I love proofs in mathematics. Getting to the heart of why something is true is a ton of fun. It’s even better when there is a nice and simple proof of a fact that is surprising. These are the proofs that I love the most, because they solve a mystery in a way that is easy to follow and makes sense the whole way.
When first learning about proofs, I proved basic facts about numbers. Often, the proofs revolved around sets or divisibility, since these are straightforward to prove. I got experience working through the algebra necessary to prove a given mathematical statement.
When you are writing up a homework assignment, the nice thing about proofs is that you don’t have to come up with the best proof. In fact, you can come up with a method that is super-convoluted, and you will still get full marks if everything is correct. The point is to follow the chain of logic until you get to a result. Nobody said you needed to take the shortest route.
However, despite enjoying the process of proving statements, I sometimes get lost in the mess of implications and items to keep track of. This is a necessary hazard for many proofs, since you cannot always get a quick-and-easy solution. Sometimes, you need to grind out the algebra, even if it is not the clearest thing to do. Some proofs are just messy.
The hope though is to find better ways to prove statements. If I come across a proof that works but I don’t understand well, I want to find another way to see the result. If not, I will always be somewhat wary of the result. This isn’t just a matter of preference either. Mathematics builds on itself, so if you don’t feel comfortable with a result, it won’t be any clearer down the road when you need to reference it. As such, I am acutely aware of the need to establish proofs that make sense to me.
The other factor to consider is that of clarity. A proof can be quite clear and still not make a lot of sense pedagogically. For example, in analysis (what is sometimes called the “rigorous” version of calculus) a lot of the proofs involve choosing certain values to establish inequalities. The proofs supplied in textbooks will often give these values first, which then makes the proof follow through smoothly. However, when a student reads the proof, it often isn’t clear as to why this specific value was chosen (and these values can get very strange). In this sense, the proof is clear, but it isn’t pedagogical.
My concern is with coming up with proofs that are pedagogical. A proof can be clear, but if it requires me to think about the topic for several hours first in order to get my bearings, I don’t want it. Instead, I want to have a proof that is pedagogical, that puts the understanding of the student first. Yes, this might mean the proof isn’t as concise as it could be, but my concern is for the student. What is the perfect balance between explanation and omission? I don’t think we need to include everything in a proof, but I do think we need to make a result as easy as possible to digest.
I have many thoughts about what makes for a great proof, but here are just a smattering of them:
- A minimum of moving parts. Sometimes you can’t escape this, but when a proof requires a bunch of different cases or ancillary details, I find myself losing focus. I’m not saying the proof has to be one line, but if you have a proof that does not depend on three different facts that you also need to prove, it’s easier to understand.
- Few dependencies. This is related to the previous point. While fewer moving parts is good, it can be taken too far. For example, many proofs follow as direct consequences of other results. If you then “prove” your result by referring to this other one, I don’t consider that to be a great proof. Sure, you may be technically proving it, but in my opinion, you are kicking the bulk of the proof down to the previous result
- Visual is better. This is more of a personal preference, but the great thing about a visual proof is that it gives people a better sense as to what is happening. I want to note something important here though: I’m not as big of a fan of proofs without words. These often fail to include enough details, which goes against the idea of being pedagogical.
- Illustrate concepts with examples. Mathematicians love to go through the most general case right off in their proofs. This is fine, but the issue in terms of education is that students often can’t follow the train of thought. By including examples as you prove a statement, the general principle can become more clear.
I think the point to keep in mind when crafting proofs is that you need to identify your motivation. Are you looking to get the result as concisely as possible, or do you want to be pedagogical? If it is the latter, I suggest making sure that your explanations are self-contained. The less outside material you rely on, the easier it is to follow what you are saying. Plus, you can spread your ideas to a wider audience.
Proofs are not always easy to follow, but I think it’s worth putting in the effort to making them as easy as possible.