Dimostrazioni indaga l'evoluzione del concetto di dimostrazione - una delle caratteristiche più significative e determinanti del pensiero matematico - attraverso episodi critici della sua storia. Dal teorema di Pitagora ai tempi moderni, e in tutte le principali discipline matematiche, John Stillwell dimostra che la dimostrazione è un concetto matematicamente vitale, che ispira l'innovazione e gioca un ruolo fondamentale nella produzione di nuova conoscenza. Stillwell inizia con Euclide e la sua influenza sullo sviluppo della geometria e dei suoi metodi di dimostrazione, per proseguire con l'algebra. Successivamente, i risultati analitici furono visti in un primo momento come "algebra infinitesimale", e l'analisi divenne un'arena per prove algebriche e computazionali piuttosto che per prove assiomatiche nello stile di Euclide. Stillwell procede con la teoria dei numeri, la geometria non euclidea, la topologia e la logica, e scruta il profondo abisso tra l'aritmetica dei numeri naturali e i numeri reali. Laggiù, Cantor, Gödel, Turing e altri scoprirono che il concetto di dimostrazione è in definitiva parte dell'aritmetica. Questo fatto sorprendente impone limiti fondamentali su quali teoremi possono essere dimostrati e quali problemi possono essere risolti.

Coniugando una visione scientifica aggiornata a una grande attenzione verso le questioni etiche e politiche connesse all'utilizzo dei test psicometrici nella società attuale, il volume introduce a tutti gli aspetti che concernono la valutazione psicometrica, comprese la storia, la costruzione e la somministrazione dei test tradizionali e le più recenti tecniche di assessment online. Dopo un'esaustiva introduzione sull'evoluzione dei metodi e delle regolamentazioni della psicometria, questa edizione (la quarta americana e la prima italiana) esamina la crescente influenza di Internet sulle analisi psicometriche che spesso vengono condotte da sistemi di intelligenza artificiale. Sono analizzate in dettaglio le conseguenze di tale cambio di paradigma, così come i temi più controversi ad esso collegati, per esempio la privacy e il micro-targeting psicografico dei messaggi online. Un lavoro di inestimabile valore sia per gli studenti universitari che per i professionisti in cerca di un testo introduttivo sui metodi psicometrici moderni.

Oltre due millenni separano la scoperta dei numeri irrazionali, a opera dei pitagorici, dalla rigorosa definizione del concetto di numero reale computabile da parte di Alan Turing. Questo volume riformula idee che hanno segnato lo sviluppo del pensiero matematico da Pitagora a Turing, mostrando come, attraverso progressivi affinamenti dell'analisi concettuale, il divario tra numeri e geometria venga a ricomporsi in un fecondo rapporto dialettico. Il suo intento è di far riflettere sul profondo legame tra matematica e filosofia, e provare l'efficacia della matematica nel definire questioni filosofiche fondamentali: la comprensione dell'infinito; l'esistenza di concetti apparentemente impossibili, come i numeri irrazionali e immaginari, i punti sull'orizzonte, gli infinitesimi, la geometria non euclidea; la diversa natura dei concetti di verità e dimostrazione. Dalle armonie musicali alla scienza dei calcolatori, la cultura dell'Occidente prende forma da una duplice matrice. Se la matematica greca deriva il suo carattere dimostrativo dalla speculazione filosofica, la filosofia moderna trova un suo alfabeto nella lingua matematica. Nei nove capitoli del presente volume, il lettore avrà modo di apprezzare come la semplicità della descrizione matematica esalti lo spessore filosofico dei concetti.

While most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light. Analysis needs the real numbers to model the line, and to support the concepts of continuity and measure. But these seemingly simple requirements lead to deep issues of set theory—uncountability, the axiom of choice, and large cardinals. In fact, virtually all the concepts of infinite set theory are needed for a proper understanding of the real numbers, and hence of analysis itself.

By focusing on the set-theoretic aspects of analysis, this text makes the best of two worlds: it combines a down-to-earth introduction to set theory with an exposition of the essence of analysis—the study of infinite processes on the real numbers. It is intended for senior undergraduates, but it will also be attractive to graduate students and professional mathematicians who, until now, have been content to "assume" the real numbers. Its prerequisites are calculus and basic mathematics.

Mathematical history is woven into the text, explaining how the concepts of real number and infinity developed to meet the needs of analysis from ancient times to the late twentieth century. This rich presentation of history, along with a background of proofs, examples, exercises, and explanatory remarks, will help motivate the reader. The material covered includes classic topics from both set theory and real analysis courses, such as countable and uncountable sets, countable ordinals, the continuum problem, the Cantor–Schröder–Bernstein theorem, continuous functions, uniform convergence, Zorn's lemma, Borel sets, Baire functions, Lebesgue measure, and Riemann integrable functions.

From a review of the second edition:"This book covers many interesting topics not usually covered in a present day undergraduate course, as well as certain basic topics such as the development of the calculus and the solution of polynomial equations. The fact that the topics are introduced in their historical contexts will enable students to better appreciate and understand the mathematical ideas involved...If one constructs a list of topics central to a history course, then they would closely resemble those chosen here."(David Parrott, Australian Mathematical Society)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 third edition includes new chapters on simple groups and new sections on alternating groups and the Poincare conjecture. Many more exercises have been added as well as commentary that helps place the exercises in context.

The 1882 Theorie der algebraischen Functionen einer Veränderlichen by Dedekind (1831-1916) and Weber (1842-1913) changed the direction of algebraic geometry and established its foundations by introducing methods from algebraic number theory. They used rings and ideals to give rigorous proofs of results that had previously been obtained in non-rigorous fashion, with the help of analysis and topology. Stillwell (mathematics, U. of San Francisco) believes that the paper still has gems for modern mathematicians that the standard commentaries do not mention. He presents the first English translation of it and provides commentary to the language and thinking of mathematics during the 19th century. Annotation ©2012 Book News, Inc., Portland, OR (booknews.com)

In a detailed and comprehensive introduction to the theory of plane algebraic curves, the authors examine this classical area of mathematics that both figured prominently in ancient Greek studies and remains a source of inspiration and a topic of research to this day. Arising from notes for a course given at the University of Bonn in Germany, “Plane Algebraic Curves” reflects the authors' concern for the student audience through its emphasis on motivation, development of imagination, and understanding of basic ideas. As classical objects, curves may be viewed from many angles. This text also provides a foundation for the comprehension and exploration of modern work on singularities.

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In the first chapter one finds many special curves with very attractive geometric presentations ? the wealth of illustrations is a distinctive characteristic of this book ? and an introduction to projective geometry (over the complex numbers). In the second chapter one finds a very simple proof of Bezout’s theorem and a detailed discussion of cubics. The heart of this book ? and how else could it be with the first author ? is the chapter on the resolution of singularities (always over the complex numbers). (…) Especially remarkable is the outlook to further work on the topics discussed, with numerous references to the literature. Many examples round off this successful representation of a classical and yet still very much alive subject.

(Mathematical Reviews)

"This sequel to the author's Yearning for the Impossible provides a readable survey of logicians' efforts to explicate the notions of truth, proof and undecidability, including the quest to find examples of `natural' undecidable statements. Leavened with historical details, it focuses on the role of infinitary considerations in the development of modern mathematics, with particular attention to the undervalued contributions of Emil Post and Gerhard Gentzen."---John W. Dawson, Jr., author of Logical Dilemmas: The Life and Work of Kurt Godel

"Stillwell has provided an accessible, scholarly treatment of all the foundational studies today's well-rounded professional mathematician ought to know, and has managed to do so in just over 200 pages. And that includes all the relevant history. I highly recommend it."---Keith Devlin, Stanford University, author of The Millennium Problems and co-author of The Computer as Crucible: An Introduction to Experimental Mathematics

"Professor Stillwell ... succeeds, in every topic treated, in bringing a fresh eye to questions even mathematicians might think have been mined in the past to boring exhaustion [and] shows there is still a lot of gold to be found, if one only thinks about things in a new way. Stillwell brings new, unorthodox insights to his writing that will stimulate readers (from high schoolers to emeritus professors) to think about old topics in new, nonstale ways."---SIAM Review

"Stillwell weaves historical details into his writing seamlessly, helping to give the reader the true feeling that mathematics is more than just a bunch of people playing games with symbols, but rather a rich and rewarding intellectual endeavor important to the human enterprise."---MAA Reviews

While many popular books have been written on the advances in our understanding of infinity, sparked by the set theory of Georg Cantor in the 1870s and incompleteness theorems of Kurt Godel in the 1930s, such books generally dwell on a single aspect of either set theory or logic. The aim of this book is to explain the whole, in which set theory interacts with logic, and both begin to affect mainstream mathematics (the latter being quite a recent development, not yet given much space in popular accounts).

In Roads to Infinity, award-winning author John Stillwell explores the consequences of accepting infinity, which are rich and surprising. The reader needs very little background beyond high school mathematics, but should have a willingness to grapple with alien ideas. Stillwell's style eases the reader into the technicalities of set theory and logic by tracing a single thread in each chapter, beginning with a natural mathematical question and following a sequence of historic responses to that question. Each response typically leads to new questions, and from them new concepts and theorems emerge. At the end of each chapter a section called "Historical Background" situates the thread in a bigger picture of mathematics and its history.

By following this path, key ideas are presented first, then revisited and reinforced by showing a wider view. Some readers, however, may be impatient to get to the core theorems and will skip the historical background sections, at least at first reading. Others, in search of a big picture from the beginning, may begin by reading the historical background and then come back to fill in details.

From a review of the second edition:"This book covers many interesting topics not usually covered in a present day undergraduate course, as well as certain basic topics such as the development of the calculus and the solution of polynomial equations. The fact that the topics are introduced in their historical contexts will enable students to better appreciate and understand the mathematical ideas involved...If one constructs a list of topics central to a history course, then they would closely resemble those chosen here."(David Parrott, Australian Mathematical Society)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 third edition includes new chapters on simple groups and new sections on alternating groups and the Poincare conjecture. Many more exercises have been added as well as commentary that helps place the exercises in context.

The ethnic composition of Britain's population is changing rapidly as the 21st century progresses. Some commentators predict a catastrophic future of increasing ethnic segregation and community breakdown, whereas others say that the benefits of ethnic pluralism and cultural diversity will lead to a more integrated society. This book addresses many of the key issues and debates associated with ethnicity and integration in Britain. It provides the reader with an enhanced understanding of ethnic population change and residential concentration, ethnic household dynamics, internal and international migration, the relationship between ethnicity and health, crime, identity and language, as well as ethnic population projections. Collectively, the findings presented here constitute an evidence base for policymakers and practitioners to draw upon when formulating solutions to the range of problems at local, regional and national level that are associated with an increasingly multi-ethnic society.

This book is the third volume of the interdisciplinary series entitled `Understanding Population Trends and Processes: Volume 1 reports the findings of research on fertility, living arrangements, care and mobility, whilst Volume 2 contains selected studies of spatial and social disparities in Britain using various secondary data sets.

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).

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).

This book provides a full overview of land-use change simulation modelling, a wide range of applications, a mix of theory and practice, a synthesis of recent research progress, and educational material for students and teachers. This volume is an indispensable guide for anyone interested in the state-of-the-art of land-use modelling, its background and its application. The CD included with the book provides demonstration versions of well-known land-use models.