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Stories are not enough, even though they are essential. And books about history, books of psychology--the best of them take us closer, but still not close enough. Maria Tumarkin's Axiomatic is a boundary-shifting fusion of thinking, storytelling, reportage and meditation. It takes as its starting point five axioms: 'Time Heals All Wounds'; 'History Repeats Itself'; 'Those Who Forget the Past are Condemned to Repeat It'; 'Give Me a Child Before the Age of Seven and I Will Show You the Woman'; and 'You Can't Enter The Same River Twice.' These beliefs--or intuitions--about the role the past plays in our present are often evoked as if they are timeless and self-evident truths. It is precisely because they are neither, yet still we are persuaded by them, that they tell us a great deal about the forces that shape our culture and the way we live.
The story of geometry is the story of mathematics itself: Euclidean geometry was the first branch of mathematics to be systematically studied and placed on a firm logical foundation, and it is the prototype for the axiomatic method that lies at the foundation of modern mathematics. It has been taught to students for more than two millennia as a mode of logical thought. This book tells the story of how the axiomatic method has progressed from Euclid's time to ours, as a way of understanding what mathematics is, how we read and evaluate mathematical arguments, and why mathematics has achieved the level of certainty it has. It is designed primarily for advanced undergraduates who plan to teach secondary school geometry, but it should also provide something of interest to anyone who wishes to understand geometry and the axiomatic method better. It introduces a modern, rigorous, axiomatic treatment of Euclidean and (to a lesser extent) non-Euclidean geometries, offering students ample opportunities to practice reading and writing proofs while at the same time developing most of the concrete geometric relationships that secondary teachers will need to know in the classroom. -- P. [4] of cover.
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.
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.
Discover the mathematical riches of 'what is diversity?' in a book that adds mathematical rigour to a vital ecological debate.
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.
This text deals with three basic techniques for constructing models of Zermelo-Fraenkel set theory: relative constructibility, Cohen's forcing, and Scott-Solovay's method of Boolean valued models. Our main concern will be the development of a unified theory that encompasses these techniques in one comprehensive framework. Consequently we will focus on certain funda mental and intrinsic relations between these methods of model construction. Extensive applications will not be treated here. This text is a continuation of our book, "I ntroduction to Axiomatic Set Theory," Springer-Verlag, 1971; indeed the two texts were originally planned as a single volume. The content of this volume is essentially that of a course taught by the first author at the University of Illinois in the spring of 1969. From the first author's lectures, a first draft was prepared by Klaus Gloede with the assistance of Donald Pelletier and the second author. This draft was then rcvised by the first author assisted by Hisao Tanaka. The introductory material was prepared by the second author who was also responsible for the general style of exposition throughout the text. We have inc1uded in the introductory material al1 the results from Boolean algebra and topology that we need. When notation from our first volume is introduced, it is accompanied with a deflnition, usually in a footnote. Consequently a reader who is familiar with elementary set theory will find this text quite self-contained.
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.
The first book to integrate axiomatic design and robust design fora comprehensive quality approach As the adoption of quality methods grows across various industries,its implementation is challenged by situations where statisticaltools are inadequate, yet the earlier a proactive quality system isintroduced into a given process, the greater the payback thesemethods will yield. Axiomatic Quality brings together two well-established theories,axiomatic design and robust design, to eliminate or reduce bothconceptual and operational weaknesses. Providing a completeframework for immediate implementation, this book guides designteams in producing systems that operate at high-quality levels foreach of their design requirements. And it shows the way towardsachieving the Six-Sigma target--six times the standard deviationcontained between the target and each side of the specificationlimits--for each requirement. This book develops an aggressive axiomatic quality approachthat: * Provides the tools to reduce conceptual weaknesses of systemsusing a framework called the conceptual design for capability * Reduces operational weaknesses of systems in terms of qualitylosses and control costs * Uses mathematical relationships to bridge the gap betweenscience-based engineering and quality methods Acclaro DFSS Light, a Java-based software package that implementsaxiomatic design processes, is available for download from a Wileyftp site. Acclaro DFSS Light is a software product of AxiomaticDesign Solutions, Inc. Laying out a comprehensive approach while working through eachaspect of its implementation, Axiomatic Quality is an essentialresource for managers, engineers, and other professionals who wantto successfully deploy the most advanced methodology to tacklesystem weaknesses and improve quality.
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.