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Elements of Mathematics: From Euclid to Godel, by John Stillwell
Get Free Ebook Elements of Mathematics: From Euclid to Godel, by John Stillwell
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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.
- Sales Rank: #65199 in eBooks
- Published on: 2016-05-31
- Released on: 2016-05-31
- Format: Kindle eBook
Review
"[Stillwell] writes clearly and engagingly. . . . [Elements of Mathematics] can appeal to various constituencies at different levels of mathematical sophistication."--Mark Hunacek, MAA Reviews
"A great exploration of elementary mathematics, its limitations, how infinity complicates things, and how various branches of mathematics fit together."--Antonio Cangiano, Math-Blog
"John Stillwell uses his broad and impressive command of mathematics to transport a reader through each topic and to a higher level of understanding and questioning. . . . Stillwell's writing is clear and direct, embellished by illustrative anecdotes and examples. . . . I consider Elements of Mathematics to be thought provoking, an informative and worthwhile read and a valuable reference."--Frank Swetz, Convergence
"An accessible read. . . . Stillwell breaks down the basics, providing both historical and practical perspectives from arithmetic to infinity."--Gemma Tarlach, Discover
"[A] sophisticated treatment of topics usually described as elementary."--John Allen Paulos
From the Back Cover
"The coherence of mathematics is on display in this masterful, beautifully written synthesis. With probing questions, lucid explanations, and fascinating examples and stories, John Stillwell reveals where the seemingly separate branches of math came from, how they intertwine, and why the hidden unifier is the deepest idea of all: infinity."--Steven Strogatz, Cornell University and author of The Joy of x
"This is a beautifully written overview and excursion through elementary mathematics, written to appeal to nonspecialists. The book is distinguished by its breadth of scope and precise details. It will be a classic in the field."--David Bressoud, Macalester College
About the Author
John Stillwell is professor of mathematics at the University of San Francisco. His many books include Mathematics and Its History and Roads to Infinity.
Most helpful customer reviews
8 of 8 people found the following review helpful.
nice panoramic view
By Oliver Knill
Like all books from Stillwell, it is a well structured and written book with a treasure trove of things. I teach a course "teaching math from a historical perspective" now for the 8th time which has a similar scope: a panoramic view over many different topics in math. I had used an other book from Stillwell (mathematics and its history) before for that class. The "elements" would be even better suited. Every chapter finishes with a section on "philosophical remarks" and a section with "historical remarks". Part of the book goes further than any other book for a more general audience: examples are "reverse mathematics" or the "Koenig infinity lemma". The book will definitely enrich any course with a broader scope.
35 of 36 people found the following review helpful.
"Elementary," but full of sophisticated insights
By J. Holt
A lovely traversal of "elementary" topics in number theory, computability, algebra, geometry, calculus, combinatorics, probability, and logic. Although I was familiar with most of the material, Stillwell's presentation was full of new insights. The connections drawn between topology and combinatorics (e.g. how the Bolzano-Weierstrass theorem is a special case of the K�nig infinity lemma) were particularly enlightening. And some more advanced topics are taken up at the end of the book (like the unsolvability of the isomorphism problem for groups). Stillwell's historical and philosophical comments were great fun, and his remarks on "reverse mathematics" (which gauges the strength of the axioms needed to prove a theorem) gave me my first taste of this relatively new field. Experts will find this book a breezy and enjoyable read, and beginners will learn a lot thanks to Stillwell's characteristically clear exposition.
4 of 5 people found the following review helpful.
Interesting and informative but at times difficult
By TWMiller
As someone who has taken many advanced calculus and introductory probability courses many years ago, I found the coverage of these and related subjects and of Euclid's Elements quite good. In several cases, particularly infinite series, the book clarified some mysteries I never did understand as a student.
On the negative side, I fault the book format, i.e., being an image of pages and not a digital Kindle, made reading very difficult at times. In particular figures were often discussed on one or rwo pages prior. Plus the author's use of notation sometimes assumed the reader was reading the book in one or two sittings -- not the case for me.
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