
    Testing the validity of knowledge requires formal expression of that knowledge. Formality of an expression is defined as the invariance, under changes of context, of the expression's meaning, i.e. the distinction which the expression represents. This encompasses both mathematical formalism and operational determination. The main advantages of formal expression are storability, universal communicability, and testability. They provide a selective edge in the Darwinian competition between ideas. However, formality can never be complete, as the context cannot be eliminated. Primitive terms, (...)
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  49.  3
    Kentaro Sato (2015). Full and Hat Inductive Definitions Are Equivalent in NBG. Archive for Mathematical Logic 54 (1-2):75-112.
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  50.  2
    Victor Pambuccian (2002). On Definitions in an Infinitary Language. Mathematical Logic Quarterly 48 (4):522-524.
    We provide the syntactic equivalent for the theorem stating that all epimorphisms of finite projective planes are isomorphisms. The definition of the inequality relation that we provide adds little to our understanding of the theorem, since its very validity can be discerned only from the validity of the model-theoretic theorem regarding epimorphisms.
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