Ellenberg and the Magnificient World of Math

Jordan Ellenberg is the prolific mathematician at University of Wisconsin-Madison, whose Slate columns and best-seller How Not To Be Wrong - The Hidden Maths of Everyday Life* has captured more than my own heart (see interviews by NPR, The Guardian and Slate.com for more media coverage). 

Before the first fifty pages were over, he had already re-awakened my teenage spirit (the hatred towards certain math problems that I probably never entirely let go of), strawmanned a blogger I adore (Dan Mitchell), discussed the Laffer curve and sparked nationalist interest by naming his first chapter "Less than Sweden". Moreover, he reminded me of something mind-blowing a friend once told me, and disproved Steve Keen's seemingly bulletproof attack on neoclassical economics (pp. 47-48) by refering to an early-19th century mathematician (Augustin-Louis Cauchy, famous for introducing the notion of limit into mathematics).

In other words, he had me at hello.

In the book, he is doing for mathematics what Freakonomics did to popularize Economics: show us normal (i.e. non-nerdy) people how math is everywhere, how it can be accessible, relevant, easy and even fun! The structure is unmistakenly like Lewitt & Dubner's: start a chapter with some unbelievable, eye-watering or exciting example, explain parts of it, digress to related topics - or in Ellenberg's case short biographies of centuries-old mathematicians and their path-breaking insights - and exactly when the reader wonders why this is relevant, you tie it all back together. He's very considerate with his readers, always meticulously explains points, numbers and graphs so that we can follow. At one point he even trigger-warns us: "A lot of numbers are about to come flying at your face!" (p. 384).

The book's raison d'être and Ellenberg's rigorous defense of mathematical thinking comes down to one simple sentence:

The mathematical approach is a formalized version of our natural mental reckonings, an extension of common sense by other means. (p. 203)

And he really does mean that. He provides fascination stories about bullet-holes in American WWII planes (and where to put the fortified armor!), if Facebook can identify your neighbour as a terrorist, how politicians and news media everyday mis-applies percentages and math, if the Bible can predict the future and how the 'Baltimore Stockbroker' has such seemingly amazing returns, why tall parents have shorter children, why the statistical significance and the scientific principle of "torturing data until it confesses" is stupid, about how playing the lottery can be rational and finally, if there really is such a thing as a public opinion. He is fairly careful not to come across as politically biased, hitting mistakes by politicians on both sides of the political spectrum, disproving and mocking topics left and right.

Sometimes the digressing nature becomes a bit too much. I admit to have doozed off and put the book down for some weeks in the midst of some yawn-provoking section on encryption and codes and satillite messages. He more than enough makes up for that, however, in a brilliant fifty-page destruction of statistical significance - the method used by most scientists (economists, social scientists as well as biologists and in medicine). My favorite example is the section titled 'Does Facebook Know You're a Terrorist?', where he illustrats the following:

Under the rules that govern the majority of contemporary science, you'd be justified in rejecting the null hypothesis and declaring your neighbor a terrorist. Except there's a 99.99% chance he's not a terrorist. (p. 169)

The best section of all, and an absolute must-read for every politician, political pundit and inequality-fanatic out there is chapter 5, neatly titled "More Pie Than Plate" (pp. 78-85). His punchline is this:

In order to give a sensible answer, you need to know more than just numbers. [...] Dividing one number by another is mere computation; figuring out what you should divide by what is mathematics. (p. 85)

The problem is this: imagine I have three business departments. The first, selling t-shirts ran a profit of $20 last month; the second, selling second-hand books also had a $20 profit last month, and lastly my hopelessly failing juice business lost me $15. My overall profit was 20+20-15 = $25. If I treat percentages like Ellenberg's examples of Republican politicians treat job growth or economists Piketty & Saez treat income gains going to the top-1%, I could easily say that my t-shirt business contributed 80% of my profits (20/25 = 0.8). But I would be equally correct in saying that my second-hand bookstore also contributed 80% of my profits - and all of a sudden I had 160% of my profits from these two sources. The example becomes even more obvious if we pretend my book store never existed; 20-15 = $5, $20 of which (400%!) came from my t-shirt store. 

Piketty & Saez makes a similar mistake when looking at the income gains over some time period; 93% of which went to the top-1%. That sounds terrible, until you realize that another 17% went to the top 10%, but not the top-1%, for a total of 110% of income gains. At this point, most people realize that there's some madness in the method; saying that 400% of my profits came from t-shirts is so obviously meaningless that I shouldn't be saying anything at all. Percentages no longer means what they normally mean. The take-away is this: don't talk about percentages of numbers when those numbers may be negative. Shit goes bananas. 

Finally, he quotes a beautiful section from the philosopher W.O.V Quine:

to believe something is to believe that it is true; therefore a reasonable person believes each of his beliefs to be true; yet experience has taught him to expect that some of his beliefs, he knows not which, will turn out to be false. A reasonable person believes, in short, that each of his beliefs is true and that some of them are false. (p. 429)

I'll finish off with Ellenbergs parting words, summarizing his wonderful book:

You are doing mathematics, the extension of common sense by other means. When are you going to use it? You've been using mathematics since you were born and you'll probably never stop. Use it well.

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* American subtitle is The Power of Mathematical Thinking.

If you wanna see Ellenberg in person, he's apparently organising a conference at the International Centre for Mathematical Sciences in Edinburgh in May 2017.