Eugenia Cheng (Profile Books)
SOME have likened the process of doing abstract mathematics to hacking your way out of some dense forest. Maryam Mirzakhani, the late mathematician and the first of only two women to win a Fields medal, once talked about feeling “lost in a jungle” and of gaining mathematical insights while trying out new tricks to extricate herself.
In mathematician Eugenia Cheng’s new book, Is Maths Real? How simple questions lead us to mathematics’ deepest truths, she uses a similar metaphor to give a glimpse into the mathematician’s mind. Each chapter starts with a seemingly innocent question – one that most adults wouldn’t dare ask for fear of looking stupid – and then shows how answering such questions seriously is “a slow process of gradually probing and pulling back the undergrowth”.
Advertisement
The questions are very basic and the answers seem obvious, except that they aren’t. For example: why does 1 + 1 = 2? Why does -(-1) = 1? And why isn’t 1 a prime number? OK, maybe that last one isn’t so simple, but that’s the idea.
Cheng shows us that what look like trivial questions may not be so simple on closer inspection. For example, does 1 + 1 equal 2 in all cases? Understanding when it does and when it doesn’t leads to a richer understanding of our underlying assumptions about numbers and their world.
Mathematics, writes Cheng, isn’t about getting the right answers to questions, but about knowing how to reason using a framework of logic (if 1 + 1 = 2, then why – or why not?). As she explains:”The solidity of maths comes from not wanting to trust things, but wanting to jump and swing and know that our framework will hold up.”
Cheng celebrates the dizziness and disorientation engendered by childlike questions, arguing that these feelings are suggestive of the right kind of bafflement, and pushing yourself to overcome them is crucial to the process. She also succeeds in making the reader feel that not understanding something in mathematics isn’t the same as being bad at it; rather, it is a clue that you are onto something deeper, the pursuit of which could reap rewards.
The book is infused with personal ruminations that lighten the load and keep the tone conversational. For example, when Cheng describes -1 as something that cancels out 1, just as antimatter obliterates matter, she writes: “When I was little I thought that pepper was the anti-matter for salt, that is, I thought that if you added too much salt to something you could add some pepper to cancel it out.”
She uses real-world examples to help the reader understand mathematical concepts. Take the commutativity of addition, which says that 2 + 4 = 4 + 2, or that the order of addition doesn’t matter. But are there situations that are non-commutative, where this isn’t the case – and what does non-commutativity mean? Cheng offers a culinary example: to make mayonnaise, start with egg yolks and add olive oil slowly; the other way around doesn’t work. “It doesn’t commute,” she writes.
Cheng wears her heart and politics on her sleeve, segueing seamlessly, and occasionally not so seamlessly, from mathematics to social concerns. Take the issue of why the square root of 2 requires thinking beyond whole numbers and fractions. “The alternative is to allow some new types of number into your life,” she writes (the new type here being the irrational numbers).
She uses this opening to call out people who “steadfastly refuse to accept new ways of being as valid (such as same-sex marriage, or non-binary gender, or women mathematicians)”. That is “not how mathematics operates,” she writes.
To the central question of her book – whether mathematics is real – Cheng’s answer sidesteps the philosophical conundrum about whether it is invented or discovered. Very pragmatically, she writes: “The concepts in abstract maths might not be part of the concrete world, but the ideas are as real as any other ideas, and… the insights we get into the real world are very real.”
Nicely parried, while providing fodder for those who want to chew on this some more.
Anil Ananthaswamy is the author of the upcoming book Why Machines Learn: The elegant math behind modern AI
