Michael: What if the secret to making WWII planes safer wasn't reinforcing where they got shot, but where they didn't? And what if the key to understanding a superstar athlete's slump isn't a jinx, but a simple, invisible mathematical force? Kevin: Okay, that's a fantastic puzzle. Reinforcing the parts that aren't damaged? My brain is already twisting. That sounds completely backward. What's this all about? Michael: It's all about the power of seeing the world through a mathematical lens. Today we’re diving into How Not to Be Wrong: The Power of Mathematical Thinking by Jordan Ellenberg. And Ellenberg is the perfect guide for this journey. Kevin: Why him specifically? Michael: Well, he's not just a world-class mathematician who was a child prodigy and won multiple gold medals at the International Mathematical Olympiad. He also has a master's degree in fiction writing. So he has this incredible, rare ability to turn abstract, complex math into genuinely compelling stories. Kevin: A mathematician and a storyteller. That's a combination you don't see every day. So he’s not just throwing formulas at us. Michael: Exactly. He argues that mathematics is like a pair of X-ray specs that let you see the hidden structures beneath the messy, chaotic surface of the world. It’s a tool for not being wrong. Kevin: I could definitely use a tool for not being wrong. Where do we start? With those airplanes?

Michael: We start with the airplanes. This is one of the most powerful stories in the book, and it takes place during World War II. The U.S. military was trying to figure out how to better protect its bombers. They were taking heavy losses over Europe. Kevin: So they wanted to add more armor. That makes sense. Michael: Right. But armor is heavy. Too much, and the plane becomes sluggish, a sitting duck. So the question was: where do you put the armor? The military had a ton of data. They analyzed all the bombers that returned from missions and meticulously mapped out where they had been hit by enemy fire. The fuselage was riddled with holes, the wings were shot up... but the engines were relatively clean. Kevin: Okay, so the obvious answer is to reinforce the fuselage and the wings, right? That’s where all the bullet holes are. Michael: That was the conclusion of the military officers. It seems like pure common sense. They brought their data to a special team of mathematicians in New York called the Statistical Research Group. And one of them, a quiet, brilliant man named Abraham Wald, looked at their data and said something that changed everything. Kevin: What did he say? Michael: He said, "The armor doesn't go where the bullet holes are. It goes where the bullet holes aren't: on the engines." Kevin: Hold on. That makes zero sense. Why would you put armor on the parts of the plane that aren't getting hit? Michael: Because Wald asked a question no one else was asking: where are the missing planes? The military's data only came from the planes that made it back home. They were looking at an incomplete dataset. Wald realized the planes that were shot in the engines… well, they never returned. The lack of bullet holes on the engines of the surviving planes was actually evidence that the engines were the most critical and vulnerable part. A shot there was catastrophic. Kevin: Whoa. So the data they had was fundamentally biased. They were only looking at the survivors, and the real story was in the ones that didn't survive. That's… that's chilling. Michael: It's a perfect example of what's called "survivorship bias." We see it everywhere. We look at a handful of college dropouts who became billionaires and think, "See, you don't need a degree to be successful!" But we're ignoring the millions of dropouts who didn't become billionaires. Their stories are invisible. Kevin: That's mind-blowing for WWII, but where else does this pop up? I feel like I'm going to start seeing this everywhere now. Michael: You will. Ellenberg gives a great modern example: mutual fund performance. You'll see ads for funds that boast, say, a 15% annual return over the last decade. Sounds amazing, right? Kevin: Yeah, I'd invest in that. Michael: But those reports often only include the funds that are still alive today. They don't include the funds that performed so badly they went out of business. When you include the data from those "dead" funds, the average return often drops dramatically, sometimes to something much more ordinary, like 8 or 9 percent. The bullet holes are the bad returns, and the planes that don't come back are the funds that fail. We only see the lucky survivors. Kevin: So we're constantly making decisions based on incomplete, biased information. That's a pretty unsettling thought. It’s like we’re all walking around with these mental blind spots. Michael: Exactly. And mathematical thinking is the tool that helps us see what's in those blind spots. It's not about the calculation; it's about asking, "What am I not seeing?"

Kevin: Okay, so survivorship bias is about seeing what's missing. What's the next big mental error math can save us from? Michael: It's an error we make every single day, and it feels so natural we don't even notice it. It's the assumption that things move in straight lines. Kevin: What do you mean, "straight lines"? Like, physically? Michael: Not just physically. I mean in our reasoning. We tend to assume that if a little of something is good, then more of it must be better. Or if a trend is going up, it will keep going up forever. Ellenberg uses this hilarious and slightly terrifying example from a real scientific paper. Kevin: I'm ready. Hit me. Michael: A few years ago, a study in the journal Obesity took the rising trend of overweight Americans, drew a straight line through the data points, and extrapolated it into the future. Their conclusion? If current trends continue, 100% of Americans will be overweight or obese by the year 2048. Kevin: (Laughs) Wait, seriously? One hundred percent? That's mathematically impossible. What happens when we hit 100%? Do we just keep getting more overweight? Do we achieve a new state of being? Michael: Exactly! It's absurd. The trend can't continue in a straight line forever. Eventually, it has to level off. Reality is almost never a straight line; it's a curve. But our brains love the simplicity of a straight line. This is called the fallacy of linear thinking. Kevin: But it feels so intuitive. If a company is growing, you project that growth forward. If a policy is working a little, you do more of it. Why is that so wrong? Michael: Let's go back to the airplane armor. Too little armor, and the plane gets shot down. But what if you add too much? Then the plane is too heavy to fly effectively, or it uses too much fuel. The relationship isn't linear. It's not "more armor is always better." There's an optimal point, a sweet spot on a curve. Deviating in either direction is bad news. Kevin: Ah, so it’s about finding the peak of the curve. Michael: Precisely. And this thinking applies directly to huge political debates. Take the Laffer Curve, for example. It's the idea that as you raise tax rates, government revenue goes up, but only to a point. If you raise taxes to 100%, revenue drops to zero because no one has any incentive to work. Kevin: Right, so there's a peak tax rate that maximizes revenue. Michael: Exactly. The relationship is a curve, not a straight line. The debate during the Reagan years was whether the U.S. was on the "good" side of the curve, where cutting taxes would actually increase revenue, or on the other side. Some readers have criticized the book for getting a bit too much like a textbook in these sections, but the real-world stakes are huge. The data suggests the tax cuts of the 80s actually led to less revenue, not more. We were on the wrong side of the curve for that particular argument to work. Kevin: So the lesson is: stop thinking in straight lines. The world is curvy. Michael: The world is very curvy. And assuming it's straight will lead you to some very wrong, and sometimes very expensive, conclusions.

Michael: So we have to watch out for missing data and faulty straight lines. The last big trap is maybe the most subtle, because it feels so much like there's a real cause. It's about mistaking randomness for a reason. Kevin, let me ask you: do you believe in the 'Sports Illustrated Jinx'? Kevin: Come on, the SI Jinx is totally real! An athlete gets on the cover, and boom, the next season is a disaster. It happens all the time. It's a curse. Michael: I'm so glad you said that. Because what you're describing isn't a curse. It's a mathematical phenomenon called "regression to the mean." And it's one of the most misunderstood concepts out there. Kevin: Okay, you're going to have to walk me through this. How is it not a jinx? Michael: Ellenberg explains it using the work of Francis Galton, a 19th-century statistician. Galton was studying the heights of parents and children. He found that extremely tall parents tended to have children who were also tall, but on average, a little bit shorter than them—a little closer to the average height. And extremely short parents had children who were short, but a little taller than them. They "regressed" toward the mean, or the average. Kevin: So... tall people are doomed to have shorter kids over time? Michael: Not exactly. The key insight is that any single outcome—like a person's height or an athlete's performance in a given season—is a combination of two things: a stable, underlying factor, like genetics or skill, and a random, unstable factor, which we can just call luck. Kevin: Skill and luck. Got it. Michael: To get an extreme outcome, like being a seven-foot-tall person or having a record-breaking season that lands you on the cover of Sports Illustrated, you almost always need both. You need incredible skill and incredible luck. You have to be at the peak of your game, and all the random bounces have to go your way. Kevin: Okay, that makes sense. Michael: Now, what happens next season? Your incredible skill is probably still there. But your incredible, off-the-charts luck? That's random. It's very unlikely to be just as good two seasons in a row. So your performance will likely be a combination of your great skill and more... well, average luck. It will still be great, but it will be a step down from the historical peak. It "regresses to the mean." Kevin: Oh, I see. So it's not a jinx that causes the decline. The "jinx" is just what we call the inevitable fading of extraordinary luck. We're assigning a cause to what is just a return to a more normal state. Michael: You've got it. It's not a curse, it's just statistics. The same thing explains the "sophomore slump" for athletes, or why a business that has a record-breaking quarter rarely has another one just like it. We are pattern-seeking animals, and we desperately want to find a cause for that drop-off. A jinx, a curse, complacency. But often, the real cause is just regression to the mean. Kevin: That's a huge insight. So how does knowing this help us? Does it mean we shouldn't praise people when they do exceptionally well, because it's just luck? Michael: Not at all! You absolutely celebrate the peak performance. But you don't panic or fire the coach when that performance returns to a more normal, sustainable level. You understand that extreme results are, by their nature, hard to repeat. It helps you distinguish between a real problem and simple statistical noise. It stops you from being wrong about why things happen.

Michael: And really, that brings us back to the core of the book. These three ideas we've talked about—survivorship bias, nonlinearity, and regression to the mean—they're all about the same fundamental thing. Kevin: What's that? Michael: They are all examples of how our built-in, "common sense" intuition can lead us astray. We look at the survivors and draw the wrong conclusion. We see a trend and assume it's a straight line. We see a peak followed by a dip and invent a cause. Kevin: And mathematics is the corrective lens for that faulty intuition. Michael: Precisely. As Ellenberg says, it's an "extension of common sense by other means." It’s not about memorizing formulas. It’s a way of thinking that forces you to be more rigorous, to question your assumptions, and to see the world with more clarity. It’s the science of not being wrong. Kevin: It makes you wonder how many "obvious truths" in our own lives are just statistical illusions we haven't noticed yet. Where are the missing bullet holes in our own thinking? Michael: That's a great question for our listeners. After hearing this, what's a 'missing bullet hole' you've realized might exist in your own field, or your own life? We'd love to hear your thoughts. Let us know. Kevin: It's a powerful and humbling idea. A reminder to stay curious and be willing to be proven wrong. Michael: Couldn't have said it better myself. Michael: This is Aibrary, signing off.