Download or read book Axiomatic Method and Category Theory written by Andrei Rodin and published by Springer Science & Business Media. This book was released on 2013-10-14 with total page 285 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Axiomatic Philosophy written by Pritam Sen and published by New Age International. This book was released on 1996 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt: # Scientific Method In Philosophy : Scientific Method, Summary Of Axiomatic Philosophies# Western Philosophies : Plato, Aristotle, Kant, Russell# Eastern Philosophies : Samkara, Bhagavadgita, Confucius, Lao Tzu# New Axioms: New Aristotelian Philosophy, A Rational Religious Philosophy# References# Index.
Download or read book Axiomatic Theories of Truth written by Volker Halbach and published by Cambridge University Press. This book was released on 2014-02-27 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: At the centre of the traditional discussion of truth is the question of how truth is defined. Recent research, especially with the development of deflationist accounts of truth, has tended to take truth as an undefined primitive notion governed by axioms, while the liar paradox and cognate paradoxes pose problems for certain seemingly natural axioms for truth. In this book, Volker Halbach examines the most important axiomatizations of truth, explores their properties and shows how the logical results impinge on the philosophical topics related to truth. In particular, he shows that the discussion on topics such as deflationism about truth depends on the solution of the paradoxes. His book is an invaluable survey of the logical background to the philosophical discussion of truth, and will be indispensable reading for any graduate or professional philosopher in theories of truth.
Download or read book Defending the Axioms written by Penelope Maddy and published by Oxford University Press. This book was released on 2011-01-27 with total page 161 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics depends on proofs, and proofs must begin somewhere, from some fundamental assumptions. The axioms of set theory have long played this role, so the question of how they are properly judged is of central importance. Maddy discusses the appropriate methods for such evaluations and the philosophical backdrop that makes them appropriate.
Download or read book Set Theory and its Philosophy written by Michael Potter and published by Clarendon Press. This book was released on 2004-01-15 with total page 362 pages. Available in PDF, EPUB and Kindle. Book excerpt: Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science.
Download or read book Axiomatic Set Theory written by Patrick Suppes and published by Courier Corporation. This book was released on 2012-05-04 with total page 265 pages. Available in PDF, EPUB and Kindle. Book excerpt: Geared toward upper-level undergraduates and graduate students, this treatment examines the basic paradoxes and history of set theory and advanced topics such as relations and functions, equipollence, more. 1960 edition.
Download or read book Principia Mathematica written by Alfred North Whitehead and published by . This book was released on 1910 with total page 696 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Axiomatic Set Theory Part 1 written by Dana S. Scott and published by American Mathematical Soc.. This book was released on 1971-12-31 with total page 482 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Axiomatic Set Theory Part 2 written by Thomas J. Jech and published by American Mathematical Soc.. This book was released on 1971 with total page 232 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Introduction to Axiomatic Set Theory written by G. Takeuti and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 251 pages. Available in PDF, EPUB and Kindle. Book excerpt: In 1963, the first author introduced a course in set theory at the University of Illinois whose main objectives were to cover Godel's work on the con sistency of the Axiom of Choice (AC) and the Generalized Continuum Hypothesis (GCH), and Cohen's work on the independence of the AC and the GCH. Notes taken in 1963 by the second author were taught by him in 1966, revised extensively, and are presented here as an introduction to axiomatic set theory. Texts in set theory frequently develop the subject rapidly moving from key result to key result and suppressing many details. Advocates of the fast development claim at least two advantages. First, key results are high lighted, and second, the student who wishes to master the subject is com pelled to develop the detail on his own. However, an instructor using a "fast development" text must devote much class time to assisting his students in their efforts to bridge gaps in the text.
Download or read book Axiomatic Utility Theory under Risk written by Ulrich Schmidt and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 216 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first attempts to develop a utility theory for choice situations under risk were undertaken by Cramer (1728) and Bernoulli (1738). Considering the famous St. Petersburg Paradox! - a lottery with an infinite expected monetary value -Bernoulli (1738, p. 209) observed that most people would not spend a significant amount of money to engage in that gamble. To account for this observation, Bernoulli (1738, pp. 199-201) proposed that the expected monetary value has to be replaced by the expected utility ("moral expectation") as the relevant criterion for decision making under risk. However, Bernoulli's 2 argument and particularly his choice of a logarithmic utility function seem to be rather arbitrary since they are based entirely on intuitively 3 appealing examples. Not until two centuries later, did von Neumann and Morgenstern (1947) prove that if the preferences of the decision maker satisfy cer tain assumptions they can be represented by the expected value of a real-valued utility function defined on the set of consequences. Despite the identical mathematical form of expected utility, the theory of von Neumann and Morgenstern and Bernoulli's approach have, however, IFor comprehensive discussions of this paradox cf. Menger (1934), Samuelson (1960), (1977), Shapley (1977a), Aumann (1977), Jorland (1987), and Zabell (1987). 2Cramer (1728, p. 212), on the other hand, proposed that the utility of an amount of money is given by the square root of this amount.
Download or read book Axiomatic Theories of Truth written by Volker Halbach and published by Cambridge University Press. This book was released on 2014-02-27 with total page 361 pages. Available in PDF, EPUB and Kindle. Book excerpt: A survey of the most important axiomatizations of truth, exploring their properties and how the logical results impinge on philosophical topics.
Download or read book Abstract Objects written by E. Zalta and published by Springer Science & Business Media. This book was released on 1983-06-30 with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book, I attempt to lay the axiomatic foundations of metaphysics by developing and applying a (formal) theory of abstract objects. The cornerstones include a principle which presents precise conditions under which there are abstract objects and a principle which says when apparently distinct such objects are in fact identical. The principles are constructed out of a basic set of primitive notions, which are identified at the end of the Introduction, just before the theorizing begins. The main reason for producing a theory which defines a logical space of abstract objects is that it may have a great deal of explanatory power. It is hoped that the data explained by means of the theory will be of interest to pure and applied metaphysicians, logicians and linguists, and pure and applied epistemologists. The ideas upon which the theory is based are not essentially new. They can be traced back to Alexius Meinong and his student, Ernst Mally, the two most influential members of a school of philosophers and psychologists working in Graz in the early part of the twentieth century. They investigated psychological, abstract and non-existent objects - a realm of objects which weren't being taken seriously by Anglo-American philoso phers in the Russell tradition. I first took the views of Meinong and Mally seriously in a course on metaphysics taught by Terence Parsons at the University of Massachusetts/Amherst in the Fall of 1978. Parsons had developed an axiomatic version of Meinong's naive theory of objects.
Download or read book Historical Epistemology and European Philosophy of Science written by Fabio Minazzi and published by Springer Nature. This book was released on 2022-04-01 with total page 415 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers a comprehensive analysis on the evolution of philosophy of science, with a special emphasis on the European tradition of the twentieth century. At first, it shows how the epistemological problem of the objectivity of knowledge and axiomatic knowledge have been previously tackled by transcendentalism, critical rationalism and hermeneutics. In turn, it analyses the axiological dimension of scientific research, moving from traditional model of science and of scientific methods, to the construction of a new image of knowledge that leverages the philosophical tradition of the Milan School. Using this historical-epistemological approach, the author rethinks the Kantian Transcendental, showing how it could be better integrated in the current philosophy of science, to answer important questions such as the relationship between science and history, scientific and social perspectives and philosophy and technology, among others. Not only this book provides a comprehensive study of the evolution of European Philosophy of Science in the twentieth century, yet it offers a new, historical and epistemological-based approach, that could be used to answers many urgent questions of contemporary societies.
Download or read book Abstract Objects written by E. Zalta and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 205 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book, I attempt to lay the axiomatic foundations of metaphysics by developing and applying a (formal) theory of abstract objects. The cornerstones include a principle which presents precise conditions under which there are abstract objects and a principle which says when apparently distinct such objects are in fact identical. The principles are constructed out of a basic set of primitive notions, which are identified at the end of the Introduction, just before the theorizing begins. The main reason for producing a theory which defines a logical space of abstract objects is that it may have a great deal of explanatory power. It is hoped that the data explained by means of the theory will be of interest to pure and applied metaphysicians, logicians and linguists, and pure and applied epistemologists. The ideas upon which the theory is based are not essentially new. They can be traced back to Alexius Meinong and his student, Ernst Mally, the two most influential members of a school of philosophers and psychologists working in Graz in the early part of the twentieth century. They investigated psychological, abstract and non-existent objects - a realm of objects which weren't being taken seriously by Anglo-American philoso phers in the Russell tradition. I first took the views of Meinong and Mally seriously in a course on metaphysics taught by Terence Parsons at the University of Massachusetts/Amherst in the Fall of 1978. Parsons had developed an axiomatic version of Meinong's naive theory of objects.
Download or read book Current Controversies in Philosophy of Science written by Shamik Dasgupta and published by Routledge. This book was released on 2020-10-27 with total page 213 pages. Available in PDF, EPUB and Kindle. Book excerpt: Current Controversies in Philosophy of Science asks twelve philosophers to debate six questions that are driving contemporary work in this area of philosophy. The questions are: I. Are Boltzmann Brains Bad? II. Does Mathematical Explanation Require Mathematical Truth? III. Does Quantum Mechanics Suggest Spacetime is Nonfundamental? IV. Is Evolution Fundamental When It Comes to Defining Biological Ontology? V. Is Chance Ontologically Fundamental? VI. Are Sexes Natural Kinds? These debates explore the philosophical foundations of particular scientific disciplines, while also examining more general issues in the philosophy of science. The result is a book that’s perfect for the advanced philosophy student, building up their knowledge of the foundations of the field and engaging with its cutting-edge questions. Preliminary descriptions of each chapter, annotated lists of further readings for each controversy, and study questions for each chapter help provide clearer and richer snapshots of active controversies for all readers.
Download or read book The Oxford Handbook of Philosophy of Mathematics and Logic written by Stewart Shapiro and published by Oxford University Press. This book was released on 2005-02-10 with total page 856 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics and logic have been central topics of concern since the dawn of philosophy. Since logic is the study of correct reasoning, it is a fundamental branch of epistemology and a priority in any philosophical system. Philosophers have focused on mathematics as a case study for general philosophical issues and for its role in overall knowledge- gathering. Today, philosophy of mathematics and logic remain central disciplines in contemporary philosophy, as evidenced by the regular appearance of articles on these topics in the best mainstream philosophical journals; in fact, the last decade has seen an explosion of scholarly work in these areas. This volume covers these disciplines in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 contributed chapters are by established experts in the field, and their articles contain both exposition and criticism as well as substantial development of their own positions. The essays, which are substantially self-contained, serve both to introduce the reader to the subject and to engage in it at its frontiers. Certain major positions are represented by two chapters--one supportive and one critical. The Oxford Handbook of Philosophy of Math and Logic is a ground-breaking reference like no other in its field. It is a central resource to those wishing to learn about the philosophy of mathematics and the philosophy of logic, or some aspect thereof, and to those who actively engage in the discipline, from advanced undergraduates to professional philosophers, mathematicians, and historians.
