Geometry: Our Cultural Heritage [Hardcover]

Audun Holme (Author)

Book Description

Publication Date: March 5, 2002

Mathematics is more important than ever, but phrases like "math avoidance" and "math anxiety" are very much in the public vocabulary. In addition to providing an invitation to mathematics in general, this book emphasizes the dynamic character of geometry and its role as part of the foundation for our cultural heritage. Aimed at an informed public and future teachers of mathematics, it seeks to heal the ills of math phobia in society.

Editorial Reviews

Review

"... 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

From the Back Cover

The book aims at future teachers of mathematics. 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. An important task is seriously attempting to heal these ills. Ills perhaps inflicted on students at an early age, through deficient or even harmful teaching practices. Thus 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. The book is divided into two parts. Part 1 is called A Cultural Heritage. The section constitutes an excursion into the history of geometry. It also contains material which is normally not included into a mathematical text. For example, we relate some of the stories told by the Greek historian, Herodotus. Apart from being good reading, some of the early geometers were very much part of these events.

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 Best Sellers Rank: #2,520,275 in Books (See Top 100 in Books)

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6 of 26 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
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This review is from: Geometry: Our Cultural Heritage (Hardcover)

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.