ugenia Cheng is on a mission to change the world for the better, using maths. Her first book, How to Bake Pi, used recipes to teach readers how to think mathematically. The Art of Logic, published in 2018, was about using the principles of mathematical logic to have more productive arguments. x + y is an even more ambitious project, the aim of which is to end the gender wars and create equality by building “a whole new theory of people”. Cheng begins by addressing why it is unhelpful to associate characteristics with gender, and explains why “leaning in” and “positive discrimination” both fail to fix inequality. She proposes a solution based on her specialist subject of category theory, which is more interested “in describing things by the role they play in a context, rather than by their intrinsic characteristics”. Mathematically, she says, “if we have two things that are not equal, we could make them equal by making the lesser one greater or by making the greater one less ... However, there is a completely different way we could do it, which is by evaluating the two things on a new dimension entirely.” Cheng insists that proper maths, the fun kind, is not about being right, but is a way of thinking differently, and that includes exploring ideas that are impossible according to existing rules. It’s a way of seeing this exhausting debate from a completely new angle. What she asks us to do is forget ideas of masculine or feminine characteristics, and instead think about types of behaviour that are either “ingressive” or “congressive”. Ingressive behaviours are competitive, adversarial and focused on the self over the community; congressive behaviours are collaborative, cooperative and focus on society over the self. It is a frustrating fact, she argues, “that although congressive behaviour is better for society, our society is set up to reward ingressive behaviour”. This book is a manifesto for switching that setup, beginning on a personal level and working all the way up to large-scale, structural change. Cheng makes reference to writers and thinkers who have explored similar subjects before, such as Angela Saini in Inferior, and Cordelia Fine in Testosterone Rex. Though she doesn’t mention his work, her coinage of “ingressive” and “congressive” is reminiscent of Simon Baron-Cohen’s “systematising” and “empathising” brains. Unlike him, she does not associate “ingressive” behaviour intrinsically with maleness, or “congressive” with femaleness, and gives several examples of congressive men and ingressive women. (One such is John Baez, “a physicist and prolific blogger who started sharing his understanding of physics way back in 1993 when the internet was barely known”.) She does acknowledge, however, that congressive behaviours are currently more common among women, and that society encourages men to be more ingressive. And, though she carefully explains the ways in which anyone can exercise congressive power and influence, a cynical reader is likely to wonder why, even in this egalitarian system, it still seems likely that ingressive people (mostly men) are going to go on making tonnes of money, while congressive people (mostly women) will be putting others first. Early in the book she gives the example of a luggage carousel at an airport. When everyone crowds forward, nobody can see their luggage; but when everyone steps back, they all benefit. But it only takes one person to step forward to ruin it for everyone: even in a congressive utopia, won’t there be an exponential stampede back to ingression? Cheng is optimistic that change is possible, and stresses that her theory, like maths, is “a ‘way of thinking’ rather than … an exercise in getting the right answer”, and that therefore “the only way to fail is not to try”. She gives concrete examples of how to neutralise ingressive energy in your own life based on anecdotes from hers, and case studies including that of Stephanie Shirley, who built her own computer programming company, staffed entirely by women, and became a phenomenal success. Among many diagrams is a sketch of Cheng’s “dream train design”, which employs two platforms, to encourage passengers to move down the carriage and let other people on and off. It clearly would be wonderful if we could start from scratch, with clever, congressive designers for everything. The book starts and ends with last-minute addenda about the coronavirus crisis, and asks whether this experience might make us all finally realise the value of individuals acting for the benefit of the group. It’s another bold and optimistic thought. Still, if we were ever going to imagine a whole new theory of people, now is as good a time as any to start. • x + y: A Mathematician’s Manifesto for Rethinking Gender is published by Profile (£16.99). To order a copy go to guardianbookshop.com. Delivery charges may apply.