Martin H. Weissman, Professor of Mathematics at University of California, Santa Cruz, has recently published a book with the AMS called An Illustrated Theory of Numbers. How does one illustrate number theory? Weissman does it in a visually appealing and pedagogically effective way. Assuming only a high school background in algebra and geometry, the book takes the reader on a journey through the classical works of Fermat, Euler and Gauss, cutting edge topics including the Riemann hypothesis and the boundedness of prime gaps, and modern applications such as data analysis. As one reviewer put it: “An Illustrated Theory of Numbers is a textbook like none other I know; and not just a textbook, but a work of practical art”.
What made you decide to write this particular book? Was there a gap in the literature you were trying to fill? Did you use existing notes from teaching?
I had taught “elementary” number theory in a variety of contexts: a course for math majors at UC Santa Cruz, a 2-week program for high-school students, various workshops for K-12 teachers. Then I took the famous one-day course with Edward Tufte, a key figure in the “Visualization of Quantitative Information.” I went on a design kick, read lots of books, picked up Python, and decided to turn my disparate number theory notes into a book.
I understood that the market for introductory number theory books was pretty crowded. There are some beautiful older books, but I thought a newer treatment was needed. Among newer books, I was unhappy with the “textbookification” I saw — bulky expensive books, with clunky layout, Wikipedia-like blurbs posing as history, and a sort of writing-by-committee voice (end-of-rant). So I thought a new book could fill the gap. And, of course, there wasn’t an illustrated number theory book!
What are your thoughts on mathematics publishing in general?
There are so many new modes of publishing, interpreted broadly. Math blogs, MathOverflow, projects like the Stacks project, and the arXiv are part of a flourishing ecosystem of mathematical communication. TeX and the internet have enabled wild openness. At the same time, I worry about the consolidation of publishing houses and neglect of math journals and books. Prices have become absurd, to the point where my library has cancelled journal subscriptions and students can’t afford their textbooks. Moreover, I don’t see the editorial or physical quality I would expect when looking at output from the megapublishers. Since I think that edited and physically printed texts are important, I’m worried. The AMS is a bright spot!
Do you have a general philosophy/approach when it comes to the dissemination of mathematics?
Be clear, concise, and correct. Respect your subject and your audience.
How did you decide on the format and style of the book? Did you consider other formats for this book? Open Source? Online Notes? Self-publication?
I was very picky about a few issues. One was the physical format of the book, since I designed it with two-page spreads (intentional left and right pages when opened), extensive marginalia, and color illustrations. Another was cost — number theory textbooks in the market cost around \$150, which I think is absurd.
Open source and self-publication would allow the production of a decent physical book at a reasonable cost (around \$60 when I researched it). But publishers like the AMS provide key feedback, editorial guidance, advertising, and a distribution network. The AMS used 4-color offset printing rather than on-demand digital printing, and I think the physical quality is superior to what I would have found through self-publishing. They also offered a reasonable cost, in my mind.
At the risk of going against the open source ethos, I do think that authors should be paid for their creative work. I think my research is supported by salary from my institution, and so it should be (and is) freely available. But this book was completed primarily on nights, weekends, and summers, and I appreciate the royalty checks. I think that nonprofits like the AMS strike a good balance, respecting the needs of the mathematical community and the needs and rights of authors.
How did you choose a publisher? What was important to you when you made the choice?
Since I had specific physical and cost requirements for the book, that immediately eliminated some large textbook publishers. Anyways, I would rather compete with McGraw-Hill, World-Scientific, and Pearson instead of joining them.
That leaves Springer, University presses like Princeton and Cambridge, and the AMS. The AMS seemed most receptive to actively working with me on the book. It was easy to talk to the AMS editors (thanks Sergei!) and production team as I made all sorts of unusual requests. Fundamentally, the AMS is dedicated to the interests of mathematicians, and that played a big role in my choice.
What was the writing process like? Did you write every day on a set schedule, or did you have periods of setting it aside?
I wrote batches of the book while teaching number theory, at UC Santa Cruz, and in Singapore at Yale-NUS College. It mostly came in bursts of days or weeks when time allowed, which is why it took close to 10 years from beginning to end. Sometimes I could set aside a few hours or a day to make an image. But mostly, I needed large blocks of time to get the sort of focus I needed to write chapters of the book. I finished the book on a family writing retreat in Cambodia and Indonesia in the summer before moving back to the U.S..
Was your writing influenced by other books? Which ones?
For layout, I was certainly influenced by Edward Tufte’s books. I used a LaTeX package called tufte-latex, which imitates his layout and fonts. I was also influenced by his principles for “graphical excellence” in the design of illustrations and the integration of graphics and text. Mathematically, I often tried to go back to the original sources and “masters”. For example, I wanted to write a really clear proof of the uniqueness of prime decomposition. I read through a lot of proofs in a lot of books on my shelf; in the end, I thought the proof in Gauss’s Disquisitiones (Art. 16) was best.
Design and mathematics share common goals of elegance under constraint. So it might be the case that learning about visual design helped me to write mathematics.
Did you find ways to get feedback while writing your book or was it a solitary effort?
The book went through some early drafts as a coursepack for students. Since undergraduate students are the target audience, their feedback was most useful. I also showed some early sections to colleagues, friends, and family. They strengthened aspects of the design, treatment of history, and more. My cat tended to sleep on printed drafts, which might qualify as feedback.
Did you have a special place where you liked to write? How did you stay motivated and focused?
I tend to filter out my surroundings, so I can write at my office or at home or a cafe. Coffee and a good Spotify playlist helped too.
What kind of feedback did you get after the book came out?
I’ve gotten lots of emails out of the blue, and I’ve appreciated every one! I’m a bit embarrassed every time someone finds a typo or error, but I track them (with acknowledgment) at the book webpage illustratedtheoryofnumbers.com. I really enjoy hearing stories from readers — some are teaching with the book, some are working through the book for enjoyment, some are sharing math problems with their kids.
What advice would you give to new authors?
If you have something to share, create something lasting and beautiful. Read blogs like this one to understand what you’re getting yourself into. A practical tip: it’s good to make and track decisions about file directories, layout, indexing, notation, etc., as early as possible. Editing a book-length manuscript is a real headache if you haven’t been consistent along the way.