Letters From a Mathematician

Several years ago I skimmed through Ian Stewart's little book, "Letters To A Young Mathematician," and it didn't register much with me, not really being a fan of "Letters to..." sorts of books. But I just re-read it, with the benefit of a few more years immersion in mathematical debates, and especially Keith Devlin's notion of "mathematical thinking," and suddenly enjoyed it immensely! Indeed, I now wonder if any of Keith's thinking was influenced by Stewart's ideas, or was the influence in the opposite direction(?), from Devlin to Stewart, so similar are many of the viewpoints expressed (Stewart's discussion of "proofs" especially mirrors a recent view voiced by Keith).

I won't review this instructive (and surprisingly rich) little volume, since it's from 2006, but will note that I think every prospective math major should ponder it before they get too far along their path. In a quick 200 pages it covers a lot of backdrop to a mathematics career.  And even though it was written almost a decade ago, it includes discussion that is pertinent to the ongoing debates in math education right now. It also contains more thoughtful passages I'll be considering for "Sunday Reflections" than almost any other book on my shelf! I especially like the whole second half of the volume.

Lest it not be clear to people, I should mention that these "letters" are fictitious missives written to a fictitious niece; which gives Stewart a lot of freedom to say what he wants in the way he wants to say it.  In fact, Stewart sees the book as a sort of update to G.H. Hardy's "A Mathematician's Apology."

For fuller reviews here are a couple of old ones from AMS and from American Scientist:
http://www.ams.org/notices/200705/rev-carberry-web.pdf
http://www.americanscientist.org/bookshelf/pub/port-and-walnuts

As one of the reviewers sums up, "'Letters to a Young Mathematician' succeeds well in opening a door into the world of mathematics and enticing the reader inside."

Ian Stewart is yet another Brit who has a knack for communicating mathematics to a public that is often resistant to it. I've enjoyed several of his past works. And funny how time changes perspectives... as this particular volume springs from being completely off my radar to now being one of my favorites from him -- at the risk of sounding like a broken record, I'll credit Keith Devlin for that, since reading him made me so much more receptive to Stewart's thoughts here.

The last chapter isn't necessarily representative of the book, but it is the most poignant chapter... I'd almost like to quote it in entirety, but will settle for some words from the final two pages:

"Our minds may indeed be just swirls of electrons in nerve cells; but those cells are part of the universe, they evolved within it, and they have been molded by Nature's deep love affair with symmetry. The swirls of electrons in our heads are not random, not arbitrary, and not -- even in a godless universe, if that is what it is -- an accident. They are patterns that have survived millions of years of Darwinian selection for congruence with reality....

"Perhaps we have created a geometer God in our own image, but we have done it by exploiting the basic simplicities that nature supplied when our brains were evolving. Only a mathematical universe can develop brains that do mathematics. Only a geometer God can create a mind that has the capacity to delude itself that a geometer God exists.
"In that sense, God is a mathematician; and She's a lot better at it than we are. Every so often, She lets us peek over her shoulder."

[On a side-note, Sol Lederman interviewed Stewart almost exactly two years ago for his "Inspired By Math" podcast series (just one of a jillion Stewart appearances on the Web!):
http://wildaboutmath.com/2013/01/11/ian-stewart-inspired-by-math-14/ ]

Ian Stewart Delivers...

Overview of Ian Stewart's "Visions of Infinity"

"Mathematics is newer, and more diverse, than most of us imagine. At a rough estimate, the world's research mathematicians number about a hundred thousand, and they produce more than two million pages of new mathematics every year...
"When we think of mathematics, what springs to mind is endless pages of dense symbols and formulas. However, those two million pages generally contain more words than symbols... As the great Carl Friedrich Gauss remarked around 1800, the essence of mathematics is 'notions, not notations'. Ideas, not symbols."
                                   -- from Ian Stewart's Preface

Ian Stewart (or one of his clones… because I refuse to be duped into believing that one person can write all these books!) has a new volume out, "Visions of Infinity." There are several popular math books available that review a variety of the most famous, intriguing often-unsolved problems in mathematics (the exact selection can vary slightly from volume to volume, though certain standards almost always show up). Anyway, this is Stewart's contribution to the genre… and… as one might expect, it is excellent. If only it had been written first, some of the other attempts might even have been unnecessary.

To be clear though, while this volume covers some of the most interesting problems in all of mathematics it is NOT a book to draw your non-mathematical friends into the math arena. Even the non-math person who wishes they could like math, and who may have enjoyed Steven Strogatz's "Joy of X,"  I think will find this particular book too heavy-going.  But for the individual already enamored of the subject, and having some familiarity with math's deepest problems, this is a fantastic read. In fact, it's probably my favorite Stewart volume of all the ones I've read.

Stewart gives enough background and depth to each discussion to hold the attention, and exercise the brain, of most math buffs, without getting so technical as to lose them in the dust (though parts, toward the end especially, are definitely tough-going). I have no idea if he was able to produce this work largely based on his past writing and knowledge, or whether it required lots of research on his part to fill in all the detail, history, and richness that are on display here... either way the book is a great accomplishment.

Fermat's Last Theorem, the Riemann Hypothesis, the four-color theorem, the Goldbach conjecture, the Poincare' conjecture, and P vs. NP, are among the classics Stewart rolls out for scrutiny (the title may be a bit misleading in so much as "infinity" is not really the central topic). He does a great job of not only explaining these mathematical conundrums in an accessible way, but also of detailing their histories, context, and relevance (when it exists) to other matters.  The first few chapters tend to warm the reader up. From there the book moves on to more complex and difficult problems as it goes along. The last couple of conjectures dealt with, the Birch--Swinnerton-Dyer conjecture and the Hodge conjecture, are the densest to follow (...but Stewart notes of the Hodge conjecture that it "is arguably more representative of real mathematics, as done by mathematicians of the twentieth and twenty-first centuries, than any other topic in this book"). Soon after those chapters Stewart follows up with a lighter chapter titled "Twelve For the Future," briefly outlining 12 more unsolved problems, some fairly well-known. This section ends with the ABC conjecture, and was apparently written before last year's announcement from Shinichi Mochizuki that he had proven the conjecture, as there is no mention of the claimed proof (which has yet to be confirmed, and indeed some say may not be comprehensible to anyone except Mochizuki!).

I suppose my favorite chapter (pardon the bias) is on the Riemann Hypothesis, where Stewart seems to capture in a chapter what others have written whole books about. Another chapter that I found especially good was on the Mass Gap Hypothesis, and more generally on particle physics, but your own interests will determine which chapters you most enjoy. They are all good. This will likely be the initial 'go-to' volume on my shelf whenever I wish to check on something regarding one of these classic problems.

The book ends with a handy 11-page glossary, an interesting set of succinct "notes," and a brief bibliography for "further reading" -- on a side note, I was pleased to see that the brief bibliography cites Matthew Watkins' "The Mystery of the Prime Numbers," a book I've been touting for quite awhile, but which I'd not yet seen referenced by any prominent author (Watkins gets a very brief mention in the body of the book as well).
This is simply a fabulous book for most mathematics enthusiasts... no matter which clone of Stewart's wrote it!