Right now, in a classroom somewhere in the world, a student is mouthing off to her math teacher. The teacher has just asked her to spend a substantial portion of her weekend computing a list of thirty definite integrals.
There are other things the student would rather do. There is, in fact, hardly anything she would not rather do.
'When am I going to use this?'
The answer given is seldom satisfying to the student. That’s because it’s normally a lie. And the teacher and the student both know it’s a lie. The number of adults who will ever make use of the integral of (1 − 3x + 4x2)−2 dx, or the formula for the cosine of 3θ, or synthetic division of polynomials, can be counted on a few thousand hands. The lie is not very satisfying to the teacher, either.
Fortunately, there’s a better answer. It goes something like this:
Mathematics is not just a sequence of computations to be carried out by rote until your patience or stamina runs out. Those integrals are to mathematics as weight training and calisthenics are to football. If you want to play football at a competitive level, you’ve got to do a lot of boring, repetitive, apparently pointless drills. Do professional players ever use those drills? Well, you won’t see anybody on the field curling a weight or zigzagging between traffic cones. But you do see players using the strength, speed, insight, and flexibility they built up by doing those drills, week after tedious week. Learning those drills is part of learning football.
You may not be aiming for a mathematically oriented career. That’s fine—most people aren’t. But you can still do maths. You probably already are, even if you don’t call it that. Maths is woven into the way we reason. Knowing mathematics is like wearing a pair of X-ray specs that reveal hidden structures underneath the messy and chaotic surface of the world. Maths is a science of not being wrong about things. With the tools of mathematics in hand, you can understand the world in a deeper, sounder, and more meaningful way.
Still unconvinced? Here is an example:
Abraham Wald and the missing bullet holes
Abraham Wald was born in 1902 in what was then the Austro-Hungarian Empire. He was the grandson of a rabbi and the son of a kosher baker, but the younger Wald was a mathematician almost from the start. He studied mathematics at the University of Vienna, but when he finished, it was the mid-1930s and Austria was deep in economic distress.
Persecution in the lead up to the second world war saw him move to the United States where he joined the Statistical Research Group (SRG), developing equations to help win the war.
So here’s one question Wald faced. You don’t want your planes to get shot down by enemy fighters, so you armour them. But armour makes the plane heavier, and heavier planes are less manoeuvrable and use more fuel. So, there must be an optimum amount of armour, what is that?
Wald was given data from officers at war – the distribution of bullet holes on planes that returned from Europe. The damage wasn’t uniformly distributed across the aircraft, with more bullet holes in the fuselage and not so many in the engines.
There was a seemingly a perfect opportunity for efficiency; you can get the same protection with less armour if you concentrate the armour on the places with the greatest need. But where that need is is not the answer most people expect.
The armour, said Wald, doesn’t go where the bullet holes are. It goes where the bullet holes aren’t.
Wald’s insight was simple – it is highly unlikely that shots from below were aimed accurately to hit the fuselage, they are almost certainly made randomly. So where were all those planes shot in the engines? 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 at all. 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.
Wald’s recommendation to put armour where the bullet holes weren’t was quickly put into effect, and was still being used by the navy and the air force through the wars in Korea and Vietnam.
The winners of wars are usually those 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 maths every step of the way.
Wald’s maths-trained habits of thought brought him an insight the officers had missed. A mathematician is always asking, “What assumptions are you making? And are they justified?” Once you recognise that you’ve been making assumptions, it can take just a moment to realise it’s wrong.
To a mathematician, the structure underlying the bullet hole problem is a phenomenon called survivorship bias. It arises again and again, in all kinds of contexts. And once you’re familiar with it, as Wald was, you’re primed to notice it wherever it’s hiding.
So, you see – you may not need a list of thirty definite integrals but the way your mind is trained through this exercise has far reaching applications into areas you had never even imagined.
This blog was adapted from 'How Not to be Wrong: The Hidden Mathematics of Everyday Life', by mathematician Jordan Ellenberg, published June 2014.
This blog was first published in June 2014 but has been updated in November 2022
