How Not to Be Wrong, the first popular math book by University of Wisconsin-Madison math professor Jordan Ellenberg, just hit the shelves. In addition to a Ph.D. in math, Ellenberg has an MFA in creative writing and has been writing about math for popular audiences for several years. Unsurprisingly, the book is witty, compelling, and just plain fun to read.

*How Not to Be Wrong* is not a popular math book about the Finonacci sequence or pi. It goes deeper. In the introduction, Ellenberg divides mathematical facts into four quadrants: they can be simple and shallow, simple and profound, complicated and shallow, or complicated and profound. The book hangs out in the “simple and profound” quadrant.

Ellenberg writes of the ideas in this quadrant, “they are not ‘mere facts,’ like a simple statement of arithmetic—they are principles, whose application extends far beyond the things you’re used to thinking of as mathematical. They are the go-to tools on the utility belt, and used properly they will help you not be wrong.”

The book explores these tools in five sections: Linearity, Inference, Expectation, Regression, and Existence. Each part contains chapters about specific mathematical techniques that fall under the section’s larger umbrella.

"Existence," my favorite section, explores the limitations of mathematics when it comes to getting the “right” answers. It's something I’ve been thinking about a lot recently. We mathematicians sometimes think of ourselves as cloistered priests who discover and protect pristine universal truths, but one of the defining features of mathematics is that we have to make the rules in the first place. We don't discover universal truths, we discover truths in the axiomatic settings we create. Sometimes, the settings seem so clearly true that we treat the truths as universal. I’m thinking of our number system. Indeed, counting and arithmetic have been wildly successful at helping us understand our world and develop new technology. It's hard to imagine math without the rules of numbers and arithmetic.

But what about Euclidean geometry? For two thousand years, it was treated as the only way to do geometry, the only source of objective geometric truth. Mathematicians spent their lives trying to prove that the parallel postulate had to be true in order for the other postulates to hold. Our intuition and experience tell us that it has to work that way, but that is only true in one axiomatic system of geometry. The “truths” we discover there are confined to that system. Euclidean geometry is very useful, but there are centuries of sailors and astronomers who will tell you that spherical geometry has some truths of its own. Euclidean geometry doesn't make spherical geometry false, and vice versa.

In the chapter called “There is no such thing as public opinion,” Ellenberg talks about Nicolas de Condorcet, a political philosopher around the time of the French Revolution who wanted to develop better ways of deciding elections. In a contest with more than three candidates, the “plurality rules” system can lead to problems (think Ross Perot or Ralph Nader). Condorcet wanted to figure out a fair voting method that would satisfy this voting "axiom:" “If the majority of voters prefer candidate A to candidate B, then candidate B cannot be the people’s choice.” It is logical, fair, and impossible. It’s easy to come up with examples of rock-paper-scissors preferences: the majority prefers candidate A to candidate B, candidate B to candidate C, and candidate C to candidate A. Under Condorcet's axiom, no candidate can win. Anarchy reigns.

Enter mathematical formalism. Formalism changes the question from getting the right answer to getting the answer that follows the rules. Ellenberg writes,

“It’s what G. H. Hardy was talking about when he remarked, admiringly, that mathematicians of the nineteenth century finally began to ask what things like1-1+1-1+…

should be

definedto be, rather than what theywere…."Hardy would certainly have recognized Condorcet’s anguish as perplexity of the most unnecessary kind. He would have advised Condorcet not to ask who the best candidate really was, or even who the public really intended to install in office, but rather which candidate we should

defineto be the public choice.”

I had never thought about it in those terms, but Ellenberg explains that elections, trials, and even baseball can be understood via mathematical formalism. In some ways this seems like cheating. It’s too hard to find the right answer, so we make up new rules instead and define "right" to mean "follows our rules." But in the final chapter, “How to be right,” Ellenberg emphasizes that it is about understanding what questions we can answer and the limitations of our methods so that we can take action wisely. We don’t just throw up our hands when things get complicated. We analyze it and weigh the options with as much information as we can get. He writes,

“People usually think of mathematics as the realm of certainty and absolute truth. In some ways that's right. We traffic in necessary facts: 2+3=5 and all that."But mathematics is also a means by which we can reason about the uncertain, taming if not altogether domesticating it….Math gives us a way of being unsure in a principled way: not just throwing up our hands and saying “huh,” but rather making a firm assertion: ‘I’m not sure, this is why I’m not sure, and this is roughly how not-sure I am.’ Or even more: ‘I’m unsure, and you should be too.’”

Ellenberg describes mathematics as “the extension of common sense by other means,” a way to go from Tony Stark to Iron Man. *How Not to Be Wrong* can help you explore your mathematical superpowers.