He simply shows that the algebra we learn in school is not the only possibility -- more a philosophical point than practical. Regarding the business of why (-) x (-) = (+), the answer in this book is "well, that's a choice, there are other possibilities," and he begins to explore that a bit. Obviously its not a bad choice, but people have been confused for eons thinking it should be more obvious somehow (of course it can be "proved" as well, given you accept certain other axioms.) Not quite at the level of math infamy as the fifth postulate, but along those lines.
>Indeed, Mazur's 'virtuous circles' ("virtuous cycles", I would believe) are only a (relatively small) portion of 'deeper understanding:
I was too hard in my words, its a good discussion in "Imagining Numbers", but he never points out the other possibilities.