The title might lead you to believe that this book is all about learning how not to be wrong. But it’s really more than that; it’s about exemplifying and promoting a certain style of thinking, about developing a kind of mathematical street smarts combined with common sense. Ellenberg notes later in his book that sometimes being wrong is the right thing to do, so it’s probably better not to concentrate too much on the right and wrong part. It’s also notable that the title says “how not to be wrong” and not “how to be right”.
The main thing you should know about this book is that it’s clever, entertaining and very well written. It covers a lot of ground. The main sections are named broadly: linearity, inference, expectation, regression and existence, but those really only give you a hint of the scope. I was surprised to see so much attention to data, statistics and basic probability, especially given that Ellenberg’s background is in number theory and algebraic geometry. But he handles those subjects very skillfully. (It’s probably no coincidence that his parents are both statisticians.) The attention to data and simple computations helps keep the book well grounded and maybe more attractive to a general reader.
- the story of consistent big wins in the Massachusetts WinFall lottery because a couple of groups paid special attention to the terms of the lottery and made a very profitable assessment of expected value;
- correlation, causation and R.A. Fisher’s contention that lung cancer caused smoking;
- the pitfalls of significance in hypothesis testing and an example where B.F. Skinner (an English major before his turn to psychology) argued that Shakespeare’s alleged skill with alliteration was a random effect; and
- choosing numbers for lottery tickets, the Fano plane and Hamming’s error correction codes.
This is just a sample.
One of the joys of the book is Ellenberg’s gift for making quite surprising connections while apparently meandering from one topic to another. There are many opportunities throughout the book for digressions, and Ellenberg happily takes advantage of them.
There are a few occasions when things don’t quite work. An early discussion of Zeno’s paradox and limits doesn’t get to a satisfying resolution of the paradox. Later in the book an extended discussion of regression to the mean never quite gets to the root of the question. There are a few other instances like this, perhaps only notable because Ellenberg’s exposition is so good just about everywhere else.
One thing that troubled me right from the beginning of the book was the sense that Ellenberg was picking a fight. This, oddly enough, occurs in an example of nonlinearity. The context is a discussion of the relationship between prosperity and “Swedishness” (which is supposed to denote something like the degree to which a country collects taxes to support social services). Ellenberg attacks a libertarian view that this relationship is linear (prosperity decreases linearly as “Swedishness” increases). With a better definition of terms, I’d agree with Ellenberg’s argument (and his politics), but the whole thing seems unnecessarily inflammatory, especially right at the beginning of the first chapter. Conservative readers might well be inclined to dismiss the book immediately as liberal propaganda. There certainly are less inflammatory examples of nonlinearity. If you’re trying to teach an important concept, best not first alienate some of the students.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.