Abstraction

If I had read How to Bake Pi before my first college math course, I probably would have gotten a lot more out of it. Eugenia Cheng, a mathematician, gifted writer, and experimental baker, somehow makes the connection between abstract mathematics and real life. In the prologue, Cheng lists common myths about math. One that she encounters a lot: “You’re a mathematician? You must be really clever.” Here’s her written response.

“Much as I like the idea that I am very clever, this very 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 problem – if 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, it also makes math easier.”

I wish I had thought about math along those lines back in college. Having had a bunch of calculus under my belt pre-college, I was credited with two semesters of calculus and therefore placed into Real Analysis. The first half of the semester was spent on Proof and Number Theory. I had no idea what was going on. I did not understand the point of proof and the seemingly strange arguments being made for things that seemed “trivial”. Except they are not. But I only came to appreciate this later in life. Needless to say I did not take another college math course for credit. (I did sit in on Linear Algebra because I was told it was useful for a Chemistry major, although not required.)

An early chapter in How to Bake Pi is the topic of Abstraction. Cheng describes abstraction as a blueprint. By ignoring some details, one can focus on the key ingredients to bake one’s pi. How does one do that? By looking at the similarities between different things. It also acts to de-clutter one’s thought process. Cheng says it can feel uncomfortable because one is “stepping away from reality for a bit” but it pays dividends when it comes to solving real-world problems as the complexity is added back bit by bit.

Cheng gives many examples of abstraction, connecting the real world to mathematics. My favorite is Road Signs. She writes: Road signs are a form of abstraction. They don’t precisely depict what is going on in the road but represent some idealized form of it.” She shows a couple of road-signs, and you can figure out what they are even if you have not encountered them often.

While actual bridges and mooses/meese might not look like their stylized images, “the benefits of this system are clear. It’s much quicker to take in a symbol than read some words while you are driving. Also it’s much easier to understand.” However, if you were to encounter a more abstract sign such as the “No Entry” (shown below), would you know what it was if you hadn’t learned it beforehand? If you do know it, however recognition is simple and immediate. Also, you’ll tend to encounter these a lot, particularly in a city center with many one-way streets. The principle here: Abstracting that which is common allows you to take in other things that require some degree of complexity and cognition. Driving in a news city can be complicated.

All this reminds me about the use of symbols in chemistry. I am going to quote Cheng but substitute “math” with “chemistry” and her explanation still works well. “Once you know what they mean, the symbols are quicker to take in, and you can reserve your chemical brainpower for the more complicated parts of the chemistry you’re supposed to be focusing on. It also makes the chemistry easier to understand across different languages – it’s surprisingly easy to read a chemistry book in a language you don’t know.” Actually that last phrase isn’t as true for chemistry as it is for math, but it’s somewhat true. As an undergraduate, I had to look up a German paper for an organic synthesis procedure. I was actually able to figure out a chunk of it, and filled in the gaps on names with the help of a German-speaking friend.

In the same chapter on Abstraction, Cheng makes a very interesting point about teaching. It’s all meta. She introduces this first by suggesting the building of a machine to do something rather than doing it by yourself, especially if it is a tedious process that will need to repeated frequently. But “in order to build [such] a machine to do something… you have to understand that thing at a different level. It’s like giving someone directions. When you walk somewhere you know well, you don’t really think about exactly what streets you’re walking on… you [go] instinctively. But when you’re telling someone else how to get there, you have to analyze more carefully how you do it, in order to explain it.” That’s why all teaching is meta. I’m trying to help my students work through a process in their minds, but I can’t think for them. Somehow I have to translate the process through designing appropriate activities and exercises, to get them to see and think chemically the way that I do as an expert in the field. This is not easy. Even more so as students may have a wide range of different misconceptions they have picked up along the way before showing up in my class. It keeps my job interesting and keeps me creative!

I’m only a quarter way through the book, and I’m looking forward to more insights! Meanwhile I end this post with an excerpt from the chapter on Generalization. If this sort of writing tickles your fancy, I highly recommend How to Bake Pi. And get yourself a pi plate!

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Here are [some] “why” questions with various levels of answer. You can ask yourself whether you find each answer inadequate, satisfying, or over the top, to see what sort of level of abstraction you like.”

Question: Why does anyone use a three-legged stool?

a) Because a three-legged stool is more stable than a four-legged stool.

b) Because if you try and put four legs down on the floor, one of them might stick up a bit more than the others, leaving a gap between it and the floor, which means the stool could wobble.

c) Because given any three points in 3-dimensional space, there is a plane that goes through them all. Whereas given any four points, there might not be a plane that goes through them all.