## Bullets, Innovation, and the dreaded "M" Word

In the intellectual equivalent of placing my hand over a candle flame, I am taking a course on complex models. And an answer your obvious question, less Cindy Crawford or Angela Lindvall and more probabilities and algebra.

I have a deep affinity and some aptitude for the English language, but I have never been an aficionado of the “M” word – mathematics.

So why dive into the deep end of the pool of models and mathematics?

Frankly, I thought the class promise of “no calculus or advanced math needed” meant I could bumble my way through the course without having to dabble in square roots, probabilities, or defining the range of Six Sigma.

But I was wrong.

But I was intrigued by the other promise – the skillset of making smarter decisions by creating, understanding and evaluating models. It has been an invaluable resource for turning nearly invisible data into visible knowledge.

Along the way, I discovered a story by Jordan Ellenberg in his wonderful (and English-major friendly) book *How Not to Be Wrong: The Power of Mathematical Thinking.*

It is the story of Abraham Wald, the grandson of a rabbi and son of a kosher baker and a genius at pure mathematics – set theory and metric spaces.

Wald spent much of World War II in Manhattan working for the Statistic Research Group -- the mathematical equivalent of The Manhattan Project – where the weapons were equations, not bombs.

According to Ellenberg, “The mathematical talent at hand was equal to the gravity of the task. In Wallis’s words, the SRG was “the most extraordinary group of statisticians ever organized, taking into account both number and quality.” Frederick Mosteller, who would later found Harvard’s statistics department, was there. So was Leonard Jimmie Savage, the pioneer of decision theory and great advocate of the field that came to be called Bayesian statistics. Norbert Wiener, the MIT mathematician and the creator of cybernetics, dropped by from time to time. This was a group where Milton Friedman, the future Nobelist in economics, was often the fourth-smartest person in the room. The smartest person in the room was usually Abraham Wald.

So I will cut to the chase. You want to add armor to your planes to protect them, but armor makes the plane heavier, and heavier planes are less maneuverable and use more fuel. So the generals came to SRG with mounds of data -- especially on where bullets were lodged in U.S. fighters.

“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 assumption was that 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.

Ellenberg continues, *“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.”

In a great analogy, Ellenberg writes, “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. “

Wald saw what the officers, who had more experience and understanding of the particulars of aerial combat, couldn’t. It comes back to his math-trained habits of thought. A mathematician is always asking, “What assumptions are you making? And are they justified?”

This can be annoying. But it can also be very productive. In this case, the officers were making an assumption unwittingly: that the planes that came back were a random sample of all the planes.

In creativity and innovation, we continually ask, “What assumptions are you making?” *And it continues to be annoying to people who “know” their subjects better than you do.*

In general, people don’t like their assumptions questioned or their biases confirmed.

And if you question them successfully, they still will carry some animus. It goes with the territory.

All of which is to say, I have not become a lover of math. I have, however, reinforced my desire to find answers and a rigorous way to find them.

Cheers.