How Not to be Wrong is an entertaining and erudite guide to some of the ways in which math features in aspects of everyday life.

A chief challenge in math pedagogy is how to make its endless drill sets and lofty theorems relatable to the average student. In this book, Jordan Ellenberg has ventured to show just how interesting and useful math training can be, not just as an abstract set of rules for manipulating numbers, but, as he puts it, an “extension of common sense by other means”.

Ellenberg’s main preoccupation is extending mathematical thinking to relatable examples in everyday life – and this causes his writing to converge around a few broad themes or sections. There’s the injunction not to fall prey to the tendency to think linearly – i.e. that more of a good thing is necessarily always better, or vice versa. When you assume that we should cut taxes because Sweden is doing so too (i.e. our Laffer curve is straight), or treat ratios and percentage figures as truths without regard for sample size or non-positive fluctuations that, summed up, make up the overall delta, we’re thinking linearly.

Then there is an extended discussion on risk and uncertainty, and the differences between them. Risk is quantifiable in terms of expected return, and that quantifiability comes from the fact that things whose risk can be measured can be iterated upon multiple times – like a lottery or successive throws of the dice. Uncertainty is when the likelihood of something cannot be quantified – like the existence of god – and one’s decisions made in under such uncertainty inevitably require that one make certain axiomatic assumptions about likelihood and perform Bayesian inference on top of it, accounting for all possible permutations of outcomes, and in this sphere, math isn’t much help.

There’s also discussion on the vast, complicated and gnarly field of statistical inference – of ascribing too many things to causation when the seeming correlative effect is just a statistical artefact. These include regression to the mean and Berkson’s fallacy, which are essentially just perceptual artifacts that stem from our cognitive biases.

Berksons’ fallacy was especially interesting to me: it is essentially the generation of spurious correlations by the unconscious elision of data points from our sample set. For example, you might wonder why trashy books are so popular, or why old music was so much better. That’s because, when thinking about them, you don’t consider the universe of trashy unpopular books or lousy old music, and so in your mind, the sample size resolves itself to the set of trashy popular books, unpopular meritorious books, and a small subset of good popular books that you regard as an outlier.

Ellenberg also touches on Condorcet’s paradox, which is basically a statement implying that group preferences can be irrational even when individual preference is rational. This points to the impossibility of creating a voting system that can perfectly capture the will of the majority – just because there is no singular will of the majority. If there are more than three choices on a particular ballot, different voting systems will lead to wildly different outcomes based on the way the systems tally the votes to produce a single winner.

This paradox segues into a broader point about math – that it provides the fundamental structure of formal reasoning to tackle more intractable questions in life, even if it can’t provide the answers. In other words, math can show you how not to be wrong, and you can use it to decide how you define how to be right. We can’t use math to demonstrate if god exists or not, but we can base our decision of what epistemological position to adopt by using the tools of Bayesian inference. We can’t use math to create a perfect voting system, but we can use Condorcet’s paradox to assure ourselves that none such is possible, and therefore define our democratic process based on which trade-offs we’re more willing to make. Math itself, or at least its formal frameworks, cannot be proven to be self-consistent, as Godel’s Theorems show. But there is scope to set the starting points for it to be useful in as wide a variety of fields as possible.

And Ellenberg is able to demonstrate how interconnected math is, in terms of the ways in which mathematicians have been able to use theorems from one field to answer questions in the most unexpected of fields. One particularly elegant example that Ellenberg invokes is the use of plane geometry – Fano plane – to determine the smallest subset of lottery ticket number combinations necessary to maximise your expected outcome in a Transylvanian lottery. The whole thing is too involved to express here but the gist is that the Fano plane is a simple geometry created from a few basic axioms – one of which is that any two line segments can share at most one point – which also can represent the set of combination of lottery numbers that cover the largest subset of expected lottery outcomes, because the essential property of those numbers is that they share as few numbers with as few other numbers as possible, in order to maximise their spread.

What I think is Ellenberg’s biggest achievement with this book is to combine his mathematical expertise with a kind of cross-disciplinary wisdom – showing how math applies to a myriad of judiciously-selected fields and examples. Ellenberg even has a keenly literary sensibility, name-dropping David Foster Wallace and Ted Chiang and often quoting beautiful passages from mathematically-minded literary greats. Ellenberg’s wide-ranging scope fills his work with a sort of erudition that puts paid to the notion that math doesn’t intersect with human experience, and his wit and style place him as an exemplar of the ability to do math and write about it in a way that captures the imagination. It does require a fair bit of work to go through, especially in some of his more involved thinking exercises, but I think that the book is better for it.

In all, How Not to be Wrong isn’t a self-help book that instructs us on its titular subject matter, but it is an arresting look into math’s utility in different fields in human experience, one that is as illuminating as it is entertaining.

I give this book: 4 out of 5 Transylvanian Lotteries