Mathematical Evolutions (Spectrum Series)

by Abe Shenitzer (Author), John Stillwell (Author)

2 customer reviews

ISBN-13: 978-0883855362

ISBN-10: 0883855364

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When the techniques of mathematics are integrated with its historical, philosophical, and critical aspects, mathematics becomes more meaningful to mathematicians and mathematical amateurs alike. In this book, its many authors provide such integration by presenting informed accounts of the mathematical ideas and issues of the past 300 years, with emphasis on developments since 1800.

The book deals with ideas that are "classical," that is, or established weight and importance, with special attention paid to those that overlap different domains of mathematics or disciplines such as physics and philosophy. The articles cover a large number of topics in analysis, algebra, geometry, probability, and logic and are grouped by subject matter.

Mathematical Evolutions makes use of the history of mathematics from the perspective of the mathematician rather than the historian. The approach is genetic: from a problem to its solution. It explores mathematical ideas, their uses, and their historical evolution.

The book is lively and readable and of very high mathematical quality. Its cobntributors include some of the finest mathematicians of the 20th century such as Atiyah, Doob, Luzin, Magnus, Tits and Weyl. The book will appeal to a broad audience, from students to professional mathematicians.

Mathematical Evolutions (Spectrum Series)

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Product Details

Series: Spectrum Series
Paperback: 304 pages
Publisher: Mathematical Assn of America (January 2001)
Language: English
ISBN-10: 0883855364
ISBN-13: 978-0883855362
Product Dimensions: 0.8 x 7 x 9.8 inches
Shipping Weight: 1.2 pounds
Average Customer Review: 4.5 out of 5 stars See all reviews (2 customer reviews)
Amazon Best Sellers Rank: #3,427,698 in Books (See Top 100 in Books)
- #2338 in Books > Science & Math > Mathematics > History
- #27013 in Books > Textbooks > Science & Mathematics > Mathematics
- #419815 in Books > History

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7 of 7 people found the following review helpful

How many of the main math ideas developed

By Charles Ashbacher HALL OF FAMETOP 500 REVIEWERVINE VOICE on February 25, 2003

Format: Paperback

Advances in mathematics are no different from those in other fields. A new result is generally the consequence of years, sometimes centuries of effort, by many people. Revolutionary results are rare, with evolutionary being a far more accurate descriptor of the movement of mathematical progress. Although mathematical progress happens faster than evolutionary changes in organisms, there are many parallels.
When a new environment becomes available, many different creatures move to occupy it and within a very short time, new species have branched off and are flourishing. Upon the development of a new area of mathematics, many branches rapidly rise and shoot off in many unexpected directions. Different creatures in isolated environments evolve to fill a common ecological niche and possess almost identical characteristics. Different mathematicians often work on the same problem independently and arrive at similar breakthroughs at roughly the same time.
With all of these similarities, it is only natural that mathematical topics be examined from an evolutionary perspective. In 1993, John Ewing, then editor of the "American Mathematical Monthly" approached one of the editors of this book (Abe Shentizer) with the idea to start a regular column "The Evolution Of . . . ", which was to chronicle the development of a mathematical idea. The columns were to be accessible to a general, albeit knowledgeable mathematical audience. This book is a collection of those columns.
The historical development of some of the basic concepts such as function, integration, optimization, rings and elliptical curves are described in great detail. While formulas are used, they appear only when necessary, which is fairly rare.Read more ›

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1 of 4 people found the following review helpful

Some excellent articles

By Viktor Blasjo on February 16, 2007

Format: Paperback

I shall indicate some features of my favourite articles. Stillwell, "Modular Miracles". Historically, modular functions grew out of the theory of elliptic functions by considering the value of the elliptic integral of 1/(1-x^2)(1-k^2x^2) as a function of the "modulus" k^2. Now, elliptic integrals are analogous to arc trigonometric integrals, and indeed modular functions satisfy equations analogous to trigonometric identities. But trig identities can be used to solve the cubic equation (reduce general cubic to 4x^3-3x=c and use 4cos^3(x)-3cos(x)=cos(3x)) and indeed Hermite discovered that modular equations can be used to solve the general quintic. Geometrically, modular functions are functions periodic with respect to the modular tessellation, which is a tessellation of the upper half-plane by congruent triangles in the sense of hyperbolic geometry; in other words they are invariant under integer linear fractional transformations with determinant 1, which is precisely the equivalence relation for lattice shapes thought of as the ratio of its two generating complex numbers, which explains the connection with elliptic functions. But lattices are also fundamental in the theory of quadratic integers as follows. The set of integers O in Q(sqrt(-D)) is a lattice because it can be formed as the closure under + and *, and so is any of its ideals. Algebraically, one has unique factorisation if every ideal is principal, i.e. of the form aO, i.e., geometrically, Q(sqrt(-D)) has unique prime factorisation if and only if all its ideal have the same shape. Perhaps, then, one could hope that modular functions will be of use in algebraic number theory.Read more ›

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