Eugenia Cheng’s How to Bake Pi was a pleasure to read. In her book, she talks about what math is, the nature of math, and category theory specifically. This book is very tangible and easy to read. Throughout her exploration of mathematics, Cheng often uses the metaphor of baking which I found very helpful.
The beginning of the book starts broadly, answering the question, what is math. According the Cheng, “mathematics is anything that obeys the rules of logic, using the rules of logic.” She describes the two ways to do mathematics. The first is abstraction. This is where we find math in seemingly unmathematical ideas. The second is generalization – where we build on things we understand.
In addition, Cheng notes that math seeks to make things simpler by noticing things which are the same, ignoring the small details. When we problem solve in math, it is crucial to be able to organize the information we have in order of significance. We need to be able to take reality, apply abstraction and bring it back to reality again. In order to prove something is true we use the mathematical method and use logic instead of evidence. In this sense, Cheng states “math is a world where the means justifies the end.” We use logic to justify our conclusions even if we can’t actually come to that conclusion using numbers. Similarly, we need to study exceptions and general bodies of behavior. Cheng points out that mathematics allows us to do this in a sense that we do not have to imagine them at all in order to study them.
Next, Cheng talks about what motivates us to do mathematics. She states, “the best mathematical inventions are the ones that make internal sense and solve some existing problem.” Where solving an existing problem may be an external motivation, we need to continually ask the questions WHY? like small children and seek to be internally motivated. Cheng points out that we often accept truth on the grounds of higher authority. We assume that they did the homework, so why should we. But “one of them aims of mathematics is to do things from scratch.” Again, we need to understand the process, and when we do “math is easy”. There is no mysterious missing steps because each step of logic is connected to a preceding one.
We have applied math and we have pure math. According to Cheng, “applied math could be thought of as the theory behind things in real life and pure math is the theory behind applied math.” Pure math is like building a complicated structure out of lego blocks (no wheel or windows). Category theory is given in pure math to group similar things in mathematics. We group things by studying the context they are in in order to learn more about them. So what allows two things to be in the same group? According to Cheng, “two things are the same if you can reverse the process to get from one to the other” (i.e. they are invertible). Category theory looks at the role a thing plays but it also looks at a role and see what plays it. In this way, category theory illuminates math and aids in our understanding.
Finally, my favorite quote in Cheng’s book is the following: “Knowledge is power, but understanding is more powerful.” Mathematics is about logic and process, and understanding is key to its power.
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