I have been busy juggling these past few weeks. But now the most
recent paper deadline is met; the presentation materials for Tuesday
are in hand; my computers are stable and mutually consistent; my
desk and my home are both clean and organized; and I lack neither
staples to attach my papers nor staples to make my meals (man does
not live on ramen alone). I've even managed to deal with most of my
technical support mail. So this evening I sit with pen and pad and
a pot of tea to finish a thought from the start of the month.
(As an aside: I wrote most of this, including the previous sentence,
on Thursday night  but I'm just now typing on Saturday morning.)
At HalfPrice books two weekends ago, I found Keith Devlin's book
The Millennium Problems sitting neglected on a corner
shelf. I have a special place in my heart for popular writing by
mathematicians like Ian Stewart, Ivars Peterson, and Martin Gardner
 I had their books checked out from the Bel Air library half the
time when I was in middle school, and I continue to find them
entertaining and interesting today. I know a mite more mathematics
now than I did then, but the writing often focuses on history,
applications, intuitions, and puzzles centered around some
mathematical idea or set of ideas; and so even when I'm familiar
with the mathematical idea, I still enjoy the reading. For the same
reason, I enjoy reading articles on science written for a popular
audicence. Devlin, who may be most familiar as the Math Guy
on NPR, has an engaging writing style, and his topic  the
Millennium Problems, seven great unsolved mathematical challenges
with milliondollar bounties on their heads  is something I wanted
to know more about. After reading the introduction, I bought the
book in happy anticipation of a good read.
Besides, the man has an extinct species of possum named after him.
What's not to like?
As I read, I found exactly what's not to like  at least, I found
points in the presentation that I very much disliked. Devlin
undertook a monumental task: not just to explain the history and the
impact of the Millennium problems, but to give a flavor for the
problems themselves  assuming a reader with only a precalculus
grounding in high school mathematics. As Devlin puts it,
Even achieving a layperson's knowledge of what [the Millennium
problems] are about takes considerable effort. I believe the
effort, however, is worthwhile. Aren't all pinnacles of human
achievement of interest?
But the challenge was too great. Devlin explained the problems at a
lay level better than I could have done; but then, I don't know how
someone could think about some of these problems without
knowing  really knowing, not just having some vague familiarity
with  some group theory, elementery number theory, and the basics
of differential calculus. Devlin gives a creditable introduction to
some of this background, but it is not enough, and by the end he is
reduced in his frustration to section titles like The Hard Stuff,
Made as Easy as I Can
and The Hodge Conjecture: Not for the
FaintHearted.
For most readers, the description of cohomology classes in the last
chapter would probably be impenetrable. For me, it was
frustrating. After all the effort of sketching in the technical
background, an effort which will likely go unappreciated by most of
your readers, why not finish the sketch for me? Just a little more
technical detail? Somehow I ceased to react to the book as the
popular exposition it was meant to be, and started to treat it as an
inadequately fleshed out technical work  which it was certainly
not meant to be.
I was irked. I was sufficiently irked that I wrote a letter to a
friend which fell somewhere between a lecture and a tirade, in which
I tried to describe what I disliked in Devlin's book. I wondered
almost as soon as I sent the letter whether I hadn't been unduly
harsh  but Yi shared an office with me for a while, and had
plenty of opportunity then (and since) to become accustomed to the
flavor of my unprompted lectures. So I expect she will take it in
stride.
After reading Devlin's book, and after spending some time grumbling,
I began to think more about the nature of mathematical intuition.
In the first chapter, Devlin has a section entitled Why Are the
Problems So Hard to Understand?
in which he argues that the
level of abstraction from everyday reality makes mathematical ideas
harder to explain to a lay audience than are ideas from any area of
science. At the same time, as he says near the end of the book,
... a trained human mind that has thought long and hard about a
particular problem frequently develops intuitions that prove to be
correct.
The problem, then, is not that there is no intuitive picture of the
ojbects of higher mathematics  just that such pictures are
inaccessible without a lot of patient thought. I disagree with
Devlin's assessment only in one point: I believe there are concepts
in modern physics and chemistry which require abstractions which are
fully as inaccessible to the lay reader as any of the Millennium
problems (which is probably why two of the Millennium problems 
mathematical understanding of the YangMills field equations and the
solution of the NavierStokes equations  come directly from modern
physics).
So why is popular science still so much more popular than
popular mathematics? And why does modern math have such a
reputation for impenetrability, particularly when  in contrast to
the experimental sciences  intuition can be had for only the cost
of time and thought? Part of the trouble, I think, is educational.
Think when the scientific method was first mentioned to you in an
elementary science class  and then think when the idea of a
mathematical proof was first introduced. Do you know more names of
physicists from the past two centuries than names of mathematicians?
Difficulty alone cannot account for the difference in perception;
nor, I believe, can the relative distance from everyday reality.
I don't have good answers to the questions why is it so hard to
understand?
and why should science seem any easier?
But
I'm still thinking on it. Maybe one day I'll gain enough
psychological intuition to understand how people gain intuition 
but I doubt it. Mathematics is easier to understand than people are.
Works considered
This started off longer, and got shorter as I realized I wasn't sure
I agreed with my own ideas. Consequently, this list is not a
collection of works cited, but rather writings that I looked at
during some version of the above.

Nature's Numbers (Ian Stewart);
Islands of Truth: A Mathematical Mystery Cruise (Ivars
Peterson) 
Peterson's book was definitely one that I kept checking out from the
library. I forget which of Stewart's books the library had, but
Nature's Numbers is a good one in any case.

Mathematical Association of
America  The MAA does a lot with mathematics education and
popularization. Both Ivars Peterson and Keith Devlin write regular
columns for the MAA, which are available from the web site.

The Honors Class: Hilbert's Problems and Their Solvers
(Paul Yandell) 
Perhaps a natural complement to a book on the Millennium Problems.
The Millennium Problems were deliberately introduced 100 years
after Hilbert introduced his list.

The Man Who Loved Only Numbers (Paul Hoffmann); A
Beautiful Mind (Sylvia Nasar)  Biographies of two modern
mathematicians. I thought both books were fascinating.
No, not all mathematicians are so nutty.

Men of Mathematics (E.T. Bell) 
Biographies of some major mathematicians up through the end of the
nineteenth century. There are more accurate historically accurate
books on the history of mathematics and on historical
mathematicians; but it is hard to find another book which is so
enthusiastically and charmingly written. Mathematicians were Bell's
heros; and he made a few of them my heros, too.

Fermat's Engima (Simon Singh) 
I have intended to finish reading this book for years now. Two of
them, actually. But I particularly wanted to mention a quote from
the dust jacket, which I repeat here in full:
Perhaps I could best describe my experience of doing mathematics
in terms of entering a dark mansion. One goes into the first room
and it's dark, really dark, and one stumbles around bumping into the
furniture. Gradually you learn where each piece of furniture is,
and finally, after six months or so, you find the light switch and
suddenly it's all illuminated and you can see exactly where you
are.
 Andre Wiles

The Value of Science: Essential Writings of Henri Poincare
(Henri Poincare; ed. Stephen Jay Gould) 
Poincare has much to say about the role of intuition and experiment
in mathematical inquiry. A profound and prolific mathematician,
Poincare turned late in life to writing for a popular audience. His
publications were deservedly successful, and are worth reading as a
model of accessible exposition, as well as for the ideas they
contain on the nature of mathematics and of science.

Linear Differential Operators (Cornelius Lanczos) 
Really, any of Lanczos's books would serve as well, but Linear
Differential Operators has a few wonderful paragraphs at the
beginning in which he describes the goal of his expository style.
In contrast to the very formal presentations of N. Bourbaki
(Bourbaki was actually the nom de plume taken by a school of French
analysts), Lanczos concentrates less on the details of the analysis
and more on highlevel intuitions. The Variational Principles
of Mechanics is another excellent text in the same style; it
was one of my primary texts in learning about classical mechanics.

Catastrophe Theory (V.I. Arnol'd) 
Usually Arnold writes more technical books, and when I found this on
the shelf at Black Oaks, I was delighted to find that it was written
for a general audience. It's short, interesting, and full of both
dry humor and intuition.