If you’d asked me ten years ago to describe a book I’d love to read that could be characterized as part of an “incredibly unlikely trend in books about math for the general public”, I might have chosen “brilliant meditations on the practice of mathematics and on mathematics at the deepest level, from first-rate mathematicians, focusing on the Langlands program, with expert-level discussion of the subject.” And yet, here we are, not much more than a year after Edward Frenkel’s Love and Math, with the publication last week of another very different but equally fascinating example of exactly this trend: Michael Harris’s Mathematics Without Apologies. If you are interested at all in what mathematics really is and what the best mathematicians really do (and you’re up for an intellectual challenge) I highly recommend that you get a copy and set some time aside for delving into this unusual book.

While Harris shares many of Frenkel’s themes and concerns, his style is very different, favoring density, indirectness, the post or post-post-modern, and deep engagement with history, philosophy and sociology. Only one of these two authors assumes a familiarity with Max Weber. Where Frenkel is ever guileless and straight-forward, Harris has a whole chapter on the “trickster”, taking some pride in being known for “Harris’s tensor product trick.” While reading, more than once one wonders whether one is really supposed to take something seriously (for instance, there’s quite a long bit about Thomas Pynchon’s novels and conic sections…).

Normally when I’m reading a book I want to later write about, my practice is to fold down the corners of pages that contain something new, unexpected, especially insightful, or something I’d really like to argue with. Then I can start writing by reviewing those pages. My problem with this book is that I ended up folding down the corners of a large fraction of the pages, so when I sat down to write, my usual method would force me to reread pretty much the entire book. Not a bad idea, since I’m convinced I missed a lot the first time through, but other tasks beckon and it’s not a quick read.

I’m not sure I can do much better here than randomly list a few of the themes of the book: the pleasures of doing mathematics, the role of pure mathematicians in society (Wall Street!) and many forms of art and culture, how best to explain number theory to an insightful actress, the philosophy of mathematics and philosophy of Mathematics (two different things), Indian Metaphysics, n-categories, the yoga of motives, Voevodsky’s univalent foundations, the life and thought of Alexander Grothendieck and Robert Langlands, etc., etc. There’s also serious doses of sex (including an extensive discussion of Frenkel’s film), drugs (from Erdos to Andreas Floer to late nights at Oberwolfach) and rock and roll (from the “Math Rock” genre which I’d never heard of before to the IAS house band “Do Not Erase”).

Harris manages to move back and forth between the deepest ideas about mathematics at the frontiers of the subject, insightful takes on the sociology of mathematical research, and a variety of topics pursued in a sometimes gonzo version of post-modern academic style. You will surely sometimes be baffled, but definitely will come away knowing about many things you’d never heard of before, and with a lot of new ideas to think about.

For some more about the book, including some early versions of some chapters, see Harris’s website here.

**Update**: Princeton University Press now has a Q and A with Harris about the book up here.

**
Update**: The book now has a blog.