Cheng - How to bake [pi]: an edible exploration of the mathematics of mathematics

Copyright 2015 by Eugenia Cheng

Published by Basic Books,

A Member of the Perseus Books Group

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A catalog record for this book is available from the Library of Congress.

Library of Congress Control Number: 2014957937

ISBN: 978-0-465-05169-4 (e-book)

10 9 8 7 6 5 4 3 2 1

To my parents and Martin Hyland

In memory of Christine Pembridge

Contents

They say mathematics is a glorious garden. I know I would certainly lose my way in it without your guidance. Thank you for walking us through the most beautiful entrance pathway.

From a students letter to the author

University of Chicago, June 2014

Here is a recipe for clotted cream.

What on earth does this have to do with math?

Math Myths

Myth: Math is all about numbers.

You might think that rice cookers are for cooking rice. This is true, but the same piece of equipment can be used for other things as well: making clotted cream, cooking vegetables, steaming a chicken. Likewise, math is about numbers, but its about many other things as well.

Myth: Math is all about getting the right answer.

Cooking is about ways of putting ingredients together to make delicious food. Sometimes its more about the method than the ingredients, just as in the recipe for clotted cream, which only has one ingredientthe entire recipe is just a method. Math is about ways of putting ideas together to make exciting new ideas. And sometimes its more about the method than the ingredients.

Myth: Math is all either right or wrong.

Cooking can go wrongyour eggs can curdle, your souffl can collapse, your chicken can be undercooked and give everyone food poisoning. But even if it doesnt poison you, some food tastes better than other food. And sometimes when cooking goes wrong you have actually accidentally invented a delicious new recipe. Fallen chocolate souffl is deliciously dark and gooey. If you forget to melt the chocolate for your cookies, you get chocolate chip cookies. Math is like this too. In high school if you write 10 + 4 = 2 you will be told that is wrong, but actually thats correct in some circumstances, such as telling the timefour hours later than 10:00 is indeed 2:00. The world of math is more weird and wonderful than some people want to tell you.

Myth: Youre a mathematician? You must be really clever.

Much as I like the idea that I am very clever, this popular myth shows that people think math is hard. The little-understood truth is that the aim of math is to make things easier. Herein lies the problemif you need to make things easier, it gives the impression that they were hard in the first place. Math is hard, but it makes hard things easier. In fact, since math is a hard thing, math also makes math easier.

Many people are afraid of math, or baffled by it, or both. Or they were completely turned off it by their classes in high school. I understand thisI was completely turned off sports in high school and have never really recovered. I was so bad at sports in high school, my teachers were incredulous that anybody so bad at sports could exist. And yet Im quite fit now and have even run the New York City Marathon. At least I now appreciate physical exercise, but I still have a horror of any kind of team sports.

Myth: How can you do research in math? You cant just discover a new number.

This book is my answer to that question. Its hard to answer it quickly at a cocktail party without sounding trite, or taking up too much of someones time, or shocking the gathered company. Yes, one way to shock people at a polite party is to talk about math.

Its true, you cant just discover a new number. So what can we discover thats new in math? In order to explain what this new math could possibly be about, I need to clear up some misunderstandings about what math is in the first place. Indeed, not only is math not just about numbers, but the branch of math Im going to describe is actually not about numbers at all. Its called

CATEGORY THEORY

and it can be thought of as the mathematics of mathematics. Its about relationships, contexts, processes, principles, structures, cakes, custard.

Yes, even custard. Because mathematics is about drawing analogies, and Im going to be drawing analogies with all sorts of things to explain how math works, including custard, cake, pie, pastry, donuts, bagels, mayonnaise, yogurt, lasagne, sushi.

Whatever you think math is... let go of it now. This is going to be different.

Math, like recipes, has both ingredients and method. And just as a recipe would be a bit useless if it omitted the method, we cant understand what math is unless we talk about the way it is done, not just the things it studies. Incidentally the method in the above recipe is quite importantthese dont cook very well in a large tray. In math the method is perhaps even more important than the ingredients. Math probably isnt whatever you studied in high school in classes called math. Yet somehow I always knew that math was more than what we did in high school. So what