- Series: Dover Books on Mathematics
- Paperback: 368 pages
- Publisher: Dover Publications; Revised ed. edition (February 1, 1995)
- Language: English
- ISBN-10: 0486284247
- ISBN-13: 978-0486284248
- Product Dimensions: 0.8 x 5 x 8.8 inches
- Shipping Weight: 13.6 ounces (View shipping rates and policies)
- Average Customer Review: 52 customer reviews
- Amazon Best Sellers Rank: #102,618 in Books (See Top 100 in Books)
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Concepts of Modern Mathematics (Dover Books on Mathematics) Revised ed. Edition
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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.