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Axiomatic Method And Category Theory

Author: Andrei Rodin
Publisher: Springer Science & Business Media
ISBN: 3319004042
Size: 40.32 MB
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This volume explores the many different meanings of the notion of the axiomatic method, offering an insightful historical and philosophical discussion about how these notions changed over the millennia. The author, a well-known philosopher and historian of mathematics, first examines Euclid, who is considered the father of the axiomatic method, before moving onto Hilbert and Lawvere. He then presents a deep textual analysis of each writer and describes how their ideas are different and even how their ideas progressed over time. Next, the book explores category theory and details how it has revolutionized the notion of the axiomatic method. It considers the question of identity/equality in mathematics as well as examines the received theories of mathematical structuralism. In the end, Rodin presents a hypothetical New Axiomatic Method, which establishes closer relationships between mathematics and physics. Lawvere's axiomatization of topos theory and Voevodsky's axiomatization of higher homotopy theory exemplify a new way of axiomatic theory building, which goes beyond the classical Hilbert-style Axiomatic Method. The new notion of Axiomatic Method that emerges in categorical logic opens new possibilities for using this method in physics and other natural sciences. This volume offers readers a coherent look at the past, present and anticipated future of the Axiomatic Method.

Logic And Probability In Quantum Mechanics

Author: Patrick Suppes
Publisher: Springer Science & Business Media
ISBN: 9401094667
Size: 62.52 MB
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During the academic years 1972-1973 and 1973-1974, an intensive sem inar on the foundations of quantum mechanics met at Stanford on a regular basis. The extensive exploration of ideas in the seminar led to the org~ization of a double issue of Synthese concerned with the foundations of quantum mechanics, especially with the role of logic and probability in quantum meChanics. About half of the articles in the volume grew out of this seminar. The remaining articles have been so licited explicitly from individuals who are actively working in the foun dations of quantum mechanics. Seventeen of the twenty-one articles appeared in Volume 29 of Syn these. Four additional articles and a bibliography on -the history and philosophy of quantum mechanics have been added to the present volume. In particular, the articles by Bub, Demopoulos, and Lande, as well as the second article by Zanotti and myself, appear for the first time in the present volume. In preparing the articles for publication I am much indebted to Mrs. Lillian O'Toole, Mrs. Dianne Kanerva, and Mrs. Marguerite Shaw, for their extensive assistance.

Studies In The Methodology And Foundations Of Science

Author: Patrick Suppes
Publisher: Springer Science & Business Media
ISBN: 940173173X
Size: 15.64 MB
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The twenty-three papers collected in tbis volume represent an important part of my published work up to the date of this volume. I have not arranged the paper chronologically, but under four main headings. Part I contains five papers on methodology concerned with models and measurement in the sciences. This part also contains the first paper I published, 'A Set of Independent Axioms for Extensive Quantities', in Portugaliae Mathematica in 1951. Part 11 also is concerned with methodology and ineludes six papers on probability and utility. It is not always easy to separate papers on probability and utility from papers on measurement, because of the elose connection between the two subjects, but Artieles 6 and 8, even though they have elose relations to measurement, seem more properly to belong in Part 11, because they are concerned with substantive questions about probability and utility. The last two parts are concerned with the foundations of physics and the foundations of psychology. I have used the term foundations rather than philosophy, because the papers are mainly concerned with specific axiomatic formulations for particular parts of physics or of psychology, and it seems to me that the termfoundations more appropriately describes such constructive axiomatic ventures. Part 111 contains four papers on the foundations of physics. The first paper deals with foundations of special relativity and the last three with the role ofprobability in quantum mechanics.

Rigor And Structure

Author: John P. Burgess
Publisher: Oxford University Press, USA
ISBN: 0198722222
Size: 33.11 MB
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John P. Burgess presents an illuminating study of the nature of mathematical rigour and of mathematical structure, and above all of the relation between them. He considers recent developments in the field including experimental mathematics and computerised formal proofs, and surveys many historical developments in mathematics, philosophy, and logic.

The Logic Of Scientific Discovery

Author: Karl Popper
Publisher: Routledge
ISBN: 1134470029
Size: 53.19 MB
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Described by the philosopher A.J. Ayer as a work of 'great originality and power', this book revolutionized contemporary thinking on science and knowledge. Ideas such as the now legendary doctrine of 'falsificationism' electrified the scientific community, influencing even working scientists, as well as post-war philosophy. This astonishing work ranks alongside The Open Society and Its Enemies as one of Popper's most enduring books and contains insights and arguments that demand to be read to this day.

Berechenbarkeit Der Welt

Author: Wolfgang Pietsch
Publisher: Springer-Verlag
ISBN: 365812153X
Size: 49.31 MB
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Der Sammelband untersucht Entwicklungen in der Wissenschaft, die in den letzten Jahren durch moderne Informationstechnologien und die zunehmende Verfügbarkeit von Daten angestoßen wurden. Aufgrund dieser neuen Ansätze werden eine Reihe von Phänomenen zum Beispiel aus den Sozial- oder den Lebenswissenschaften berechenbar, die zuvor als zu komplex für eine wissenschaftliche Analyse galten. Der Band versammelt Texte aus einer Vielzahl unterschiedlicher Perspektiven, von der Physik über die Sozial- und Kulturwissenschaften bis hin zur Informatik. Neben Beiträgen, die die aktuellen Entwicklungen direkt thematisieren, untersuchen andere Aufsätze zentrale Begriffe in unmittelbarem Zusammenhang wie Komplexität oder Information.

Language As Calculus Vs Language As Universal Medium

Author: Maren Kusch
Publisher: Springer Science & Business Media
ISBN: 9780792303336
Size: 26.31 MB
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I first became interested in Husserl and Heidegger as long ago as 1980, when as an undergraduate at the Freie Universitat Berlin I studied the books by Professor Ernst Tugendhat. Tugendhat's at tempt to bring together analytical and continental philosophy has never ceased to fascinate me, and even though in more recent years other influences have perhaps been stronger, I should like to look upon the present study as still being indebted to Tugendhat's initial incentive. It was my good fortune that for personal reasons I had to con tinue my academic training from 1981 onwards in Finland. Even though Finland is a stronghold of analytical philosophy, it also has a tradition of combining continental and Anglosaxon philosophical thought. Since I had already admired this line of work in Tugendhat, it is hardly surprising that once in Finland I soon became impressed by Professor Jaakko Hintikka's studies on Husserl and intentionality, and by Professor Georg Henrik von Wright's analytical hermeneu tics. While the latter influence has-at least in part-led to a book on the history of hermeneutics, the former influence has led to the present work. My indebtedness to Professor Hintikka is enormous. Not only is the research reported here based on his suggestions, but Hintikka has also commented extensively on different versions of the manuscript, helped me to make important contacts, found a publisher for me, and-last but not least-was a never failing source of encouragement.

Constructive Adpositional Grammars

Author: Federico Gobbo
Publisher: Cambridge Scholars Publishing
ISBN: 144383128X
Size: 59.13 MB
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This book presents a new paradigm of natural language grammar analysis, based on adposition as the key concept, considered a general connection between two morphemes – or group of morphemes. The adpositional paradigm considers the morpheme as the basic unit to represent morphosyntax, taken as a whole, in terms of constructions, while semantics and pragmatics are treated accordingly. All linguistic observations within the book can be described through the methods and tools of Constructive Mathematics, so that the modelling becomes formally feasible. A full description in category-theoretic terms of the formal model is provided in the Appendix. A lot of examples taken from natural languages belonging to different typological areas are offered throughout the volume, in order to explain and validate the modeling – with special attention given to ergativity. Finally, a first real-world application of the paradigm is given, i.e., conversational analysis of the transcript of therapeutic settings in terms of constructive speech acts. The main goal of this book is to broaden the scope of Linguistics by including Constructive Mathematics in order to deal with known topics such as grammaticalization, children’s speech, language comparison, dependency and valency from a different perspective. It primarily concerns advanced students and researchers in the field of Theoretical and Mathematical Linguistics but the audience can also include scholars interested in applications of Topos Theory in Linguistics.

A Mathematical Prelude To The Philosophy Of Mathematics

Author: Stephen Pollard
Publisher: Springer
ISBN: 3319058169
Size: 39.71 MB
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This book is based on two premises: one cannot understand philosophy of mathematics without understanding mathematics and one cannot understand mathematics without doing mathematics. It draws readers into philosophy of mathematics by having them do mathematics. It offers 298 exercises, covering philosophically important material, presented in a philosophically informed way. The exercises give readers opportunities to recreate some mathematics that will illuminate important readings in philosophy of mathematics. Topics include primitive recursive arithmetic, Peano arithmetic, Gödel's theorems, interpretability, the hierarchy of sets, Frege arithmetic and intuitionist sentential logic. The book is intended for readers who understand basic properties of the natural and real numbers and have some background in formal logic.

The Transport System And Transport Policy

Author: Bert van Wee
Publisher: Edward Elgar Publishing
ISBN: 0857936905
Size: 75.96 MB
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ÔThis very interesting book provides an excellent multi-disciplinary introduction into the functioning of transport systems and the interaction with their environments.Õ Ð Erik Verhoef, VU University Amsterdam, The Netherlands ÔThe editors of this important book have clearly identified that few writings on transport treat the transport system as a whole. Implicit in this is a need for a genuinely multidisciplinary approach. An impressive list of contributors ensures that the book draws on the latest research whilst providing new insights into some of the key challenges facing transport students and researchers, transport providers and policy makers.Õ Ð Roger Vickerman, University of Kent, UK ÔSince ancient times transportation has brought our world together. But the need for connectivity and accessibility in a spatially differentiated world has prompted the emergence of very complex transportation systems. This book offers a fresh and operational contribution to a better understanding of the complexity and manageability of a mobile world, by addressing in a balanced way both conceptual and applied or policy aspects of modern transportation systems.Õ Ð Peter Nijkamp, Free University of Amsterdam, The Netherlands Transport impacts on people and businesses in many different ways, and presents some of the key problems that decision-makers need to address. This comprehensive textbook introduces the transport system in a holistic and multidisciplinary way, bringing together the myriad components of transport. This textbook is written for an international readership of undergraduate and postgraduate students in transport and related subjects, as well as for professionals and policy decision-makers across both public and private sectors. Key features include: ¥ Discussion of the importance of transport accessibility and the impacts of transport on the environment and safety ¥ Policy issues relating to all of the discussed issues and prescribed future options. ¥ Transport evaluation methods and modelling approaches. ¥ Examples to highlight the linkages between components of the transport system Ð for example infrastructures, land-use, vehicle technologies Ð and the relevance of these linkages for decision making.