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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.
Mathematicians like to give names to the phenomena our common sense describes: instead of saying, “This thing added to that thing is the same thing as that thing added to this thing,” we say, “Addition is commutative.” Or, because we like our symbols, we write: For any choice of a and b, a + b= b + a.
Mathematics is the study of things that come out a certain way because there is no other way they could possibly be.
Without the rigorous structure that math provides, common sense can lead you astray.
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 difference between the two pictures is the difference between linearity and nonlinearity,
Usually, when someone announces they’re a “nonlinear thinker” they’re about to apologize for losing something you lent them.
Scatter brained?
Nonlinear thinking means which way you should go depends on where you already are.
Laffer curve,
if the universe hands you a hard problem, try to solve an easier one instead, and hope the simple version is close enough to the original problem that the universe doesn’t object.
straight locally, curved globally.
And the slope of this line is what Newton called the fluxion, and what we’d now call the derivative.
our intuition about time and motion is formed by the phenomena we observe in the world.
0.33333. . . . .= 1/3. Multiply both sides by 3 and you’ll see 0.99999. . . .= 3/3= 1.
In the real world, you can never have infinitely many heaps. What’s the numerical value of an infinite sum? It doesn’t have one—until we give it one.
In the mathematical context, the good choices are the ones that settle unnecessary perplexities without creating new ones.
define the value of the infinite sum to be 1.
We’re untroubled by the fact that the English language sometimes uses two different strings of letters (i.e., two words) to refer synonymously to the same thing in the world; in the same way, it’s not so bad that two different strings of digits can refer to the same number.
all interesting theories of infinite sums either give it the value 1/2 or decline, like Cauchy’s theory, to give it any value at all.*
I have some good news. We’re not all going to be overweight in the year 2048. Why? Because not every curve is a line.
But every curve, as we just learned from Newton, is pretty close to a line. That’s the idea that drives linear regression,
it works on any data set at all.
That’s a weakness as well as a strength. You can do linear regression without thinking about whether the phenomenon you’re modeling is actually close to linear.
One gets such wholesale returns of conjecture out of such a trifling investment of fact.
From Mark Twain
purely mechanical, your calculator can carry them out, and it is very dangerous to use them inattentively.
Working an integral or performing a linear regression is something a computer can do quite effectively. Understanding whether the result makes sense—or deciding whether the method is the right one to use in the first place—requires a guiding human hand. When we teach mathematics we are supposed to be explaining how to be that guide. A math course that fails to do so is essentially training the student to be a very slow, buggy version of Microsoft Excel.
Unfortunately how most math classes were taught to me
But current trends will not continue. They can’t! If they did, by 2060, a whopping 109% of Americans would be overweight.
See the problem? If all Americans are supposed to be overweight in 2048, where are those one in five future black men without a weight problem supposed to be? Offshore?
When there are two men left in the bar at closing time, and one of them coldcocks the other, it is not equivalent in context to 150 million Americans getting simultaneously punched in the face.
An important rule of mathematical hygiene: when you’re field-testing a mathematical method, try computing the same thing several different ways. If you get several different answers, something’s wrong with your method.
The Better Angels of Our Nature,
Add to goodreads
Law of Large Numbers.
the more coins you flip, the more and more extravagantly unlikely it is that you’ll get 80% heads.
Scoring by raw number of heads gives the Big team an insuperable advantage; but using percentages slants the game just as badly in favor of the Smalls.
A few child prodigies or a few third-grade slackers can swing a small school’s average wildly; in a large school, the effect of a few extreme scores will simply dissolve into the big average, hardly budging the overall number.
the typical discrepancy* is governed by the square root of the number of coins you toss.
As a proportion of the total number of tosses, the discrepancy shrinks as the number of coins grows, because the square root of the number of coins grows much more slowly than does the number of coins itself.
The way the overall proportion settles down to 50% isn’t that fate favors tails to compensate for the heads that have already landed; it’s that those first ten flips become less and less important the more flips we make.
That’s how the Law of Large Numbers works: not by balancing out what’s already happened, but by diluting what’s already happened with new data,
Don’t talk about percentages of numbers when the numbers might be negative.
“Seventy-five percent” sounds like it means “almost all,” but when you’re dealing with numbers that could be either positive or negative, like profits, it might mean something very different.
The combination of positive and negative allows you, if you’re not careful, to tell a fake story,
93% and 17% add up to more than 100%; how does this make sense? It makes sense because the bottom 90% actually had lower average income in 2011 than they did in 2010, recovery or no recovery. Negative numbers in the mix make percentages act wonky.
We have no idea how many jobs were created and how many destroyed over the three-year period; only that the difference of those two numbers is 740,000. The net job loss is positive sometimes, and negative other times, which is why taking percentages of it is a dangerous business.
It’s the right answer to the wrong question.
But the calculator enters only once you’ve figured out what calculation you want to do.
Dividing one number by another is mere computation; figuring out what you should divide by what is mathematics.
In fact, you can compute the odds on the nose: if the duffer has a 50% chance of getting each prediction right, then the chance of his getting the first two predictions right is half of half, or a quarter, his chance of getting the first three right is half of that quarter, or an eighth, and so on.
reality TV being where we go for parables
interesting comparison
you’ve been swayed by the impressive results, but you don’t know how many chances the broker had to get those results.
