As a student in mathematics and physics, I’m part of two different worlds. On the one hand, proofs and abstraction come from the side of mathematics. On the other hand, physics is where concrete examples and applications are the norm. In physics (at least, within the scope of undergraduate education), we only care about the mathematical tools that we can apply to a given problem.
Of the two camps, I find myself identifying more with the mathematics side. That’s just because I like abstracting past one application and finding the link between a lot of concepts. I realize that we do this in physics as well, but I find myself at home within the mathematical proofs versus the handwaving that often happens in physics (which isn’t always a bad thing!). I’m definitely not a pure mathematician. I still like to link what I’ve learned with the world, so perhaps I fit best into the “applied mathematics” camp.
The reason I bring this up is because I’ve been thinking of the way we introduce examples in mathematics. If you’re in an advanced mathematics class, chances are the professor will first go over some definitions and perhaps a few theorems before giving examples. Even then, the examples only illustrate the idea, so they aren’t the focus of the class. Instead, the focus is on the proofs and how to go from argument to argument.
This makes sense. After all, advanced mathematics tends to involve abstracting past examples and capturing the general case. This means the examples are less important than the underlying characteristics they share. This of course is the strength of mathematics. When you can look past the immediate features of a specific example and see what many examples have in common, you can come up with theorems that apply more generally.
This came to light in my abstract algebra class. There, I learned about a mathematical object called a ring. The specifics aren’t important, but what is important is that a ring is something we are all familiar with. The integers under addition and multiplication form a ring. The real numbers form a ring. If we look at linear algebra, nxn matrices form a ring. The point is that even if you have no idea of what a ring is, you have likely worked within a ring in your mathematical journey.
As such, the underlying structure of a ring was there, beneath your feet, all this time. Instead though, you studied arithmetic and linear algebra as separate concepts. It turns out though that they are both examples of rings. Therefore, by studying the properties of rings, we can capture the core ideas of matrices and integers or real numbers in one go.
That being said, there’s a downside to abstracting to higher and higher levels. Each time you go up in abstraction, you lose certain features. If you climb a mountain, the landscape of trees you see from the summit will look smooth. Descend the mountain though, and you find that there’s a staggering amount of diversity present within the forest. What looked uniform from above can break up into unique parts upon closer inspection.
I think we can make a similar comparison with abstraction and examples. Sure, abstraction is great in the sense that it captures everything we want in one sweep. But the price we pay for it is that we don’t have specific examples in mind when working through the mathematics. This might not sound like a bad thing, but it makes it difficult to apply our knowledge to specific scenarios.
This is something I also learned in my abstract algebra class. As great as it is to study rings, integral domains, ideals, and fields, it’s also important to find examples for these specific objects. Examples clarify definitions and make the abstractions we study easier to visualize. Without them, it’s difficult to attach meaning to our objects of study.
It’s tempting while studying mathematics to jump straight to the abstraction. I’ve been there, and I know the feeling. You want to do it because you figure that knowing the general case will be a lot better than any specific one. However, this ends up not being true at all. In fact, I would venture to say that specific examples provide the footholds necessary to be comfortable with abstraction. When I feel at ease with a concept, it’s usually because I’ve internalized a specific example and I can see how that example stems from the general case.
Unfortunately, advanced mathematics courses often prioritize proofs and abstraction over concrete examples. This is emphasized through the kinds of problems that are assigned and the amount of class time dedicated to examples.
On the one hand, going through a bunch of examples can seen repetitive in class. Furthermore, problems which involve examples tend to be easier and don’t involve proofs. This isn’t a bad thing, but it does mean that the students don’t get to practice abstract thinking as much. I would venture to say that this is a big reason why professors don’t assign these problems as often.
On the other hand, I find that focusing only on the abstractions prevents you from playing with an idea in specific settings. I found the lack of examples in my classes a hindrance when it came to working on problems that dealt with specific examples. It might seem like applying the general knowledge you know to a specific example would be easy, but I can assure you it’s not. Furthermore, it’s once you succeed in applying the general knowledge to a concrete case that the idea becomes familiar. I would argue that knowing abstract knowledge without being able to apply it anywhere is next to useless. You need to have a balance of both the abstract and the concrete to thrive.
This is why I’ve begun thinking about my “bag of examples”. It’s great to know important mathematical results, but if I can’t illustrate them with an example, it becomes difficult to communicate them. Plus, working through a specific example tends to be the easiest way to grasp an idea.
Good examples become fertile ground for experimentation. When you have a concrete example, you can see how the result you’ve proved works in this case. The best examples can even inform the general result.
I know that my bag of examples is very empty at the moment. However, I want to build it up. Knowing mathematical proofs is great, but having a bag of good examples that you can pull out at any time is an under-appreciated asset.
Finally, I don’t want to forget about the related category of counterexamples. These can be just as important as examples, because they remind us that mathematics can be misleading if we look at just a few cases. Counterexamples force us to be more careful in our hasty generalizations and to remember that the final arbiter of truth is through a proof.
As such, my goal for now is to start amassing examples and counterexample stop illustrate various mathematical ideas. I want to find the shining jewel of examples for any idea. It’s great to keep on learning new material, but sometimes it’s worth pausing and building up a bag of examples.