Numbers, Sets and Axioms

The Apparatus of Mathematics

Author: A. G. Hamilton

Publisher: Cambridge University Press

ISBN: 9780521287616

Category: Mathematics

Page: 255

View: 4089

Following the success of Logic for Mathematicians, Dr Hamilton has written a text for mathematicians and students of mathematics that contains a description and discussion of the fundamental conceptual and formal apparatus upon which modern pure mathematics relies. The author's intention is to remove some of the mystery that surrounds the foundations of mathematics. He emphasises the intuitive basis of mathematics; the basic notions are numbers and sets and they are considered both informally and formally. The role of axiom systems is part of the discussion but their limitations are pointed out. Formal set theory has its place in the book but Dr Hamilton recognises that this is a part of mathematics and not the basis on which it rests. Throughout, the abstract ideas are liberally illustrated by examples so this account should be well-suited, both specifically as a course text and, more broadly, as background reading. The reader is presumed to have some mathematical experience but no knowledge of mathematical logic is required.

The Foundations of Mathematics

Author: Ian Stewart,Professor of Math and Gresham Professor of Geometry Ian Stewart,David Tall,David Orme Tall

Publisher: Oxford University Press on Demand

ISBN: 9780198531654

Category: Fiction

Page: 263

View: 7101

"There are many textbooks available for a so-called transition course from calculus to abstract mathematics. I have taught this course several times and always find it problematic. The Foundations of Mathematics (Stewart and Tall) is a horse of a different color. The writing is excellent and there is actually some useful mathematics. I definitely like this book."--The Bulletin of Mathematics Books

Simon Stevin

Author: Simon Stevin

Publisher: N.A


Category: Mathematics

Page: N.A

View: 370

Great Events from History

The 20th century, 1901-1940

Author: Robert F. Gorman

Publisher: Salem PressInc


Category: History

Page: 3453

View: 6297

Presents essays arranged in chronological order on key world events that occurred in such areas as politics, science, medicine, communications, literature, music, philosophy, and international affairs during the first forty years of the twentieth century.


Author: N.A

Publisher: N.A


Category: Academic libraries

Page: N.A

View: 8083

An Outline of Mathematical Logic

Fundamental Results and Notions Explained with All Details

Author: A. Grzegorczyk

Publisher: Springer Science & Business Media

ISBN: 9401022046

Category: Philosophy

Page: 596

View: 2867

Recent years have seen the appearance of many English-Ianguage hand books of logie and numerous monographs on topieal discoveries in the foundations of mathematies. These publications on the foundations of mathematies as a whole are rather difficult for the beginners or refer the reader to other handbooks and various pieeemeal eontribu tions and also sometimes to largely conceived "mathematical fol klore" of unpublished results. As distinct from these, the present book is as easy as possible systematic exposition of the now classical results in the foundations of mathematics. Henee the book may be useful especially for those readers who want to have all the proofs carried out in full and all the concepts explained in detail. In this sense the book is self-contained. The reader's ability to guess is not assumed, and the author's ambition was to reduce the use of sueh words as evident and obvious in proofs to aminimum. This is why the book, it is believed, may be helpful in teaehing or learning the foundation of mathematics in those situations in which the student cannot refer to a parallel lecture on the subject. This is also the reason that I do not insert in the book the last results and the most modem and fashionable approaches to the subjeet, which does not enrich the essential knowledge in founda tions but ean discourage the beginner by their abstract form. A. G.

Sets: Naïve, Axiomatic and Applied

A Basic Compendium with Exercises for Use in Set Theory for Non Logicians, Working and Teaching Mathematicians and Students

Author: D. Van Dalen,H. C. Doets,H. De Swart

Publisher: Elsevier

ISBN: 1483150399

Category: Mathematics

Page: 360

View: 4514

Sets: Naïve, Axiomatic and Applied is a basic compendium on naïve, axiomatic, and applied set theory and covers topics ranging from Boolean operations to union, intersection, and relative complement as well as the reflection principle, measurable cardinals, and models of set theory. Applications of the axiom of choice are also discussed, along with infinite games and the axiom of determinateness. Comprised of three chapters, this volume begins with an overview of naïve set theory and some important sets and notations. The equality of sets, subsets, and ordered pairs are considered, together with equivalence relations and real numbers. The next chapter is devoted to axiomatic set theory and discusses the axiom of regularity, induction and recursion, and ordinal and cardinal numbers. In the final chapter, applications of set theory are reviewed, paying particular attention to filters, Boolean algebra, and inductive definitions together with trees and the Borel hierarchy. This book is intended for non-logicians, students, and working and teaching mathematicians.

Multidimensional Geographic Information Science

Author: Jonathan Raper

Publisher: CRC Press

ISBN: 9780748405060

Category: Technology & Engineering

Page: 332

View: 8230

The way people normally view a GIS is 2-dimensional, a greatly limiting form. However, as developments occur within the field, researchers and practitioners are finding ways to make a GIS 3-dimensional, and in some instances even 4-dimensional. Being able to view a GIS in more than 2 dimensions greatly enhances its usability. This forward-looking text, looks at the ways in which 3- and 4-dimensional (multidimensional) GIS can be incorporated into the area in the future using a variety of programming techniques. The author of this unique book also discusses current examples and uses of multidimensional GIS in the field and shows the way forward for users in the coming years.

Lectures in Logic and Set Theory: Volume 1, Mathematical Logic

Author: George Tourlakis

Publisher: Cambridge University Press

ISBN: 9781139439428

Category: Mathematics

Page: N.A

View: 2035

This two-volume work bridges the gap between introductory expositions of logic or set theory on one hand, and the research literature on the other. It can be used as a text in an advanced undergraduate or beginning graduate course in mathematics, computer science, or philosophy. The volumes are written in a user-friendly conversational lecture style that makes them equally effective for self-study or class use. Volume 1 includes formal proof techniques, a section on applications of compactness (including nonstandard analysis), a generous dose of computability and its relation to the incompleteness phenomenon, and the first presentation of a complete proof of Godel's 2nd incompleteness since Hilbert and Bernay's Grundlagen theorem.

Introduction to Set Theory, Third Edition, Revised and Expanded

Author: Karel Hrbacek,Thomas Jech

Publisher: CRC Press

ISBN: 9780824779153

Category: Mathematics

Page: 310

View: 2435

Thoroughly revised, updated, expanded, and reorganized to serve as a primary text for mathematics courses, Introduction to Set Theory, Third Edition covers the basics: relations, functions, orderings, finite, countable, and uncountable sets, and cardinal and ordinal numbers. It also provides five additional self-contained chapters, consolidates the material on real numbers into a single updated chapter affording flexibility in course design, supplies end-of-section problems, with hints, of varying degrees of difficulty, includes new material on normal forms and Goodstein sequences, and adds important recent ideas including filters, ultrafilters, closed unbounded and stationary sets, and partitions.

The Autopoiesis of Architecture, Volume II

A New Agenda for Architecture

Author: Patrik Schumacher

Publisher: John Wiley & Sons

ISBN: 1119940478

Category: Architecture

Page: 760

View: 1655

This is the second part of a major theoretical work by Patrik Schumacher, which outlines how the discipline of architecture should be understood as its own distinct system of communication. Autopoeisis comes from the Greek and means literally self-production; it was first adopted in biology in the 1970s to describe the essential characteristics of life as a circular self-organizing system and has since been transposed into a theory of social systems. This new approach offers architecture an arsenal of general comparative concepts. It allows architecture to be understood as a distinct discipline, which can be analyzed in elaborate detail while at the same time offering insightful comparisons with other subject areas, such as art, science and political discourse. On the basis of such comparisons the book insists on the necessity of disciplinary autonomy and argues for a sharp demarcation of design from both art and engineering. Schumacher accordingly argues controversially that design as a discipline has its own sui generis intelligence – with its own internal logic, reach and limitations. Whereas the first volume provides the theoretical groundwork for Schumacher’s ideas – focusing on architecture as an autopoeitic system, with its own theory, history, medium and its unique societal function – the second volume addresses the specific, contemporary challenges and tasks that architecture faces. It formulates these tasks, looking specifically at how architecture is seeking to organize and articulate the complexity of post-fordist network society. The volume explicitly addresses how current architecture can upgrade its design methodology in the face of an increasingly demanding task environment, characterized by both complexity and novelty. Architecture’s specific role within contemporary society is explained and its relationship to politics is clarified. Finally, the new, global style of Parametricism is introduced and theoretically grounded.

From Kant to Hilbert

Author: William Bragg Ewald

Publisher: OUP Oxford

ISBN: 9780191523106

Category: Mathematics

Page: 709

View: 3444

Immanuel Kant's Critique of Pure Reason is widely taken to be the starting point of the modern period of mathematics while David Hilbert was the last great mainstream mathematician to pursue important nineteenth cnetury ideas. This two-volume work provides an overview of this important era of mathematical research through a carefully chosen selection of articles. They provide an insight into the foundations of each of the main branches of mathematics—algebra, geometry, number theory, analysis, logic and set theory—with narratives to show how they are linked. Classic works by Bolzano, Riemann, Hamilton, Dedekind, and Poincare are reproduced in reliable translations and many selections from writers such as Gauss, Cantor, Kronecker and Zermelo are here translated for the first time. The collection is an invaluable source for anyone wishing to gain an understanding of the foundation of modern mathematics.

Decision Making and Change in Human Affairs

Proceedings of the Fifth Research Conference on Subjective Probability, Utility, and Decision Making, Darmstadt, 1–4 September, 1975

Author: H. Jungermann,G. De Zeeuw

Publisher: Springer Science & Business Media

ISBN: 9401012768

Category: Social Science

Page: 528

View: 1678

It is only just recently that people have the tools to judge how well they are doing when making decisions. These tools were conceptualized in the seventeenth century. Since then many people have worked to sharpen the concepts, and to explore how these can be applied further. The problems of decision-making and the theory developed correspondingly have drawn the interest of mathematicians, psychologists, statisticians, economists, philosophers, organizational experts, sociologists, not only for their general relevance, but also for a more intrinsic fascination. There are quite a few institutionalized activities to disseminate results and stimulate research in decision-making. For about a decade now a European organizational structure, centered mainly around the psy chological interest in decision-making. There have been conferences in Hamburg, Amsterdam, Uxbridge, Rome and Darmstadt. Conference papers have been partly published+. The organization has thus stabilized, and its re latively long history makes it interesting to see what kind of developments occurred, within the area of interest.