"HOW NOT TO BE WRONG: THE POWER OF MATHEMATICAL THINKING" by Jordan Ellenberg

• Math is woven into the way we reason. And math makes you better at things.
• With the tools of mathematics in hand, you can understand the world in a deeper, sounder, and more meaningful way.
• A mathematician is always asking, “What assumptions are you making? And are they justified?” This can be annoying. But it can also be very productive.
• Mathematics is the study of things that come out a certain way because there is no other way they could possibly be.
• The specialized language in which mathematicians converse with one another is a magnificent tool for conveying complex ideas precisely and swiftly.
• Without the rigorous structure that math provides, common sense can lead you astray.
• Nonlinear thinking means which way you should go depends on where you already are.
• A basic rule of mathematical life: 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.
• Every smooth curve, when you zoom in enough, looks just like a line.
• 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.
• Anytime a lot of people in a small country come to a bad end, editorialists get out their slide rules and start figuring: how much is that in dead Americans?
• 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.
• 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, until the past is so proportionally negligible that it can safely be forgotten.
• Great Big Book of Horrible Things,
• Don’t talk about percentages of numbers when the numbers might be negative.
• Negative numbers in the mix make percentages act wonky.
• Improbable things happen a lot.
• Trial and error is a very powerful weapon.
• The point is, reverse engineering is hard.
• Many scientific questions can be boiled down to a simple yes or no: Is something going on, or not?
• The “does nothing” scenario is called the null hypothesis. That is, the null hypothesis is the hypothesis that the intervention you’re studying has no effect.
• So here’s the procedure for ruling out the null hypothesis, in executive bullet-point form: Run an experiment. Suppose the null hypothesis is true, and let p be the probability (under that hypothesis) of getting results as extreme as those observed. The number p is called the p-value. If it is very small, rejoice; you get to say your results are statistically significant. If it is large, concede that the null hypothesis has not been ruled out.
• New things require new vocabulary. There are two ways to go. You can cut new words from fresh cloth, as we do when we speak of cohomology, syzygies, monodromy, and so on; this has the effect of making our work look forbidding and unapproachable. More commonly, we adapt existing words for our own purposes, based on some perceived resemblance between the mathematical object to be described and a thing in the so-called real world.
• twice a tiny number is a tiny number. How good or bad it is to double something depends on how big that something is!
• Impossible things never happen. But improbable things happen a lot.
• The logarithm of a positive number N, called log N, is the number of digits it has. Wait, really? That’s it? No. That’s not really it. We can call the number of digits the “fake logarithm,” or flogarithm. It’s close enough to the real thing to give the general idea of what the logarithm means in a context like this one.
• Data is messy, and inference is hard.
• The purpose of a court is not truth, but justice.
• In the Bayesian framework, how much you believe something after you see the evidence depends not just on what the evidence shows, but on how much you believed it to begin with.
• Bayes’s Theorem can be seen not as a mere mathematical equation but as a form of numerically flavored advice.
• A reasonable person believes, in short, that each of his beliefs is true and that some of them are false.”
• This sounds weird, but as a logical deduction it’s irrefutable; drop one tiny contradiction anywhere into a formal system and the whole thing goes to hell.
• when you’re working hard on a theorem you should try to prove it by day and disprove it by night.
• 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.