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He is not a true man of science who does not bring some sympathy to his studies, and expect to learn something by behavior as well as by application. It is childish to rest in the discovery of mere coincidences, or of partial and extraneous laws. The study of geometry is a petty and idle exercise of the mind, if it is applied to no larger system than the starry one. Mathematics should be mixed not only with physics but with ethics; that is mixed mathematics. The fact which interests us most is the life of the naturalist. The purest science is still biographical.
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Negative Math: How Mathematical Rules can be Positively Bent, Alberto A. Martinez, 2005, 280 pp., 32 illustrations. Cloth, $24.95, ISBN 0-691-123098-8. Princeton University Press, Princeton, NJ, 08540 http://pup.princeton.edu.
The author states that the main goals of his book are to make the reader aware that: "some aspects of traditional numerical algebra do not correspond to everyday experience; new mathematics can be devised by modifying traditional rules and new mathematics can describe aspects of the physical world.” He stresses that the focus of the book is on elementary numerical algebra and that its contents will be accessible to any reader familiar with basic high school mathematics.
With this perspective, and as a mathematics educator, I thought perhaps that this book might primarily be a resource for teaching and continued reading. I found the contents interesting but it was not an easy read. The book contains much material. It is information dense and includes the work of almost every great mathematician that ever considered negative numbers and how mathematics developed from their work. The author quotes Francis Maseres as saying "modern historians of mathematics have virtually ignored the logical foundation of negative numbers through the middle of the Nineteenth century." Many readers might be surprised to learn how abhorrent negative numbers were to many mathematicians. For example, as late as 1803, Lazare Carnot spoke of the idea of a negative quantity by arguing "the theory is completely false and invincibly conduces to error."
I found Chapter 7, "Making a Meaningful Mathematics", particularly interesting. Here the author leads the reader towards devising new systems of rules and manipulating arbitrary symbols, techniques most of us did not learn at the university. Martinez has certainly undertaken thorough research on the subject of negative numbers; however, he ignores the work of non-western contributors to this field. The book contains extensive references and a suggested “Further Reading” list.
Personally as a secondary teacher and mathematical historian, I found many ideas that could be adapted to classroom teaching. But I do not feel other secondary school teachers who do not have a similar background in the history of mathematics would benefit from the reading as I have. While many teachers enjoy using historical vignettes in their classrooms, they prefer the history and pedagogy already combined. This book does not suit such needs. On the tertiary level, the book is not an appropriate text for a history of mathematics course. However, Negative Math has the potential to serve as a resource for a seminar course.
As a final note to a more mathematically inclined reader, even though Martinez intended the level of mathematics in his book to be for a general reader, this does not lessen the value of this book for other readers. Rather, because the material examines the ideas and work of great mathematicians, it should appeal to a wide audience.