Readers who (like the author) persist in wondering what I was trying to say after they have finished the book may find it useful to take the book’s subtitle more literally. If you believe mathematics is a problematic vocation, it doesn’t necessarily follow that you believe that the problems have solution, much less that the book’s author has found them. Just identifying the problems may help to clear up misunderstandings (for example, that certain questions necessarily have answers). Assigning problems to appropriate categories may be even more helpful. With this in mind, here is a short but far from exhaustive list of some of the problems examined (but not solved) in MWA, divided among four categories: ethical problems (taking a stance on one of these problems entails an ethical commitment, and it is difficult to avoid taking a stance); historical problems (what appears to be a question about some intrinsic feature of mathematics is better addressed by investigating the questions’ history; best left to historians); linguistic problems (the imaginative resources we can apply to understanding the problem are limited by our language); and other.
1. Is mathematics elitist and/or hierarchical, and must it be? (Mainly addressed in Chapter 2)
2. Who should pay for mathematics? (Mainly addressed in Chapters 3, 4, and 10)
I wrote a three-part post about this last spring after realizing that what I wrote in the book lent itself to misinterpretation. But the misinterpretations continue, in part as a result of some of the most recent reviews so I will write another post on this topic that I hope (but don’t expect) will settle it once and for all.
3. Are mathematicians responsible for the uses to which our work is put? What are the implications of “Faustian bargains” with funders? (Mainly addressed in Chapters 4 and 10)
4. How to explain number theory (or topology, or dynamical systems) at a dinner party? (Mainly addressed in the obvious place)
5. Must mathematicians have bodies? (Addressed briefly in Chapter 6 and even more briefly in Chapter 7)
1. Does mathematics belong to high or low culture? (Mainly addressed in Chapter 8)
2. Should mathematics be seen as an Art or a Science, or both? (Mainly addressed in Chapters 3 and 10)
3. How are Foundations of mathematics chosen? (Mainly addressed in Chapters 3 and 7)
1. Does mathematics have a beginning and an end? (Mainly addressed in the short first and last chapters, and in Chapter 7)
2. Is mathematics created or discovered? (Mainly addressed in Chapter 7; also in Chapter 3. See also realism vs. nominalism, etc.)
1. Is the image of mathematics in popular culture accurate, and if not, what can be done about it? (Mainly addressed in Chapters 6 and 8, as well as in the “How to Explain” dialogues)
2. What does mathematics have (structurally and socially) in common with the arts, especially the visual arts? (Mainly addressed in Chapter 3 and Chapter 10 and near the end of Chapter 8)
3. What should mathematicians write or think about the motivations of literary authors, specifically (but not exclusively) authors of fiction, who make allusion to sophisticated mathematics in their writings? (Mainly addressed in Bonus Chapter 5, but also in the Science Wars)
I was inspired to write this list by some of Mike Shulman’s recent comments (this one, for example), and more directly after reading a recent post on the always-engaging MathTango website. The inimitable Shecky Riemann showed excellent taste in choosing Siobhan Roberts’s biography of John Conway as his “favorite popular math book” of 2015. And I can’t fault his taste in picking MWA for the second slot. Shecky writes:
Harris, more than any mathematician I’ve read, has a knack for saying things that sound interesting, but that are just vague or ambiguous enough to leave one uncertain of what his exact point is. That sounds like a criticism, but in some perverse way it makes his writing all the more thought-provoking and engaging…
Perhaps the exact point is that I’m uncertain (and maybe you should be too).