What is math? How exactly does it work? And what do three siblings trying to share a cake have to do with it? In

*How to Bake Pi*, math professor Eugenia Cheng provides an accessible introduction to the logic and beauty of mathematics, powered, unexpectedly, by insights from the kitchen. We learn how the béchamel in a lasagna can be a lot like the number five, and why making a good custard proves that math is easy but life is hard. At the heart of it all is Cheng’s work on category theory, a cutting-edge “mathematics of mathematics,” that is about figuring out how math works.

Combined with her infectious enthusiasm for cooking and true zest for life, Cheng’s perspective on math is a funny journey through a vast territory no popular book on math has explored before. So, what is math? Let’s look for the answer in the kitchen.

Melt the butter and chocolate, stir together, and allow to cool a little. 2. Whisk the eggs and the sugar together until fluffy. 3. Beat the chocolate into the egg mixture slowly. 4. Fold in the potato flour. 5. Bake in very small individual cupcake liners at 350◦ F for about 10 minutes. 7 8 How to Bake π Math, like recipes, has both ingredients and method. And just as a recipe would be a bit useless if it omitted the method, we can’t understand what math is unless we talk about the way it

chapter. Secondly, there’s the process of generalization: we work out how to build more complicated things out of the things we’ve already understood. This is like making a cake in your blender, and making the frosting in your blender, and then piling it all up.† In math this is how we get things like polynomials and matrices, complicated shapes, four-dimensional space, and so on, out of simpler things like numbers, triangles, and our everyday world. We’ll look into this in Chapter 5. These two

to the different notions of sameness that go best with different contexts. The notion of sameness that we’ve introduced for topology is the playdough type, which is called homotopy equivalence. So technically we say that the Möbius strip is homotopy equivalent to a circle. This is useful but unsatisfactory, because a Möbius strip is much more exciting than a mere circle. One way this can be expressed is by a more sophisticated mathematical structure called a vector bundle. Remember earlier on

we can draw on a page, as we’ll see later. We’re going to see that category theory works by picking what relationships between things we are interested in, and emphasizing those. We’ll even Context 181 generalize the notion of relationship to encompass things that at first sight didn’t look very much like relationships, so that we can study more and more situations using the same way of thinking. This is the subject of the next chapter. Chapter 11 Relationships Porridge Ingredients 1 cup

just the essence is captured. Not every humpbacked bridge looks exactly like this:† † Road sign images are Crown Copyright and reproduced under the Open Government Licence. Abstraction 23 but this captures the essence of humpbacked-bridge-ness. Similarly, not all children crossing the road look exactly like this: Nevertheless, the benefits of this system are clear. It’s much quicker to take in a symbol than read some words while you are driving. Also it’s much easier for foreigners to