The issue of mathematics teaching and its impact on learners' attainments in this subject has continuously been on the public agenda. The anthology of chapters in this book consists of varied up-to-date studies of some of the best mathematics education researchers and mathematics teaching experts, exploring the varied aspects of this essential. The book depicts the elementary school mathematics teachers' world while relating to three aspects which comprise the professional environment of mathematics teachers: Teachers' education and teachers' knowledge, Teaching and Teachers' personality. The chapters are written on a level which addresses and might interest a wide readership: researchers, in-service teachers, pre-service teachers, parents and learners.

In the four decades since Imre Lakatos declared mathematics a "quasi-empirical science," increasing attention has been paid to the process of proof and argumentation in the field -- a development paralleled by the rise of computer technology and the mounting interest in the logical underpinnings of mathematics. Explanantion and Proof in Mathematics assembles perspectives from mathematics education and from the philosophy and history of mathematics to strengthen mutual awareness and share recent findings and advances in their interrelated fields. With examples ranging from the geometrists of the 17th century and ancient Chinese algorithms to cognitive psychology and current educational practice, contributors explore the role of refutation in generating proofs, the varied links between experiment and deduction, the use of diagrammatic thinking in addition to pure logic, and the uses of proof in mathematics education (including a critique of "authoritative" versus "authoritarian" teaching styles).

A sampling of the coverage:

• The conjoint origins of proof and theoretical physics in ancient Greece.
• Proof as bearers of mathematical knowledge.
• Bridging knowing and proving in mathematical reasoning.
• The role of mathematics in long-term cognitive development of reasoning.
• Proof as experiment in the work of Wittgenstein.
• Relationships between mathematical proof, problem-solving, and explanation.

Explanation and Proof in Mathematics is certain to attract a wide range of readers, including mathematicians, mathematics education professionals, researchers, students, and philosophers and historians of mathematics.

This unique book presents a particularly beautiful way of looking at special relativity. The author encourages students to see beyond the formulas to the deeper structure. The unification of space and time introduced by Einstein's special theory of relativity is one of the cornerstones of the modern scientific description of the universe. Yet the unification is counterintuitive because we perceive time very differently from space. Even in relativity, time is not just another dimension, it is one with different properties The book treats the geometry of hyperbolas as the key to understanding special relativity. The author simplifies the formulas and emphasizes their geometric content. Many important relations, including the famous relativistic addition formula for velocities, then follow directly from the appropriate (hyperbolic) trigonometric addition formulas. Prior mastery of (ordinary) trigonometry is sufficient for most of the material presented, although occasional use is made of elementary differential calculus, and the chapter on electromagnetism assumes some more advanced knowledge. Changes to the Second Edition The treatment of Minkowski space and spacetime diagrams has been expanded. Several new topics have been added, including a geometric derivation of Lorentz transformations, a discussion of three-dimensional spacetime diagrams, and a brief geometric description of "area" and how it can be used to measure time and distance. Minor notational changes were made to avoid conflict with existing usage in the literature. Table of Contents Preface 1. Introduction. 2. The Physics of Special Relativity. 3. Circle Geometry. 4. Hyperbola Geometry. 5. The Geometry of Special Relativity. 6. Applications. 7. Problems III. 8. Paradoxes. 9. Relativistic Mechanics. 10. Problems II. 11. Relativistic Electromagnetism. 12. Problems III. 13. Beyond Special Relativity. 14. Three-Dimensional Spacetime Diagrams. 15. Minkowski Area via Light Boxes. 16. Hyperbolic Geometry. 17. Calculus. Bibliography. Author Biography Tevian Dray is a Professor of Mathematics at Oregon State University. His research lies at the interface between mathematics and physics, involving differential geometry and general relativity, as well as nonassociative algebra and particle physics; he also studies student understanding of "middle-division" mathematics and physics content. Educated at MIT and Berkeley, he held postdoctoral positions in both mathematics and physics in several countries prior to coming to OSU in 1988. Professor Dray is a Fellow of the American Physical Society for his work in relativity, and an award-winning teacher.

This volume is intended for researchers, curriculum developers, policy makers, and classroom teachers who want comprehensive information on what students at grades 4, 8, and 12 (the grades assessed by NAEP) can and cannot do in mathematics. After two introductory chapters on the design of NAEP, the volume contains a chapter on the challenges in analyzing NAEP data at the item level followed by five chapters that report 2005 through 2013 student performance on specific assessment items. These chapters are organized by content area and then by topic (e.g., understanding of place value, knowledge of transformations, ability to use metric and U.S. systems of measurement) and thus provide baseline data on the proportion of students who are able to complete the mathematics tasks currently used in the upper elementary, middle, and high?school mathematics curriculum. Additional chapters focus on student reasoning, U.S. performance on international assessments, and using construct analysis rather than percent correct on clusters of items to understand student knowledge on specific mathematics topics. Several themes emerge from the volume. One is that while the rate of improvement in mathematics learning in grades 4 and 8 has slowed in recent years, it has slowed more on some topics than others. Another is that relatively minor changes in wording can have significant effects on student performance and thus it is difficult to be specific about what students can do without knowing exactly what questions they were asked. A third theme is that changes in performance over time can sometimes but not always be understood in terms of what students are taught. For example, there were substantial gains on several grade 4 items requiring understanding of fractions and that is probably because the amount of instruction on fractions in grades 3 and 4 has been increasing. In contrast, while relatively few twelfth?grade students have ever been good at factoring trinomials, performance on this skill seems to be decreasing. This suggests that while more students are completing advanced mathematics courses in high school, these courses are not helping in the area of factoring trinomials. Finally, there are limitations to using NAEP as a measure of student performance on the Common Core State Standards. To the extent that NAEP can be used, however, the NAEP data show a substantial gap between expectations and performance.

* Original and up-to-date contribution that highlights key research perspectives * Presents a deep synthesis of the research in the field of university mathematics * Brings together the insights from leading experts as well as early career reserachers from a range of national and institutional backgrounds * Draws on the work of INDRUM, an international network that gathers researchers in univeristy mathematics eduation from around the world

'What I like best about this fascinating book is the detail. Brian Butterworth doesn't just tell us stories of animals with numerical abilities: he tells us about the underlying science. Elegantly written and a joy to read' - Professor Ian Stewart, author of What's the Use? and Taming the Infinite 'Full of thought-provoking studies and animal observations' - Booklist 'Enlightening and entertaining' - Publishers Weekly The Hidden Genius of Animals: Every pet owner thinks their own dog, cat, fish or hamster is a genius. What makes CAN FISH COUNT? so exciting is the way it unveils just how widespread intelligence is in nature. Pioneering psychologist Brian Butterworth describes the extraordinary numerical feats of all manner of species ranging from primates and mammals to birds, reptiles, fish and insects. Whether it's lions deciding to fight or flee, frogs competing for mates, bees navigating their way to food sources, fish assessing which shoal to join, or jackdaws counting friends when joining a mob - every species shares an ability to count. Homo Sapiens may think maths is our exclusive domain, but this book shows that every creature shares a deep-seated Darwinian ability to understand the intrinsic language of our universe: mathematics CAN FISH COUNT? is that special sort of science book - a global authority in his field writing an anecdotally-rich and revelatory narrative which changes the way you perceive something we take for granted.

This volume is a result of a meeting which took place in June 1986 at 'll Ciocco" in Italy entitled 'Deformation theory of algebras and structures and applications'. It appears somewhat later than is perhaps desirable for a volume resulting from a summer school. In return it contains a good many results which were not yet available at the time of the meeting. In particular it is now abundantly clear that the Deformation theory of algebras is indeed central to the whole philosophy of deformations/perturbations/stability. This is one of the main results of the 254 page paper below (practically a book in itself) by Gerstenhaber and Shack entitled "Algebraic cohomology and defor mation theory". Two of the main philosphical-methodological pillars on which deformation theory rests are the fol lowing * (Pure) To study a highly complicated object, it is fruitful to study the ways in which it can arise as a limit of a family of simpler objects: "the unraveling of complicated structures" . * (Applied) If a mathematical model is to be applied to the real world there will usually be such things as coefficients which are imperfectly known. Thus it is important to know how the behaviour of a model changes as it is perturbed (deformed).

The book discusses receiving signals that most electrical engineers detect and study. The vast majority of signals could never be detected due to random additive signals, known as noise, that distorts them or completely overshadows them. Such examples include an audio signal of the pilot communicating with the ground over the engine noise or a bioengineer listening for a fetus' heartbeat over the mother's. The text presents the methods for extracting the desired signals from the noise. Each new development includes examples and exercises that use MATLAB to provide the answer in graphic forms for the reader's comprehension and understanding.

This book is devoted to the study of nonlinear evolution and difference equations of first and second order governed by a maximal monotone operator. This class of abstract evolution equations contains not only a class of ordinary differential equations, but also unify some important partial differential equations, such as the heat equation, wave equation, Schrodinger equation, etc. In addition to their applications in ordinary and partial differential equations, this class of evolution equations and their discrete version of difference equations have found many applications in optimization. In recent years, extensive studies have been conducted in the existence and asymptotic behaviour of solutions to this class of evolution and difference equations, including some of the authors works. This book contains a collection of such works, and its applications. Key selling features: Discusses in detail the study of non-linear evolution and difference equations governed by maximal monotone operator Information is provided in a clear and simple manner, making it accessible to graduate students and scientists with little or no background in the subject material Includes a vast collection of the authors' own work in the field and their applications, as well as research from other experts in this area of study

What is terrorism? What can we learn and what cannot we learn from terrorism data? What are the perspectives and limitations of the analysis of terrorism data? Over the last decade, scholars have generated unprecedented insight from the statistical analysis of ever-growing databases on terrorism. Yet their findings have not reached the public. This book translates the current state of knowledge on global patterns of terrorism free of unnecessary jargon. Readers will be gradually introduced to statistical reasoning and tools applied to critically analyze terrorism data within a rigorous framework. Debunking Seven Terrorism Myths Using Statistics communicates evidence-based research work on terrorism to a general audience. It describes key statistics that provide an overview of the extent and magnitude of terrorist events perpetrated by actors independent of state governments across the world. The books brings a coherent and rigorous methodological framework to address issues stemming from the statistical analysis of terrorism data and its interpretations. Features Uses statistical reasoning to identify and address seven major misconceptions about terrorism. Discusses the implications of major issues about terrorism data on the interpretation of its statistical analysis. Gradually introduces the complexity of statistical methods to familiarize the non-statistician reader with important statistical concepts to analyze data. Use illustrated examples to help the reader develop a critical approach applied to the quantitative analysis of terrorism data. Includes chapters focusing on major aspects of terrorism: definitional issues, lethality, geography, temporal and spatial patterns, and the predictive ability of models.

This book provides an accessible introduction to the mathematical methods of quantum optics. Starting from first principles, it reveals how a given system of atoms and a field is mathematically modelled. The method of eigenfunction expansion and the Lie algebraic method for solving equations are outlined. Analytically exactly solvable classes of equations are identified. The text also discusses consequences of Lie algebraic properties of Hamiltonians, such as the classification of their states as coherent, classical or non-classical based on the generalized uncertainty relation and the concept of quasiprobability distributions. A unified approach is developed for determining the dynamics of a two-level and a three-level atom interacting with combinations of quantized fields under certain conditions. Simple methods for solving a variety of linear and nonlinear dissipative master equations are given.

Algebra is the language that must be mastered for any course that uses math because it is the gateway for entry into any science, technology, engineering, and mathematics (STEM) discipline. This book fosters mastery of critical math and algebraic concepts and skills essential to all of the STEM disciplines and some of the social sciences. This book is written by practitioners whose primary teaching subject is not math but who use math extensively in their courses in STEM disciplines, social science statistics, and their own research. Moreover, in the writing of this book, the authors have used the teaching principles of anchoring, overlearning, pruning the course to its essentials, and using simple and familiar language in word problems.

The issue of how to provide equitable schooling for ethnic and linguistic minorities has come to the forefront of education. This volume brings together top researchers to examine equity from the standpoint of mathematics education--an excellent forum for the topic, since the results are quantifiable. The first essays address broad cultural issues, such as how social class and our notion of merit enter into education. The second section of the book analyzes gender issues in math learning, and the final section examines language and mathematics. A number of themes cut across these three groupings. For example, a critique of the reform movement surfaces in several chapters; many of the chapters look closely at teachers and the dynamics of the classroom, and chapters in different sections address issues of teacher empowerment and skill upgrading. Researchers, students, and policy makers in education, psychology, and cognitive science will want to read these provocative contributions.

Incorporates mixed-effects modeling techniques for more powerful and efficient methods
This book presents current and effective nonparametric regression techniques for longitudinal data analysis and systematically investigates the incorporation of mixed-effects modeling techniques into various nonparametric regression models. The authors emphasize modeling ideas and inference methodologies, although some theoretical results for the justification of the proposed methods are presented.
With its logical structure and organization, beginning with basic principles, the text develops the foundation needed to master advanced principles and applications. Following a brief overview, data examples from biomedical research studies are presented and point to the need for nonparametric regression analysis approaches. Next, the authors review mixed-effects models and nonparametric regression models, which are the two key building blocks of the proposed modeling techniques.
The core section of the book consists of four chapters dedicated to the major nonparametric regression methods: local polynomial, regression spline, smoothing spline, and penalized spline. The next two chapters extend these modeling techniques to semiparametric and time varying coefficient models for longitudinal data analysis. The final chapter examines discrete longitudinal data modeling and analysis.
Each chapter concludes with a summary that highlights key points and also provides bibliographic notes that point to additional sources for further study. Examples of data analysis from biomedical research are used to illustrate the methodologies contained throughout the book. Technical proofs are presented in separate appendices.
With its focus on solving problems, this is an excellent textbook for upper-level undergraduate and graduate courses in longitudinal data analysis. It is also recommended as a reference for biostatisticians and other theoretical and applied research statisticians with an interest in longitudinal data analysis. Not only do readers gain an understanding of the principles of various nonparametric regression methods, but they also gain a practical understanding of how to use the methods to tackle real-world problems.

Graph Theory: An Introduction to Proofs, Algorithms, and Applications Graph theory is the study of interactions, conflicts, and connections. The relationship between collections of discrete objects can inform us about the overall network in which they reside, and graph theory can provide an avenue for analysis. This text, for the first undergraduate course, will explore major topics in graph theory from both a theoretical and applied viewpoint. Topics will progress from understanding basic terminology, to addressing computational questions, and finally ending with broad theoretical results. Examples and exercises will guide the reader through this progression, with particular care in strengthening proof techniques and written mathematical explanations. Current applications and exploratory exercises are provided to further the reader's mathematical reasoning and understanding of the relevance of graph theory to the modern world. Features The first chapter introduces graph terminology, mathematical modeling using graphs, and a review of proof techniques featured throughout the book The second chapter investigates three major route problems: eulerian circuits, hamiltonian cycles, and shortest paths. The third chapter focuses entirely on trees - terminology, applications, and theory. Four additional chapters focus around a major graph concept: connectivity, matching, coloring, and planarity. Each chapter brings in a modern application or approach. Hints and Solutions to selected exercises provided at the back of the book. Author Karin R. Saoub is an Associate Professor of Mathematics at Roanoke College in Salem, Virginia. She earned her PhD in mathematics from Arizona State University and BA from Wellesley College. Her research focuses on graph coloring and on-line algorithms applied to tolerance graphs. She is also the author of A Tour Through Graph Theory, published by CRC Press.

Graph Theory: An Introduction to Proofs, Algorithms, and Applications Graph theory is the study of interactions, conflicts, and connections. The relationship between collections of discrete objects can inform us about the overall network in which they reside, and graph theory can provide an avenue for analysis. This text, for the first undergraduate course, will explore major topics in graph theory from both a theoretical and applied viewpoint. Topics will progress from understanding basic terminology, to addressing computational questions, and finally ending with broad theoretical results. Examples and exercises will guide the reader through this progression, with particular care in strengthening proof techniques and written mathematical explanations. Current applications and exploratory exercises are provided to further the reader's mathematical reasoning and understanding of the relevance of graph theory to the modern world. Features The first chapter introduces graph terminology, mathematical modeling using graphs, and a review of proof techniques featured throughout the book The second chapter investigates three major route problems: eulerian circuits, hamiltonian cycles, and shortest paths. The third chapter focuses entirely on trees - terminology, applications, and theory. Four additional chapters focus around a major graph concept: connectivity, matching, coloring, and planarity. Each chapter brings in a modern application or approach. Hints and Solutions to selected exercises provided at the back of the book. Author Karin R. Saoub is an Associate Professor of Mathematics at Roanoke College in Salem, Virginia. She earned her PhD in mathematics from Arizona State University and BA from Wellesley College. Her research focuses on graph coloring and on-line algorithms applied to tolerance graphs. She is also the author of A Tour Through Graph Theory, published by CRC Press.

This classic textbook has been used successfully by instructors and students for nearly three decades. This timely new edition offers minimal yet notable changes while retaining all the elements, presentation, and accessible exposition of previous editions. A list of updates is found in the Preface to this edition. This text is based on the author's experience in teaching graduate courses and the minimal requirements for successful graduate study. The text is understandable to the typical student enrolled in the course, taking into consideration the variations in abilities, background, and motivation. Chapters one through six have been written to be accessible to the average student, w hile at the same time challenging the more talented student through the exercises. Chapters seven through ten assume the students have achieved some level of expertise in the subject. In these chapters, the theorems, examples, and exercises require greater sophistication and mathematical maturity for full understanding. In addition to the standard topics the text includes topics that are not always included in comparable texts. Chapter 6 contains a section on the Riemann-Stieltjes integral and a proof of Lebesgue's t heorem providing necessary and sufficient conditions for Riemann integrability. Chapter 7 also includes a section on square summable sequences and a brief introduction to normed linear spaces. C hapter 8 contains a proof of the Weierstrass approximation theorem using the method of aapproximate identities. The inclusion of Fourier series in the text allows the student to gain some exposure to this important subject. The final chapter includes a detailed treatment of Lebesgue measure and the Lebesgue integral, using inner and outer measure. The exercises at the end of each section reinforce the concepts. Notes provide historical comments or discuss additional topics.

Developed in cooperation with the International Baccalaureate (R) Enable students to construct, communicate and justify correct mathematical arguments, with a range of activities and examples of maths in the real world. - Engage and excite students with examples and photos of maths in the real world, plus inquisitive starter activities to encourage their problem-solving skills - Build mathematical thinking with our 'Toolkit' and mathematical exploration chapter, along with our new toolkit feature of questions, investigations and activities - Develop understanding with key concepts and applications integrated throughout, along with TOK links for every topic - Prepare your students for assessment with worked examples, and extended essay support - Check understanding with review exercise midway and at the end of the coursebook Follows the new 2019 IB Guide for Mathematics: analysis and approaches Standard Level Available in the series Mathematics for the IB Diploma: Analysis and approaches SL Student Book ISBN: 9781510462359 Student Book Boost eBook ISBN: 9781398334304 Exam Practice Workbook Mathematics for the IB Diploma: Analysis and approaches SL 9781398321182 Exam Practice Workbook Mathematics for the IB Diploma: Analysis and approaches SL Boost eBook 9781398342316 Mathematics for the IB Diploma: Analysis and approaches HL Student Book ISBN: 9781510462366 Student Book Boost eBook ISBN: 9781398334311 Exam Practice Workbook Mathematics for the IB Diploma: Analysis and approaches HL 9781398321878 Exam Practice Workbook Mathematics for the IB Diploma: Analysis and approaches HL Boost eBook 9781398342361 SL & HL Boost Subscription: 9781398341265

Never before in the history of mathematics has there been an individual theorem whose proof has required 10,000 journal pages of closely reasoned argument. Who could read such a proof, let alone communicate it to others? But the classification of all finite simple groups is such a theorem-its complete proof, developed over a 30-year period by about 100 group theorists, is the union of some 500 journal articles covering approximately 10,000 printed pages. How then is one who has lived through it all to convey the richness and variety of this monumental achievement? Yet such an attempt must be made, for without the existence of a coherent exposition of the total proof, there is a very real danger that it will gradually become lost to the living world of mathematics, buried within the dusty pages of forgotten journals. For it is almost impossible for the uninitiated to find the way through the tangled proof without an experienced guide; even the 500 papers themselves require careful selection from among some 2,000 articles on simple group theory, which together include often attractive byways, but which serve only to delay the journey.

X Marks the Spot is written from the point of view of the users of mathematics. Since the beginning, mathematical concepts and techniques (such as arithmetic and geometry) were created as tools with a particular purpose like counting sheep and measuring land areas. Understanding those purposes leads to a greater understanding of why mathematics developed as it did. Later mathematical concepts came from a process of abstracting and generalizing earlier mathematics. This process of abstraction is very powerful, but often comes at the price of intuition and understanding. This book strives to give a guided tour of the development of various branches of mathematics (and what they're used for) that will give the reader this intuitive understanding. Features Treats mathematical techniques as tools, and areas of mathematics as the result of abstracting and generalizing earlier mathematical tools Written in a relaxed conversational and occasionally humorous style making it easy to follow even when discussing esoterica. Unravels how mathematicians think, demystifying math and connecting it to the ways non-mathematicians think and connecting math to people's lives Discusses how math education can be improved in order to prevent future generations from being turned off by math.

X Marks the Spot is written from the point of view of the users of mathematics. Since the beginning, mathematical concepts and techniques (such as arithmetic and geometry) were created as tools with a particular purpose like counting sheep and measuring land areas. Understanding those purposes leads to a greater understanding of why mathematics developed as it did. Later mathematical concepts came from a process of abstracting and generalizing earlier mathematics. This process of abstraction is very powerful, but often comes at the price of intuition and understanding. This book strives to give a guided tour of the development of various branches of mathematics (and what they're used for) that will give the reader this intuitive understanding. Features Treats mathematical techniques as tools, and areas of mathematics as the result of abstracting and generalizing earlier mathematical tools Written in a relaxed conversational and occasionally humorous style making it easy to follow even when discussing esoterica. Unravels how mathematicians think, demystifying math and connecting it to the ways non-mathematicians think and connecting math to people's lives Discusses how math education can be improved in order to prevent future generations from being turned off by math.

This monograph evolved over the past five years. It had its origin as a set of lecture notes prepared for the Ninth Summer School of Mathematical Physics held at Ravello, Italy, in 1984 and was further refined in seminars and lectures given primarily at the University of Colorado. The material presented is the product of a single mathematical question raised by Dave Kassoy over ten years ago. This question and its partial resolution led to a successful, exciting, almost unique interdisciplinary col laborative scientific effort. The mathematical models described are often times deceptively simple in appearance. But they exhibit a mathematical richness and beauty that belies that simplicity and affirms their physical significance. The mathe matical tools required to resolve the various problems raised are diverse, and no systematic attempt is made to give the necessary mathematical background. The unifying theme of the monograph is the set of models themselves. This monograph would never have come to fruition without the enthu siasm and drive of Dave Eberly-a former student, now collaborator and coauthor-and without several significant breakthroughs in our understand ing of the phenomena of blowup or thermal runaway which certain models discussed possess. A collaborator and former student who has made significant contribu tions throughout is Alberto Bressan. There are many other collaborators William Troy, Watson Fulks, Andrew Lacey, Klaus Schmitt-and former students-Paul Talaga and Richard Ely-who must be acknowledged and thanked."

The book intends to give a modern presentation of the classical Markov and Lagrange spectrum, which are fundamental objects from the theory of Diophantine approximations and of their several generalizations related to Dynamical Systems and Differential Geometry. Besides presenting many classical results, the book includes several topics of recent research on the subject, connecting several fields of Mathematics - Number Theory, Dynamical Systems and Fractal Geometry.It includes topics as:

Advanced Regression Models with SAS and R exposes the reader to the modern world of regression analysis. The material covered by this book consists of regression models that go beyond linear regression, including models for right-skewed, categorical and hierarchical observations. The book presents the theory as well as fully worked-out numerical examples with complete SAS and R codes for each regression. The emphasis is on model accuracy and the interpretation of results. For each regression, the fitted model is presented along with interpretation of estimated regression coefficients and prediction of response for given values of predictors. Features: Presents the theoretical framework for each regression. Discusses data that are categorical, count, proportions, right-skewed, longitudinal and hierarchical. Uses examples based on real-life consulting projects. Provides complete SAS and R codes for each example. Includes several exercises for every regression. Advanced Regression Models with SAS and R is designed as a text for an upper division undergraduate or a graduate course in regression analysis. Prior exposure to the two software packages is desired but not required. The Author: Olga Korosteleva is a Professor of Statistics at California State University, Long Beach. She teaches a large variety of statistical courses to undergraduate and master's students. She has published three statistical textbooks. For a number of years, she has held the position of faculty director of the statistical consulting group. Her research interests lie mostly in applications of statistical methodology through collaboration with her clients in health sciences, nursing, kinesiology, and other fields.

'Et moi, ..., si j'avait su comment en revenir, One service mathematics has rendered the human race. It has put common sense back je n'y serais point aile.' where it belongs, on the topmost shelf next Jules Verne to the dusty canister labelled 'discarded n- sense'. The series is divergent; therefore we may be Eric T. Bell able to do something with it. O. Hcaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics seNe as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One seIVice topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'j tre of this series."