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:
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!):