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  1. Reflecting on the 3x+1 Mystery. Outline of a Scenario - Improbable or Realistic ?Edward G. Belaga - manuscript
    Guessing the outcome of iterations of even most simple arithmetical functions could be an extremely hazardous experience. Not less harder, if at all possible, might be to prove the veracity of even a "sure" guess concerning iterations : this is the case of the famous 3x+1 conjecture. Our purpose here is to study and conceptualize some intuitive insights related to the ultimate (un)solvability of this conjecture.
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  2. Algunos tópicos de Lógica matemática y los Fundamentos de la matemática.Franklin Galindo - manuscript
    En este trabajo matemático-filosófico se estudian cuatro tópicos de la Lógica matemática: El método de construcción de modelos llamado Ultraproductos, la Propiedad de Interpolación de Craig, las Álgebras booleanas y los Órdenes parciales separativos. El objetivo principal del mismo es analizar la importancia que tienen dichos tópicos para el estudio de los fundamentos de la matemática, desde el punto de vista del platonismo matemático. Para cumplir con tal objetivo se trabajará en el ámbito de la Matemática, de la Metamatemática y (...)
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  3. Nota: ¿CUÁL ES EL CARDINAL DEL CONJUNTO DE LOS NÚMEROS REALES?Franklin Galindo - manuscript
    ¿Qué ha pasado con el problema del cardinal del continuo después de Gödel (1938) y Cohen (1964)? Intentos de responder esta pregunta pueden encontrarse en los artículos de José Alfredo Amor (1946-2011), "El Problema del continuo después de Cohen (1964-2004)", de Carlos Di Prisco , "Are we closer to a solution of the continuum problem", y de Joan Bagaria, "Natural axioms of set and the continuum problem" , que se pueden encontrar en la biblioteca digital de mi blog de Lógica (...)
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  4. Principles and Philosophy of Linear Algebra: A Gentle Introduction.Paul Mayer - manuscript
    Linear Algebra is an extremely important field that extends everyday concepts about geometry and algebra into higher spaces. This text serves as a gentle motivating introduction to the principles (and philosophy) behind linear algebra. This is aimed at undergraduate students taking a linear algebra class - in particular engineering students who are expected to understand and use linear algebra to build and design things, however it may also prove helpful for philosophy majors and anyone else interested in the ideas behind (...)
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  5. Styles of Argumentation in Late 19th Century Geometry and the Structure of Mathematical Modernity.Moritz Epple - forthcoming - Boston Studies in the Philosophy of Science.
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  6. Logical perspectives on the foundations of probability.Jürgen Landes & Hykel Hosni - 2023 - Open Mathematics 21 (1).
    We illustrate how a variety of logical methods and techniques provide useful, though currently underappreciated, tools in the foundations and applications of reasoning under uncertainty. The field is vast spanning logic, artificial intelligence, statistics, and decision theory. Rather than (hopelessly) attempting a comprehensive survey, we focus on a handful of telling examples. While most of our attention will be devoted to frameworks in which uncertainty is quantified probabilistically, we will also touch upon generalisations of probability measures of uncertainty, which have (...)
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  7. Working Toward Solutions in Fluid Dynamics and Astrophysics: What the Equations Don’t Say.Lydia Patton & Erik Curiel (eds.) - 2023 - Springer Verlag.
    Systems of differential equations are used to describe, model, explain, and predict states of physical systems. Experimental and theoretical branches of physics including general relativity, climate science, and particle physics have differential equations at their center. Direct solutions to differential equations are not available in many domains, which spurs on the use of creative mathematics and simulated solutions.
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  8. El Axioma de elección en el quehacer matemático contemporáneo.Franklin Galindo & Randy Alzate - 2022 - Aitías 2 (3):49-126.
    Para matemáticos interesados en problemas de fundamentos, lógico-matemáticos y filósofos de la matemática, el axioma de elección es centro obligado de reflexión, pues ha sido considerado esencial en el debate dentro de las posiciones consideradas clásicas en filosofía de la matemática (intuicionismo, formalismo, logicismo, platonismo), pero también ha tenido una presencia fundamental para el desarrollo de la matemática y metamatemática contemporánea. Desde una posición que privilegia el quehacer matemático, nos proponemos mostrar los aportes que ha tenido el axioma en varias (...)
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  9. Introduction to the n-SuperHyperGraph - the most general form of graph today.Florentin Smarandache - 2022 - Neutrosophic Sets and Systems 48 (1):483-485.
    We recall and improve our 2019 and 2020 concepts of n-SuperHyperGraph, Plithogenic nSuperHyperGraph, n-Power Set of a Set, and we present some application from the real world. The nSuperHyperGraph is the most general form of graph today and it is able to describe the complex reality we live in, by using n-SuperVertices (groups of groups of groups etc.) and nSuperHyperEdges (edges connecting groups of groups of groups etc.).
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  10. On the correctness of problem solving in ancient mathematical procedure texts.Mario Bacelar Valente - 2020 - Revista de Humanidades de Valparaíso 16:169-189.
    It has been argued in relation to Old Babylonian mathematical procedure texts that their validity or correctness is self-evident. One “sees” that the procedure is correct without it having, or being accompanied by, any explicit arguments for the correctness of the procedure. Even when agreeing with this view, one might still ask about how is the correctness of a procedure articulated? In this work, we present an articulation of the correctness of ancient Egyptian and Old Babylonian mathematical procedure texts – (...)
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  11. Un teorema sobre el Modelo de Solovay.Franklin Galindo - 2020 - Divulgaciones Matematicas 21 (1-2): 42–46.
    The objective of this article is to present an original proof of the following theorem: Thereis a generic extension of the Solovay’s model L(R) where there is a linear order of P(N)/fin that extends to the partial order (P(N)/f in), ≤*). Linear orders of P(N)/fin are important because, among other reasons, they allow constructing non-measurable sets, moreover they are applied in Ramsey's Theory .
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  12. Tópicos de Ultrafiltros.Franklin Galindo - 2020 - Divulgaciones Matematicas 21 (1-2):54-77.
    Ultrafilters are very important mathematical objects in mathematical research [6, 22, 23]. There are a wide variety of classical theorems in various branches of mathematics where ultrafilters are applied in their proof, and other classical theorems that deal directly with ultrafilters. The objective of this article is to contribute (in a divulgative way) to ultrafilter research by describing the demonstrations of some such theorems related (uniquely or in combination) to topology, Measure Theory, algebra, combinatorial infinite, set theory and first-order logic, (...)
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  13. Algunas notas introductorias sobre la Teoría de Conjuntos.Franklin Galindo - 2019 - Apuntes Filosóficos: Revista Semestral de la Escuela de Filosofía 18 (55):201-232.
    The objective of this document is to present three introductory notes on set theory: The first note presents an overview of this discipline from its origins to the present, in the second note some considerations are made about the evaluation of reasoning applying the first-order Logic and Löwenheim's theorems, Church Indecidibility, Completeness and Incompleteness of Gödel, it is known that the axiomatic theories of most commonly used sets are written in a specific first-order language, that is, they are developed within (...)
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  14. 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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  15. Philosophy of Mathematics and Economics: Image, Context and Perspective.Thomas A. Boylan & Paschal F. O'Gorman - 2018 - Routledge.
    Economic methodology has been dominated by developments in the philosophy of science. This book's central thesis is that a great deal can be gained by refocusing attention on developments in the philosophy of mathematics, in particular those that took place over the course of the twentieth century. In this book the authors argue that a close examination of the major developments in the philosophy of mathematics both deepens and enriches our understanding of the formalisation of economics, while also offering novel (...)
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  16. Algebraic structures of neutrosophic triplets, neutrosophic duplets, or neutrosophic multisets. Volume I.Florentin Smarandache, Xiaohong Zhang & Mumtaz Ali - 2018 - Basel, Switzerland: MDPI. Edited by Florentin Smarandache, Xiaohong Zhang & Mumtaz Ali.
    The topics approached in the 52 papers included in this book are: neutrosophic sets; neutrosophic logic; generalized neutrosophic set; neutrosophic rough set; multigranulation neutrosophic rough set (MNRS); neutrosophic cubic sets; triangular fuzzy neutrosophic sets (TFNSs); probabilistic single-valued (interval) neutrosophic hesitant fuzzy set; neutro-homomorphism; neutrosophic computation; quantum computation; neutrosophic association rule; data mining; big data; oracle Turing machines; recursive enumerability; oracle computation; interval number; dependent degree; possibility degree; power aggregation operators; multi-criteria group decision-making (MCGDM); expert set; soft sets; LA-semihypergroups; single valued (...)
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  17. Why Axiomatize?Mario Bunge - 2017 - Foundations of Science 22 (4):695-707.
    Axiomatization is uncommon outside mathematics, partly for being often viewed as embalming, partly because the best-known axiomatizations have serious shortcomings, and partly because it has had only one eminent champion, namely David Hilbert. The aims of this paper are to describe what will be called dual axiomatics, for it concerns not just the formalism, but also the meaning of the key concepts; and to suggest that every instance of dual axiomatics presupposes some philosophical view or other. To illustrate these points, (...)
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  18. Discrete and continuous: a fundamental dichotomy in mathematics.James Franklin - 2017 - Journal of Humanistic Mathematics 7 (2):355-378.
    The distinction between the discrete and the continuous lies at the heart of mathematics. Discrete mathematics (arithmetic, algebra, combinatorics, graph theory, cryptography, logic) has a set of concepts, techniques, and application areas largely distinct from continuous mathematics (traditional geometry, calculus, most of functional analysis, differential equations, topology). The interaction between the two – for example in computer models of continuous systems such as fluid flow – is a central issue in the applicable mathematics of the last hundred years. This article (...)
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  19. High School Teachers' Level of Knowledge and Skills in Applying Mathematics in Real Life: An Assessment towards Teacher's Preparedness for Senior High School Mathematics.Melanie Gurat, Dominga Valtoribio, Cesar Medula Jr & Rommel de Gracia - 2017 - Uhd-Ctu Annual Economicsand Business Conference Proceedings -2017 1 (1):168-179.
    It is the role of the teacher to teach the students how to develop basic competencies and skills in solving, creative and critical thinkin. Thus, it is the teacher that is considered as one of the factors that could affect the students. This study measured high school teacher’s knowledge and skills in applying mathematics in real life using standardized questionnaires. Result revealed that the teachers of schools that will offer academic track have moderate knowledge and below level 1 skill in (...)
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  20. Philosophy of the Matrix.A. C. Paseau - 2017 - Philosophia Mathematica 25 (2):246-267.
    A mathematical matrix is usually defined as a two-dimensional array of scalars. And yet, as I explain, matrices are not in fact two-dimensional arrays. So are we to conclude that matrices do not exist? I show how to resolve the puzzle, for both contemporary and older mathematics. The solution generalises to the interpretation of all mathematical discourse. The paper as a whole attempts to reinforce mathematical structuralism by reflecting on how best to interpret mathematics.
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  21. Oppositions and Paradoxes: Philosophical Perplexities in Science and Mathematics.John L. Bell - 2016 - Peterborough, Ontario, Canada: Broadview Press.
    Since antiquity, opposed concepts such as the One and the Many, the Finite and the Infinite, and the Absolute and the Relative, have been a driving force in philosophical, scientific, and mathematical thought. Yet they have also given rise to perplexing problems and conceptual paradoxes which continue to haunt scientists and philosophers. In _Oppositions and Paradoxes_, John L. Bell explains and investigates the paradoxes and puzzles that arise out of conceptual oppositions in physics and mathematics. In the process, Bell not (...)
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  22. Identity in Homotopy Type Theory: Part II, The Conceptual and Philosophical Status of Identity in HoTT.James Ladyman & Stuart Presnell - 2016 - Philosophia Mathematica:nkw023.
  23. Solving Prior’s Problem with a Priorean Tool.Martin Pleitz - 2016 - Synthese 193 (11):3567-3577.
    I will show how a metaphysical problem of Arthur Prior’s can be solved by a logical tool he developed himself, but did not put to any foundational use: metric logic. The broader context is given by the key question about the metaphysics of time: Is time tenseless, i.e., is time just a structure of instants; or is time tensed, because some facts are irreducibly tensed? I take sides with Prior and the tensed theory. Like him, I therefore I have to (...)
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  24. The nature of the topological intuition.L. B. Sultanova - 2016 - Liberal Arts in Russia 5 (1):14.
    The article is devoted to the nature of the topological intuition and disclosure of the specifics of topological heuristics in the framework of philosophical theory of knowledge. As we know, intuition is a one of the support categories of the theory of knowledge, the driving force of scientific research. Great importance is mathematical intuition for the solution of non-standard problems, for which there is no algorithm for such a solution. In such cases, the mathematician addresses the so-called heuristics, built on (...)
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  25. The History of Mathematical Proof in Ancient Traditions.Jochen Brüning - 2015 - Common Knowledge 21 (3):524-525.
  26. Christopher Hollings. Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups. xi + 441 pp., figs., tables, notes, app., bibl., index. Providence, R.I.: American Mathematical Society, 2014. $109. [REVIEW]Emily Redman - 2015 - Isis 106 (4):980-981.
  27. Frege Meets Aristotle: Points as Abstracts.Stewart Shapiro & Geoffrey Hellman - 2015 - Philosophia Mathematica:nkv021.
    There are a number of regions-based accounts of space/time, due to Whitehead, Roeper, Menger, Tarski, the present authors, and others. They all follow the Aristotelian theme that continua are not composed of points: each region has a proper part. The purpose of this note is to show how to recapture ‘points’ in such frameworks via Scottish neo-logicist abstraction principles. The results recapitulate some Aristotelian themes. A second agenda is to provide a new arena to help decide what is at stake (...)
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  28. Berkeleys Kritik am Leibniz´schen calculus.Horst Struve, Eva Müller-Hill & Ingo Witzke - 2015 - Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 46 (1):63-82.
    One of the most famous critiques of the Leibnitian calculus is contained in the essay “The Analyst” written by George Berkeley in 1734. His key argument is those on compensating errors. In this article, we reconstruct Berkeley's argument from a systematical point of view showing that the argument is neither circular nor trivial, as some modern historians think. In spite of this well-founded argument, the critique of Berkeley is with respect to the calculus not a fundamental one. Nevertheless, it highlights (...)
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  29. The Parable of the Three Rings.Claudio Tugnoli - 2015 - Philosophy Study 5 (3).
  30. Interpolation property and homogeneous structures.Z. Gyenis - 2014 - Logic Journal of the IGPL 22 (4):597-607.
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  31. Solving ordinary differential equations by working with infinitesimals numerically on the Infinity Computer.Yaroslav Sergeyev - 2013 - Applied Mathematics and Computation 219 (22):10668–10681.
    There exists a huge number of numerical methods that iteratively construct approximations to the solution y(x) of an ordinary differential equation (ODE) y′(x) = f(x,y) starting from an initial value y_0=y(x_0) and using a finite approximation step h that influences the accuracy of the obtained approximation. In this paper, a new framework for solving ODEs is presented for a new kind of a computer – the Infinity Computer (it has been patented and its working prototype exists). The new computer is (...)
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  32. How to Read and Do Proofs: An Introduction to Mathematical Thought Processes.Daniel Solow - 2013 - Hoboken, New Jersey: Wiley.
    This text makes a great supplement and provides a systematic approach for teaching undergraduate and graduate students how to read, understand, think about, and do proofs. The approach is to categorize, identify, and explain (at the student's level) the various techniques that are used repeatedly in all proofs, regardless of the subject in which the proofs arise. How to Read and Do Proofs also explains when each technique is likely to be used, based on certain key words that appear in (...)
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  33. Reasoning About Truth in First-Order Logic.Claes Strannegård, Fredrik Engström, Abdul Rahim Nizamani & Lance Rips - 2013 - Journal of Logic, Language and Information 22 (1):115-137.
    First, we describe a psychological experiment in which the participants were asked to determine whether sentences of first-order logic were true or false in finite graphs. Second, we define two proof systems for reasoning about truth and falsity in first-order logic. These proof systems feature explicit models of cognitive resources such as declarative memory, procedural memory, working memory, and sensory memory. Third, we describe a computer program that is used to find the smallest proofs in the aforementioned proof systems when (...)
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  34. Representing Popov v Hayashi with dimensions and factors.T. J. M. Bench-Capon - 2012 - Artificial Intelligence and Law 20 (1):15-35.
    Modelling reasoning with legal cases has been a central concern of AI and Law since the 1980s. The approach which represents cases as factors and dimensions has been a central part of that work. In this paper I consider how several varieties of the approach can be applied to the interesting case of Popov v Hayashi. After briefly reviewing some of the key landmarks of the approach, the case is represented in terms of factors and dimensions, and further explored using (...)
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  35. What is The Reason to Use Clifford Algebra in Quantum Cognition? Part I: “It from Qubit” On The Possibility That the Amino Acids Can Discern Between Two Quantum Spin States.Elio Conte - 2012 - Neuroquantology 10 (3):561-565.
    Starting with 1985, we discovered the possible existence of electrons with net helicity in biomolecules as amino acids and their possibility to discern between the two quantum spin states. It is well known that the question of a possible fundamental role of quantum mechanics in biological matter constitutes still a long debate. In the last ten years we have given a rather complete quantum mechanical elaboration entirely based on Clifford algebra whose basic entities are isomorphic to the well known spin (...)
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  36. Rhetorical uses of mathematical harmonics in Philo and Plutarch.David Creese - 2012 - Studies in History and Philosophy of Science Part A 43 (2):258-269.
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  37. Husserl on Geometry and Spatial Representation.Jairo José da Silva - 2012 - Axiomathes 22 (1):5-30.
    Husserl left many unpublished drafts explaining (or trying to) his views on spatial representation and geometry, such as, particularly, those collected in the second part of Studien zur Arithmetik und Geometrie (Hua XXI), but no completely articulate work on the subject. In this paper, I put forward an interpretation of what those views might have been. Husserl, I claim, distinguished among different conceptions of space, the space of perception (constituted from sensorial data by intentionally motivated psychic functions), that of physical (...)
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  38. On some putative graph-theoretic counterexamples to the Principle of the Identity of Indiscernibles.Rafael De Clercq - 2012 - Synthese 187 (2):661-672.
    Recently, several authors have claimed to have found graph-theoretic counterexamples to the Principle of the Identity of Indiscernibles. In this paper, I argue that their counterexamples presuppose a certain view of what unlabeled graphs are, and that this view is optional at best.
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  39. Simplicius on Tekmeriodic Proofs.Orna Harari - 2012 - Studies in History and Philosophy of Science Part A 43 (2):366-375.
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  40. Husserl on Geometry and Spatial Representation.Jairo José Silva - 2012 - Axiomathes 22 (1):5-30.
    Husserl left many unpublished drafts explaining (or trying to) his views on spatial representation and geometry, such as, particularly, those collected in the second part of Studien zur Arithmetik und Geometrie (Hua XXI), but no completely articulate work on the subject. In this paper, I put forward an interpretation of what those views might have been. Husserl, I claim, distinguished among different conceptions of space, the space of perception (constituted from sensorial data by intentionally motivated psychic functions), that of physical (...)
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  41. Dual Choice and Iteration in an Abstract Algebra of Action.Kim Solin - 2012 - Studia Logica 100 (3):607-630.
    This paper presents an abstract-algebraic formulation of action facilitating reasoning about two opposing agents. Two dual nondeterministic choice operators are formulated abstract-algebraically: angelic (or user) choice and demonic (or system) choice. Iteration operators are also defined. As an application, Hoare-style correctness rules are established by means of the algebra. A negation operator is also discussed.
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  42. Set Theory and Logic.Robert Roth Stoll - 2012 - San Francisco and London: Courier Corporation.
    Explores sets and relations, the natural number sequence and its generalization, extension of natural numbers to real numbers, logic, informal axiomatic mathematics, Boolean algebras, informal axiomatic set theory, several algebraic theories, and 1st-order theories.
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  43. Theory of Computation.George Tourlakis - 2012 - Hoboken, N.J.: Wiley.
    In addition, this book contains tools that, in principle, can search a set of algorithms to see whether a problem is solvable, or more specifically, if it can be solved by an algorithm whose computations are efficient.
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  44. Introduction to mathematical logic.Micha? Walicki - 2012 - Hackensack, NJ: World Scientific.
    A history of logic -- Patterns of reasoning -- A language and its meaning -- A symbolic language -- 1850-1950 mathematical logic -- Modern symbolic logic -- Elements of set theory -- Sets, functions, relations -- Induction -- Turning machines -- Computability and decidability -- Propositional logic -- Syntax and proof systems -- Semantics of PL -- Soundness and completeness -- First order logic -- Syntax and proof systems of FOL -- Semantics of FOL -- More semantics -- Soundness and (...)
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  45. Francesco Berto. There's Something about Gödel. Malden, Mass., and Oxford: Wiley-Blackwell, 2009. ISBN 978-1-4051-9766-3 ; 978-1-4051-9767-0 . Pp. xx + 233. English translation of Tutti pazzi per Gödel! : Critical Studies/Book Reviews. [REVIEW]Vann Mcgee - 2011 - Philosophia Mathematica 19 (3):367-369.
    There's Something about Gödel is a bargain: two books in one. The first half is a gentle but rigorous introduction to the incompleteness theorems for the mathematically uninitiated. The second is a survey of the philosophical, psychological, and sociological consequences people have attempted to derive from the theorems, some of them quite fantastical.The first part, which stays close to Gödel's original proofs, strikes a nice balance, giving enough details that the reader understands what is going on in the proofs, without (...)
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  46. Thermoscopes, thermometers, and the foundations of measurement.David Sherry - 2011 - Studies in History and Philosophy of Science Part A 42 (4):509-524.
    Psychologists debate whether mental attributes can be quantified or whether they admit only qualitative comparisons of more and less. Their disagreement is not merely terminological, for it bears upon the permissibility of various statistical techniques. This article contributes to the discussion in two stages. First it explains how temperature, which was originally a qualitative concept, came to occupy its position as an unquestionably quantitative concept (§§1–4). Specifically, it lays out the circumstances in which thermometers, which register quantitative (or cardinal) differences, (...)
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  47. Introduction to proof in abstract mathematics.Andrew Wohlgemuth - 2011 - Mineola, N.Y.: Dover Publications.
    Originally published: Philadelphia: Saunders College Pub., c1990.
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  48. Proceedings of International Conference on Challenges and Applications of Mathematics in Science and Technology: CAMIST, January 11-13, 2010.Snehashish Chakraverty (ed.) - 2010 - Delhi: Macmillan Publishers India.
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  49. Perfect set properties in models of ZF.Franklin Galindo & Carlos Di Prisco - 2010 - Fundamenta Mathematicae 208 (208):249-262.
    We study several perfect set properties of the Baire space which follow from the Ramsey property ω→(ω) ω . In particular we present some independence results which complete the picture of how these perfect set properties relate to each other.
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  50. The Gödel hierarchy and reverse mathematics.Stephen G. Simpson - 2010 - In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: Essays for His Centennial. Association for Symbolic Logic.
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1 — 50 / 172