What do pure mathematicians do, and why do they do it? Looking beyond the conventional answers—for the sake of truth, beauty, and practical applications—Michael Harris offers an eclectic panorama of the lives and values and hopes and fears of mathematicians in the twenty-first century, assembling material from a startlingly diverse assortment of scholarly, journalistic, and pop culture sources.
Princeton University Press catches up with Michael Harris, author of Mathematics without Apologies, to talk about the culture of math and what writing has to do with the pace of innovation.
PUP: What is the book about?
MH: The preface claims the book is “about how hard it is to write a book about mathematics.” This becomes less self-referential and paradoxical if the sentence is completed: “… without introducing distortions that transform the book into one about certain conventional images of mathematics.” One thing I had to learn when I started trying to explain what it means to be a mathematician was that the point of an activity like mathematics doesn’t speak for itself through the products of the activity. If you try to find a simple definition of mathematics you’ll see it’s not so easy. As a first approximation we might say that “mathematics is what mathematicians do, plus the stories that are told about that.” The book is then about mathematics in that sense, with an emphasis on the stories, and not only the conventional ones, nor only the stories told by mathematicians.
Why did you write this book?
MH: For a long time I have been hoping to see a book about mathematics, for the non-specialist public, that broke with stereotypes and clichés and a predictable stock of references, and instead reflected the values to which mathematicians refer when we talk to one another. At the same time, I hoped the book, while not being a historical study, would at least acknowledge that these values have a history, and would take seriously the idea that mathematics also belongs to cultural history, by exploring the roots of some of the notions and habits of thought that mathematicians take for granted, using the tools of cultural analysis—but without adopting the elevated tone that is too common in this kind of exercise.
I have written a few book reviews and articles with these hopes in mind, waiting for someone to take the hint. In recent years several mathematicians have made a valiant effort to challenge stereotypes by writing about mathematics as a living activity, and a few writers have examined mathematics through the lens of cultural criticism; but it’s still sadly the case that when mathematicians write the word “culture” the reader can nearly always expect a dose of uplift. Soon enough I realized I would have to write the book myself.
There’s a more selfish reason as well: I thought it would be prudent to develop a second skill, to prepare for the dire moment when the pace of new developments in my mathematical specialty began to outstrip my ability to keep up with them, and I would need to find a different way to keep my brain occupied. Writing was the only plausible option. Strangely enough, when I reached the end of the book I found I could still function reasonably well as a mathematician, even though the pace of innovation in my field has suddenly accelerated—but that’s another story.
The text refers to any number of controversies and polemics, historical or contemporary. But you don’t come down clearly in favor of a solid position on anything. Is this a “postmodern” book?
MH: 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.
What’s with all the endnotes?
MH: Two of the blurbs describe the author as “erudite,” which is a kind thing to write but is unfortunately far from the truth. It’s amazing how easy the internet has made it to look well-read; it helps to think of asking questions different from the ones that are usually asked. The endnotes and the extensive bibliography are there, in the first place, to convince the reader, that mathematics really does deal intimately with an extraordinarily varied range of experience. I hope in particular that genuine scholars can use this material to expand their sense of what’s relevant in writing about mathematics.
In the second place, the notes are there to convince the reader that I didn’t make things up. But please don’t get the impression that I actually read more than a few pages of most of the references quoted.
The notes are also a convenient hiding place for the author’s true opinions. But what do they matter?
Describe your writing process. How long did it take you to finish your book? Where do you write?
MH: Each chapter started with a clear-cut theme, though some of them led me in unexpected directions. Chapter 8, for example, was supposed to be an exploration of why it’s so important for mathematics to appear to be serious, and specifically why so much is written about the supposed affinity between mathematics and classical music. The “trickster” theme was supposed to serve as an indirect way of introducing the question of mathematical seriousness. But mathematical “tricks” turned out to have such a rich and unfamiliar history that they tricked themselves into the chapter’s main theme.
Each chapter’s theme evolved as I collected relevant material. Some of the material organized itself into a plausible narrative outline. Then the actual writing began. The individual paragraphs were easy enough to complete, but assembling them in a coherent order often enough presented an impossible mathematical problem: I need to talk about B before I can explain C, and B is incomprehensible until I talk about A; but it makes no sense to bring up A without having already mentioned C. Resolving this kind of problem is what took up most of the time between when I started writing in early 2011 and when I submitted a completed manuscript three years later. Usually it was only possible in a state of total isolation, which I could only maintain for a few days at most.
At the end I found myself discarding enough material for at least two books the same length. But there’s no reason to write them, because they would say the same thing!
Who do you see as the audience for this book?
MH: Anyone who is willing to take seriously the idea that mathematics deserves respect, not only because it can be used to provide efficient solutions to practical problems (though that is eminently worthy of respect), but also as a living community, a cultural form, an autonomous domain of experience.
Check out the introduction to Mathematics without Apologies here. The book was recently reviewed at Library Journal and Peter Woit’s Not Even Wrong.