Jordan Ellenberg: How Not to be Wrong
If you’re not moving into a mathematically oriented career, that’s fine. But you can still do math. Math is woven into the way we reason. And math makes you better at things.
Math is a science of not being wrong about things, its techniques and habits hammered out by centuries of hard work and argument.
The problems we think through everyday are shot through with mathematics.
A mathematician is always asking: “What assumptions are you making? And are they justified?”
Mathematics is the study of things that come out a certain way because there is no other way they could possibly be.
Looking at the dots of a stereo, you can flip your mind back and forth between seeing columns and seeing rows.
Regardless of whether you see 8 columns of 6 or 6 rows of 8, you have the same number of dots on the stereo.
It cannot be any other way and this is why multiplication is commutative.
Mathematics is like an atomic robotic arm to attach to your common sense.
Fascinating: is your problem linear or nonlinear?
Linear means the highest point will be at one of the ends; one of the extremes.
Nonlinear means more complicated. There is nuance to the extreme, it is somewhere in the middle.
Ex. Both of these can be right, and give different stories, based on whether you assume linearity or nonlinearity.
If you believe there’s such a thing as too much welfare state and such a thing as too little, the linear picture is not accurate.
To answer: Should Obama move the US towards a greater social welfare state (like Sweden)?
Nonlinear thinking means that the way you should go depends on where you already are.
Ex. Sweden should decrease size of government and US should increase.
Linear reasoning is thinking that if something is good to have, having more of it is better.
Beware the thoughtlessness of linear extraction, especially in the social sciences.
The law of large numbers - large populations are much more likely to be average than small populations.
A team of 100,000 people flipping 100 coins will likely have a head probability of 50%. A team of 10 people flipping coins could have 70% of their 100 coin flips as heads.
The fraction of heads converges inexorably to 50% as if squeezed by a vice.
Small samples are thin reeds whipped around by winds of chance. Large samples are grand old oaks that barely bend.
This is because the more and more coins you flip, the less the first ten coin flips matter.
Always remember that what happened in the past, in terms of flipping the coin, has no impact on the probability in the future. The coin has no memory.
The law of large numbers works be diluting what’s happened with new data, not by balancing out what’s already happened.
A partially ordered set means that come pairs can be meaningfully compared, and others cannot.
Ex. With historical casualties.
More pie than plate.
If there are negative numbers in play, like jobs lost, then be careful.
Ex. Wisconsin had 9k job growth in one month, which was 50% of the 18k jobs added nationally. Texas had 75% of the jobs added nationally. How? Because other states had big job losses.