Ambiguity + Paradox + Contradiction = Mathematics
Continuing with books to read during the summer, I next suggest William Byers' How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics (Princeton University Press, 2007).
The book has just been released and I have just started reading it.....yet am ready to recommend it! For example, one interesting goal of the author is to expand our view of mathematics and the culture of mathematics beyond the single dimensions of logic and rigor...into the creative world of the "translogical" which is not the same as either nonlogical or illogical. Interested?
A primary focus of the book is a fascinating examination of the role of ambiguity in mathematics, giving special attention to the revealing effects of contradiction and paradox. Then, bypassing logical structure, the author focuses on ideas in mathematics as fundamental building blocks...claiming that "some of the most profound ideas in mathematics arise out of situations that are characterized not by logical harmony but by a form of extreme conflict." The book then concludes with a discussion of the "mystery that lies at the heart of the relationship between mathematics and truth."
This all sounds heavy.....but the author uses accessible examples to illustrate his points and support his arguments. And, the examples are all useable with secondary students...re-introducing the "conflicts" of infinity, visual illusions, non-Euclidean geometry, converging series, perspective, Cantor Set, and fractals.
Admittedly, the book is not an easy read.......if you are willing to pause and reflect on the author's arguments. But, he does develop the necessary mathematics (and metamathematics) along the way....helping reader's remember some things they once knew...perhaps.