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The Logic of It All

Mathematician and musician Eugenia Cheng exposes the patterns at play in the kitchen, the concert hall, and everywhere else.

The bones of musical compositions, the tools for dissecting an argument, and the mathematical concepts embodied by cakes and bagels are as essential to Eugenia Cheng's lessons as are raw figures and calculations. The British mathematician and pianist teaches math at the School of the Art Institute of Chicago and, in talks for the public, draws from her expertise in category theory—the study of math as a conceptual system, or the "mathematics of mathematics." Her first book, How to Bake Pi , offers an introduction to the subject that leverages her knowledge of desserts. Her latest, The Art of Logic in an Illogical World , deploys ideas from her discipline to help readers grapple with the types of thorny questions that don't appear in math textbooks.

 University of Sheffield
OBJECT LESSON: Eugenia Cheng uses examples from baking, music, and social issues to illustrate math concepts.
Photo: University of Sheffield

What do you say to people who think they can get by in everyday life without math?

That is completely true—but I think you can get by better with some math. Not long division and solving equations, necessarily, but the principles of abstraction and logic are things that we can all use. Abstraction is how you get to the core of what an argument is really saying and make good analogies between things. I think that's what empathy is about: analogies between people. If you can draw an analogy between yourself and somebody else, then you can empathize with them, even if you're not actually in their situation and have never experienced it.

What can thinking like a mathematician bring to a contentious conversation?

Math has a clear framework for how you unravel an argument back to its beginnings—which, in life, are your fundamental beliefs. Instead of saying one person is right and one is wrong, we can ask what it is about an argument that is right, and what starting points make that happen.

You argue that logic and emotions should work together. Do others in your line of work agree?

Mathematicians I talk to acknowledge that we use instinct and emotions, especially at the beginning of research, when we're just trying to find a way forward. I often talk about things that make me feel ill. In category theory, you try not to make arbitrary choices, because if you do, you're imposing yourself on a situation instead of finding its natural structure. I can tell that I've made an arbitrary choice more quickly if I feel a bit sick to my stomach. Another time we use emotions is when we're trying to explain things to other people. We often use language that anthropomorphizes mathematical concepts: "This is really trying to be an equivalence. How can we help it?"

Math concepts like 4-D shapes transcend what's possible in the real world. Does art do that too?

Picasso depicted people's bodies in unreal positions, but that work gives us some feeling about an aspect of humanity or somebody's character that you can't see in real life. One of the reasons that opera is probably my favorite art form is that you can encapsulate somebody's character so quickly with a little phrase of music.

As a musician, how do you find that a grasp of song structure relates to mathematical understanding?

We are awfully prone to compartmentalizing, and it may be partly because of the education system: There is a subject called History, a subject called Music, a subject called Math. But if a person can recognize that a piece of music is organized into sections A, B, and then A again, that is math. Just seeing that the abstract structure is there and that it has symmetry means that you can save brain power.

What gives you hope for the power of logical thinking?

People get stuck in their ways as they get older—I do, too—so it's hopeful when new generations are not afraid to do things differently and point out what the older generations have been missing. I think it's wonderful that the younger generations care about other people, about the environment, and about injustice, and that they can see a bigger picture than just their own lives as individuals. One of the big problems of the world is the boundaries that we've imposed—abstract boundaries between subjects, such as the idea that math is only useful for certain things, but also boundaries between communities.

You've used bagels and wine glasses as teaching aids. How can fun help convey complex ideas?

Keeping people amused is one way of keeping them interested. It can also make the material more memorable. I've been to plenty of boring talks in my life, but people have particularly low expectations for a math talk. So they are very ready to laugh. I feel that the first thing we should care about in teaching is that students are having a good time. There's a backlash against that idea, but if learning is not fun, then students are going to hate it, and if they hate it, they're not going to learn anything in the long run. They might retain it temporarily, under pressure, but nothing's going to stick.

You jokingly describe mathematicians—including yourself—as "lazy," because math concepts can make reasoning less work.

When I taught at the University of Chicago, it struck me that the students were not nearly lazy enough. They were used to drilling huge problem sets and would often go through the most laborious calculations instead of finding elegant ways to get there. To me, elegance in mathematics is about avoiding tedious busy work. If I find myself doing even two things that feel similar, then there must be some abstract explanation for both. I have always liked the idea of using the same thing in slightly different ways. I once became very cross with a particularly utilitarian chair I had—I just couldn't find more than one way of sitting in it.