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.

Good to hear… I just saw this volume in a bookstore last weekend (having seen no pre-publicity for it?) and, leafing through it, thought it looked darn good… and different. Hope to start reading it next week, and now even more anxious to do so!

I was never sure what was going on with this book. It is serious or a parody (like in the Pynchon chapter you mention) and we’re being made fun of? Does he dislike people who make tons of money or is Jim Simons OK? What about the quants, are they cogs in the machine or morally responsible? And what about Frenkel? I don’t think he likes Frenkel.

Ninguem,

I don’t think it’s ever a parody, mostly it’s completely serious, but not always. Harris is not a black and white thinker, and has a well-developed sense of humor. I can’t speak for him, but I can assure you it’s possible to have a more complicated view of Jim Simons and the money coming into math research from the finance industry than just “it’s bad” or “it’s good”. I don’t see any evidence at all that he dislikes Frenkel. In terms of his mathematical research I know he’s quite interested in geometric Langlands and the things Frenkel has worked on.

Michael Harris has told me that a series of questions and answers about the book will soon appear at the Princeton University Press blog site. One of them addresses Ninguem’s question:

3. The text refers to any number of controversies and polemics, historical or contemporary. But the author doesn’t come down clearly in favor of a solid position on anything. Is this a “postmodern” book? Or does the author just not care?

I am certainly opinionated about a great many things, and it is my considered opinion that most of the sharpest controversies — like platonism vs. nominalism, or positions on what Wigner called “the unreasonable effectiveness of mathematics” — miss the features that make it really interesting to be a mathematician. To avoid distracting the reader with pointless polemics, I consciously chose to present those features with a minimum of ideological adornment, and to allude to controversies only obliquely. I’m told there’s a risk that some will find it disorienting to read a book about mathematics that doesn’t tell them what to think; but it’s a risk I’m willing to take.

[Regarding the controversy over the role of mathematics in finance: the allusions in the book are admittedly hardly oblique, but I tried to limit myself to reporting positions that have already been expressed in the press. I have my own opinions, of course, but why should anyone care what I think?]

It appears Harris’ title is referencing (refuting?) Hardy’s famous essay. Is there more to the relationship than the cover?

Art,

Yes, it’s definitely a reference to the Hardy book, and there’s quite a lot of discussion of the Hardy book in Harris’s. In some sense, the Hardy book (while very different) is the closest historical analog of this one, in the sense of having many of the same themes: why mathematicians do mathematics, the role of mathematics in the wider culture, etc.

Thanks very much. It’s even nicer to have two treats in store. (I never got around to reading Hardy; I see it’s now free.) Although with your notes and the MIT QM course life will be busy…

Thanks again for reviewing another popular math book.

Love and Math is one of the best books on the subject I’ve read in several years, especially since Frenkel touches on ideas of mathematical realism and maths amazing ability to uncover the deepest structures of our universe. Now that you say this one is on par with it, I’ll definitely have to make a stop at the local bookstore.