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William Byers
How mathematicians think
The book gives plenty of examples of results that didn't seem to make sense. There's the irrationality of √2, the invention of nonEuclidean geometry and the paradoxes of selfreference. Even the introduction of zero to the number system took some getting used to. And there are any number of weird ideas that come from thinking about infinity. Byers argues that these caused consternation when they were first seen, and can still cause confusion when encountered by students today. These problems weren't aren't much solved, rather the decision is made to accept a workable resolution, and this leads to the opening up of new areas of mathematics.
Towards the end of the book Byers moves away from actual examples and towards a more philosophical discussion. I felt that the book lost its way a bit at this stage. In particular, in the last chapter Byers argues that computers will never be able to do mathematics  but he starts off assuming this, so the chapter seems a bit pointless. However, you shouldn't let the later parts put you off. I would recommend the book both to those interested in mathematics, who want a peek at some of the stranger parts of the subject, and to those teaching mathematics (including those teaching themselves), who will see that getting over some of the stumbling blocks of the subject involved more than an appeal to logic.