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# Five Golden Rules: Great Theories of 20th Century Mathematics and Why They Matter

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on 3 August 2001
To make difficult concepts accessible, inevitably you have to be economical with the truth. You have to remain aware of what your audience does and doesn't know, and what lies you have told. Maths teachers do this at every level, so there are some people who are well practised at it.
I had to reach for graduate-level textbooks to work out what Casti was trying to tell me. It ought to be the other way around.
This book is written at a level for the scientifically/mathematically literate reader. So he cannot be allowed to get away with the sloppiness I will now demonstrate.
Casti discusses the concept of convexity in relation to topological spaces, without telling us that convexity is an algebraic or geometric property, not a topological property. Moreover the non-topological nature of convexity is plain to the reader who has understood what went before, since straight lines are not an admissible concept in "rubber sheet" topology. The reader with a little mathematical education (many of the target audience) will further realise that convexity is a property of sets within a larger space, not of a space itself, so to refer to a "convex topological space" is a contradiction in terms.
To illustrate convexity (surely unnecessary for the target readership), he draws some convex and non-convex sets. But some of the non-convex sets plainly possess the Brouwer property that is asserted only for convex sets. There is no explanation of this contradiction. He then asserts that the Brouwer property is held by the surface of a sphere, a set which is non-convex and cannot even be topologically deformed to a convex set.
If the reader was to gain any insight into the Brouwer property, then surely it is to obtain an intuitive understanding of why Brouwer's theorem is true for the surface of a sphere but false for a ring. I would love to understand this. Casti does not even try.
The problems illustrated here are present, to a greater or lesser extent, throughout the rest of the book.
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on 28 November 2015
Received a hardbound perfect copy in quick time
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