Math and Science Book Review
Eugenia Cheng takes on the ambitious task of trying to bring the excitement of math to the lay person through the study of category theory. This is a long shot that requires truckloads of metaphors. Luckily, she has a deft hand at metaphor and she does an admirable job of bringing her excitement about the subject matter to the page. She starts simple with, “Mathematics is the study of anything that obeys the rules of logic, using the rules of logic.” For you Vulcans out there, this means that math can study damn near anything, except for maybe teenage girls. One of the really cool aspects of math is that it is based on that very simple premise, math can take us to crazy new places. Another one of her great metaphors, “Do you know that feeling of climbing to the top of a hill, only to find that you can now see all the higher hills beyond it? Math is like that too. The more it progresses, the more things it comes up with to study.” This certainly brings to mind an internal exploration of logic and reason that can seem almost limitless in it’s scope and breadth. As long as you have the tools, there really are no limits to where math can take you and that really is exciting.
Two of the major tools to start this exploration are abstraction and generalization. Abstraction begs the budding mathematician to ignore some details so that the situation becomes easier to understand. In Cheng’s words, “Abstraction is like preparing to cook something and putting away the equipment and ingredients that you don’t need for this recipe, so that your kitchen is less cluttered. it is the process of putting away the ideas you don’t need for the present purposes, so that your brain is less cluttered.” She goes on to acknowledge that abstraction is also dangerous and where we lose a lot of people in the weeds. “Abstraction is the key to understanding what mathematics is. Abstraction is also at the heart of why mathematics can seem removed from ‘real life’. That detachment from reality is where math derives its strength, but also its limitations.” She emphasizes that one of the keys to the mathematical method is being very clear about your assumptions. This reminded me a lot of Nate Silver’s book, The Signal and The Noise, that dove deep into mathematical modeling. One of his major premises was that mathematical models need sunlight shone on their assumptions from the very beginning to be successful. You can tell that Cheng loves the process of abstraction because it opens up so many crazy doors, “Part of the process of abstraction is like using your imagination. Mathematical abstraction takes us into an imaginary world where anything is possible as long is it’s not contradictory. Can you imagine transparent Lego blocks?…Four dimensional Lego blocks? Invisible Lego blocks?”
Before she dives into generalization she spends some time on mathematical rigor. This is another area where I, as a physics major, always had a problem with math in school, mathematicians seemed to get all tied up in the minutia of the process which, in my mind, didn’t relate to the real world. Cheng poses that math really is all about the process and that’s what makes it interesting, “Math is a world in which the end does not justify the means: quite the reverse. The means justify the end; That’s the whole reason it’s there. It’s called mathematical proof,…” I’m starting to come around.
Generalization is the other major tool in a mathematicians toolbox. “This is the point of generalization in mathematics as well – you start with a familiar situation, and you modify it a bit so that it can become useful in more situations. It’s called generalization because it makes a concept more general, so that the notion of ‘cake’ can encompass some other things that aren’t exactly a cakes but are close.”
With these tools, imagination starts to become a mathematician’s friend. “The key in math is that things exist as soon as you imagine them, as long as they don’t cause a contradiction….Do you think it’s cheating to solve a problem by inventing a whole new concept and declaring it to be the answer? For me this is one of the most exciting aspects of math. As long as your new idea doesn’t cause a contradiction, you are free to invent it.” How many art majors would have become mathematicians with a teacher like Cheng? It’s a shame that we teach math in this painful, rote methodology that squeezes all of the excitement out of what rigor and logic have to offer. Math has been called a lot of things, but creative is rarely one of those things. How do we bring this excitement back to the classroom?
She never gets into the nitty gritty of category theory, but instead she alludes to it through many different examples. Here is one of them, “Whatever mathematics does to the world, category theory does for mathematics. It’s a sort of meta-mathematics like Lego Lego…Category theory is also an organizing principle, just one that operates inside the world of mathematics. It serves to organize mathematics.” I never truly understood category theory from the book, just that it is worth looking into in more detail.
I thoroughly enjoyed reading the book, especially the actually mathematical examples that took me out of my comfort zone. Those things that make you think beyond what you can see are what seem to bring true knowledge and this book is full of those things. Overall though, it’s her joy of the subject matter that makes the book work. It’s worth your while.