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Books : reviews

Eugenia Cheng.
Cakes, Custard, and Category Theory: easy recipes for understanding complex maths (== How to Bake Pi).
Profile. 2015

rating : 2.5 : great stuff
review : 22 August 2015

What is maths all about? At what on earth do cake and custard have to do with it?

Möbius bagels, Euclid’s flourless chocolate cake and apple π – this is maths, but not as you know it. In Cakes, Custard & Category Theory, mathematical crusader and star baker Eugenia Cheng has rustled up a batch of delicious culinary insights into everything from simple numeracy to category theory (‘the mathematics of mathematics’), via Fermat, Poincaré and Riemann.

Maths is much more than simultaneous equations and πr2: it is an incredibly powerful tool for thinking about the world around us. And once you learn how to think mathematically, you’ll never think about anything – cakes, custard, bagels or doughnuts; not to mention fruit crumble, kitchen clutter and Yorkshire puddings – the same way again.

The purpose of mathematics is to make difficult things easier; the purpose of category theory is to make difficult mathematics easier.

So argues research mathematician Eugenia Cheng in this excellent book. She starts off gently, with relatively simple mathematics, and oodles of real world examples, many based, unsurprisingly given the title, on cooking. These culinary examples serve both to illuminate the concepts, and to demonstrate her thesis: for example, finding out how much icing a cake needs is made easier using mathematics.

The first half of the book is about mathematics in general, and what it can and can't do. There are some lovely descriptions of the role of abstraction and generalisation, and the process of doing mathematics. By the end of this part we are confidently reading about axiomatisation. The second half then delves into the promised category theory. This covers the role of relationships and structure, along with a discussion of sameness. This is all achieved with a lightness of touch, whilst covering some quite profound ideas.

By the end, Cheng has explored a broad range of concepts, illuminating a lot about the philosophical stance of mathematicians, and the relationships of mathematics to the world. And now I want some cake.

Eugenia Cheng.
Beyond Infinity: an expedition to the outer limits of the mathematical universe.
Profile. 2018

rating : 3 : worth reading
review : 21 March 2025

How can one little symbol – ∞ – hold the biggest idea of all? In Beyond Infinity, Eugenia Cheng explores the inner workings of this powerful, counterintuitive concept to shed light on the fundamental truths at the heart of mathematics.

Cheng explains mathematical concepts of infinity with great clarity. She uses the example of the Hilbert Hotel to explain cardinal infinity, and then changes this to the Hilbert Queue for ordinal infinity. I love the Hilbert Queue with the books of numbered cloakroom tickets. Previously the explanations I have seen have involved lines of telegraph poles disappearing into the distance; the queues are much more intuitive.

The first half of the book establishes the basics, then the second half covers a variety of examples, including infinitesimals. We also get a little bit of category theory, and how it is at the top of the tower of abstraction:

p.190. This is how category theory arose, as a new piece of maths to study maths. In a way category theory is an ultimate abstraction. To study the world abstractly you use science; to study science abstractly you use maths; to study maths abstractly you use category theory. Each step is a further level of abstraction. But to study category theory abstractly you use category theory.

Along the way, we get gentle explanations of why maths can be fun despite being hard: after all, other things like hiking can be fun but hard, and even fun because it’s hard. And there’s also cake.

Eugenia Cheng.
Is Maths Real: how simple questions lead us to mathematicians' deepest truths.
Profile. 2023

To many, maths feels like an unmapped wilderness. Between abstract concepts like imaginary numbers and infinity, it can sometimes feel like a lot of maths is just made up. Why, for example, is 1 not a prime? Why do two negatives cancel each other out? Where does trigonometry come from? Is maths even real?

Abstract mathematician Eugenia Cheng shows that curiosity is the best teacher. Is Maths Real? takes us on a scintillating tour of the simple questions that provoke mathematics’ deepest insights.

Eugenia Cheng.
The Joy of Abstraction: an exploration of math, category theory, and life.
CUP. 2023

Mathematician and popular science author Eugenia Cheng is on a mission to show you that mathematics can be flexible, creative, and visual. This joyful journey through the world of abstract mathematics into category theory will demystify mathematical thought processes and help you develop your own thinking, with no formal mathematical background needed. The book brings abstract mathematical ideas down to earth using examples of social justice, current events, and everyday life – from privilege to COVID-19 to driving routes. The journey begins with the ideas and workings of abstract mathematics, after which you will gently climb toward more technical material, learning everything needed to understand category theory, and then key concepts in category theory like natural transformations, duality, and even a glimpse of ongoing research in higher-dimensional category theory. For fans of How to Bake Pi, this will help you dig deeper into mathematical concepts and build your mathematical background.