Cakes, Custard + Category Theory
Eugenia Cheng is a senior mathematics lecturer at Sheffield University (UK) whose domain is higher-dimensional category theory. She has gained some popularity from her YouTube videos where she mixes her love for cooking and for mathematics to show the analogy between both and to show that knowledge of one can help understanding the other. This is exactly what she also wants to achieve in this book as the subtitle promises: Easy recipes for understanding complex maths. She is also active as a pianist, but that is not used much in this context. This illustrates that she is a very enthusiast communicative and talkative ambassador for the popularization of mathematics. She definitely wants to convince people that it is not mathematics that is difficult, but that it is life that is complicated and mathematics is just there to simplify it and make it much easier to solve problems.
In this book she engages in the task to explain what category theory is to mathematical lay people, which is certainly not an easy or obvious choice. I doubt that a mathematical illiterate after reading the book will be able to tell you what category theory really is. But they will have gotten at least a vague idea. Fortunately, Cheng starts from scratch and is meandering along many other topics along the way. In fact, there are two parts: the first explains what mathematics is about, and the second part explains what category theory is: the mathematics of mathematics. The two parts are not much different. Cheng is following the mathematical river of concepts flowing to its estuary of understanding. She also tells about the many brooks, streamlets, and bourns that feed it. The recipes that she starts each chapter with, are not really essential in my opinion. Of course cookery programs are currently very popular and it is a kind of a opening sentence to start a discussion about something that really matters. The recipes sum up the ingredients and give a brief description of the method, and you will get some ideas of how to deal with certain allergies in your cooking, but I believe you should know something about cooking if you want to really use them since not many details are given. More or less the same holds for the mathematics. The most elementary topics of mathematics are explained, but it is advisable that you know a bit of mathematics to keep apace with Cheng. You do learn that the concept of a number is not that obvious, you learn about logic, what a proof is, how one arrives at an axiomatic system by repeatedly asking `why?', you learn about complex numbers, and a group, about the unsuccessful attempts to prove the fifth axiom of Euclidean geometry, you are convinced that distance is not always the same as a Euclidean distance, and you are introduced to topology. That's a whole lot if you only have secondary school mathematics in you backpack, and certainly if it has been a while since you needed it. All this is wrapped up in much story telling featuring Fermat, Poincaré, and Riemann, and a lot of foody and cookery stuff. And this is just the mathematics part.
In the category part, relations (morphisms) represented as arrows connecting objects become important. The example of genetic and mathematical family ties (e.g. the Erdős number), are examples. It is all about structures and removing as much as possible to keep the simplest skeleton. Some of the properties of the mappings are explained and simple examples are given, but a clear and strict axiomatic definition is not really given. However you learn about what it can mean to say that structures are `the same', what a monoid or a universal property is, and even what a colimit is. And again the wrapping consists of many stories e.g. about Nelson's last message to his fleet before the battle of Trafalgar, the three domes of St. Paul's Cathedral and Battenberg cakes. I find the discussion in the concluding chapter about truth most interesting. It is about different gradations or meanings of `truth' depending on (1) what we know, (2) what we understand, and (3) what we believe. The most `secure' truth is what is in the intersection of the three.
The enthusiasm of Cheng is contagious, and she knows how to take the reader along on her hiking tour (not really a stroll in the park). Do not expect that after reading the book you will be ready to start reading current research in category theory. Even the reader that is a mathematician may be somewhat confused because it is too different from the top-down axiomatic and much less verbose books that he probably is more used to. But I do not think professional mathematicians are the first targets that Cheng had in mind when writing this book. Nevertheless, it is entertaining reading stuff that the professional and the non-professional will appreciate.
To avoid some confusion, let me finally point out that this book is published in UK by Profile Books, but that the same book is available in the US under a different title How to bake pi: an edible exploration of the mathematics of mathematics published by Basic Books.