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Concepts of Modern Mathematics (Dover Books on Mathematics) Revised ed. Edition

by Ian Stewart (Author)
4.5 4.5 out of 5 stars 230 ratings
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Some years ago, "new math" took the country's classrooms by storm. Based on the abstract, general style of mathematical exposition favored by research mathematicians, its goal was to teach students not just to manipulate numbers and formulas, but to grasp the underlying mathematical concepts. The result, at least at first, was a great deal of confusion among teachers, students, and parents. Since then, the negative aspects of "new math" have been eliminated and its positive elements assimilated into classroom instruction.
In this charming volume, a noted English mathematician uses humor and anecdote to illuminate the concepts underlying "new math": groups, sets, subsets, topology, Boolean algebra, and more. According to Professor Stewart, an understanding of these concepts offers the best route to grasping the true nature of mathematics, in particular the power, beauty, and utility of pure mathematics. No advanced mathematical background is needed (a smattering of algebra, geometry, and trigonometry is helpful) to follow the author's lucid and thought-provoking discussions of such topics as functions, symmetry, axiomatics, counting, topology, hyperspace, linear algebra, real analysis, probability, computers, applications of modern mathematics, and much more.
By the time readers have finished this book, they'll have a much clearer grasp of how modern mathematicians look at figures, functions, and formulas and how a firm grasp of the ideas underlying "new math" leads toward a genuine comprehension of the nature of mathematics itself.

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  1. ISBN-10
    9780486284248
  2. ISBN-13
    978-0486284248
  3. Edition
    Revised ed.
  4. Publisher
    Dover Publications
  5. Publication date
    February 1, 1995
  6. Language
    English
  7. Dimensions
    5 x 0.75 x 8.75 inches
  8. Print length
    368 pages
  9. See all details
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Product details

  • ASIN ‏ : ‎ 0486284247
  • Publisher ‏ : ‎ Dover Publications; Revised ed. edition (February 1, 1995)
  • Language ‏ : ‎ English
  • Paperback ‏ : ‎ 368 pages
  • ISBN-10 ‏ : ‎ 9780486284248
  • ISBN-13 ‏ : ‎ 978-0486284248
  • Item Weight ‏ : ‎ 2.31 pounds
  • Dimensions ‏ : ‎ 5 x 0.75 x 8.75 inches
  • Customer Reviews:
    4.5 4.5 out of 5 stars 230 ratings

About the author

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Ian Stewart

Ian Stewart

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Ian Stewart FRS is Emeritus Professor of Mathematics at the University of

Warwick and a leading popularizer of mathematics. He is author or coauthor of

over 200 research papers on pattern formation, chaos, network dynamics, and

biomathematics. He has been a Fellow of the Royal Society since 2001, and has

served on Council, its governing body. He has five honorary doctorates.

He has published more than 120 books including Why Beauty is Truth, Professor

Stewart's Cabinet of Mathematical Curiosities, Calculating the Cosmos,

Significant Figures, and the four-volume series The Science of Discworld with

Terry Pratchett and Jack Cohen. He has also written the science fiction novels

Wheelers and Heaven with Jack Cohen, and The Living Labyrinth and Rock Star with

Tim Poston.

He wrote the Mathematical Recreations column for Scientific American from 1990

to 2001. He has made 90 television appearances and 450 radio broadcasts, most of

them about mathematics for the general public, and has delivered hundreds of

public lectures on mathematics.

His awards include the Royal Society’s Faraday Medal, the Gold Medal of the

Institute of Mathematics and Its Applications, the Zeeman Medal (IMA and London

Mathematical Society), the Lewis Thomas Prize (Rockefeller University), and the

Euler Book Prize (Mathematical Association of America).

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4.5 out of 5 stars
4.5 out of 5
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230 global ratings

Top reviews from the United States

charles L. schwennesen
5.0 out of 5 stars What did Professor Stewart say?
Reviewed in the United States on March 3, 2014
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My review joins the chorus of praise for Dr. Stewart's Concepts of Modern Mathematics ("Concepts"). It has been many years since I put down my slide rule and picked up my math diploma. I wanted a refresher and turned to Concepts for help. Its stated purpose is to spread the concepts underlying the "New Math" to a non-specialized audience. Concepts was the perfect choice.

Concepts is organized into twenty short chapters. Dr. Stewart's writing style is informal, enthusiastic, and conversational with a spot of humor. He stresses the primacy of intuition, the role of experimentation in proving utility and the need for rigor when complexity multiplies. In mixing these factors, Dr. Stewart reveals glimpses of how a mathematician works. Abstract algebra, topology and analysis are the "cornerstones" of modern mathematics. The first nine chapters are devoted to an introduction and to teaching axiomatic set-theory at the heart of the "New Math". As Dr. Stewart quips, to study French literature, one must know the French language. Topology is covered in four chapters, real analysis in only one, and applications in five.

Dr. Stewart employs his "toolkit" of "...aims, methods, problems and applications..." to illuminate an imposing intellectual structure. For example, complex numbers and infinite sets can cloud the brightest minds. Wielding his tools skillfully, Dr. Stewart demystifies complex numbers by recasting as modular arithmetic and makes cardinals plausible using bijections with natural numbers.

Sadly, "Foundations", the final chapter, casts doubt on the reliability of modern mathematics. Dr. Stewart visualizes mathematics as an inverted pyramid balancing precariously on only a few assumptions that cannot be confirmed. The chapter includes outlines of Godels famous theorems and Dr. Stewart's opinion that "...it proves the impossibility of an arithmetical proof of the consistency of arithmetic". The remedy, says Dr. Stewart, is to trust theorems that should be true and to hope for logic that someday fulfills its promise. Finally, Concepts is just the beginning of a "hard technical grind" for the serious student.
20 people found this helpful
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Aaa
5.0 out of 5 stars Excellent and interesting overview of modern mathematics
Reviewed in the United States on April 25, 2013
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Ian Stewart's "Concepts of Modern Mathematics" is the best general book that I have read about mathematics. The writing is interesting and clear and reading the book is a pleasure. Stewart walks the reader through the major breakthroughs of mathematics in the last two hundred years by explaining things from a fundamental viewpoint. Ideas are developed with examples, then made general by abstraction, and finally used to prove interesting ideas about the topic.

For example, in the section on axiomatics, he starts with Euclid's axioms and the question "Is Euclid's sixth axiom really necessary?". Then he explains what consistency means in the case of axioms, and finally builds an explanation to show that Euclid was right after all.

If all of this sounds complicated, esoteric, or technical, you should know that it's not. Stewart goes through each idea slowly and with plenty of explanation so that the reader understands ideas as they come. As a math major in college, a lot of the ideas I already knew, but it was still interesting to go over them again and see things I never covered before, like algebraic topology and graph theory. I would recommend this book to anyone who is interested in mathematics or science.
10 people found this helpful
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Vincent Poirier
4.0 out of 5 stars A layman's guide to the basic branches of mathematics
Reviewed in the United States on November 26, 2011
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What is mathematics about? We all learn math in grade school and high school, but that's just arithmetic and very basic algebra, with a dash of set theory, logic, geometry and trigonometry. In this book, Ian Stewart goes over the main branches of mathematics and these go far beyond what we learn in school.

He touches on topics such as abstract algebra, real analysis, topology, logic, axiomatics, and geometry. He explains what groups, rings, and fields are; he explains what continuity is, what functions are, and what counting is.

My favorite section of the book comes at the end of the chapter introducing abstract algebra where Stewart presents a proof that a game of solitaire, where the goal is to eliminate pegs from a board until you are left with one in the center, has only a certain number of possible solutions. Here, Stewart does an excellent job of explaining enough of the topic to allow us to follow through a simple and convincing proof of something pretty interesting.

Apart from that, it's not a bad book by any means but it should have been better. For instance, in the opening paragraph of this review, I wrote "just arithmetic and very basic algebra". Just arithmetic. Think about it: it's amazing that today, everyone can count, can add and multiply, can divide. We all understand percentages, we can all do very basic algebra in our heads. (What's x in 4+x=5?) Shouldn't we feel a sense of wonder at this?

In general Stewart's bird's eye view of mathematics today left me... I'm at a loss for words here, so I'll just say it left me.

Vincent Poirier, Tokyo
6 people found this helpful
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Amazonian
5.0 out of 5 stars A must read for all math undergrads!
Reviewed in the United States on October 2, 2022
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Beautiful written. Professor Ian Stewart makes the difficult easy to absorb and comprehend. His humor and engaging style makes reading mathematics delightful. I truly could not put the book down!
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Roberto Rigolin F Lopes
5.0 out of 5 stars The math majesty in a grand parade of abstractions for layperson
Reviewed in the United States on October 27, 2018
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We are in 1975, Ian is teasing us with bits of modern mathematics. He warms up remembering us that mathematicians are juggling concepts (instead of numbers) mostly for intellectual satisfaction. Impressive jugglers playing with axioms, functions, sets and so on. Everything connected together by powerful and enjoyable concepts from the "queen of the sciences". Here, we must put emphasis on enjoyable because our math majesty is introduced to layperson by a very good teacher. The whole thing is well illustrated with examples and plots in an engaging parade of abstractions; majestic indeed.
4 people found this helpful
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Alberto
5.0 out of 5 stars Excelente libro
Reviewed in Mexico on October 2, 2021
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Redacción sencilla y completa para una introducción al algebra moderna
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Nicolas Ives
5.0 out of 5 stars Um excelente texto para quem busca uma compreensão maior sobre as ideias matemáticas
Reviewed in Brazil on November 18, 2018
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Ian Stewart é um verdadeiro gênio quando se trada da comunicação de idéias e conceitos matemáticos. Neste livro o autor pincela os varias conceitos que fundamentam a matemática moderna, desde as mais fundamentais como conjuntos, números e funções até as mais complexas como topologia e hiperespaços. O autor consegue ainda introduzir um toque de elegância e sofisticação que tornam a leitura satisfatória e compreensiva,até para alguém que já tenha tido contato com a matemática mais avançada que a ginasial.
É importante ressaltar que este livro cumpre um papel análogo à livros de divulgação cientifica, isto é, ele apenas apresenta os conceitos e trabalha os conceitos da matemática moderna de forma ilustrativa e simples, sem utilizar o rigor dos livros que buscam instruir de forma mais profunda sobre os assuntos abordados.
2 people found this helpful
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Amazon Customer
5.0 out of 5 stars Requires a pen and paper
Reviewed in India on October 15, 2018
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If you want to get the maximum the bedroom is not the right place to enjoy this one. Its in between a popular science and a text book. If you consider yourself too poor in math I wouldn't recommend this for you. That said its a great work and got me curious to explore more serious stuff.
4 people found this helpful
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Curtis
5.0 out of 5 stars Clearly written so it is easy to read
Reviewed in Canada on January 10, 2017
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Clearly written so it is easy to read. You should have some familiarity with Maths if you want to read this and get the most out of it.
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ab..c
5.0 out of 5 stars sound accumulation of early degree math syllabus
Reviewed in the United Kingdom on May 20, 2010
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This book is a 'Dover' paperback which, if your have not bought from this publisher before, they tend to re-publish math books previously sold from around the world. Their budget-priced books tend to be in b&w and graphs in grayscale.

First Impressions

What stands out the most is the VERY attractive, organised exploration of linked math topics from the earliest chapters and throughout the book. Its clear the author encourages some theoretical issues before 'labouring' with calculations.

For example, i am attracted at the quick and early linking between 'Sets', 'Functions', 'Group Theory' and (MOD), and 'axiom' mathematics.The reason being if you are up to understanding these connections with the first three areas, then the way the axiom stuff is handled gives a VERY clear way in answering many math problems that are a model for throughout your future studies! The other linear algebraic stuff is (i.m.h.o) less attractive - compaired to other authors - but still well-worth the effort to digest it.

Harder topics

To me, the areas that seemed rather difficult and laboured being the 'Topology' aress. But i can say that the many graphs give a 'gut-feeling' of the topic. The graphs did help and i read the whole book over a long week-end.

Summary

This book has a very pleasent and stimulating style in its explanations, that will be of use for whatever math studies you may continue to follow.
11 people found this helpful
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