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{
Review of Dave Bressoud's ``Proofs and Confirmations'' 
}
\rm
\bigskip
\centerline{ {\it 
Doron ZEILBERGER
}\footnote{$^1$}
{\eightrm  \raggedright
Department of Mathematics, Temple University,
Philadelphia, PA 19122, USA. 
%\break
{\eighttt zeilberg@math.temple.edu} \hfill \break
{\eighttt http://www.math.temple.edu/\Tilde zeilberg/   .}
April  11, 2000.  Supported in part by the NSF.
} 
}
 
In {\it Proofs and Confirmations: the Story of the
Alternating Sign Matrix Conjecture}, Dave Bressoud has
created a beautiful new genre of mathematical exposition.
It is neither popular mathematics, nor textbook, nor 
research monograph, nor problem book. It is
all these and much more: a historical novel, 
a detective story, and, implicitly,
a philosophical manifesto. 
Yet the mathematics is deep, and all the proofs are complete.
 
Dave Bressoud belongs to that tiny set of mathematicians---Polya 
and Knuth come to mind---who are
first-rate researchers {\it and} masters of expository writing.
Bressoud is also very passionate
about teaching and very active in mathematics education. 
He is one of the few  moderate and sober voices in the fierce battle
between extreme reformists (like Gleason and Hughes Hallet)
and hard-nosed counter-reformists (like Andrews and Askey,
to whom the book is dedicated).
He believes that we need change, but that we should not
{\it lose the very essence of mathematics, which is inextricably
tied to proofs}.
 
Unfortunately, proofs have recently acquired a bad name.
They smack of authoritarian despotism, and their quaint claim for
absolute certainty seems anachronistic in this postmodern
age of relativism and deconstruction.
Of course, this is partly our fault. We mathematicians just keep
teaching the same old way and losing the best and the brightest,
first to physics and now to computer science. 
According to some people such as
Gregory Chaitin, proofs are indeed {\it pass\'e}, and
we should start doing quasi-empirical mathematics; that is, we humans can
only prove {\it trivial} theorems, and if we want to get a glimpse
of non-trivial mathematical worlds, we have to make do with
quasi-certainty.
 
Most scientists do not worry or care about (mathematically rigorous) proofs,
and they often ridicule mathematicians as eccentric and pedantic
fanatics.
Flourishing areas of physics like quantum field theory
and the renormalization group method are non-rigorous, and physicists
view with indifference and mild amusement efforts by mathematicians
to make them rigorous. In recent years, 
physics has helped mathematics much more than vice-versa by
infusing new ideas and techniques, for example, the Seiberg-Witten
equations that have revolutionized lower-dimensional topology.
Another great discovery by physicists that was very influential
in mathematics is the Yang-Baxter equation.
One of its consequences, the Izergin-Korepin formula, turned out to be crucial
for the proof of the Alternating Sign Matrix Conjecture.
 
But, for better or for worse, proofs are {\it the very essence of mathematics},
and if we want to keep the mathematical flame alive
and attract to our profession the most mathematically talented,
rather than the computer-science drop-out, we have to find
ways to present proofs, and the {\it activity of proving}, in a way
that will excite students and make them want to become mathematicians.
That's why we desperately need books like this one.
It conveys not only the mathematics but the excitement of
doing the two major mathematical activities: conjecturing and
proving. Bressoud chose the Alternating Sign Matrix
Conjecture and its proof as a medium to {\it prove} his message:
that mathematics {\it and proofs} are lively and exciting activities.
The story itself and the mathematics along the way are charmingly
told.
You will encounter many interesting main characters
with their even more interesting mathematics: Euler, Jacobi, Sylvester,
MacMahon, Schur, Andrews, Mills-Robbins-Rumsey,  Stanley, Baxter,
Izergin-Korepin, Kuperberg, and, of course, Zeilberger.
You will also get to know the work of supporting actors
like Dodgson, Gessel, Viennot, Askey, and Krathenthaler.
Also, very modestly and understatedly, Bressoud's own technical
contributions  make cameo appearances.
 
I almost forgot---what are alternating sign matrices, and what
is the conjecture about them?  ASM's were discovered by Howard
Rumsey and Dave Robbins in their elegant extension of Dodgson's
method for evaluating determinants.
In an ASM, every entry is either $0$, $-1$ or $1$, and 
in every row and column the non-zero entries alternate,
starting and ending at $1$, i.e. a line can either have just
one $1$ and all the rest $0$, or have the pattern 
$1,-1,1$, interspersed with zeros, or the pattern
$1,-1,1,-1,1$ sprinkled with zeros, etc. 
Here is an example of an ASM:
$$
\pmatrix{
0&0&1&0&0\cr
0&1&-1&0&1\cr
1&-1&0&1&0\cr
0&1&0&0&0\cr
0&0&1&0&0\cr
} 
$$
If there are no $-1$'s, then
we have permutation matrices, of which there are
$n!$. In 1980, 
William Mills, Dave Robbins, and Howard Rumsey conjectured that
the number of $n \times n$ ASMs is
$$
{{1!4!7! \cdots (3n-2)!} \over {n!(n+1)! \cdots (2n-1)!}} .
$$
All of this and much more, 
including the ultimate proof, is grippingly described in this book.
 
The book is also visually very attractive. In particular,
Greg Kuperberg's
beautiful rendition of an ASM as a zigsaw puzzle
adorns the cover, there are many clear diagrams, and the photographs
of the main players in the drama add a warm human touch.
 
 
This book is destined to be a classic.  It should be
on the shelf of every mathematician, regardless of specialty,
since the larger message that it conveys is universal.
It is also ideal as a textbook
for an innovative course for math majors or graduate students,
and is already used very successfully as such, for example, at
the University of Wisconsin.
Make sure to buy a copy
while the first edition lasts, and have Dave sign it next time
you see him at a National meeting (he usually shows up). One day
you, or your heirs, will thank me for this advice.
\bye


