So here’s the question. You don’t want your planes to get
shot down by enemy fighters, so you armor them. But armor makes the plane
heavier, and heavier planes are less maneuverable and use more fuel. Armoring
the planes too much is a problem; armoring the planes too little is a problem.
Somewhere in between there’s an optimum. The reason you have
a team of mathematicians socked away in an apartment in New York City is to
figure out where that optimum is.
The military came to the SRG with some data they thought
might be useful. When American planes came back from engagements over Europe,
they were covered in bullet holes. But the damage wasn’t uniformly distributed
across the aircraft. There were more bullet holes in the fuselage, not so many
in the engines.
The officers saw an opportunity for efficiency; you can get
the same protection with less armor if you concentrate the armor on the places
with the greatest need, where the planes are getting hit the most. But exactly
how much more armor belonged on those parts of the plane? That was the answer
they came to Wald for. It wasn’t the answer they got.
The armor, said Wald, doesn’t go where the bullet holes are.
It goes where the bullet holes aren’t: on the engines.
Wald’s insight was simply to ask: where are the missing
holes? The ones that would have been all over the engine casing, if the damage
had been spread equally all over the plane? Wald was pretty sure he knew. The
missing bullet holes were on the missing planes. The reason planes were coming
back with fewer hits to the engine is that planes that got hit in the engine
weren’t coming back. Whereas the large number of planes returning to base with
a thoroughly Swiss-cheesed fuselage is pretty strong evidence that hits to the
fuselage can (and therefore should) be tolerated. If you go the recovery room
at the hospital, you’ll see a lot more people with bullet holes in their legs
than people with bullet holes in their chests. But that’s not because people
don’t get shot in the chest; it’s because the people who get shot in the chest
don’t recover.
Here’s an old mathematician’s trick that makes the picture
perfectly clear: set some variables to zero. In this case, the variable to
tweak is the probability that a plane that takes a hit to the engine manages to
stay in the air. Setting that probability to zero means a single shot to the
engine is guaranteed to bring the plane down. What would the data look like
then? You’d have planes coming back with bullet holes all over the wings, the
fuselage, the nose—but none at all on the engine. The military analyst has two
options for explaining this: either the German bullets just happen to hit every
part of the plane but one, or the engine is a point of total vulnerability.
Both stories explain the data, but the latter makes a lot more sense. The armor
goes where the bullet holes aren’t.
Wald’s recommendations were quickly put into effect, and
were still being used by the navy and the air force through the wars in Korea
and Vietnam. I can’t tell you exactly how many American planes they saved,
though the data-slinging descendants of the SRG inside today’s military no
doubt have a pretty good idea. One thing the American defense establishment has
traditionally understood very well is that countries don’t win wars just by
being braver than the other side, or freer, or slightly preferred by God. The
winners are usually the guys who get 5% fewer of their planes shot down, or use
5% less fuel, or get 5% more nutrition into their infantry at 95% of the cost.
That’s not the stuff war movies are made of, but it’s the stuff wars are made
of. And there’s math every step of the way.
**
If your acquaintance with mathematics
comes entirely from school, you have been told a story that is very limited,
and in some important ways false. School mathematics is largely made up of a
sequence of facts and rules, facts which are certain, rules which come from a
higher authority and cannot be questioned. It treats mathematical matters as
completely settled.
Mathematics is not
settled. Even concerning the basic objects of study, like numbers and
geometric figures, our ignorance is much greater than our knowledge. And the
things we do know were arrived at only after massive effort, contention, and
confusion. All this sweat and tumult is carefully screened off in your
textbook.
**
Mathematical facts can be simple or complicated, and they
can be shallow or profound. This divides the mathematical universe into four
quadrants:
“Basic arithmetic facts, like 1 + 2 = 3, are simple and
shallow. So are basic identities like sin(2x) = 2 sin x cos x or the
quadratic formula: they might be slightly harder to convince yourself of than 1
+ 2 = 3, but in the end they don’t have much conceptual heft.
Moving over to complicated/shallow, you have the problem of
multiplying two ten-digit numbers, or the computation of an intricate definite
integral, or, given a couple of years of graduate school, the trace of
Frobenius on a modular form of conductor 2377. It’s conceivable you might, for
some reason, need to know the answer to such a problem, and it’s undeniable
that it would be somewhere between annoying and impossible to work it out by
hand; or, as in the case of the modular form, it might take some serious
schooling even to understand what’s being asked for. But knowing those answers
doesn’t really enrich your knowledge about the world.
The complicated/profound quadrant is where professional
mathematicians like me try to spend most of our time. That’s where the
celebrity theorems and conjectures live: the Riemann Hypothesis, Fermat’s Last
Theorem,* the Poincaré Conjecture, P vs. NP, Gödel’s
Theorem . . . Each one of these theorems involves ideas of deep
meaning, fundamental importance, mind-blowing beauty, and brutal technicality,
and each of them is the protagonist of books of its own.
But not this book. This book is going to hang out in the
upper left quadrant: simple and profound. The mathematical ideas we want to
address are ones that can be engaged with directly and profitably, whether your
mathematical training stops at pre-algebra or extends much further. And they
are not “mere facts,” like a simple statement of arithmetic—they are
principles, whose application extends far beyond the things you’re used to
thinking of as mathematical. They are the go-to tools on the utility belt, and
used properly they will help you not be wrong.
**
The more abstract and distant from lived experience my
research got, the more I started to notice how much math was going on in the
world outside the walls. Not Galois representations or cohomology, but ideas
that were simpler, older, and just as deep—the northwest quadrant of the
conceptual foursquare. I started writing articles for magazines and newspapers
about the way the world looked through a mathematical lens, and I found, to my
surprise, that even people who said they hated math were willing to read them.
It was a kind of math teaching, but very different from what we do in a
classroom.
