Apart from alliteration, what on earth do cakes, custard and category theory have in common? As a recent winner of the Best Mathematical Cake prize at MathsJam, the recreational mathematics conference, I feel I am fairly qualified to understand the connections that mathematician Eugenia Cheng illustrates here.
Cheng – the only female category theorist in South Yorkshire, she quips – has written this deliciously lively text with the aim of showing that “mathematics is there to make difficult things easy”. It is a book of two halves: the first explains the mathematical concepts needed to understand category theory, and the second describes the rudiments of category theory itself, a branch of abstract mathematics often described as the “mathematics of mathematics”. She combines these definitions to deduce that “category theory is there to make difficult mathematics easy” – in a neat contrast to the warning in the title of Carl E. Linderholm’s memorably mischievous look at the subject, Mathematics Made Difficult (1972).
Satisfyingly, most of the chapters begin with a recipe, artfully designed to illustrate a culinary process or strategy that highlights a particular mathematical procedure or theory. It is Cheng’s delightful descriptions of her gastronomic adventures that bring the mathematics to life. It’s easy to identify with her description of searching for suitable recipes with the express intention of using a new kitchen gadget, or adapting a favourite recipe for a friend trying to avoid gluten or high-fat food.
Sometimes in cooking, recipes can be changed so much that the result is a long way from the original. In describing an intriguing recipe for olive oil plum cake, which is wheat-free, sugar-free and dairy-free, Cheng says that while it looks like and behaves similarly to a cake, it is obviously not quite the same as a cake. She calls this a “generalisation of a cake” and goes on to discuss the analogous, important, concept in mathematics. Sadly, most of her recipes are not written for anyone who, like me, is allergic to chocolate. Her conference chocolate pudding, inspired by a rather tipsy conference dinner, is filled with melted chocolate and sounds divine, but alas, eating it would give me a migraine.
Near the end of the book, Cheng serves up the recipe for raw chocolate cookies: low in fat, suitable for vegans, sugar-free, gluten-free and raw, they have remarkably few similarities to a standard cookie recipe. However, this leads on to an interesting discussion of sameness: how triangles with sides of different lengths but having the same angles are classed as similar, and the way that expressions such as 4 × 3 = 3 × 4 are similar, but not quite the same, when they refer to three bags containing four apples or four bags containing three apples.
Much of the mathematical part of the book is concerned with the more abstract logical side of maths. We are told about the axioms for defining groups (the mathematical structures that help us to analyse symmetry) and other exciting areas of mathematics, such as topology (which regards bagels and doughnuts as essentially the same as mugs of coffee). Cheng explains how mathematicians can make 7 + 7 = 2, how they “invented” imaginary numbers, how they define certain mathematical abstractions using relationships, and how they can prove that something is true by showing that it cannot be false – proof by contradiction.
The culinary example provided for proof by contradiction is a demonstration that a cake can’t be made without flour or that bread can’t be made without yeast. But as Cheng points out, since flourless cakes and unleavened bread exist, this is an example where things don’t quite work out as expected. Such occurrences happen in mathematics; proof by contradiction has led to mathematical discoveries such as elliptical and hyperbolic geometries.
Custard finally makes its appearance in a demonstration of the tree diagrams used in category theory. When combining ingredients for custard (using eggs rather than custard powder), the order of the operations matters. This is true in most areas of mathematics and can be represented diagrammatically, which is simpler than defining something algebraically.
Cheng is keen to dispel the myths that maths is concerned only with whether the answer is right or wrong and is about numbers alone. Her examples are clear and straightforward, based on everyday activities: cooking, shopping, map-reading and so forth. Her engaging style draws you in so that you empathise with her over broken relationships, laugh at her tale of failing to recognise the entire Arsenal team as they walked through the hotel bar she was sitting in, and admire her determination, aged 16, to learn welding in order to improve her understanding of how a car works, even though it didn’t yield the desired outcome.
But this is not just an idiosyncratic account of interesting mathematics. Cheng touches on many challenges faced when learning maths: thinking that you’re stupid because you don’t understand something; feeling demoralised when you keep getting the wrong answer but don’t know why; and (as I frequently tell my students) the importance of understanding the principle behind the process rather than simply memorising the process. This is a fast-paced book packed with so many examples and analogies that it is easy to lose sight of where it is taking you. But I don’t think this matters: the reader is left with the overall impression that mathematics is fun and exciting – and is no harder than making custard.
Despite the fact that the number of students taking A-level maths has risen in recent years and that girls outperform boys at GCSE, the number of girls taking A-level mathematics is proportionally much lower. This book has the potential to play a significant role in convincing girls that they can be mathematicians. Written by a successful young woman who openly admits that there were times when she did not “get it”, and who uses examples that girls, in particular, can relate to (such as discussions about relationships), this is as girl-friendly as maths gets. Importantly, however, the maths is not watered down. Category theory is an advanced topic in mathematics that is more often encountered in graduate than undergraduate studies, and it is a remarkable achievement to be able to present it so accessibly for a lay and student readership.
The final chapter, “What category theory is”, brings it all together. It begins with a Venn diagram that links belief, knowledge and understanding – which makes your brain ache as you try to work out whether it is possible to believe something without knowing or understanding it – and concludes with a short discourse on mathematics education in which Cheng laments that only up to a point do we encourage schoolchildren to ask why mathematics works. She urges us not to stifle young people’s quest for the illumination of mathematical principles, but rather to seek to explain and demystify, helping to lay the groundwork for future mathematical discoveries to occur.
This book will inspire learners who are intrigued by mathematics but unsure whether to take it beyond A level, will encourage parents keen to help their children to learn maths, and could well benefit those working in higher education who want to understand a little more about what their mathematical colleagues do and why mathematics is so often described as being different from other disciplines. And if it doesn’t succeed in exciting you about mathematics, it will certainly change the way you approach baking. As long as you aren’t allergic to chocolate.
Noel-Ann Bradshaw is principal lecturer in mathematics and operational research, University of Greenwich.
Cakes, Custard and Category Theory: Easy Recipes for Understanding Complex Maths
By Eugenia Cheng
Profile Books, 304pp, ?12.99
ISBN 9781781252871 and 9781782830825 (e-book)?
Published 4 June 2015
<榴莲视频>The author
“I was born in Hampshire but my family moved to Sussex when I was one, and?that's where I spent the rest of my childhood,” says Eugenia Cheng, senior lecturer in pure mathematics at the University of Sheffield.?
“Sussex is still my favourite part of the UK. To me, gently rolling hills are much more beautiful than more dramatic landscapes, and the sea is my favourite part of nature. I really miss the sea.”
Her interest in her discipline began early. “My mother is the one who got me excited about mathematics when I was very young, by showing me mathematical concepts that intrigued me, unlike the things we did at school.?My father encouraged me to be imaginative and make things up for myself. He rarely does what he's told and neither do I!
“The biggest influence on my mind, other than my parents, was my piano teacher, who is the third dedicatee of this book.?She taught me not just about piano, but about all of music, and life.?Her dedication to passing on everything she knew inspires me to want to do the same.”
Cheng was, she says, “a very serious child, though not exactly studious, as I found school very boring.?I practised the piano a lot, every day from the age of four, because I loved it so much. I also devoured books and would typically sit in a chair without moving until I finished the book, no matter how long it was.?(I still do that, usually through the night, and end up with awful pins and needles.)?I was also very serious about eating and would keep eating until there was no more food in sight.?My parents never had to persuade me to practise the piano – they usually had to persuade me to stop because it was dinner time.”
It was perhaps inevitable, then, that this book should involve food, and a keen-eyed understanding of how to engage the less numerate. Perhaps it is the author’s palpable delight in incorporating Pi pie, Battenberg cake, bagels and cream teas in explorations of her discipline that allows her to forgive journalists for being for being so much more interested in mathematics when there’s an edible angle.
“I think it’s completely understandable. Maths can be awfully dry and can seem irrelevant when presented in the wrong way.?I’m glad I’ve found something that piques people's interest in maths, and am happy to use that as much as I can as a way to help people into this world that I love so much.”
A frequent media commentator on subjects such as “maths phobia” – with and without clotted cream as a prop – Cheng observes that it is not a universal phenomenon. “In my experience there’s much less of it in Hong Kong (where I still have family) or in France, where people seem to revere mathematics and mathematicians.?That's why I'm hopeful that it can be eradicated (or at least, reduced) in the UK and US as well.”
Cheng took her undergraduate and postgraduate degrees at the University of Cambridge. “I felt very at home there.?I loved being surrounded by brilliant and energetic people and finally being able to specialise in maths.?I didn’t find it easy and had to study extremely hard to get my firsts, especially for finals, which I still remember with some horror.?I also did tons of music, playing in concerts and organising concerts, and made many friends with whom I am still close. Many of those people have gone on to brilliant careers especially in classical music and I'm so proud to have ‘grown up’ with them.”
On the subject of her college, Gonville and Caius, she adds, “I'm very loyal. I found it to be a very nurturing, appreciative and stimulating community, and being surrounded by brilliant people in all fields pushed me to try to be brilliant in multiple fields as well.”
She has also found much to love at Sheffield and at scholarly posts in the US and France.
“My first stay in Chicago [at the University of Chicago] was a postdoctoral position from 2004 to 2006.?I thought I was going to hate it, but accepted the position because it was obviously the most prestigious offer I had had and would be good for my future. I turned out to love it, as I could be both a mathematician and a musician.
“I think universities always gain from having academics go on sabbatical. First of all, it kick-starts research projects if they embed themselves in a different environment with different collaborators.?But also it means we can bring back new ideas about the educational environment.?It's easy to get stuck in the ways of a single institution and forget that there are other ways of doing things.?If people stay teaching in the same place for 20 or 30 years it can get very stagnant. And in my case, it meant I had time to write two books.”
Cheng was also a Marie Curie research fellow at Université de Nice-Sophie Antipolis. “Again, it's important for researchers to go to different institutions, work with different people, and see different ways of doing things.?The Marie Curie programme I was on was specifically to encourage people to go to a different country in the European Union – in fact, you can only apply to take up the fellowship in a different country from your home country.?The internet has helped a huge amount with collaboration across the globe, but?there's still nothing like staying in a place for a while and sharing ideas in a department every day.?There's a romantic view of mathematicians that imagines solitary people working by themselves in a corner for years on a problem, possibly in secret, but that's not really representative.?Or healthy.”
And if she had to choose between the cuisine of the Midwest and that of the Cote d’Azur?
“Ha! I don't want to offend an entire city.?Chicago has great food of all kinds, but the portion sizes are enormous.?I love French food, too, and there were some wonderful things in Nice, like the ice cream place with 96 flavours including things like black olive, tomato and rosemary.?My favourite place for food, though, is Paris.?Last time I spent a month there on a research trip; I ate delicious food every day and yet effortlessly lost weight.?I still haven't worked that one out.”
What gives her hope? “Small children: their natural compassion, openness, excitement and curiosity. We expect children to learn from us, but I think we have a lot to learn from them, too.”
Karen Shook
Review originally published as: Cooking the numbers (4 June 2015)