Geometry: Our Cultural Heritage

It is unfortunate that today, at a time when mathematics is more important than ever, phrases like math avoidance and math anxiety are very much in the public vocabulary. Making a serious effort to heal these ills is an essential task. Thus the book also aims at an informed public, interested in making a new beginning in math For the 2nd edition, some of the historical material has been expanded and numerous illustrations have been added, as has a chapter on polyhedra and tessellations and their symmetries. A large number of exercises with some suggestions for solutions is also included.

Geometry: Our Cultural Heritage: Audun Holme: av-store.com: Books

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Please try again later. 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. 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.


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A familiar counterexample is in asking "Did the husband stop or not stop beating his wife? The concerned Law of Excluded Middle is accordingly restricted.

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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.


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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".

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.

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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.


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If the barber shaves ALL those not shaving themselves, he cannot be among them, since he would contradictorily shave himself.

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