Geometry: Our Cultural Heritage (Hardcover)

~ (Author) "Mankind must have possessed knowledge about geometric phenomena as far back as our historical records take us, and undoubtedly even much further back into the..." (more)
Key Phrases: continued proportionality, verging problem, affine restriction, Roman Empire, Euclid's Elements, Catastrophe Theory (more...)

Editorial Reviews

Review

From the reviews:

"... The book can be recommended as an excellent textbook for several possible courses (such as Historical topics in geometry, or Introduction to modern geometry). It can be recommended to historians of mathematics, and to professional mathematicians, teachers and students wanting to understand the origins of modern geometry."

European Mathematical Society Newsletter, p. 30, June 03

"... As the author states, the book aims at working resp. future teachers but also at the "informed public", and this readership will really profit from it."

G.Kawohl, Monatshefte für Mathematik 140, Heft 4 (2003)

"This book, or at least the first half of it, should really be compulsory reading for everyone who ever teaches mathematics. It contains a wealth of (his)stories that would make a course in mathematics, in particular in geometry, interesting and exciting. ...In conclusion I would say that the book is really about mathematical culture; interesting things that every mathematical teacher should know are well explained. ..."

Hendrik Van Maldeghem, Belgian Mathematical Society - Simon Stevin Bulletin, 2003, p. 159-160

"This is a wonderful book based on lectures on geometry given by the author to undergraduate students at the University of Bergen, Norway. ...

Summarizing: I think that this book is a beautiful treatment of geometry. ... I strongly recommend this book to instructors who want to find a textbook for a comprehensive undergraduate geometry course, or for lecturers considering a presentation on geometry for the general public. The book will be very useful also for high-school and undergraduate interested in studying geometry. A receptive and informed general public interested in mathematics will find in the first (and, maybe, second) part of the book a very readable and fun approach to geometry."

Mihaela Poplicher, The MAA ONLINE book review column, for complete review see http://www.maa.org/reviews/holmegeom.html

"Holme .. offers a highly readable survey of geometry. ..."

D.P.Turner, Choice October 2002

"... To my mind this book delivers both a broad survey of the history of geometry and - at a few well-chosen points - a deeper insight. It impressively works out the importance of geometry as one of the gresat achievements of human culture. The way this book is written makes it capable of conveying this idea to many culturally interested people who do not shy away from getting involved into mathematics. I especially recommend this book to students and teachers in mathematics and related fields."

J.Lang, Internationale Mathematische Nachrichten 2002, Vol. 56, Issue 191

"... Im Gegensatz zu vielen anderen Geometriebüchern wird in dem vorliegenden Band kein Aufgebot an speziellen Bezeichnungen verwandt. So kann man eigentlich in jedem Kapitel anfangen zu lesen, bzw. das Buch auch als Nachschlagewerk verwenden. Damit wird es auch dem vom Autor sich selbst gesetzten Ziel gerecht, die Leser nicht durch eine pedantische und formalistische Presentation von der Dynamik und Schönheit der Geometrie abzulenken. Wie Holme in seiner Einleitung schreibt, will er insbesondere unseren künftigen Lehrern und der Mathematik interessierten Gemeinschaft in seinem Buch ein umfangreiches und abgerundetes Bild der Geometrie präsentieren. Gleichzeitig soll die Monographie auch als Grundlage für Vorlesungen über Geometrie dienen. Meiner Ansicht nach ist der Autor seinem Ziel gerecht geworden."

Ch.Birkenhake, Jahresberichte der DMV, JB 106. Band (2004), Heft 3

"... Wer sich einen im historischen Kontext verankerten Überblick über die Geometrie verschaffen möchte, ist mit diesem Buch sehr gut bedient. ... es kann durchaus als Klassiker gelten."

H.Walser, Elemente der Mathematik 2004, Vol. 59

"This book discusses central themes from the history of classical and modern geometry … . The book can be recommended as an excellent textbook for several possible courses (such as Historical topics in geometry, or Introduction to modern geometry). It can be recommended to historians of mathematics, and to professional mathematicians, teachers and students wanting to understand the origins of modern geometry." (EMS , June 2003)

"Audun Holme has written an elementary book on geometry … aimed primarily at American mathematics teachers in both elementary and high school, as well as at 'an informed public interested in making a new beginning in mathematics.' … As a popular account of one area of mathematics, [Holmes] has some attractive new features. … it provides a sense of the drama of mathematical developments and their role in world culture. This could be a valuable contribution." (H. Wu, Notices of the AMS, Vol. 51 (5), 2004)

"This is an introduction in selected parts of geometry together with a detailed view on the importance of geometry in the history of mathematics. … As the author states, the book aims at working resp. future teachers but also at the 'informed public', and this readership will really profit a lot from it." (G. Kowol, Monatshefte für Mathematik, Vol. 140 (4), 2003)

"Holme (Univ. of Bergen, Norway) offers a highly readable survey of geometry. … Readability is enhanced by the inclusion of more than 160 figures. … Upper-division undergraduates through faculty." (D. P. Turner, CHOICE, October 2002)

"This book … should really be compulsory reading for everyone who ever teaches mathematics. It contains a wealth of (his)stories that would make a course in mathematics, in particular in geometry, interesting and exciting. … I must admit I learned a lot reading these chapters. … In conclusion I would say that the book is really about mathematical culture; interesting things that every mathematical teacher should know are well explained." (Hendrik van Maldeghem, Belgian Mathematical Society, 2003)

"This is a wonderful book based on lectures on geometry given by the author to undergraduate students at the University of Bergen, Norway. … I think that this book is a beautiful treatment of geometry. It might very well serve as textbook for geometry courses … . I strongly recommend this book to instructors who want to find a textbook for a comprehensive undergraduate geometry course … . The book will be very useful also for high-school and undergraduate interested in studying geometry." (Mihaela Poplicher, MAA, 25.01.2003)

"To my mind this book delivers both a broad survey of the history of geometry and –at a few well chosen points – a deeper insight. It impressively works out the importance of geometry as one of the great achievements of human culture. The way this book is written makes it capable of conveying this idea to many culturally interested people who do not shy away from getting involved into mathematics. I especially recommend this book to students and teachers in mathematics and related fields." (J. Lang, Internationale Mathematische Nachrichten, Vol. 56 (191), 2002)

"It has of course never been possible to give a full and final account of geometry as such, at an undergraduate level: A carefully considered selection has always been necessary. The present book constitutes the main central themes of these selections. … One of the groups I am aiming at, is future teachers of mathematics. … the book also aims at an informed public, interested in making a new beginning in math. And in doing so, learning more about this part of our cultural heritage." (Wordtrade, 2009)

Product Description

This book, aimed at future teachers of mathematics, contains selected topics from the history of geometry, with "modern" proofs of some of the results, as well as a fully modern treatment of selected basic issues in geometry. All too often the geometry which goes into the syllabus for teacher-students presents the material as pedantic and formalistic, suppressing its dynamic character and its role as part of the foundation for our common cultural heritage. The motivation for the book is to open up these aspects of the field. Another motivation is to provide an invitation to mathematics in general. It is an unfortunate fact that today, at a time when mathematics and knowledge of mathematics are more important than ever, phrases like math avoidance and math anxiety are very much in the public vocabulary.

Product Details

• Hardcover: 410 pages
• Publisher: Springer; 1 edition (March 5, 2002)
• Language: English
• ISBN-10: 3540419497
• ISBN-13: 978-3540419495
• Product Dimensions: 9.4 x 6.4 x 1.1 inches
• Shipping Weight: 1.6 pounds (View shipping rates and policies)
• Average Customer Review: 3.0 out of 5 stars  See all reviews (1 customer review)
• Amazon.com Sales Rank: #1,312,854 in Books (See Bestsellers in Books)

Customer Reviews

Most Helpful Customer Reviews

4 of 17 people found the following review helpful:

3.0 out of 5 stars A good classroom account of geometry past and present, November 19, 2004

By	Paul Vjecsner (New York, NY United States) - See all my reviews (REAL NAME)

Amazon Verified Purchase

Noticeable weaknesses are the typos and lapses in grammar on virtually every page. Otherwise the history and current state of geometry are nicely presented, and my objections are again to failings I find to pervade recent geometry and mathematics in general.

I will focus here on the author's discussion of foundations of mathematics (pp.170-176) preceding his more detailed presentations. He speaks characteristically of a high level of precision in present-day logic and set theory, yet I see numerous misconceptions.

He describes logic in terms of "boolean functions", by which a statement P is either true or false, and "No other alternatives exist". This has been contested since Aristotle. A familiar counterexample is in asking "Did the husband stop or not stop beating his wife?" about a husband who never beat her. The concerned Law of Excluded Middle is accordingly restricted.

The author in that vein also assigns to a statement P customary "truth-values", T and F, applied in "truth tables" for composite statements like "P implies Q". Briefly, that statement is held, e.g., true if P is false and Q is true. Beside this conflict with the common understanding of implication, also phrased as "if P then Q", the example conflictingly supposes P true in the implication and false in the truth-value.

One other failing I might consider in that part of the book is the misunderstanding (by Russell also) and consequent widespread influence of "Russell's paradox". It posits

THE SET OF ALL SETS NOT MEMBERS OF THEMSELVES.

For instance, the set of all chairs (not itself a chair) qualifies, but not the set of all non-chairs. It is then observed that if the posited set is a member of itself then, since not in the set, it is contradictorily not a member of itself, and if it is not then, since it qualifies, it is.

A like paradox has been noted in considering

A MAN WHO AS BARBER IN A VILLAGE SHAVES ALL MEN IN IT NOT SHAVING THEMSELVES.

It has been tried to dismiss this, in efforts to view the paradox as a problem in set theory. But the problem is merely an unrelated apparent contradiction.

If the barber shaves ALL those not shaving themselves, he cannot be among them, since he would contradictorily shave himself. And if he shaves ONLY those not shaving themselves, he must be among them, since he would otherwise again contradictorily shave himself. Hence he cannot both shave all and only those in question, but either the first or the second.

Likewise, the concerned set cannot include both all and only those in question, but either one or the other.