Michael Spivak: A Memorial (Notices of the AMS)

Extracting Fernando Gouvêa's article on Spivak's Calculus book. I, like many others, used this book as my first introduction to "real" math, and it still has an important place in my heart. But the rest of the memorial is definitely worth reading. I had no idea that Spivak was gay, and it is good to see that representation.

Michael Spivak’s Calculus

By Fernando Q. Gouvêa

Let me begin with some autobiography. I was introduced to calculus during my first year at the University of São Paulo, in Brazil. At that time, we didn’t use a textbook; instead, we used references and small booklets of notes, prepared by one of the teachers. The references were a variety of books that our professors felt could be consulted with profit. Among them was Spivak’s Calculus. I didn’t buy a copy at first, because I was fairly happy with the notes and with the books I already had. But the name stuck in my head.

A couple of months into the course, I was strolling through the aisles of my favorite bookstore, and saw a big gray calculus textbook. I think it was the first edition. It was definitely in soft cover, perhaps an edition for sale in third world markets. I recognized the author’s name, bought the book, took it home, and set to reading it.

It was love at first sight. I’ll admit that I wasn’t doing all—or even most—of the problems. (There are lots of very hard problems; instructors will be glad that there is an answer book.) But Spivak’s account of what calculus is all about, his careful but precise account of the theoretical underpinnings of the material, his chapter on the “hard theorems” (the ones that require an understanding of the completeness of the reals), his pictures, even his asides…This was calculus as an intellectual adventure, deep, compelling, and beautiful. I read the whole book, making my way far beyond what we had covered in class by that point. I even read, with some small level of understanding, the sections on complex power series and the explanation of Dedekind cuts.

It will give you a measure of Spivak’s impact on me to note that I took quite seriously the annotated suggested reading list given in the back. I sought out many of them, read a good chunk of the ones I bought, and understood a few of the ones I read.

I never forgot some bits of the book. Spivak’s comment that “mathematicians like to pretend that they can’t even add, but most of them can when they have to” (on page 179) stuck in my mind. I’ve used it in class many times. His chapter on integration in finite terms is another I remembered, and especially problem 7 in that chapter (described as “Potpourri. No holds barred.”). I remember “the world’s sneakiest substitution” $t=\tan (x/2)$. One of the problems was marked as “An exercise to convince you that this substitution should be used only as a last resort.” The next problem had the note: “a last resort.”

The treatment of power series also stuck with me; as a result, I’ve been a fan of series ever since. I’m grateful that the edition I read didn’t include the section on Kepler’s laws, added in the third edition, because I don’t think I could have followed it. The reading list, alas, was never updated.

A calculus textbook that does everything honestly, even if gently enough so that good first-year students can follow it, is not for everyone. Many people would want to see more applications or more history. In addition, there is no multivariable calculus, which is a pity; I’d have loved to see what Spivak could do with that at this level. (His Calculus on Manifolds is, of course, a classic, but it is so terse as to be impenetrable for most students.)

When I use it in class, students always find the book very hard to read and understand. But they also understand that they are getting the real thing, that they are being treated as intellectual adults. Spivak’s achievement was to produce a textbook that has the potential to lead talented students into an appreciation for both the subject and the methods of mathematics. There must be many mathematicians today who cut their teeth on this book; may there be many more.