John Stillwell

This book offers a collection of historical essays detailing a large variety of mathematical disciplines and issues; it’s accessible to a broad audience. This second edition includes new chapters on Chinese and Indian number theory, on hypercomplex numbers, and on algebraic number theory. Many more exercises have been added as well as commentary that helps place the exercises in context.

Mathematics and Its History

An exciting look at the world of elementary mathematics Elements of Mathematics takes readers on a fascinating tour that begins in elementary mathematics--but, as John Stillwell shows, this subject is not as elementary or straightforward as one might think. Not all topics that are part of today's elementary mathematics were always considered as such, and great mathematical advances and discoveries had to occur in order for certain subjects to become "elementary." Stillwell examines elementary mathematics from a distinctive twenty-first-century viewpoint and describes not only the beauty and scope of the discipline, but also its limits. From Gaussian integers to propositional logic, Stillwell delves into arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, and logic. He discusses how each area ties into more advanced topics to build mathematics as a whole. Through a rich collection of basic principles, vivid examples, and interesting problems, Stillwell demonstrates that elementary mathematics becomes advanced with the intervention of infinity. Infinity has been observed throughout mathematical history, but the recent development of "reverse mathematics" confirms that infinity is essential for proving well-known theorems, and helps to determine the nature, contours, and borders of elementary mathematics. Elements of Mathematics gives readers, from high school students to professional mathematicians, the highlights of elementary mathematics and glimpses of the parts of math beyond its boundaries.

From Euclid to Gödel

Elements of Mathematics

How the concept of proof has enabled the creation of mathematical knowledge The Story of Proof investigates the evolution of the concept of proof—one of the most significant and defining features of mathematical thought—through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge. Stillwell begins with Euclid and his influence on the development of geometry and its methods of proof, followed by algebra, which began as a self-contained discipline but later came to rival geometry in its mathematical impact. In particular, the infinite processes of calculus were at first viewed as “infinitesimal algebra,” and calculus became an arena for algebraic, computational proofs rather than axiomatic proofs in the style of Euclid. Stillwell proceeds to the areas of number theory, non-Euclidean geometry, topology, and logic, and peers into the deep chasm between natural number arithmetic and the real numbers. In its depths, Cantor, Gödel, Turing, and others found that the concept of proof is ultimately part of arithmetic. This startling fact imposes fundamental limits on what theorems can be proved and what problems can be solved. Shedding light on the workings of mathematics at its most fundamental levels, The Story of Proof offers a compelling new perspective on the field’s power and progress.

Logic and the History of Mathematics

The Story of Proof

In this new textbook, acclaimed author John Stillwell presents a lucid introduction to Lie theory suitable for junior and senior level undergraduates. In order to achieve this, he focuses on the so-called "classical groups'' that capture the symmetries of real, complex, and quaternion spaces. These symmetry groups may be represented by matrices, which allows them to be studied by elementary methods from calculus and linear algebra. This naive approach to Lie theory is originally due to von Neumann, and it is now possible to streamline it by using standard results of undergraduate mathematics. To compensate for the limitations of the naive approach, end of chapter discussions introduce important results beyond those proved in the book, as part of an informal sketch of Lie theory and its history. John Stillwell is Professor of Mathematics at the University of San Francisco. He is the author of several highly regarded books published by Springer, including The Four Pillars of Geometry (2005), Elements of Number Theory (2003), Mathematics and Its History (Second Edition, 2002), Numbers and Geometry (1998) and Elements of Algebra (1994).

Naive Lie Theory

This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse Mathematics, John Stillwell gives a representative view of this field, emphasizing basic analysis—finding the “right axioms” to prove fundamental theorems—and giving a novel approach to logic. Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers. The second was the twentieth-century arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation. Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets. Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the “right axiom” to prove it. By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics.

Proofs from the Inside Out

Reverse Mathematics

This book is unique in that it looks at geometry from 4 different viewpoints - Euclid-style axioms, linear algebra, projective geometry, and groups and their invariants Approach makes the subject accessible to readers of all mathematical tastes, from the visual to the algebraic Abundantly supplemented with figures and exercises

The Four Pillars of Geometry

the surprising truths of mathematics

Yearning for the impossible

This Element aims to present an outline of mathematics and its history, with particular emphasis on events that shook up its philosophy. It ranges from the discovery of irrational numbers in ancient Greece to the nineteenth- and twentieth-century discoveries on the nature of infinity and proof. Recurring themes are intuition and logic, meaning and existence, and the discrete and the continuous. These themes have evolved under the influence of new mathematical discoveries and the story of their evolution is, to a large extent, the story of philosophy of mathematics.

A Concise History of Mathematics for Philosophers .

Presents the papers of Beltrami, Klein, and Poincare that brought hyperbolic geometry into the mainstream of mathematics.

Sources of Hyperbolic Geometry

The geometry of surfaces is an ideal starting point for learning geometry, for, among other reasons, the theory of surfaces of constant curvature has maximal connectivity with the rest of mathematics. This text provides the student with the knowledge of a geometry of greater scope than the classical geometry taught today, which is no longer an adequate basis for mathematics or physics, both of which are becoming increasingly geometric. It includes exercises and informal discussions.

Geometry of Surfaces

Henk J. Scholten

In August 1989, a Summer Institute was held at the Academie van Bouwkunst, the seventeenth century home of Amsterdam's School of Architecture, Town Planning and Landscape. The meeting brought together experts in Geographical Information Systems from throughout the world to address an international audience of planners. The contents of this book reflect many of the themes that were presented and discussed at the conference. The Summer Institute, let alone this volume, would not have been possible without the support of the International Association for the Development and Management of Existing and New Towns (INTNAIVN), the International Society of City and Regional Planners (ISoCaRP), The National Physical Planning Agency of the Netherlands (RPD) and the Berlage Studio. We wish to acknowledge the assistance provided by these organisations and by the various sponsors: The Ministry of Housing, Physical Planning and Environment, the Municipality of Amsterdam, Logisterion b.v., ESRI, UNISYS, MABON b.v., SPSS, PRIME Computer Inc., PANDATA. The provision of hardware facilities by the various computer companies allowed immensely valuable 'hands on' experience to be gained by all the participants.

Geographical Information Systems for Urban and Regional Planning

Stan Openshaw

Stan Geertman

The 1990s have seen some remarkable changes in geographical information (GI) provision and computer technology that have impacted on many of the activities that constitute planning in all its different forms. However, relatively few texts in the field of geographical information systems (GIS) and planning have been published since Henk Scholten and John Stillwell edited Geographical Information Systems for Urban and Regional Planning in 1990. This volume seeks to redress the balance by showing how GI of various types is being used in urban, physical, environmental, socio-economic and business planning contexts at local, regional and national scales with the assistance of GIS and modelling methods, and how the uses of GI and GI technologies have evolved over the last decade. During this period, a number of meetings took place in Europe in different locations organised initially by European Geographical Information Systems (EGIS, 1990- 94) and more recently by the Joint European Conference and Exhibition (JEC) on Geographical Information (1995-97). These meetings brought together members of the GI community from across the world to discuss GI research and GIS applications. One of the Special Interest Groups associated with the JEC gatherings was that on 'Geographical Information and Planning' and several of the contributions in this book have their origins in papers presented to the group's meetings.

European Perspectives

Geographical Information and Planning

Graham Clarke

Only applications-driven book dealing with commerically-sponsored spatial analysis research. Focuses on business and public sector planning case studies, offering readers a snapshot of the use of spatial analysis across a broad range of areas. Internationally-renowned editors and contributors present a broad variety of global applications, and demonstrate GIS components and spatial methodologies in practice.

Applied GIS and Spatial Analysis

The collection of reliable and comprehensive data on the magnitude, composition and distribution of a country’s population is essential in order for governments to provide services, administer effectively and guide a country’s development. The primary source of basic demographic statistics is frequently a population census, which provides hugely important data sets for policy makers, practitioners and researchers working in a wide range of different socio-demographic contexts. The Routledge Handbook of Census Resources, Methods and Applications provides a comprehensive and authoritative guide to the collection, processing, quality assessment and delivery of the different data products that constitute the results of the population censuses conducted across the United Kingdom in 2011. It provides those interested in using census data with an introduction to the collection, processing and quality assessment of the 2011 Census, together with guidance on the various types of data resources that are available and how they can be accessed. It demonstrates how new methods and technologies, such as interactive infographics and web-based mapping, are now being used to visualise census data in new and exciting ways. Perhaps most importantly, it presents a collection of applications of census data in different social and health science research contexts that reveal key messages about the characteristics of the UK population and the ways in which society is changing. The operation of the 2011 Census and the use of its results are set in the context of census-taking around the world and its historical development in the UK over the last 200 years. The results of the UK 2011 Census are a unique and reliable source of detailed information that are immensely important for users from a wide range of public and private sector organisations, as well as those working in Population Studies, Human Geography, Migration Studies and the Social Sciences more generally.

Unlocking the UK 2011 Census

The Routledge Handbook of Census Resources, Methods and Applications

Joseph Ferreira, Jr.

Robert Goodspeed

This book is a selection of the best and peer-reviewed articles presented at the CUPUM (Computers in Urban Planning and Urban Management) conference, held in the second week of July 2015 at MIT in Boston, USA. The contributions provide state-of the art overview of the availability and application of Planning Support Systems (PSS) in the framework of Smart Cities.

Planning Support Systems and Smart Cities

Chris Pettit

Andrew Allan

This book offers a selection of the best articles presented at the CUPUM (Computers in Urban Planning and Urban Management) Conference, held in the second week of July 2017 at the University of South Australia in Adelaide. It provides a state-of-the-art overview of the availability and application of planning support systems (PSS) in the context of smart cities, big data, and urban futures. Rapid advances in computing, information, communication and web-based technologies are reaching into all facets of urban life, creating new and exciting urban futures. With the universal adoption of networked computing technologies, data generation is now so massive and all pervasive in society that it offers unprecedented technological solutions for planning and managing urban futures. These technologies are essential to effective urban planning and urban management in an increasingly challenging world, with socially disruptive changes, more complex and sophisticated urban lives and the need for resilience to deal with the possibility of adverse future environmental events and climate change. The book discusses examples of these technologies which encompass, inter alia: ‘smart urban futures’, where cities with myriad sensors are networked with communication technologies that enable the city planners to monitor well-being and be responsive to citizens' needs to allow dynamic management in real-time; PSS that encompass new hardware, develop new indicators, applications and innovative ways of facilitating public and community involvement in the management and planning of urban areas; and urban modelling that draws on theory and the richness of data from the growing range of urban sensing and communication technologies to build a better understanding of urban dynamics, trends and 'what-if' scenario investigations, and to provide better tools for planning and policymaking.

Planning Support Science for Smarter Urban Futures

Yearning for the Impossible: The Surprising Truth of Mathematics, Second Edition explores the history of mathematics from the perspective of the creative tension between common sense and the "impossible" as the author follows the discovery or invention of new concepts that have marked mathematical progress. The author puts these creations into a broader context involving related "impossibilities" from art, literature, philosophy, and physics. This new edition contains many new exercises and commentaries, clearly discussing a wide range of challenging subjects.

The Surprising Truths of Mathematics, Second Edition

Yearning for the Impossible

Solutions of equations in integers is the central problem of number theory and is the focus of this book. The amount of material is suitable for a one-semester course. The author has tried to avoid the ad hoc proofs in favor of unifying ideas that work in many situations. There are exercises at the end of almost every section, so that each new idea or proof receives immediate reinforcement.

Elements of Number Theory

Martin Clarke

Although the human population growth rate of the world has been declining since peaking in the early 1960s, the populations of individual countries are changing at different rates. Population dynamics at national level are partly determined by levels of fertility and mortality, but the impact of international migration is playing an increasingly important role. Moreover, internal migration plays a major part in population change at the sub-national level. This fourth volume in the series “Understanding Population Trends and Processes” is a celebration of the work of Professor Philip Rees. It contains chapters by contributors who have collaborated with Phil Rees on research or consultancy projects or as postgraduate students. Several chapters demonstrate the technical nature of population projection modelling and simulation methods while others illustrate issues relating to data availability and estimation. This book demonstrates the application of theoretical and modelling methods and addresses key issues relating to contemporary demographic patterns and trends.

John Stillwell

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