Michael Spivak is one of my favourite mathematicians and expositors of mathematics. It's a pity that it's difficult to find biographical information about him. I only know him through his *wonderful* expositions of mathematical analysis, differential geometry, and topology. The non-mathematical audience may also have some acquaintance with him through his Spivak pronouns.

### A Figure Clad in Mystery

Spivak has no biographical entry in the popular web resource of St Andrew's college for biographies of mathematicians. He apparently sired no student's PhD thesis, because the Mathematics Geneaology Project at this address does not list any descendants, though we do know that Spivak earned his PhD in Princeton in 1964. When searching for papers written by him at MathSciNet, (which requires a registration that most major universities have), all that comes up is a couple of papers on differential geometry and differential topology, published on 1975 and 1993 respectively. I'm surprised to also find a presentation of some aspects of classical mechanics from a mathematician's viewpoint published in 2004. I'm not even sure this is even the same Michael Spivak, but I don't see why not. Besides these scant papers, the only other things he published in the mathematical journals is his PhD thesis in 1967. He is still alive as of this noding; he must be in his sixties or so. A Google search also brings up a certain Michael *Lee* Spivak found at this webpage who is interested in mathematical education and gender studies. This might be the same person or perhaps a son of Michael Spivak. Michael Lee Spivak seems to be much too young to be the same person as Michael Spivak, since he lists a defunct link to a personal student page at the University of Illinois, Urbana-Champaign.

The point is that Michael Spivak did not pursue a career as a research mathematician. He did not publish much in the journals. I have read somewhere that at some point he was part of the faculty of the mathematics department of Brandeis University, though right now he is not even listed as a professor emeritus in their webpage. He has a publishing firm that I presume he founded himself, called Publish or Perish, Inc. found at this web address. Reading accounts of him and his Yellow Pigs (of which I will say more later) found at this other web address, it seems that Spivak was a promising young mathematician and undoubtedy a bright fellow. Besides English, he at least speaks German well enough to translate original papers from eighteenth and nineteenth century German into English, and possibly also speaks Russian, for in one of his books he makes a joke involving Cyrillic.

I surmise that what happened is something like this: he began his studies of mathematics under an eminence such as John Milnor. Then he began to run the gauntlet towards becoming a tenured professor (a postdoctoral fellowship, a tenure-track position, publishing papers regularly, etc.), but at some point became completely fed up with the system and its "publish or perish" philosophy. At that point, he left academia, founded his own publishing firm (appropriately called "Publish or Perish") and dedicated the rest of his career to promoting mathematical education and writing textbooks. Not that many text books, not as many as that nut Serge Lang puts out. Rather, Spivak's motto seems to be few but ripe. It is said that Michael Spivak now lives somewhere in Houston, Texas, for at least that's the address given for Publish or Perish, Inc.

### A Text Book Author

Michael Spivak has written the following books, all of which have become highly influential.

*Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus*, © 1965. He seems to have written this book just after he obtained his PhD degree, and he hasn't published a new edition ever since. His exposition here is wonderful as always, though it is noticeably a lot "greener" than what we can see in his later books. Here he presents theorems like the implicit function theorem and its cousin the inverse function theorem. Differential forms are also an imortant stop during this quick tour of differentiation and integration on manifolds. The entire point of this little book (it's short compared to this other works) is to present Stokes' Theorem for manifolds as quickly as possible and with perfect rigour. For celerity, he only considers manifolds as embedded subsets of Euclidean space. There is no loss of generality in doing this, since by Whitney's Theorem, we know that any abstract manifold can be embedded into Euclidean space of sufficiently high dimension. This book requires lots of careful thought, since here Spivak prefers to let the reader do the work rather than doing it himself, although he knows how to assign just the "right" kind of work. He did a good job of stripping down the preliminaries of integration on manifolds and differential forms to the bare essential without requiring too much from the reader except a willingness to undertake some hard work.

*The Joy of TeX: A Gourmet Guide to Typesetting with the AMS-TeX Macro package*, © 1990. This is the book that introduces Spivak pronouns in its preface, and then makes relatively little use of them throughout the book. Note that Spivak capitalises all instances of his pronouns, and also that although Wikipedia lists the earlier *Calculus on Manifolds* as another place where Spivak pronouns appear, I cannot find them anywhere in that book. In fact, I cannot find them anywhere but in *The Joy of TeX*.

Of course, the purpose of this book isn't to write a dissertation on gender, but rather to talk about the typesetting language that has become a *lingua franca* in the mathematical community, Donald Knuth's TeX. Spivak has some very precise instructions and recommendations about typesetting mathematics. In fact, the American Mathematical Society's LaTeX package is based on Spivak's recommendations, and is the most standard addition that mathematicians use with Leslie Lamport's macro package LaTeX for TeX.

*Calculus* 3rd edition © 1994. With answer book. This book is a gem. It's called *Calculus*, but in the preface to the second edition he admits that he often hears that it should be called *An Introduction to Analysis*. I agree with his choice to keep the original title, because this book is written in such a way to present the classical elementary calculus without getting into advanced topics but using all the modern rigour and hindsight that the analysts of the nineteenth and twentieth century have given us. It was the first Spivak text that I read, and I still pick it up sometimes for bedtime reading and browse through it. If I ever feel tired or bored with mathematics, I always have a leisurely stroll throught its pages and have my faith restored. This book is *that* pleasurable to read.

What this book does is present the notions of numbers (yes, it begins almost with a definition of number), functions, continuity, limits, derivatives, integrals, then moves on to topics like sequences, series, and a brief foray into complex analysis in order to have a complete treatement of analytic functions as far as elementary calculus is concerned. It concludes with an epilogue detaling a construction of the real numbers in order to fully close the gap on rigour according to modern standards. The accompanying answer book is often described more as a hint book, because it provides solutions to the problems in broad strokes rather than giving detailed answers.

It's not the subject that makes this book unique. Heck, there are hundreds of calculus textbooks out there. I used to run a small undergraduate mathematical library that consisted mostly of donations, and almost half of our library was calculus texts. Throw a rock and you'll hit one. Rather, it's the trademark Spivak presentation where this book really shines. He's very chatty, and you can tell that he is having *fun* writing the mathematics. He hides Easter eggs for the reader throughout his prose in order to keep your interest. He uses classical techniques of narrating a story, like foreshadowing and conflict. It's almost like reading a novel. Approximately half of this book consists of insightful problems connected with the body of the text. Sometimes important developments are found in the problems themselves, but Spivak guides you through these problems expertly without actually doing all of the work for you and leaving some quite difficult quandaries for the reader to ponder. He will point out to the literature and always connects each problem back to the bigger picture. Nothing is suprefluous, and you can really appreciate how the entire subject is weaved together.

*The Hitchhiker's Guide to Calculus*, © 1995. I have not read this book, although I understand it's of course written in the signature Spivak Style. It's supposed to be a shorter and more elementary introduction to the subject than his *Calculus*, covering subjects in a different order and more in accord with the traditional calculus curriculum in North American secondary schools. As the title suggests, this book is more intended for the casual beginner than *Calculus* is.

*A Comprehensive Introduction to Differential Geometry* 3rd edition, © 1999. Five volumes. This is Spivak's magnum opus, and a treasure trove of information for differential geometry. As he states in the preface, he set out to write "The Great American Differential Geometry Text", and as far as I'm concerned, he succeeded. Only relatively recent in 1999 did this book get properly typeset and bound according to his own standards of presentation. Before that, the book could be found in university libraries written with a typewriter and lots of additional symbols and drawings put in there by hand. The latest 3rd edition printing has some exquisite decorative drawings on the covers alluding to various aspects of differential geometry. These books are widely celebrated and almost surely cited in the references of any modern treatment of differential geometry. They have become classics.

These books assume a more solid background in mathematical analysis than his other books. He explicitly states that his *Calculus on Manifolds* is partially intended to give preparation for *A Comprehensive Introduction to Differential Geometry*. The first two books form the core of any introductory graduate or advanced undergraduate course on Riemannian manifolds and integration of differential forms.

The second book contains a rare treat: two highly influential papers by Karl Friedrich Gauss and Bernhard Riemann, translated, annotated, and commented by Spivak himself. Nowhere else can you find such a clear explanation of the historical development of ideas. The third volume builds up surfaces towards the remarkable Gauss-Bonnet theorem linking the total curvature of a surface with its topological properties. The next book gives some theory of the classical calculus of variations as it connects with differential geometry. Finally, the last and fifth book brings partial differential equations into the picture and concludes with a grand generalisation of the Gauss-Bonnet Theorem, and points the way towards further developments without going into great depth.

I insist again: these books stand out not precisely by what they say (well, except for *A Comprehensive Introduction*), but for *how* they say it. There's a very particular Spivak Style for presenting mathematics which makes these books fun and educational to read. These books have become popular and recognisable enough for translations of them into several other languages to appear. I have seen translations into Czech, Russian, and Spanish. None of them, except perhaps *The Hitchhiker's Guide*, are particularly suited for the layperson, however. I would recommend them only to dedicated people who want to see the real beauty of mathematics and one of the best ways of presenting it.

### Seventeen and the Yellow Pigs

There's a running gag in his books that started somewhere in a bar near Princeton university in the sixties when he was getting his PhD. It involves the number 17 and the Yellow Pigs. It seems that Mike Spivak (as he was known back then) and his friend David Kelly decided to start a Cult of 17, where they would list interesting properties of the number 17, in a frenzy of tongue-in-cheek numerology (e.g. the most famous mathematical constant π is the 17th letter of the Greek alphabet, or the sum of the squares of the first primes up to 17 adds up to the number of the beast, etc.) . In the beginning, rival factions would spring up with mystical numbers of their and special properties of their own, but Kelly and Spivak would counter with two special properties of 17 for every property that their rivals could produce. Eventually, Kelly and Spivak won this competition and declared that 17 is a most special number indeed.

I don't quite understand what Yellow Pigs have to do with it, although there is a bit of an explanation at this webpage. The thing is that this was turned into quite a long-running joke. There is even a certain Sara Smollett who runs a personal website called yellowpigs.net, although I doubt that she is a direct acquaintance of Michael Spivak; perhaps just a fan. Michael Spivak since then has hidden at least one reference to the Yellow Pigs in all of his books (one of them has a Chinese policeman, for example) or the number 17, often both. It's a small bit of inside humour for those who know to look for it.

The Yellow Pigs and 17 highlight another aspect of the Spivak Style: don't take it all too seriously. Have fun writing your mathematics. Put little jokes here and there, even if you're sticking to deadly serious business such as mathematics. Enjoy.

I wonder what Spivak is up to nowadays. I wonder if I'll ever get to meet him. I hope I do. I also wonder if he'd like to be a noder. In all sobriety, probably not, although there is something about his personal philosophy that would make an excellent noder. I always have been endeavouring to imitate his style in my own mathematical nodes, because I believe that Spivak would be a great Everythingian.

**Sources**: *Besides Spivak's books that I mention above, I also consulted the following web resources:*

**Michael Spivak's Entry on the Mathematics Geneaology Project**

http://www.genealogy.math.ndsu.nodak.edu/html/id.phtml?id=15162

*This shows when Spivak got his PhD from John Milnor*

Accessed August 6, 2005

**MathSciNet**

http://www.ams.org/mathscinet

*Requires registration, and gives reviews of the scant papers that Spivak published*

Accessed August 6, 2005

**Michael Lee Spivak**

http://www.mste.uiuc.edu/users/spivak/

*May or may not have anything to do with Michael Spivak. Perhaps his son? *

Accessed August 6, 2005

**17 and the yellow pigs **

http://www.vinc17.org/yp17/index.en.html

*Notes on the origins of the running joke in Spivak's texts*

Accessed August 6, 2005

**Sara Smollett's Yellow Pigs personal webpage **

http://yellowpigs.net

*Looks like a page inspired by Spivak's personal joke*

Accessed August 6, 2005