A few posts back I wrote that I had finally figured out how to answer that question. My solution: to make a list of the main themes. Initially I thought the list would be really long, maybe 20 items. But it turns out the most important questions addressed by the book fit under one of five headings:
If not goodness, truth, and beauty, then what?
In other words, what do (pure) mathematicians see as the aim of their practice? What gives it value? And how far can these values be shared with people outside the profession?
Is mathematics democratic? Is it elitist?
This includes some of the really difficult questions, including the gender question in mathematics (which I largely avoided, but which I will have to address in the next few days), as well as everything related to social and cultural diversity (which the book addressed indirectly). The book is more explicit in discussing (especially in Chapter 2) how the meaning of mathematics is intimately tied with a hierarchical attitude to the evaluation of mathematical results. More on this later.
How to guarantee access to “external goods” while protecting the “internal goods” of the practice?
This refers both to questions of the social responsibility of mathematicians and to the pressure to adopt the entrepreneurial mindset.
The last two are self-explanatory:
The gap between the image and reality of mathematics: in popular as well as “legitimate” culture, and among mathematicians themselves.
The communication question (“How to explain number theory at a dinner party”).