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Category: Books - Mathematics

The 21^st century is beginning with a renaissance in mathematics. The previous century was dominated by mathematicians who reveled in the esoteric nature of the new mathematical languages. It was a matter of pride to write papers that only an elite few could access.

Suddenly, in 2003, bookshelves are full of exciting new works written in common English. These books are making mathematics accessible to people who love math, but have not mastered the new mathematical languages. This is a trend worth supporting. I highly recommend any of the following works: (You can view this list of titles at Powells City of Books)

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The Road to Reality; Roger Penrose.

In a comprehensive volume, Roger Penrose takes a look at the mathematical and physical description of the cosmos. Published in 2004, the work includes a nice summary of current thought on the mathematical description of the universe. Fortunately, Penrose neither holds back on his use of mathematics, nor does he fall into the terse indecipherable symbolism that makes too much of modern mathematics inaccessible to the masses. In other words it is the perfect combination of math and prose for the lay reader.

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Isaac Newton; James Gleick, pages 267.

I must give credit to James Gleick for bringing magic back to the study of calculus. Isaac Newton is a short, fun biography of a man who is unquestionably one of the most important natural philosophers of the Western tradition. This short work does an excellent job providing a basic biography of Isaac Newton while introducing us to the many disputes, and intrigues that surrounded Newton's development of The Calculus and modern physics. The book is easy to read. I highly recommend it to anyone who loves mathematics or philosophy.

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Mathematics: From the Birth of Numbers; Jan Goldberg, pages 1093.

Weighing in at almost eleven hundred pages, this tome by Jan Goldberg is one of the most impressive collection of basic mathematical knowledge on the market. The book belongs on the shelves of all math lovers. This is a great gift for high school and college students as it includes in one reference a very concise and readable over view of undergraduate level mathematics. I consider it to be the single most valuable reference in my library.

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Mathematics: The Loss of Certainty; Morris Kline, pages 376.

The Loss of Certainty by Morris Kline does a superb job of presenting the conflicts surrounding the diagonal method and the other odd mathematical and philosophical systems that were thought to deliver absolute certainty. It is one of the best written, accessible mathematical treatise written in the last half century, and looks smart on any library shelf.

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Gamma: Exploring Euler's Constant; Julian Havil & Freeman Dyson, pages 266.

This is probably the hardest reading in my list of popular math books. Julian Havel has no compunctions about jumping into mathematical equations, and does not always provide a text translation of the equation; however, the mathematics is absolutely fascinating. I regret tremendously that gamma was not covered in my mathematical education. Gamma unlocks the mysteries of factorial. I would recommend this book for people who've read Prime Obsession and are looking for more detailed mathematics.

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Abel's Proof; Peter Presic, pages 211.

Abel and Galois developed the foundations for Group Theory in their efforts to prove that the quintic equation could not be solved with an algebraic equation. Group Theory is one of the prizes of modern mathematics and is used frequently in Physics for vectors and other equations.

Like Peter Presic, I was dismayed in my Group Theory classes that, while my professors did a great job in presenting Group Theory, they never went the full distance to demonstrate that the quintic equation was not solvable. I thought it strange, Group Theory was developed in an effort to address the solubility of algebraic equations. The theory proved extremely valuable in physics. The odd result is that a large number of people hearn the theory, but do not learn how to apply it to the actual problem it was designed to solve.

Presic addresses this situation. Rather than looking at Group Theory and the quintic equation in its modern context, Presic talks about the works of Ruffini, Abel and Galois in their historical context. The result is a book that can give the reader a general over view as to why the quintic is not solvable. I would recommend the book to anyone interested in higher mathematics. The book is a must read for any student of physics or mathematics who is studying group theory, as it develops Group Theory in its historical context.

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Prime Obsession; John Derbyshire, pages 304.

John Derbyshire gives an interesting presentation on the history of the Riemann Hypothesis and the ongoing mathematical obsession with the prime numbers. To add a human interest, Derbyshire alternates chapters. He presents a chapter with mathematics followed by a chapter on history. We get to see both the mathematics of great thinkers like Gauss and Euler as well as the context in which they wrote their works.

The author has made great strides to keep the mathematics at a readable level. He has targetted an audience with a precalculus understanding of mathematics. The work introduces concepts such as diverging and converging sequences, exponentials, etc.. I believe the author has done a phenomenal job of combining difficult mathematical ideas in plain readable text. I highly recommend the work to anyone interesting in the history of ideas. The work was published in March 2003.

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Sacred Geometry; Miranda Lundy, pages 64.

Sacred Geometry by Miranda Lundy would be a terrific gift item for students or the intellectually inclined. This short books looks at several different mathematical shapes starting with the simple point and line, and growing to some historically important and elegant geometrical shapes. While from from a precise mathematical treatise, the book does a great job of stirring curiosity and presenting terminology used in mathematics.

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Dialogue on Two New Sciences; Galileo Galilei, pages 300.

It is wise to acquaint yourself with the classic works of Science. Galileo introduces multiple original thoughts on the nature of science through a Socratic Dialog. The work provides insights on the emerging subjects of physics and mathematics. My interests lie in the Galileo's Paradox of the infinite. However, you will find a wealth of knowledge on a variety of subjects.

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Everything and More; David Foster Wallace, pages 319.

David Foster Wallace is a popular modern fiction writer. In Everything and More, Mr. Wallace turns his pen to the metaphysics surrounding modern set theory. Everthing and More does an excellent job of portraying the history of mathematics seen through the veil of modern dialectics. The great villian in the work is Aristotle and logic, the great hero is Zeno of Elea.

In reading the book, you should realize that transfinite theory is still quite controversial. It is both the source and justification for "New Math," and is the primary reason that schools stopped teaching logic. Transfinite Theory is based on the same dialectical methods as Marx's communism, Hegel's historicism and Freud's psychoanalsys. Each of these systems try to build a foundation for a subject by developing a dichotomy between terms. In Cantor's case, he drives a dictomy between the rationals and reals...that is, the rationals are "denumerable" and the reals are not. The catharsis of this thesis/antithesis couple is transfinite theory.

The theory is currently in crisis. Cantor's metaphysical view of space was largely disproven with Quantum Electrical Dynamics and Relativity Theory. In the computer age, as people learn to master digital technologies and learn the intracies of large sets, they are likely to realize that, like Freud, Cantor based his views on a false dichotomy. However, the great debates that swirle around the diagonal method and transfinite theory gave us set theory and have helped us develop our understanding of the irrational numbers.

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