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Axiomatic Method And Category Theory

Author: Andrei Rodin
Publisher: Springer Science & Business Media
ISBN: 3319004042
Size: 19.55 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.

Studies In The Methodology And Foundations Of Science

Author: Patrick Suppes
Publisher: Springer Science & Business Media
ISBN: 940173173X
Size: 34.98 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: OUP Oxford
ISBN: 019103360X
Size: 73.10 MB
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While we are commonly told that the distinctive method of mathematics is rigorous proof, and that the special topic of mathematics is abstract structure, there has been no agreement among mathematicians, logicians, or philosophers as to just what either of these assertions means. John P. Burgess clarifies the nature of mathematical rigor and of mathematical structure, and above all of the relation between the two, taking into account some of the latest developments in mathematics, including the rise of experimental mathematics on the one hand and computerized formal proofs on the other hand. The main theses of Rigor and Structure are that the features of mathematical practice that a large group of philosophers of mathematics, the structuralists, have attributed to the peculiar nature of mathematical objects are better explained in a different way, as artefacts of the manner in which the ancient ideal of rigor is realized in modern mathematics. Notably, the mathematician must be very careful in deriving new results from the previous literature, but may remain largely indifferent to just how the results in the previous literature were obtained from first principles. Indeed, the working mathematician may remain largely indifferent to just what the first principles are supposed to be, and whether they are set-theoretic or category-theoretic or something else. Along the way to these conclusions, a great many historical developments in mathematics, philosophy, and logic are surveyed. Yet very little in the way of background knowledge on the part of the reader is presupposed.

The Logic Of Scientific Discovery

Author: Karl Popper
Publisher: Routledge
ISBN: 1134470029
Size: 67.34 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: 47.86 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.

Foundations Of Statistical Natural Language Processing

Author: Christopher D. Manning
Publisher: MIT Press
ISBN: 9780262133609
Size: 33.34 MB
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An introduction to statistical natural language processing (NLP). The text contains the theory and algorithms needed for building NLP tools. Topics covered include: mathematical and linguistic foundations; statistical methods; collocation finding; word sense disambiguation; and probalistic parsing.

The Q Theory Of Finite Semigroups

Author: John Rhodes
Publisher: Springer Science & Business Media
ISBN: 0387097813
Size: 40.74 MB
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This comprehensive, encyclopedic text in four parts aims to give the reader — from the graduate student to the researcher/practitioner — a detailed understanding of modern finite semigroup theory, focusing in particular on advanced topics on the cutting edge of research. The q-theory of Finite Semigroups presents important techniques and results, many for the first time in book form, thereby updating and modernizing the semigroup theory literature.

Constructive Adpositional Grammars

Author: Federico Gobbo
Publisher: Cambridge Scholars Publishing
ISBN: 144383128X
Size: 31.55 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.

Working With Objects

Author: Trygve Reenskaug
Publisher: Prentice Hall
ISBN: 9780134529301
Size: 52.17 MB
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Working With Objects is the authoritative account of the highly successful OOram method for software analysis, design, development, maintenance and reuse. OOram has been fine-tuned over a decade on hundreds of projects by Trygve Reenskaug and his collaborators. It is supported by available CASE tools, training, and consulting. Book jacket.

Language As Calculus Vs Language As Universal Medium

Author: Maren Kusch
Publisher: Springer Science & Business Media
ISBN: 9780792303336
Size: 48.10 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.