Category: Book Review

Polya’s three rules of style

In G. Polya‘s book “How to Solve It”, one of the shortest sections is called “Rules of style”. This section contains Polya’s three rules of style, which are worth repeating.

“The first rule of style”, writes Polya, “is to have something to say”.

“The second rule of style is to control yourself when, by chance, you have two things to say; say first one, then the other, not both at the same time”.

Polya’s third rule of style is: “Don’t say what does not need to be said” or maybe “Don’t say the obvious”. I am not sure of the exact formulation, because Polya doesn’t write the third rule down – that would be a violation of the rule!

Polya’s three rules are excellent and one is advised to follow them if one strives for good style when writing mathematics. However, style is not the only criterion by which we measure mathematical writing. There is a tradeoff between succinct and elegant style, on the one hand, and clarity and precision, on the other.

“Don’t say the obvious” – sure! But what is obvious? And to whom? A careful writer leaving a well placed exercise in a textbook is one thing. An author of a long and technical paper that leaves an exercise to the poor, overworked referee, is something different. And, of course, a mathematician leaving cryptic notes to his four-months-older self, is the most annoying of them all.

“Don’t say the obvious” – sure, sure! But is it even true? I think that all the mistakes that I am responsible for publishing have originated by an omission of an “obvious” argument. I won’t speak about actual mistakes made by others, but I do have the feeling that some people have gotten away with not explaining something non-trivial, and were lucky that things turned out to be as their intuition suggested (granted, having the correct intuition is also a non-trivial achievement).

I disagree with Polya’s third rule of style. And you see, to reject it, I had to formulate it. QED.

Tapioca on page 49

To my long camping vacation this year I took the book “Topological Vector Spaces” by Alex and Wendy Robertson. I “inherited” this book (together with a bunch of other classics) from an old friend after he officially decided to leave academic mathematics and go into high-tech. The book is a small and thin hard-cover, with pages of high quality that are starting to become a delicious cream color.

Hal-moss, not Hal-mush

The title of this post is a small service to Paul Halmos. I recently read his book “I Want to be a Mathematician”, subtitled “an Automathography”, where I found this:

Do all readers know that I reject ‘Hal-mush’ – some people’s notion of the “right” way to pronounce me? Please, please, say ‘Hal-moss’.

Sure, as you wish (I occasionally used to say “Hal-mosh”. No more).

Halmos was an influential mathematician who was born a hundred years ago, and died ten years ago. He worked in several areas (measure and ergodic theory, logic, operator theory) and wrote many successful books. He is considered to be a superb expositor. [His Hilbert Space Problem Book is the most refreshing, provocative and captivating book that I ever found accidentally on the library shelf (browsing with no definite goal, when I was a TA in a course on functional analysis). A Hilbert Space Problem Book is not only a beautiful and original idea, it is also executed to perfection and thus very useful.]

Perhaps I will take the opportunity of his 100 hundredth birthday (a few months from now) to write about one or some of his classic papers. But now I want to write about the book “I Want to be a Mathematician”.

The book (the automathography) is a kind of professional autobiography, omitting almost everything in personal life, and concentrating on his life as a mathematician, and that includes almost every aspect of the profession. Halmos has some interesting and definite opinions on various matters, and he believes that they should be expressed unequivocally: “I must not waffle and shilly-shally. It’s better to be wrong sometimes than to equivocate…”. The readers follow Halmos’s career, and every mathematician who crossed his way (including himself) is given a supposedly fair yet ruthless treatment. This is great book.

Halmos is happy to sort mathematicians into ranks: Gauss and Archimedes are mathematicians of the first rank, Klein and MacLane the second, Mackey, Tarski and Zygmund the third. He puts himself in the fourth rank, together with Birkhoff and Kuratowski. Immediately after discussing ranks, he introduces Fomin, a mathematician who he met in Moscow: “As a mathematician, he was perhaps of rank five”.

The final chapter of the book is called “How to be a mathematician”. I am guessing (a wild guess) that Halmos considered this as a possibility for the title of the book, but realised that it’s the wrong title. A more precise title for the book would be “How to be Halmos”. In the ruthless spirit of the author, one might also suggest: “How to be a great mathematician without really being one”.

Perhaps that’s precisely what makes the book so interesting to me. It is written by an unquestionably human mathematician. Smart, innovative, talented, idiosyncratic, hard working, ambitious – yes, but still human. An important mathematician, but not a Great One. I recommend it, it is fun to read whether or not you agree with what he has to say. I have a lot of criticism on his views, but man does he know how to write!

(Well, the book is perhaps too long and has it’s ups and downs. But one is free to skip the boring “funny” stories on the incompetent waiter in Moscow, or adventures in Uruguay).

I cannot resist objecting loudly to two pieces of advice that Halmos gives.

Halmos writes “…to stay  young, you have to change fields every five years.” Watch out (everyone except Terry, yes?): that is dangerous advice!

I personally love to branch out and work on different kinds of problems, and to learn things in different fields, but if you are interested in reaching into the deep you have to focus on some concentrated part of mathematics for a long time, for years. I have no regrets, but my experience taught me a few things that one should take into account. When you switch fields the expertise which you acquired becomes pretty much useless and you have to invent or learn new techniques from scratch. To become a reliable scholar in a new area you have to pay an expensive entrance fee by learning the literature, and your investment in the literature of the previous field goes to waste, at least in some sense. From a pragmatic point of view, it will be hard to get good letters for your promotion if you don’t stick long enough in one field to make an impact. And you may receive invitations to workshops and conferences that are no longer very relevant to you, while you are not yet recognised by the people organising workshops that you would like to go to.

It is very hard to be a true expert, a learned scholar, and to make an impact even in one field. Halmos worked on measure theory, ergodic theory, probability, statistics, operator theory, and logic. It is very very unusual, and I don’t know if Halmos is really an exception, for someone who is even very strong to make deep contributions in logic as well as in operator theory. Well, at least in this time and age it is very unusual – remember that Halmos was born 100 years ago, and mathematics has changed since the 40s and 50s quite a lot. But I think that changing fields dramatically and often was bad advice even when Halmos was active. Would his contributions to operator theory been deeper if he had not left it for several years to work on logic?

Of course, if an opportunity to branch out comes along, if your heart pulls you to a different subject, if one problem leads you naturally into a different field, then go for it! But changing fields is not an item on your checklist. Contrary to what Halmos writes, “if a student writes a thesis on the calculus of variations when he is 25, and keeps publishing papers on the calculus of variations till he is 65”, he certainly may be a first rate mathematician.

The second piece of Halmos wisdom I wish to denounce is something that appears in the chapter “How to be a mathematician”, a piece which has appeared separately and which I bumped into already many years ago, and has annoyed me even then.

Halmos writes: “[to be a mathematician] you must love mathematics more than anything else”. He goes on:

To be a mathematician you must love mathematics more than family, religion, money, comfort, pleasure, glory.

What!? More than your children? Well, Halmos did not have any children, and he probably would not have written that line if he did. But even if you don’t have children, really? Do you love mathematics more than love? More than making love? I reject this point of view altogether.

Sure, it’s not just “a job”. You shouldn’t (and couldn’t) be a mathematician if you are not thrilled by it, if it does not captivate your thoughts sometimes to the point of obsession. And you won’t succeed unless you are very devoted, unless you work with joy and work very hard. But if math is more important to you than everything else, then you are simply nuts. It can’t be more important to you more than everything else, because it’s not. In any case, there are many counter examples to the above assertion; many (all?) great mathematicians had loves, devotions, or callings, bigger than mathematics.

In fact, I believe that Halmos himself is a counter example to his claim. You can find the proof in the first and last few paragraphs of the book. These are among the most touching passages in the book, so I will just leave it at that.

*****

Apropos Halmos’s book, I take this opportunity to NOT recommend – meaning recommend not to read – Hardy’s book “A Mathematician’s Apology” (Prof. Hardy: apology not accepted!) together with Littlewood’s “A Mathematical Miscellany” (who cares?). I read Hardy’s book because a friend recommended it very highly, and I read Littlewood’s book as a possible compensation, or better: retaliation, for reading Hardy’s book. My verdict: bad books, don’t waste your time with either of these!