Jeremy Côté

Slick Versus Pedagogical Proofs

In mathematics, there’s almost always an opportunity to make proofs more concise. For example, when you first learn a concept, it might take a while to prove a result using the definitions that were developed. The reason it’s longer to do is because the definitions require you to spell out ideas explicitly. As a result, you might get to the end of the proof, but it takes a bunch of little intermediate steps to do so.

On the other hand, if you can develop some extra machinery, then you can leverage that machinery to make proofs easier. These new ideas act as shortcuts, allowing you to bypass the long definitions and go straight to the result. What this ends up doing is giving you a way to formulate a proof without needing to dig into too much of the details.

It’s important to note here that this doesn’t mean you don’t know the details. Rather, you spent the time building up machinery which then lets you apply the definitions without having to actually use them.

A clear example of this comes from calculus. When taking a course in calculus, you learn how to take limits. If you had a teacher that cares about sharing the definitions of such things, you might have seen something with the symbols ε and δ. It’s likely though that this was shown only once or twice. You then learned about limit laws, and the rest of the time you dealt with limits involved using those laws.

However, limit laws are not the basic building blocks of limits. Instead, it’s the very definition of a limit. What the limit laws allow us to do is bypass the definition in specific cases because we know how the general case works. Therefore, if we want to prove a result using limits, one doesn’t normally use the definition of a limit. Instead, we use this extra machinery to make the proof smoother.

At this point, you may be glimpsing the coming problem. At what point does a proof lose all of its clarity by offloading the key ideas of a proof to other machinery?

This is something I think about a lot with respect to education. When learning a new mathematical topic, everything is novel. This means that it takes extra effort when referencing proofs and results seen in class. Students will likely be most comfortable with basic definitions, since those were introduced the earliest. If you ask a student what kind of proof is “better”, I’m guessing they would answer with the long and convoluted one that uses the definitions. Not because it is concise, but because it is transparent.

I like to think of proofs as falling into roughly two categories: slick proofs and pedagogical proofs. A slick proof is one that achieves the result without writing out every single detail starting from the basic definitions. Slick proofs often reference other proofs, making the total length of the proof much shorter than if nothing was referenced.

A pedagogical proof is one that is transparent, laying out all of the gory details for the reader to see. If one was proving a result about limits, this would involve taking out the definition and picking the right δ to satisfy the constraints. Each step would be laid out, and there would be little confusion to a reader about where a specific choice of this or that symbol came from.

For this reason, a pedagogical proof will run longer than your average slick proof. However, that’s not a problem, because a pedagogical proof is often self-contained. Whereas a slick proof will make use of other results to shorten the work needed, a pedagogical proof embraces the extra work.

If you give me a choice between the two, the slick proof is definitely the “fun” way to go about seeing a result. However, the thing people don’t mention about slick proofs is that they carry with them a lot of baggage. If you want to really understand what’s going on with a proof, you need to understand all of the results a slick proof might reference.

This is why I will go to the pedagogical proof when I’m first learning a subject. I aspire to get to the slick proof, but I know that a slick proof will only bury the knowledge I need to understand. Therefore, even if I might “get” a slick proof, without understanding the background deeply, it will be difficult to fully comprehend it.

I’ve been thinking lately about establishing a collection of pedagogical and slick proofs. I think both are important when learning mathematics. You could think of them as being the beginner or advanced version of a result. When you’re starting off, the pedagogical proof will often be more informative. However, once you’ve mastered the basics, the slick proof will offer a concise way to think about a result.

It’s worth remembering that people aren’t all at the same level in mathematics. As such, it’s good to have a variety of proofs that can appeal to different levels of mathematicians.