Eugenia Cheng, author of the cleverly-titled *x + y: A Mathematician’s Manifesto for Rethinking Gender*, is—goes Wikipedia—“a British mathematician, concert pianist, and an honorary fellow of pure mathematics at the University of Sheffield. Her mathematical interests include higher-dimensional category theory, and as a pianist she specialises in lieder and art song.” Her current gig is Scientist in Residence at the School of the Art Institute of Chicago. Even more daunting, her website features a stint on *The Late Show* with Stephen Colbert.

The male reviewer risks (arguably empirically credible) accusations of “mansplaining”; in this case, perhaps also “mathsplaining”, the combination of which is surely worse. Fortunately, Cheng is clear, focused and precise. Despite it’s nominally nerdy take, *x + y* reads considerably faster than most books of equivalent length.

It’s an injustice to summarize the book in a sentence or two, but Cheng sets out to debunk the arguments that justify gender inequality, a key example of which is explaining that gender imbalance in her own field of mathematics is somehow due to innate differences between men and women.

Cheng employs formal logic—which I recall learning in something other than a math class—to do this, or rather, she points how the arguments trotted out in favor of gender inequality are logically flawed. She gives the example:

##### Men’s brains tend to be stronger in systemizing than empathizing, and systemizing is important in mathematics, so it is to be expected that there are more men than women mathematicians.

One doesn’t actually need formal logic to work out what is wrong with this, but it provides a methodology that can be applied to statements of this type. There is (much) more to the book than this, but to some extent, when she’s finished with this, she’s done.

The book explores whether a mathematical frame of mind can help untangle complex social problems.

*x + y* can also—and usefully—be read as an example-based introduction to mathematical principles with gender inequality as a case study. In addition to “category theory” (something which arose after my time), she runs through imaginary numbers, reflections, rotations, averages, statistical distributions and much else in a user-friendly way (a skill learned, she believably says, from teaching math to art students).

The book is also a foray into the broader question of whether a mathematical frame of mind (as opposed to necessarily mathematics *per se*) can help untangle (if not necessarily solve) complex social problems. Cheng is a “pure mathematician” and approaches the problem rather differently that I—who studied such things as modelling and statistics—might have. And from this perspective, Cheng has somewhat buried her lede. It is not until a quarter of the way in that she mentions the “null hypothesis”, the default assumption one would make in the absence of any evidence or data. Drugs, for example, are assumed to have no effect until it is demonstrated that they do. The null hypothesis here would be that (innate) mathematical ability is not linked to gender. Proving otherwise would be difficult, for gender is linked to a great many social conventions. But having a high burden of proof for something that affects half the population is probably no bad thing.

So how is that so many people with mathematical leanings, and who one would have thought should have known better, seem to accept the proposition that there is something about men that makes them better at math? Cheng points out that there is an alternate null hypothesis: that (and here I paraphrase) we know men and women are different, so the burden of proof is on the claim that in mathematical ability they are the same. Aha! Cheng puts the blame on science—“science is supposed to be impartial,” she writes. “This is one way in which science has historically been used to hold women back.” But this isn’t “science”; it’s intellectual legerdemain.

Cheng is not arguing that all people are the same with the same abilities but takes issue with the division of character traits into “masculine” and “feminine”. She proposes instead a spectrum of character using two neologisms: “ingressive” (“going into things”) and “congressive” (“bringing things together”). That “ingressive” sounds like “aggressive” is perhaps coincidence, but Cheng makes it clear she prefers traits that lie on the “congressive” axis.

This perhaps shows some limitations of the process. Math has neither objectives nor ethics: the results just are. But society does: it is trying to optimize something and people will disagree, quite legitimately in many cases, about both what that something is and what is optimum. Cheng writes that “congressive behavior is better for society”; I happen to agree out of both principle and practicality, but “better” is not itself a mathematical concept. Cheng’s “whole new theory of people” requires inclusion of non-mathematical considerations which are to some extent judgmental.

That being said, *x + y *is a rare combination of precision, passion and optimism. Whether via math or just clear and rigorous thinking, knotty issues benefit from discussions like this.