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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.
In the last five decades various attempts to formulate theories of quantum gravity have been made, but none has fully succeeded in becoming the quantum theory of gravity. One possible explanation for this failure might be the unresolved fundamental issues in quantum theory as it stands now. Indeed, most approaches to quantum gravity adopt standard quantum theory as their starting point, with the hope that the theory’s unresolved issues will get solved along the way. However, these fundamental issues may need to be solved before attempting to define a quantum theory of gravity. The present text adopts this point of view, addressing the following basic questions: What are the main conceptual issues in quantum theory? How can these issues be solved within a new theoretical framework of quantum theory? A possible way to overcome critical issues in present-day quantum physics – such as a priori assumptions about space and time that are not compatible with a theory of quantum gravity, and the impossibility of talking about systems without reference to an external observer – is through a reformulation of quantum theory in terms of a different mathematical framework called topos theory. This course-tested primer sets out to explain to graduate students and newcomers to the field alike, the reasons for choosing topos theory to resolve the above-mentioned issues and how it brings quantum physics back to looking more like a “neo-realist” classical physics theory again.
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.
The World Wide Web has enabled the creation of a global information space comprising linked documents. As the Web becomes ever more enmeshed with our daily lives, there is a growing desire for direct access to raw data not currently available on the Web or bound up in hypertext documents. Linked Data provides a publishing paradigm in which not only documents, but also data, can be a first class citizen of the Web, thereby enabling the extension of the Web with a global data space based on open standards - the Web of Data. In this Synthesis lecture we provide readers with a detailed technical introduction to Linked Data. We begin by outlining the basic principles of Linked Data, including coverage of relevant aspects of Web architecture. The remainder of the text is based around two main themes - the publication and consumption of Linked Data. Drawing on a practical Linked Data scenario, we provide guidance and best practices on: architectural approaches to publishing Linked Data; choosing URIs and vocabularies to identify and describe resources; deciding what data to return in a description of a resource on the Web; methods and frameworks for automated linking of data sets; and testing and debugging approaches for Linked Data deployments. We give an overview of existing Linked Data applications and then examine the architectures that are used to consume Linked Data from the Web, alongside existing tools and frameworks that enable these. Readers can expect to gain a rich technical understanding of Linked Data fundamentals, as the basis for application development, research or further study. Table of Contents: List of Figures / Introduction / Principles of Linked Data / The Web of Data / Linked Data Design Considerations / Recipes for Publishing Linked Data / Consuming Linked Data / Summary and Outlook
Antifragile is a standalone book in Nassim Nicholas Taleb’s landmark Incerto series, an investigation of opacity, luck, uncertainty, probability, human error, risk, and decision-making in a world we don’t understand. The other books in the series are Fooled by Randomness, The Black Swan, and The Bed of Procrustes. Nassim Nicholas Taleb, the bestselling author of The Black Swan and one of the foremost thinkers of our time, reveals how to thrive in an uncertain world. Just as human bones get stronger when subjected to stress and tension, and rumors or riots intensify when someone tries to repress them, many things in life benefit from stress, disorder, volatility, and turmoil. What Taleb has identified and calls “antifragile” is that category of things that not only gain from chaos but need it in order to survive and flourish. In The Black Swan, Taleb showed us that highly improbable and unpredictable events underlie almost everything about our world. In Antifragile, Taleb stands uncertainty on its head, making it desirable, even necessary, and proposes that things be built in an antifragile manner. The antifragile is beyond the resilient or robust. The resilient resists shocks and stays the same; the antifragile gets better and better. Furthermore, the antifragile is immune to prediction errors and protected from adverse events. Why is the city-state better than the nation-state, why is debt bad for you, and why is what we call “efficient” not efficient at all? Why do government responses and social policies protect the strong and hurt the weak? Why should you write your resignation letter before even starting on the job? How did the sinking of the Titanic save lives? The book spans innovation by trial and error, life decisions, politics, urban planning, war, personal finance, economic systems, and medicine. And throughout, in addition to the street wisdom of Fat Tony of Brooklyn, the voices and recipes of ancient wisdom, from Roman, Greek, Semitic, and medieval sources, are loud and clear. Antifragile is a blueprint for living in a Black Swan world. Erudite, witty, and iconoclastic, Taleb’s message is revolutionary: The antifragile, and only the antifragile, will make it. Praise for Antifragile “Ambitious and thought-provoking . . . highly entertaining.”—The Economist “A bold book explaining how and why we should embrace uncertainty, randomness, and error . . . It may just change our lives.”—Newsweek “Revelatory . . . [Taleb] pulls the reader along with the logic of a Socrates.”—Chicago Tribune “Startling . . . richly crammed with insights, stories, fine phrases and intriguing asides . . . I will have to read it again. And again.”—Matt Ridley, The Wall Street Journal “Trenchant and persuasive . . . Taleb’s insatiable polymathic curiosity knows no bounds. . . . You finish the book feeling braver and uplifted.”—New Statesman “Antifragility isn’t just sound economic and political doctrine. It’s also the key to a good life.”—Fortune “At once thought-provoking and brilliant.”—Los Angeles Times From the Hardcover edition.
In this compilation of essays written over a fifteen-year period, the distinguished anthropologist explains his view of culture and its symbolic dimensions
One of the main themes that has emerged from behavioral decision research during the past three decades is the view that people's preferences are often constructed in the process of elicitation. This idea is derived from studies demonstrating that normatively equivalent methods of elicitation (e.g., choice and pricing) give rise to systematically different responses. These preference reversals violate the principle of procedure invariance that is fundamental to all theories of rational choice. If different elicitation procedures produce different orderings of options, how can preferences be defined and in what sense do they exist? This book shows not only the historical roots of preference construction but also the blossoming of the concept within psychology, law, marketing, philosophy, environmental policy, and economics. Decision making is now understood to be a highly contingent form of information processing, sensitive to task complexity, time pressure, response mode, framing, reference points, and other contextual factors.

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