Math Book Club: “How Not to Be Wrong”
I love book clubs, especially as an idea. I love talking about books, and watching whole new worlds open up from a different reading of the same text. However, absent a really good guide or leader, book talk can be: “I liked that book.” “Admiral Harte is sure a jerk!” “That was boring.” This is actually pretty fun; chatting lightly about shared experiences strikes me as a pretty major human need and creator of community. I get a spark from realizing someone else heard the same song on the radio as I did, let alone experienced something as lengthy and sometimes heavy as reading the same book. The problem is exactly that, though: some books have a lot in them, and it’s frustrating to stay on the surface of something that feels very substantial. But who knows where to even start? Hence the need for awesome English professors. Plus in some book clubs, coolness prevails in that it is somehow a little shameful to talk about meaning in public. Even though you’re in book club, ostensibly to talk about the book, saying anything heavy is just slightly in bad taste, pretentious, oversharing. Why is it so weird to talk about meaning?!? Maybe because it’s hard, and so we find ourselves saying things that aren’t quite what we want to say. Which is kind of embarrassing. Or maybe we are all kind of insecure and afraid we didn’t really get it. I am in awe of how really good literature teachers can work a discussion, both in providing ways to look at a text and in overcoming all of this social crap that we put in the way of talking about meaning. Anyway, my experience in book clubs has been that people talk for a few about characters they liked and didn’t, and then everybody has some wine and book club becomes essentially planned social drinking! Which is always cool with me. Sign me up for book club.
However, I don’t have one now, and I have been reading a lot of mathy books lately. If it’s hard to find a great book club, how about a great book club that wants to talk about math? So, I guess that’s one reason to have a blog—it can be your pretend math book club! Perhaps we should all pour a glass of wine. Okay, all set on this end. This week we are (um, I am) discussing Jordan Ellenberg’s recent book How Not to Be Wrong.
First, the quick and surface pre-wine book club version of my thoughts: “Super awesome book! Math is cool! Jordan Ellenberg is freaking smart.” Okay, now in case you didn’t read the book, (also a hazard of book club), HNTBW is sort of like a tour of great mathematical ideas (subtitle: The Power of Mathematical Thinking). It goes everywhere. Like, scatterplots, the Baltimore stockbroker con, philosophy in mathematics and statistics, error correcting codes, game theory, perspective, voting paradoxes. Picture a book that picks up where Darrell Huff’s How to Lie with Statistics left off and then tours through the country of Burger and Starbird’s The Heart of Mathematics, with the historical color of E.T. Bell’s Men of Mathematics, but with the advantage that the history (I’m pretty sure) is all real. Ellenberg sketches out ideas and stories that are essential to modern scientific thinking. The anecdotes and examples are contemporary, bizarre, funny, and very relevant
The most important thing I learned: statistical significance is shady, shady business. Statistics is how we make sense out of data, so statistics lies at the base of all science. I think of myself as a logical thinker, a scientist, and I like to use scientific studies to help me make decisions. So how is it that I have never thought carefully about this business. When I gleefully read read the aforementioned How to Lie with Statistics as a kid (yeah, I was pretty nerdy), I thought I knew all the tricks and pitfalls. So wrong. The second section of HNTBW, entitled “Inference,” was a lesson in what p-values really mean. If we gather some data from an experiment, and compare two quantities (like height and shoe size), we would think it likely that they could be connected if as one quantity changes, the other seems to change in a predictable way as well. We are “allowed” to say that the quantities are correlated (the data shows a statistically significant correlation) if there is a probability (p-value) of less than 5% that we would see this relationship in the data by chance if the quantities were actually totally independent (the null hypothesis).
That sounds pretty good—if the two quantities are not correlated, there is less than a 5% chance that we would say they were statistically significantly correlated by looking at a particular data set. But I never really thought about it in this light: say we gathered data on a lot of entirely uncorrelated quantities. Out of every hundred pairs of uncorrelated quantities, five of the pairs of data sets would show statistically significant correlation entirely by chance. Of course if we gathered more new data on these the two seemingly-correlated quantities, there is only a 5% chance this false correlation would show up again. So our false correlation would very likely be debunked. But first there might be a news story about it, because positive results are exciting! And then, when another scientist could not replicate our result, there might be a follow-up piece, but then again maybe not, because negative results are boring. Their non-replication might not be published at all, even in a scientific journal. In fact, it could be that someone else had done the same study years earlier and found no correlation, which would be sort of a pre-non-replication of our study, but we would never have known because this earlier group couldn’t publish their negative results.
Now, despite the beauty and validity of the butterfly effect, I believe that most pairs of quantities in the universe are not meaningfully and predictably correlated. So I’m ready to dismiss claims (based on fMRI images of one out of many, many scanned “subjects”) that dead fish can correctly assess human emotions from photographs (Chapter 7), even though the correlation was statistically significant, based on, well, yeah. However, with complicated biological systems, how can we really apply common sense to say two things are not related? A study that finds a connection between Alzheimer’s and a particular gene is exciting, important, and not at all an implausible connection on the surface. One study finds a statistically significant correlation, and millions of people will factor that in to their decision making. The dead fish example is particularly striking given how much people will believe a story if accompanied by fMRI images.
My mind was totally blown by this previously unconsidered pitfall. Apparently people who were not blind to this have been advocating for the publication of negative results for years. It appears that the advent of open-access online journals may have finally made this possible.
The last part of this book is called “Existence.” If you had any doubts about whether Jordan Ellenberg is willing to talk about meaning, you can lay them to rest right now. I really like the final few chapters of this book. We get voting theory, slime molds and non-Euclidean geometry. And then, the question: what can we say about the truth, and about being right? Drawing on Condorcet’s paradox, that a group’s preferences for various candidates can result in unresolvably cyclical rankings, Ellenberg raises a question. If there is no one true preference of the people, what are elections doing? What can it mean when there is no one right answer, one correct winner for an election, one consistent geometry? If there is no one right answer, then what the hell is math doing? Is it searching for the truth? What if there is no right answer, or if all of the study and data tells you with great certainty that you can not be certain. What is the point of all this, anyway?
This launches a discussion of what mathematical thinking could bring to every person’s daily struggle to decide what is true, and of what it means to be a creative mathematician. He does not shy away from discussing meaning, both in the careful sense of statistical and mathematical meaning and in the more philosophical sense of questions like, why do we do these things? And, if we can choose how to think about the world, why should we think about it like that? He talks about rigorous uncertainty, the inherent contradictions we face in math and in life, the destructive idealization of genius, and how math comes from a whole community. Math is like football? You mean that in a good way? Yeah, in the team sport way (luckily not in the concussion and sweaty dude way), and this actually gave me a really warm feeling about math. Plus there is poetry, David Foster Wallace, my favorite Philadelphian Benjamin Franklin on how annoying mathematicians are, Terry Tao deflating the cult of genius… also F. Scott Fitzgerald, The Housemartins, and Captain Kirk. This could seem like showing off, or just a ham-fisted attempt to synthesize everything and force a moral into mathematics. It does not. Ellenberg really, ahem, scores a touchdown. This is the best discussion of what it means to be a mathematician, and how that relates to big questions in life that I have read. I think this is a book that we should have our first-year math majors read. Or all the first-year students—which would probably mean more math majors, for better or worse.
The problem of audience—David Foster Wallace brought this up in his essay “Rhetoric and the Math Melodrama.” What reader is this book written for? I can’t un-love math, so I of course I can’t say how this book would speak to people who don’t love math. I would guess that it is aiming for an interested but not specifically mathematical audience, and that it will speak incredibly well. Ellenberg has a great knack for being casual and conversational but never sloppy, and he starts with topics that people care about for their own sake (slipping in the cool math and the hard, unanswered questions after your interest is solid). He comes across as a hell of a guy to have in your book club. If he had time for book clubs, that is, what with a family and a billion speaking engagements and writing for the New York Times and like, proving stuff. But, hey, if he was in book club, we could look forward to more lines like this:
“The moment, early in the Italian Renaissance, at which painters understood perspective was the moment visual representation changed forever, the moment when European paintings stopped looking like your kid’s drawings on the refrigerator door (if your kid mostly drew Jesus dead on the cross) and started looking like the things they were paintings of.”
Conspiracy theories are “like the multi-drug-resistant E. coli of the information ecosystem. In a weird way you have to admire them.”
On a study finding that women were more likely to say they would vote for Mitt Romney during the most fertile part of their ovulatory cycle: “I was disappointed to find that this study has not yet spawned any conspiracy videos coming that Obama’s support of birth control coverage was aimed at suppressing women’s biological drive to vote GOP during ovulation. Get on the stick, conspiracy video producers!”
“It’s not wrong to say that Hilbert was a genius. But it’s more right to say that what Hilbert accomplished was genius. Genius is a thing that happens, not a kind of person.”
“If I’m going to pay a college kid to superimpose dancing cutlery on all my pages, I want to know not only whether it works, but how well.”
“Here, ‘Swedishness’ refers to ‘quantity of social services and taxation,’ not to other features of Sweden such as ‘ready availability of herring in dozens of different sauces,’ a condition to which all nations should obviously aspire.”
And all good book clubs, as well.