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  • Philosophy (General)
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  • 1
    Online Resource
    Online Resource
    Cham : Springer International Publishing
    ISBN: 9783319034522
    Language: English
    Pages: Online-Ressource (XVIII, 220 p, online resource)
    Series Statement: Synthese Library, Studies in Epistemology, Logic, Methodology, and Philosophy of Science 370
    Series Statement: SpringerLink
    Series Statement: Bücher
    Parallel Title: Druckausg. Montano, Ulianov Explaining beauty in mathematics: an aesthetic theory of mathematics
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    Keywords: Philosophy (General) ; Aesthetics ; Logic ; Logic, Symbolic and mathematical ; Philosophy ; Philosophy (General) ; Aesthetics ; Logic ; Logic, Symbolic and mathematical ; Schönheit ; Mathematik ; Ästhetik
    Abstract: This book develops a naturalistic aesthetic theory that accounts for aesthetic phenomena in mathematics in the same terms as it accounts for more traditional aesthetic phenomena. Building upon a view advanced by James McAllister, the assertion is that beauty in science does not confine itself to anecdotes or personal idiosyncrasies, but rather that it had played a role in shaping the development of science. Mathematicians often evaluate certain pieces of mathematics using words like beautiful, elegant, or even ugly. Such evaluations are prevalent, however, rigorous investigation of them, of mathematical beauty, is much less common. The volume integrates the basic elements of aesthetics, as it has been developed over the last 200 years, with recent findings in neuropsychology as well as a good knowledge of mathematics. The volume begins with a discussion of the reasons to interpret mathematical beauty in a literal or non-literal fashion, which also serves to survey historical and contemporary approaches to mathematical beauty. The author concludes that literal approaches are much more coherent and fruitful, however, much is yet to be done. In this respect two chapters are devoted to the revision and improvement of McAllister’s theory of the role of beauty in science. These antecedents are used as a foundation to formulate a naturalistic aesthetic theory. The central idea of the theory is that aesthetic phenomena should be seen as constituting a complex dynamical system which the author calls the aesthetic as process theory. The theory comprises explications of three central topics: aesthetic experience (in mathematics), aesthetic value and aesthetic judgment. The theory is applied in the final part of the volume and is used to account for the three most salient and often used aesthetic terms often used in mathematics: beautiful, elegant and ugly. This application of the theory serves to illustrate the theory in action, but also to further discuss and develop some details and to showcase the theory’s explanatory capabilities
    Description / Table of Contents: IntroductionPart 1. Antecedents -- Chapter 1. On Non-literal Approaches -- Chapter 2. Beautiful, Literally -- Chapter 3. Ugly, Literally -- Chapter 4. Problems of the Aesthetic Induction -- Chapter 5. Naturalizing the Aesthetic Induction -- Part 2. An Aesthetics of Mathematics -- Chapter 6. Introduction to a Naturalistic Aesthetic Theory -- Chapter 7. Aesthetic Experience -- Chapter 8. Aesthetic Value -- Chapter 9. Aesthetic Judgement I: Concept -- Chapter 10. Aesthetic Judgement II: Functions -- Chapter 11. Mathematical Aesthetic Judgements -- Part 3. Applications -- Chapter 12. Case Analysis I: Beauty -- Chapter 13. Case Analysis II: Elegance -- Chapter 14. Case Analysis III: Ugliness, Revisited -- Chapter 15. Issues of Mathematical Beauty, Revisited.
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  • 2
    Online Resource
    Online Resource
    Dordrecht : Springer Netherlands
    ISBN: 9789400770584
    Language: English
    Pages: Online-Ressource (XVII, 291 p. 16 illus, online resource)
    Series Statement: Logic, Epistemology, and the Unity of Science 32
    Series Statement: SpringerLink
    Series Statement: Bücher
    Parallel Title: Druckausg. Friend, Michèle Pluralism in mathematics
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    Keywords: Philosophy (General) ; Logic ; Science Philosophy ; Logic, Symbolic and mathematical ; Philosophy ; Philosophy (General) ; Logic ; Science Philosophy ; Logic, Symbolic and mathematical ; Logic ; Logic, Symbolic and mathematical ; Philosophy ; Philosophy (General) ; Science Philosophy ; Pluralismus ; Mathematik
    Abstract: This book is about philosophy, mathematics and logic, giving a philosophical account of Pluralism which is a family of positions in the philosophy of mathematics. There are four parts to this book, beginning with a look at motivations for Pluralism by way of Realism, Maddy’s Naturalism, Shapiro’s Structuralism and Formalism. In the second part of this book the author covers: the philosophical presentation of Pluralism; using a formal theory of logic metaphorically; rigour and proof for the Pluralist; and mathematical fixtures. In the third part the author goes on to focus on the transcendental presentation of Pluralism, and in part four looks at applications of Pluralism, such as a Pluralist approach to proof in mathematics and how Pluralism works in regard to together-inconsistent philosophies of mathematics. The book finishes with suggestions for further Pluralist enquiry. In this work the author takes a deeply radical approach in developing a new position that will either convert readers, or act as a strong warning to treat the word ‘pluralism’ with care.
    Description / Table of Contents: IntroductionPart I. Motivating the Pluralist Position from Familiar Positions -- Chapter 1. Introduction. The Journey from Realism to Pluralism -- Chapter 2. Motivating Pluralism. Starting from Maddy’s Naturalism -- Chapter 3. From Structuralism to Pluralism -- Chapter 4. Formalism and Pluralism Co-written with Andrea Pedeferri -- Part II. Initial Presentation of Pluralism.- Chapter 5. Philosophical Presentation of Pluralism -- Chapter 6. Using a Formal Theory of Logic Metaphorically -- Chapter 7. Rigour in Proof Co-written with Andrea Pedeferri -- Chapter 8. Mathematical Fixtures -- Part III. Transcendental Presentation of Pluralism -- Chapter 9. The Paradoxes of Tolerance and the Transcendental Paradoxes -- Chapter 10. Pluralism Towards Pluralism -- Part IV. Putting Pluralism to Work. Applications -- Chapter 11. A Pluralist Approach to Proof in Mathematics -- Chapter 12. Pluralism and Together-Inconsistent Philosophies of Mathematics -- Chapter 13. Suggestions for Further Pluralist Enquiry -- Conclusion.
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  • 3
    Online Resource
    Online Resource
    Dordrecht : Springer
    ISBN: 9789400760912
    Language: English
    Pages: Online-Ressource (XV, 389 p. 35 illus, online resource)
    Series Statement: Logic, Argumentation & Reasoning, Interdisciplinary Perspectives from the Humanities and Social Sciences 1
    Series Statement: SpringerLink
    Series Statement: Bücher
    Parallel Title: Druckausg. Cellucci, Carlo, 1940 - Rethinking logic
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    Keywords: Philosophy (General) ; Logic ; Computer science ; Philosophy ; Philosophy (General) ; Logic ; Computer science ; Computer science ; Logic ; Philosophy ; Philosophy (General) ; Logik ; Interdisziplinäre Forschung
    Abstract: This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice. Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically
    Description / Table of Contents: PrefaceChapter 1. Introduction -- Part I. Ancient Perspectives -- Chapter 2. The Origin of Logic -- Chapter 3. Ancient Logic and Science -- Chapter 4. The Analytic Method -- Chapter 5. The Analytic-Synthetic Method -- Chapter 6. Aristotle's Logic: The Deductivist View -- Chapter 7. Aristotle's Logic: The Heuristic View -- Part II. Modern Perspectives -- Chapter 8. The Method of Modern Science -- Chapter 9. The Quest for a Logic of Discovery -- Chapter 10. Frege's Approach to Logic -- Chapter 11. Gentzen's Approach to Logic -- Chapter 12. The Limitations of Mathematical Logic -- Chapter 13. Logic, Method, and the Psychology of Discovery -- Part III: An Alternative Perspective -- Chapter 14. Reason and Knowledge -- Chapter 15. Reason, Knowledge and Emotion -- Chapter 16. Logic, Evolution, Language and Reason -- Chapter 17. Logic, Method and Knowledge -- Chapter 18. Classifying and Justifying Inference Rules -- Chapter 19. Philosophy and Knowledge -- Part IV: Rules of Discovery -- Chapter 20. Induction and Analogy -- Chapter 21. Other Rules of Discovery -- Chapter 22. Conclusion -- References -- Name Index -- Subject Index.
    Note: Includes bibliographies and index
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  • 4
    Online Resource
    Online Resource
    Dordrecht : Springer
    ISBN: 9789400753570 , 1283936097 , 9781283936095
    Language: English
    Pages: Online-Ressource (VIII, 215 p. 23 illus, digital)
    Series Statement: Synthese Library, Studies in Epistemology, Logic, Methodology, and Philosophy of Science 362
    Series Statement: SpringerLink
    Series Statement: Bücher
    Parallel Title: Buchausg. u.d.T. Bayesian argumentation
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    Keywords: Philosophy (General) ; Genetic epistemology ; Computer simulation ; Applied linguistics ; Social sciences Methodology ; Applied psychology ; Philosophy ; Philosophy (General) ; Genetic epistemology ; Computer simulation ; Applied linguistics ; Social sciences Methodology ; Applied psychology ; Reasoning (Psychology) ; Congresses ; Logic ; Congresses ; Thought and thinking ; Congresses ; Probabilities ; Congresses ; Bayesian statistical decision theory ; Congresses ; Konferenzschrift ; Argumentationstheorie ; Bayes-Entscheidungstheorie
    Abstract: Relevant to, and drawing from, a range of disciplines, the chapters in this collection show the diversity, and applicability, of research in Bayesian argumentation. Together, they form a challenge to philosophers versed in both the use and criticism of Bayesian models who have largely overlooked their potential in argumentation. Selected from contributions to a multidisciplinary workshop on the topic held in Lund, Sweden, in autumn 2010, the authors count legal scholars and cognitive scientists among their number, in addition to philosophers. They analyze material that includes real-life court cases, experimental research results, and the insights gained from computer models.The volume provides a formal measure of subjective argument strength and argument force, robust enough to allow advocates of opposing sides of an argument to agree on the relative strengths of their supporting reasoning. With papers from leading figures such as Mike Oaksford and Ulrike Hahn, the book comprises recent research conducted at the frontiers of Bayesian argumentation and provides a multitude of examples in which these formal tools can be applied to informal argument. It signals new and impending developments in philosophy, which has seen Bayesian models deployed in formal epistemology and philosophy of science, but has yet to explore the full potential of Bayesian models as a framework in argumentation. In doing so, this revealing anthology looks destined to become a standard teaching text in years to come.
    Description / Table of Contents: Bayesian Argumentation; Foreword; Contents; Bayesian Argumentation: The Practical Side of Probability; 1 Introduction; 2 The Bayesian Approach to Argumentation; 3 Chapter Overview; 3.1 The Bayesian Approach to Argumentation; 3.2 The Legal Domain; 3.3 Modeling Rational Agents; 3.4 Theoretical Issues; References; Part I: The Bayesian Approach to Argumentation; Testimony and Argument: A Bayesian Perspective; 1 Introduction; 2 Testimony, Argumentation and the `Third Way´; 3 Some Problems for MAXMIN; 4 A Bayesian Perspective; 5 Message Content and Message Source: Exploring Norms and Intuitions
    Description / Table of Contents: 6 Rehousing Argumentation Schemes Within a Bayesian Framework7 Concluding Remarks; References; Why Are We Convinced by the Ad Hominem Argument?: Bayesian Source Reliability and Pragma-Dialectical Discussion Rules; 1 Types of the Argumentum Ad Hominem; 2 The Pragma-Dialectical Approach; 3 The Bayesian Approach; 4 An Experiment on the Argument Ad Hominem; 5 Method; 6 Results and Discussion; 7 Conclusion; Appendix: Experimental Materials; Abusive; Circumstantial; Tu Quoque; Control; References; 1 Introduction; 2 Survey of Relevant Uncertainties; Part II: The Legal Domain
    Description / Table of Contents: A Survey of Uncertainties and Their Consequences in Probabilistic Legal Argumentation2.1 The Example Case; 2.2 Factual Uncertainty; 2.3 Normative Uncertainty; 2.4 Moral Uncertainty; 2.5 Empirical Uncertainty; 2.6 Interdependencies; 3 Desirable Attributes for a Probabilistic Argument Model to Assist Litigation Planning; 3.1 Assessment of Utilities; 3.2 Easy Knowledge Engineering; 3.3 Conflict Resolution and Argument Weights; 4 Sample Assessment of Graphical Models; 4.1 A Graphical Structure of the Analysis; 4.2 Casting the Example into a Graphical Model; 4.3 Generic Bayesian Networks
    Description / Table of Contents: 5 Carneades5.1 A Brief Introduction to the Carneades Model; 5.2 Carneades Bayesian Networks; 5.3 Carneades Bayesian Networks with Probabilistic Assumptions; 5.4 Introduction to Argument Weights; 6 Extension of Carneades to Support Probabilistic Argument Weights; 7 Desiderata for Future Developments; 7.1 Weights Subject to Argumentation; 7.2 Inform Weights from Values; 8 Conclusions and Future Work; References; Was It Wrong to Use Statistics in R v Clark? A Case Study of the Use of Statistical Evidence in Criminal Courts; 1 Introduction; 2 Factual Background; 3 Existing Explanations
    Description / Table of Contents: 3.1 The Flaws in Meadow´s Calculation3.2 The Psychological Effect of the Statistical Evidence; 3.3 The Prosecutor´s Fallacy; 3.4 Bayes´ Theorem; 3.5 The Insignificance of the SIDS Statistics; 4 The Contrastive Explanation; 5 Conclusion; References; Part III: Modeling Rational Agents; A Bayesian Simulation Model of Group Deliberation and Polarization; 1 Introduction; 2 The Laputa Simulation Framework; 3 The Underlying Bayesian Model; 4 Interpreting Laputa; 5 Do Bayesian Inquirers Polarize?; 6 Conclusion and Discussion; Appendix; References
    Description / Table of Contents: Degrees of Justification, Bayes´ Rule, and Rationality
    Description / Table of Contents: Introduction: Frank Zenker.​- Part 1 -- The Bayesian Approach to Argumentation -- Chapter 1. Testimony and Argument: A Bayesian Perspective: Ulrike Hahn, Mike Oaksford and Adam J.L. Harris -- Chapter 2. Why are we convinced by the Ad Hominem Argument?: Source Reliability or Pragma-Dialectics: Mike Oaksford and Ulrike Hahn.- Part 2. The Legal Domain.-Chapter 3. A survey of uncertainties and their consequences in Probabilistic Legal Argumentation: Matthias Grabmair and Kevin D. Ashley -- Chapter 4. What went wrong in the case of Sally Clark? A case-study of the use of Statistical Evidence in Court: Amid Pundik -- Part 3. Modeling Rational Agents -- Chapter 5. A Bayesian Simulation Model of Group Deliberation: Erik J. Olsson -- Chapter 6. Degrees of Justification, Bayes' Rule, and Rationality: Gregor Betz -- Chapter 7. Argumentation with (Bounded) Rational Agents: Robert van Rooij and Kris de Jaeghery -- Part 4. Theoretical Issues -- Chapter 8. Reductio, Coherence, and the Myth of Epistemic Circularity: Tomoji Shogenji -- Chapter 9. On Argument Strength: Niki Pfeiffer -- Chapter 10 -- Upping the Stakes and the Preface Paradox: Jonny Blamey -- References.​.
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  • 5
    Online Resource
    Online Resource
    Dordrecht : Springer Science+Business Media B.V
    ISBN: 9789048135424
    Language: English
    Pages: Online-Ressource (XX, 5p, digital)
    Additional Information: Rezensiert in Baldasso, Renzo [Rezension von: Goulding, Robert, Defending Hypatia: Ramus, Savile, and the Renaissance Rediscovery of Mathematical History] 2013
    Series Statement: Archimedes, New Studies in the History and Philosophy of Science and Technology 25
    Series Statement: SpringerLink
    Series Statement: Bücher
    Parallel Title: Buchausg. u.d.T. Goulding, Robert Defending Hypatia
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    Keywords: Philosophy (General) ; Science Philosophy ; Mathematics ; Humanities ; Philosophy ; Humanities ; Mathematics ; Philosophy (General) ; Science Philosophy ; Ramus, Petrus 1515-1572 ; Mathematikgeschichte ; Savile, Henry 1549-1622 ; Mathematikgeschichte ; Mathematik ; Geschichte
    Abstract: Why should mathematics, the purest of sciences, have a history? Medieval mathematicians took little interest in the history of their discipline. Yet in the Renaissance the history of mathematics flourished. This book explores how Renaissance scholars recovered and reconstructed the origins of mathematics by tracing its invention in prehistoric Antiquity, its development by the Greeks, and its transmission to modern Europe via the works of Euclid, Theon and Proclus. The principal architects of this story -- the French philosopher and University of Paris reformer Peter Ramus, and his critic, the young Oxford astronomy lecturer Henry Savile worked out diametrically opposed models for the development of the mathematical arts, models of historical progress and decline which mirrored each scholar's larger convictions about the nature of mathematical thinking, the purpose of the modern university, and the potential of the human mind. In their hands, the obscure story of mathematical history became a site of contention over some of the most pressing philosophical and pedagogical debates of the sixteenth century.
    Description / Table of Contents: Defending Hypatia; Acknowledgements; Contents; Introduction; Structure of the Book; Drowning by Numbers; 1 Lineages of Learning; 2 Ramus and the History of Mathematics; 3 From Plato to Pythagoras: The Scholae mathematicae; 4 ``To Bring Alexandria to Oxford:'' Henry Savile's 1570 Lectures on Ptolemy; 5 The Puzzling Lives of Euclid; 6 Rending Hypatia: The Body of the Elements; Contents of Savile's History of Mathematics; Evidence for the Extent of Savile's Lectures; References; Index;
    Note: Includes bibliographical references (p. 191-197) and index
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  • 6
    Online Resource
    Online Resource
    Dordrecht : Springer Science+Business Media B.V
    ISBN: 9789048137299 , 1282927612 , 9781282927612
    Language: English
    Pages: Online-Ressource (XXVII, 216p, digital)
    Series Statement: Phaenomenologica, Published Under the Auspices of the Husserl-Archives 195
    Series Statement: SpringerLink
    Series Statement: Bücher
    Parallel Title: Buchausg. u.d.T. Phenomenology and mathematics
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    Keywords: Philosophy (General) ; Logic ; Phenomenology ; Science Philosophy ; Mathematics_$xHistory ; Philosophy ; Philosophy (General) ; Logic ; Phenomenology ; Science Philosophy ; Mathematics_$xHistory ; Aufsatzsammlung ; Phänomenologie ; Mathematik ; Husserl, Edmund 1859-1938 ; Phänomenologie ; Mathematik ; Phänomenologie ; Mathematik ; Husserl, Edmund 1859-1938
    Abstract: During Edmund Husserl,s lifetime, modern logic and mathematics rapidly developed toward their current outlook and Husserl,s writings can be fruitfully compared and contrasted with both 19th century figures (Boole, Schroder, Weierstrass) as well as the 20th century characters (Heyting, Zermelo, Godel). Besides the more historical studies, the internal ones on Husserl alone and the external ones attempting to clarify his role in the more general context of the developing mathematics and logic, Husserl,s phenomenology offers also a systematically rich but little researched area of investigation
    Description / Table of Contents: PHENOMENOLOGY AND MATHEMATICS; Contents; Acknowledgements; Contributors; List of Abbreviations; Introduction; I Mathematical Realism and Transcendental Phenomenological Idealism; I. Standard Simple Formulations of Realism and Idealism (Anti-Realism) About Mathematics; I. Introduction; I. Introduction; II. Mathematical Realism; II. Benacerrafs Dilemma and Some Negative or Skeptical Solutions; II. R-Structured Wholes; III. Transcendental Phenomenological Idealism; IV. Mind-Independence and Mind-Dependence in Formulations of Mathematical Realism; IV. Meaningless Symbols in PA
    Description / Table of Contents: V. Compatibility or Incompatibility?V. Categorial Intuition; V. Logical Systems; III. Benacerrafs Dilemma and Kantian Structuralism; VI. Brief Interlude: Where to Place Gdel, Brouwer, and Other Mathematical Realists and Idealists in our Schematization?; VII. A Conclusion and an Introduction; VI. Imaginary Elements: Earlier Treatment; VII. Imaginary Elements: Later Treatment; IV. The HW Theory; V. Conclusion: Benacerrafs Dilemma Again and Recovered Paradise; References; II Platonism, Phenomenology, and Interderivability; I. Introduction; II. Phenomenology, Constructivism and Platonism
    Description / Table of Contents: III. InterderivabilityIV. Situations of Affairs: Historical Preliminaries; V. Situations of Affairs: Systematic Treatment; VI. Conclusion; VII. Appendix; References; III husserl on axiomatization andarithmetic; I. Introduction; II. Husserls Initial Opposition to the Axiomatization of Arithmetic; III. Husserls VOLTE-FACE Volte-Face; IV. Analysis of the Concept of Number; V. Calculating with Concepts and Propositions; VI. Three Levels of Logic; VII. Manifolds and Imaginary Numbers; VIII. Mathematics and Phenomenology; VIII. Formal Ontology; IX. What Numbers Could Not Be For Husserl
    Description / Table of Contents: IX. Critical ConsiderationsX. The Problem of Symbolic Knowledge in the Development of Husserls Philosophy; X. Conclusion; References; IV Intuition in Mathematics: on the Function of Eidetic Variation in Mathematical Proofs; I. Some Basic Features of Husserls Theory of Knowledge; II. The Method of Seeing Essences in Mathematical Proofs; 1. The Eidetic Method (Wesensschau) Used for Real Objects; 1. Pre-emptive Negative or Skeptical Solutions; 1. Preliminaries; 2. Eidetics in Material Mathematical Disciplines; 2. Concessive Negative or Skeptical Solutions; 2. The Part-of Relation
    Description / Table of Contents: 3. Eidetics in Formal-Axiomatic Contexts3. One Sort of Structured Wholes: R-Structured Wholes; References; V How Can a Phenomenologist Have a Philosophy of Mathematics?; References; VI The Development of Mathematics and the Birth of Phenomenology; I. Weierstrass and Mathematics as Rigorous Science; II. Husserl in Weierstrasss Footsteps; III. Philosophy of Arithmetic as an Analysis of the Concept of Number; IV. Logical Investigations and the Axiomatic Approach; VI. Aristotle or Plato (and Which Plato)?; VII. Platonism of the Eternal, Self-Identical, Unchanging Objectivities
    Description / Table of Contents: VIII. Platonism as an Aspiration for Reflected Foundations
    Note: Includes bibliographical references and index
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  • 7
    ISBN: 9781402093845
    Language: English
    Pages: Online-Ressource , v.: digital
    Edition: Online-Ausg. 2009 Springer eBook Collection. Humanities, Social Science and Law Electronic reproduction; Available via World Wide Web
    Series Statement: Logic, Epistemology, and the Unity of Science 14
    Parallel Title: Print version From a Geometrical Point of View
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    Keywords: Algebraic topology ; Logic, Symbolic and mathematical ; Science Algebra ; Mathematics_$xHistory ; Algebra ; Philosophy (General) ; Kategorientheorie ; Geschichte
    Abstract: From a Geometrical Point of View explores historical and philosophical aspects of category theory, trying therewith to expose its significance in the mathematical landscape. The main thesis is that Klein’s Erlangen program in geometry is in fact a particular instance of a general and broad phenomenon revealed by category theory. The volume starts with Eilenberg and Mac Lane’s work in the early 1940’s and follows the major developments of the theory from this perspective. Particular attention is paid to the philosophical elements involved in this development. The book ends with a presentation of categorical logic, some of its results and its significance in the foundations of mathematics. From a Geometrical Point of View aims to provide its readers with a conceptual perspective on category theory and categorical logic, in order to gain insight into their role and nature in contemporary mathematics. It should be of interest to mathematicians, logicians, philosophers of mathematics and science in general, historians of contemporary mathematics, physicists and computer scientists.
    Description / Table of Contents: Introduction; Category Theory and Klein's Erlangen Program; Introducing Categories, Functors and Natural Transformations; Categories as Spaces, Functors as Transformations; Discovering Fundamental Categorical Transformations: Adjoint Functors; Adjoint Functors: What They are, What They Mean; Invariants in Foundations: Algebraic Logic; Invariants in Foundations: Geometric Logic; Conclusion.
    Note: Includes bibliographical references and index , Electronic reproduction; Available via World Wide Web
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  • 8
    ISBN: 9781402086229
    Language: English
    Pages: Online-Ressource , v.: digital
    Edition: Online-Ausg. Springer eBook Collection. Humanities, Social Science and Law Electronic reproduction; Available via World Wide Web
    Series Statement: Trends in Logic, Studia Logica Library 27
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    Keywords: Algebra ; Computational complexity ; Data mining ; Genetic epistemology ; Logic ; Logic, Symbolic and mathematical ; Philosophy (General)
    Abstract: A Geometry of Approximation addresses Rough Set Theory, a field of interdisciplinary research first proposed by Zdzislaw Pawlak in 1982, and focuses mainly on its logic-algebraic interpretation. The theory is embedded in a broader perspective that includes logical and mathematical methodologies pertaining to the theory, as well as related epistemological issues. Any mathematical technique that is introduced in the book is preceded by logical and epistemological explanations. Intuitive justifications are also provided, insofar as possible, so that the general perspective is not lost. Such an approach endows the present treatise with a unique character. Due to this uniqueness in the treatment of the subject, the book will be useful to researchers, graduate and pre-graduate students from various disciplines, such as computer science, mathematics and philosophy. It features an impressive number of examples supported by about 40 tables and 230 figures. The comprehensive index of concepts turns the book into a sort of encyclopaedia for researchers from a number of fields. A Geometry of Approximation links many areas of academic pursuit without losing track of its focal point, Rough Sets.
    Description / Table of Contents: Preface; CONTENTS; List of Figures; Notation; Abbreviations; Introduction; 1 Perception and Concepts: A Phenomenological Approach; 2 Monological Approach to Perception and Concepts; 3 Phenomenology and Logic; 4 The Logico-Algebraic Interpretation of Rough Set Systems; 5 Equivalence Classes, Abstraction and Meaning; 6 Rough Sets and Logic; 7 Concluding Remarks; I A Mathematics of Perception; 1 Observations, Noumena and Phenomena; 2 Concrete and Formal Information Constructions; 3 Pre-Topological and Topological Approximation Operators; 4 Frames (Part I)
    Description / Table of Contents: II The Logico-Algebraic Theory of Rough Sets5 Logic and Rough Sets: An Overview; 6 Basic Logico-Algebraic Structures; 7 Local Validity, Grothendieck Topologies and Rough Sets; 8 Approximation and Algebraic Logic; 9 A Logico-Philosophic Excursus; 10 Frames (Part II); III The Modal Logic of Rough Sets; 11 Modality and Knowledge; 12 Modalities and Relations; 13 Modalities, Topologies and Algebras; 14 The Propositional Modal Logic of Rough Sets; 15 Frames (Part III); 16 Mathematical Toolkits; Bibliography; Index
    Note: Includes bibliographical references and index , Electronic reproduction; Available via World Wide Web
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  • 9
    Online Resource
    Online Resource
    Dordrecht : Springer Netherlands
    ISBN: 9781402050879
    Language: English
    Pages: Online-Ressource (XIII, 206 p, online resource)
    Series Statement: Synthese Library 335
    Series Statement: SpringerLink
    Series Statement: Bücher
    Parallel Title: Druckausg. Atten, Mark van, 1973 - Brouwer meets Husserl
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    Keywords: Mathematics ; Metaphysics ; Ontology ; Phenomenology ; Mathematical logic ; Metaphysics ; Ontology ; Phenomenology ; Philosophy (General) ; Logic, Symbolic and mathematical ; Phänomenologie ; Wahlfolge ; Brouwer, Luitzen E. J. 1881-1966 ; Wahlfolge ; Phänomenologie ; Intuitionistische Mathematik ; Husserl, Edmund 1859-1938 ; Brouwer, Luitzen E. J. 1881-1966 ; Husserl, Edmund 1859-1938
    Abstract: An Informal Introduction -- The Argument -- The Original Positions -- The Phenomenological Incorrectness of the Original Arguments -- The Constitution of Choice Sequences -- Application: An Argument for Weak Continuity -- Concluding Remarks.
    Abstract: Can the straight line be analysed mathematically such that it does not fall apart into a set of discrete points, as is usually done but through which its fundamental continuity is lost? And are there objects of pure mathematics that can change through time? The mathematician and philosopher L.E.J. Brouwer argued that the two questions are closely related and that the answer to both is "yes''. To this end he introduced a new kind of object into mathematics, the choice sequence. But other mathematicians and philosophers have been voicing objections to choice sequences from the start. This book aims to provide a sound philosophical basis for Brouwer's choice sequences by subjecting them to a phenomenological critique in the style of the later Husserl. "It is almost as if one could hear the two rebels arguing their case in a European café or on a terrace, and coming to a common understanding, with both men taking their hat off to the other, in admiration and gratitude. Dr. van Atten has convincingly applied Husserl's method to Brouwer's program, and has equally convincingly applied Brouwer's intuition to Husserl's program. Both programs have come out the better." Piet Hut, professor of Interdisciplinary Studies, Institute for Advanced Study, Princeton, U.S.A.
    Description / Table of Contents: CONTENTS; Preface; Acknowledgements; 1 An Informal Introduction; 2 Introduction; 2.1 The Aim; 2.2 The Thesis; 2.3 Motivation; 2.4 Method, and an Assumption; 2.5 The Literature; 3 The Argument; 3.1 Presentation; 3.2 Comments; 4 The Original Positions; 4.1 The Incompatibility of Husserl's and Brouwer's Positions; 4.2 Two Sources of Mutual Pressure; 4.3 Resolving the Conflict: The Options, and a Proposal; 5 The Phenomenological Incorrectness of the Original Arguments; 5.1 The Phenomenological Standard for a Correct Argument in Ontology; 5.2 Husserl's Weak Revisionism
    Description / Table of Contents: 5.3 Husserl's Implied Strong Revisionism5.4 The Incompleteness of Husserl's Argument; 5.5 The Irreflexivity of Brouwer's Philosophy; 6 The Constitution of Choice Sequences; 6.1 A Motivation for Choice Sequences; 6.2 Choice Sequences as Objects; 6.3 Choice Sequences as Mathematical Objects; 7 Application: An Argument for Weak Continuity; 7.1 The Weak Continuity Principle; 7.2 An Argument That Does Not Work; 7.3 A Phenomenological Argument; 8 Concluding Remarks; Appendix: Intuitionistic Remarks on Husserl's Analysis of Finite Number in the Philosophy of Arithmetic; Notes; References
    Description / Table of Contents: Name and Citation IndexSubject Index
    Note: Includes bibliographical references and index
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  • 10
    Online Resource
    Online Resource
    Dordrecht : Springer
    ISBN: 9781402055874
    Language: English
    Pages: Online-Ressource , v.: digital
    Edition: Online-Ausg. Springer eBook Collection. Humanities, Social Science and Law Electronic reproduction; Available via World Wide Web
    RVK:
    Keywords: Science Logic, Symbolic and mathematical ; Computer science ; Artificial intelligence ; Geometry ; Topology ; Philosophy (General) ; Logic, Symbolic and mathematical ; Raum ; Mathematische Logik
    Abstract: Johan Benthem
    Abstract: A spatial logic is a formal language interpreted over any class of structures featuring geometrical entities and relations, broadly construed. In the past decade, spatial logics have attracted much attention in response to developments in such diverse fields as Artificial Intelligence, Database Theory, Physics, and Philosophy. The aim of this handbook is to create, for the first time, a systematic account of the field of spatial logic. The book comprises a general introduction, followed by fourteen chapters by invited authors. Each chapter provides a self-contained overview of its topic, descr
    Description / Table of Contents: CONTENTS; Preface; Contributing Authors; Second Readers xxi; Chpater 1 What is Spatial Logic?; Chapter 2 First-Order Mereotopology; Chapter 3 Axioms, Algebras and Topology; Chpater 4 Qualitative Spatial Reasoning Using Constraint Calculi; Chapter 5 Modal Logics of Space; Chapter 6 Topology and Epistemic Logic; Chapter 7 Logical Theories for Fragments of Elementary Geometry; Chapter 8 Locales and Toposes as Spaces; Chapter 9 Spatial Logic + Temporal Logic = ?; Chapter 10 Dynamic Topological Logic; Chapter 11 Logic of Space-Time and Relativity Theory; Chapter 12 Discrete Spatial Models
    Description / Table of Contents: Chapter 13 Real Algebraic Geometry and Constriaint DatabasesChapter 14 Mathematical Morphology; Chapter 15 Spatial Resoning and Ontology: Parts, Wholes, and Locations; Index
    Note: Includes bibliographical references and index , Electronic reproduction; Available via World Wide Web
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  • 11
    Online Resource
    Online Resource
    Dordrecht : Springer-Science+Business Media, B.V.
    ISBN: 9789401577311
    Language: English
    Pages: 1 Online-Ressource (xiv, 186 Seiten)
    Edition: Springer eBook Collection. Humanities, Social Sciences and Law
    Series Statement: Synthese library Volume182
    Series Statement: Springer eBook Collection
    Parallel Title: Erscheint auch als Detlefsen, Michael Hilbert's program
    RVK:
    RVK:
    RVK:
    Keywords: Philosophy (General) ; Logic, Symbolic and mathematical ; Logic ; Mathematics. ; History. ; Mathematical logic. ; Mathematik ; Philosophie ; Hilbert, David 1862-1943 ; Mathematik ; Grundlage ; Hilbertsche Probleme
    Abstract: I: The Philosophical Fundamentals of Hilbert’s Program -- II: A Closer Look at the Problems -- III: The Gödelian Challenge -- IV: The Stability Problem -- V: The Convergence Problem and the Problem of Strict Instrumentalism -- Appendix: Hilbert’s Program and the First Theorem -- References.
    Abstract: Hilbert's Program was founded on a concern for the phenomenon of paradox in mathematics. To Hilbert, the paradoxes, which are at once both absurd and irresistible, revealed a deep philosophical truth: namely, that there is a discrepancy between the laws accord­ ing to which the mind of homo mathematicus works, and the laws governing objective mathematical fact. Mathematical epistemology is, therefore, to be seen as a struggle between a mind that naturally works in one way and a reality that works in another. Knowledge occurs when the two cooperate. Conceived in this way, there are two basic alternatives for mathematical epistemology: a skeptical position which maintains either that mind and reality seldom or never come to agreement, or that we have no very reliable way of telling when they do; and a non-skeptical position which holds that there is significant agree­ ment between mind and reality, and that their potential discrepan­ cies can be detected, avoided, and thus kept in check. Of these two, Hilbert clearly embraced the latter, and proposed a program designed to vindicate the epistemological riches represented by our natural, if non-literal, ways of thinking. Brouwer, on the other hand, opted for a position closer (in Hilbert's opinion) to that of the skeptic. Having decided that epistemological purity could come only through sacrifice, he turned his back on his classical heritage to accept a higher calling.
    Note: Literaturverzeichnis: Seite 179-186
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