Inconsistent Mathematics

Author: C.E. Mortensen
Publisher: Springer Science & Business Media
ISBN: 9401584532
Category: Mathematics
Page: 158
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without a properly developed inconsistent calculus based on infinitesimals, then in consistent claims from the history of the calculus might well simply be symptoms of confusion. This is addressed in Chapter 5. It is further argued that mathematics has a certain primacy over logic, in that paraconsistent or relevant logics have to be based on inconsistent mathematics. If the latter turns out to be reasonably rich then paraconsistentism is vindicated; while if inconsistent mathematics has seri ous restriytions then the case for being interested in inconsistency-tolerant logics is weakened. (On such restrictions, see this chapter, section 3. ) It must be conceded that fault-tolerant computer programming (e. g. Chapter 8) finds a substantial and important use for paraconsistent logics, albeit with an epistemological motivation (see this chapter, section 3). But even here it should be noted that if inconsistent mathematics turned out to be functionally impoverished then so would inconsistent databases. 2. Summary In Chapter 2, Meyer's results on relevant arithmetic are set out, and his view that they have a bearing on G8del's incompleteness theorems is discussed. Model theory for nonclassical logics is also set out so as to be able to show that the inconsistency of inconsistent theories can be controlled or limited, but in this book model theory is kept in the background as much as possible. This is then used to study the functional properties of various equational number theories.

An Introduction to the Philosophy of Mathematics

Author: Mark Colyvan
Publisher: Cambridge University Press
ISBN: 1107377005
Category: Science
Page: N.A
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This introduction to the philosophy of mathematics focuses on contemporary debates in an important and central area of philosophy. The reader is taken on a fascinating and entertaining journey through some intriguing mathematical and philosophical territory, including such topics as the realism/anti-realism debate in mathematics, mathematical explanation, the limits of mathematics, the significance of mathematical notation, inconsistent mathematics and the applications of mathematics. Each chapter has a number of discussion questions and recommended further reading from both the contemporary literature and older sources. Very little mathematical background is assumed and all of the mathematics encountered is clearly introduced and explained using a wide variety of examples. The book is suitable for an undergraduate course in philosophy of mathematics and, more widely, for anyone interested in philosophy and mathematics.

Philosophy of Mathematics

Author: N.A
Publisher: Elsevier
ISBN: 9780080930589
Category: Philosophy
Page: 733
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One of the most striking features of mathematics is the fact that we are much more certain about the mathematical knowledge we have than about what mathematical knowledge is knowledge of. Are numbers, sets, functions and groups physical entities of some kind? Are they objectively existing objects in some non-physical, mathematical realm? Are they ideas that are present only in the mind? Or do mathematical truths not involve referents of any kind? It is these kinds of questions that have encouraged philosophers and mathematicians alike to focus their attention on issues in the philosophy of mathematics. Over the centuries a number of reasonably well-defined positions about the nature of mathematics have been developed and it is these positions (both historical and current) that are surveyed in the current volume. Traditional theories (Platonism, Aristotelianism, Kantianism), as well as dominant modern theories (logicism, formalism, constructivism, fictionalism, etc.), are all analyzed and evaluated. Leading-edge research in related fields (set theory, computability theory, probability theory, paraconsistency) is also discussed. The result is a handbook that not only provides a comprehensive overview of recent developments but that also serves as an indispensable resource for anyone wanting to learn about current developments in the philosophy of mathematics. -Comprehensive coverage of all main theories in the philosophy of mathematics -Clearly written expositions of fundamental ideas and concepts -Definitive discussions by leading researchers in the field -Summaries of leading-edge research in related fields (set theory, computability theory, probability theory, paraconsistency) are also included

Philosophy of Mathematics

Set Theory, Measuring Theories, and Nominalism
Author: Gerhard Preyer,Georg Peter
Publisher: Walter de Gruyter
ISBN: 3110323680
Category: Philosophy
Page: 184
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One main interest of philosophy is to become clear about the assumptions, premisses and inconsistencies of our thoughts and theories. And even for a formal language like mathematics it is controversial if consistency is acheivable or necessary like the articles in the firt part of the publication show. Also the role of formal derivations, the role of the concept of apriority, and the intuitions of mathematical principles and properties need to be discussed. The second part is a contribution on nominalistic and platonistic views in mathematics, like the "indispensability argument" of W. v. O. Quine H. Putnam and the "makes no difference argument" of A. Baker. Not only in retrospect, the third part shows the problems of Mill, Frege's and the unity of mathematics and Descartes's contradictional conception of mathematical essences. Together, these articles give us a hint into the relationship between mathematics and world, that is, one of the central problems in philosophy of mathematics and philosophy of science.

Understanding Inconsistent Science

Author: Peter Vickers
Publisher: OUP Oxford
ISBN: 0191662976
Category: Science
Page: 288
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In recent years philosophers of science have urged that many scientific theories are extremely useful and successful despite being internally inconsistent. Via an investigation of eight alleged 'inconsistent theories' in the history of science, Peter Vickers urges that this view is at best overly simplistic. Most of these cases can only be described as examples of 'inconsistent science' if we employ reconstructions of science which depart from the real (history of) science to an unacceptable degree. And where we do find genuine inconsistency he argues that the nature of—and correct response to—the inconsistency differs dramatically depending on the details of the science in question. Thus we are warned against making overly general claims about 'science': what are all called 'theories' in the history of science are actually significantly different entities, which work in different ways and react to inconsistency in different ways. Vickers argues that the traditional goal of philosophy to make substantial, fully general claims about 'how science works' is misguided, and can be significantly circumvented if we re-frame our debates such that reference to 'theories' is eliminated. In this way one is not tempted to think of the history of science as a history of instances of the same kind—theory—about which one could hope to say something substantial and general. And in addition eliminating theory means that we avoid fruitless debates about the 'real' nature and content of 'theories'. Vickers' account leads to a particularist philosophy of science, where the reader is urged to appreciate the often dramatic differences between the different 'inconsistencies in science' which have been identified.

The Best Writing on Mathematics 2010

Author: Mircea Pitici
Publisher: Princeton University Press
ISBN: 9780691148410
Category: Literary Collections
Page: 407
View: 7863

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This anthology brings together the year's finest writing on mathematics from around the world. Featuring promising new voices alongside some of the foremost names in mathematics, The Best Writing on Mathematics makes available to a wide audience many articles not easily found anywhere else--and you don't need to be a mathematician to enjoy them. These writings offer surprising insights into the nature, meaning, and practice of mathematics today. They delve into the history, philosophy, teaching, and everyday occurrences of math, and take readers behind the scenes of today's hottest mathematical debates. Here readers will discover why Freeman Dyson thinks some mathematicians are birds while others are frogs; why Keith Devlin believes there's more to mathematics than proof; what Nick Paumgarten has to say about the timing patterns of New York City's traffic lights (and why jaywalking is the most mathematically efficient way to cross Sixty-sixth Street); what Samuel Arbesman can tell us about the epidemiology of the undead in zombie flicks; and much, much more. In addition to presenting the year's most memorable writing on mathematics, this must-have anthology also includes a foreword by esteemed mathematician William Thurston and an informative introduction by Mircea Pitici. This book belongs on the shelf of anyone interested in where math has taken us--and where it's headed.

Relevant Logics and Their Rivals

A continuation of the work of Richard Sylvan, Robert Meyer, Val Plumwood and Ross Brady
Author: Richard Sylvan,Ross Brady
Publisher: Ashgate Pub Limited
Category: Philosophy
Page: 425
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The aim of the first volume of this set was to enhance and expand on the Routley-Meyer semantics for relevant logic. The key author, Richard Routley died in 1996, but the editor has remained true to Routley's direction in volume two.

Mathematical Ideas and Sociological Theory

Current State and Prospects
Author: Thomas J. Fararo
Publisher: Taylor & Francis
ISBN: 9780677166353
Page: 185
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First Published in 1984. Routledge is an imprint of Taylor & Francis, an informa company.

Teaching and Learning Mathematics

A Teacher's Guide to Recent Research and Its Application
Author: Marilyn Nickson
Publisher: A&C Black
ISBN: 0826472370
Category: Education
Page: 226
View: 980

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This is a summary of the research in all the major topics of interest and concern to teachers of mathematics, from primary (elementary) to secondary (high) schools. It is directed towards students, in-service teachers, maths advisers and tutors.

Numerical Mathematics and Advanced Applications

Proceedings of ENUMATH 2005 the 6th European Conference on Numerical Mathematics and Advanced Applications, Santiago de Compostela, Spain, July 2005
Author: Alfredo Bermúdez de Castro,Dolores Gómez,Peregrina Quintela,Pilar Salgado
Publisher: Springer Science & Business Media
ISBN: 3540342885
Category: Mathematics
Page: 1232
View: 6442

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These proceedings collect lectures given at ENUMATH 2005, the 6th European Conference on Numerical Mathematics and Advanced Applications held in Santiago de Compostela, Spain in July, 2005. Topics include applications such as fluid dynamics, electromagnetism, structural mechanics, interface problems, waves, finance, heat transfer, unbounded domains, numerical linear algebra, convection-diffusion, as well as methodologies such as a posteriori error estimates, discontinuous Galerkin methods, multiscale methods, optimization, and more.

Topics in Mathematical Analysis and Differential Geometry

Author: Nicolas K. Laos
Publisher: World Scientific
ISBN: 9789810231804
Category: Mathematics
Page: 559
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This book studies the interplay between mathematical analysis and differential geometry as well as the foundations of these two fields. The development of a unified approach to topological vector spaces, differential geometry and algebraic and differential topology of function manifolds led to the broad expansion of global analysis. This book serves as a self-contained reference on both the prerequisites for further study and the recent research results which have played a decisive role in the advancement of global analysis.

A Study of Braids

Author: Kunio Murasugi,B. Kurpita
Publisher: Springer Science & Business Media
ISBN: 9780792357674
Category: Computers
Page: 272
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This book provides a comprehensive exposition of the theory of braids, beginning with the basic mathematical definitions and structures. Among the many topics explained in detail are: the braid group for various surfaces; the solution of the word problem for the braid group; braids in the context of knots and links (Alexander's theorem); Markov's theorem and its use in obtaining braid invariants; the connection between the Platonic solids (regular polyhedra) and braids; the use of braids in the solution of algebraic equations. Dirac's problem and special types of braids termed Mexican plaits are also discussed. Audience: Since the book relies on concepts and techniques from algebra and topology, the authors also provide a couple of appendices that cover the necessary material from these two branches of mathematics. Hence, the book is accessible not only to mathematicians but also to anybody who might have an interest in the theory of braids. In particular, as more and more applications of braid theory are found outside the realm of mathematics, this book is ideal for any physicist, chemist or biologist who would like to understand the mathematics of braids. With its use of numerous figures to explain clearly the mathematics, and exercises to solidify the understanding, this book may also be used as a textbook for a course on knots and braids, or as a supplementary textbook for a course on topology or algebra.

Inconsistency, Asymmetry, and Non-Locality

A Philosophical Investigation of Classical Electrodynamics
Author: Mathias Frisch
Publisher: Oxford University Press
ISBN: 0199883777
Category: Science
Page: 222
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Mathias Frisch provides the first sustained philosophical discussion of conceptual problems in classical particle-field theories. Part of the book focuses on the problem of a satisfactory equation of motion for charged particles interacting with electromagnetic fields. As Frisch shows, the standard equation of motion results in a mathematically inconsistent theory, yet there is no fully consistent and conceptually unproblematic alternative theory. Frisch describes in detail how the search for a fundamental equation of motion is partly driven by pragmatic considerations (like simplicity and mathematical tractability) that can override the aim for full consistency. The book also offers a comprehensive review and criticism of both the physical and philosophical literature on the temporal asymmetry exhibited by electromagnetic radiation fields, including Einstein's discussion of the asymmetry and Wheeler and Feynman's influential absorber theory of radiation. Frisch argues that attempts to derive the asymmetry from thermodynamic or cosmological considerations fail and proposes that we should understand the asymmetry as due to a fundamental causal constraint. The book's overarching philosophical thesis is that standard philosophical accounts that strictly identify scientific theories with a mathematical formalism and a mapping function specifying the theory's ontology are inadequate, since they permit neither inconsistent yet genuinely successful theories nor thick causal notions to be part of fundamental physics.

Hans Reichenbach: Logical Empiricist

Author: M.H. Salmon
Publisher: Springer Science & Business Media
ISBN: 9789027709585
Category: Science
Page: 793
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Logical empiricism - not to be confused with logical positivism (see pp. 40-44) - is a movement which has left an indelible mark on twentieth century philosophy; Hans Reichenbach (1891-1953) was one of its found ers and one of its most productive advocates. His sudden and untimely death in 1953 halted his work when he was at the height of his intellectual powers; nevertheless, he bequeathed to us a handsome philosophical inheritance. At the present time, twenty-five years later, we can survey our heritage and see to what extent we have been enriched. The present collection of essays constitutes an effort to do just that - to exhibit the scope and unity of Reichenbach's philosophy, and its relevance to current philosophical issues. There is no Nobel Prize in philosophy - the closest analogue is a volume in The Library of Living Philosophers, an honor which, like the Nobel Prize, cannot be awarded posthumously. Among 'scientific philosophers,' Rudolf Carnap, Albert Einstein, Karl Popper, and Bertrand Russell have been so honored. Had Reichenbach lived longer, he would have shared the honor with Carnap, for at the time of his death a volume on Logical Empiricism, treating the works of Carnap and Reichenbach, was in its early stages of preparation. In the volume which emerged, Carnap wrote, "In 1953, when Reichenbach's creative activity was suddenly ended by his premature death, our movement lost one of its most active leaders.

The Semantic Web: Research and Applications

6th European Semantic Web Conference, ESWC 2009 Heraklion, Crete, Greece, May 31– June 4, 2009 Proceedings
Author: Lora Aroyo,Paolo Traverso,Fabio Ciravegna,Philipp Cimiano,Tom Heath,Eero Hyvönen,Riichiro Mizoguchi,Eyal Oren,Marta Sabou,Elena Simperl
Publisher: Springer Science & Business Media
ISBN: 3642021204
Category: Computers
Page: 961
View: 697

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This book constitutes the refereed proceedings of the 6th European Semantic Web Conference, ESWC 2009, held in Heraklion, Crete, Greece, in May/June 2009. The 45 revised full papers of the research track presented together with the abstracts of 4 keynote lectures were carefully reviewed and selected from more than 250 submissions. The papers are organized in topical sections on applications, evaluation and benchmarking, ontologies and natural language, ontology alignment, ontology engineering, query processing, reasoning, search and identities, semantic Web architectures, semantic Web services, and tagging and annotation. In addition to the technical research track, this book presents 8 contributions to the ESWC 2009 PhD symposium, 24 system demo papers, as well as 8 contributions to the semantic Web in-use track.

Type Theory and Formal Proof

An Introduction
Author: Rob Nederpelt,Herman Geuvers
Publisher: Cambridge University Press
ISBN: 1316061086
Category: Computers
Page: N.A
View: 360

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Type theory is a fast-evolving field at the crossroads of logic, computer science and mathematics. This gentle step-by-step introduction is ideal for graduate students and researchers who need to understand the ins and outs of the mathematical machinery, the role of logical rules therein, the essential contribution of definitions and the decisive nature of well-structured proofs. The authors begin with untyped lambda calculus and proceed to several fundamental type systems, including the well-known and powerful Calculus of Constructions. The book also covers the essence of proof checking and proof development, and the use of dependent type theory to formalise mathematics. The only prerequisite is a basic knowledge of undergraduate mathematics. Carefully chosen examples illustrate the theory throughout. Each chapter ends with a summary of the content, some historical context, suggestions for further reading and a selection of exercises to help readers familiarise themselves with the material.

Neutrosophic Sets and Systems, Vol. II

book series
Author: Florentin Smarandache,Mumtaz Ali
Publisher: Infinite Study
ISBN: 1599733110
Page: N.A
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This book is a collection of eleven papers, written by different authors and co-authors (listed in the order of the papers): S. Alkhazaleh, E. Marei, S. Broumi, F. Smarandache, R. Sahin, A. A. Salama, V. Kroumov, K. Perez-Taruel , M. Leyva-Vazquez, A. A. A. Agboola, B. Davvaz, W. B. V. Kandasamy, J. Ye, Q. Zhang, M. Ali, M. Shabir, M. Naz, S. Pramanik, T. K. Roy, P. Biswas and B. C. Giri. In first paper, the author proposed Mappings on Neutrosophic Soft Classes. On Neutrosophic Implications is proposed in the second paper. Hierarchical Clustering Algorithms are studied in third paper. In fourth paper Neutrosophic Crisp Sets and Neutrosophic Crisp Topological Spaces are introduced. Similarly in fifth paper, Neutrosophic Logic for Mental Model Elicitation and Analysis is discussed. In paper six, On Neutrosophic Hypergroups and Neutrosophic Hyperrings is study conducted by the authors. Neutrosophic Lattices are given in seventh paper. Paper eight is about Single Valued Neutrosophic Similarity Measures for Multiple Attribute Decision Making. In the next paper Soft Neutrosophic Bigroups and Soft Neutrosophic N-groups are discussed. In the paper, Neutrosophic Game Theoretic Approach to Indo-Pak Conflict over Jammu-Kashmir is proposed. The authors introduced Entropy Based Grey Relational Analysis Method for Multi-Attribute Decision Making under Single Valued Neutrosophic Assessments in the last paper.