How Not to Be Wrong: The Power of Mathematical Thinking
Jordan Ellenberg (2014, The Penguin Press)
Jordan Ellenberg wants to teach us to love math because it’s a super-power. “Math is like an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength.” He’s not just talking about algebra, or Euclidian proofs, though he’d cheerfully confess to the beauty and utility of those things. In How Not to Be Wrong, he’s talking about how we look at the world, and how we understand what we see.
The world, as it turns out, is full of bad math, because people employ the tools and gadgets of math without the common sense. Ellenberg deconstructs a study that extrapolates from four decades of increasing obesity to conclude that all Americans will be obese by 2048, if current trends continue. “But current trends will not continue. They can’t! If they did, by 2060, a whopping 109% of Americans would be overweight.” As it turns out, when looked at from the appropriate distance, many straight lines are actually curves.
Probability is another area where mathematics helps make sense of our intuition. The probability that a large number of coin flips will come up heads half the time is too taxing to grasp, though we have to guard against believing that the coin remembers its previous results. But, says Ellenberg, what about expressing tomorrow’s chance of rain in percentage terms? “Tomorrow only happens once; it’s not an experiment we can repeat like a coin flip again and again.”
Still, we use the tools we have, and Ellenberg wants us to use them wisely, or at least sensibly. The modern practice of statistics relies on the null hypothesis significance test, familiar from discussions of drug trials, economic theory, and psychological experiments. If you set up a null hypothesis, that such and such a thing has no effect, running tests, and finding the null hypothesis comes up less than five per cent of the time, you can say that the thing has a statistically significant chance of being true. Ellenberg points out, in the first place, that “the significance test that scientists use doesn’t measure importance,” though it sounds like it would.
“If you make the test more sensitive–by increasing the size of the studied population, for example–you enable yourselves to see ever-smaller effects.” Just because something that almost never happens is three times likelier to happen doesn’t make it significant in the ordinary English sense, over-heated headlines notwithstanding. The significance test is a cousin to the reductio ad absurdum, in which mathematicians set up an assumption to disprove. But, Ellenberg warns, “impossible and improbable are not the same–not even close. Impossible things never happen. But improbable things happen a lot.”
How Not to Be Wrong is a delightfully approachable book, though there’s plenty of real math in it. The reader comes away knowing more about probability theory, encryption algorithms, alternative geometries, and why elections with more than two candidates are an unsolvable problem.
And, Ellenberg hopes, we will find a taste for using reason in a structured way: “I find it’s a good habit to put pressure on all your beliefs, social, political, scientific, and philosophical. Believe what 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.”
Doesn’t that sound like a wonderful thing?
E-mail edition, October 1, 2014