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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?
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.
“What assumptions are you making? And are they justified?”
which way you should go depends on where you already are.
If you’ve ever used America’s most popular sort-of-illegal psychotropic substance, you know what it feels like to have too-flat priors. Every single stimulus that greets you, no matter how ordinary, seems intensely meaningful. Each experience grabs hold of your attention and demands that you take notice. It’s a very interesting mental state to be in. But it’s not conducive to making good inferences.
This is a property most successful crackpot theories have in common; they’re encased in just enough protective stuff that they’re equally consistent with many possible observations, making them hard to
dislodge.
When you’re faced with a math problem you don’t know how to do, you’ve got two basic options. You can make the problem easier, or you can make it harder.
you replace the problem with a simpler one, solve that, and then hope that the understanding gained by solving the easier problem gives you some insight about the actual problem you’re trying to solve.
If anything, I’ve found that in moments of emotional extremity there is nothing like a math problem to quiet the complaints the rest of the psyche serves up.
Von Neumann and Morgenstern,* in their foundational book The Theory of Games and Economic Behavior
Burton Malkiel’s A Random Walk down Wall Street
But the ability to work hard—to keep one’s whole attention and energy focused on a problem, systematically turning it over and over and pushing at everything that looks like a crack, despite the lack of outward signs of progress—is not a skill everybody has. Psychologists nowadays call it “grit,”
What you learn after a long time in math—and I think the lesson applies much more broadly—is that there’s always somebody ahead of you, whether they’re right there in class with you or not.
Genius is a thing that happens, not a kind of person.
It is not the critic who counts; not the man who points out how the strong man stumbles, or where the doer of deeds could have done them better. The credit belongs to the man who is actually in the arena, whose face is marred by dust and sweat and blood; who strives valiantly; who errs, who comes short again and again, because there is no effort without error and shortcoming; but who does actually strive to do the deeds; who knows great enthusiasms, the great devotions; who spends himself in a worthy cause; who at the best knows in the end the triumph of high achievement, and who
at the worst, if he fails, at least fails while daring greatly, so that his place shall never be with those cold and timid souls who neither know victory nor defeat.
the survival of civilization depends on the triumph of the bold, commonsensical, and virile against the soft, intellectual, and infertile.*
Proving by day and disproving by night is not just for mathematics. I find it’s a good habit to put pressure on all your beliefs, social, political, scientific, and philosophical. Believe whatever you believe by day; but at night, argue against the propositions you hold most dear. Don’t cheat! To the greatest extent possible you have to think as though you believe what you don’t believe. And if you can’t talk yourself out of your existing beliefs, you’ll know a lot more about why you believe what you believe. You’ll have come a little closer to a proof.
Ever tried. Ever failed. No matter. Try again. Fail again. Fail better.
