Definitions matter. They seem pedantic, but they do matter.
What is a square? Fours equal sides, four right angles… right?
Well this is a square too isn’t it? obviously not. We can see that this image isn’t a square, but what if we couldn’t. What if we were dealing with some object that wasn’t so easy to understand, and we just never realised we were wrong.
Anyone who has done any olympiad mathematics has probably had this exact same experience: You’re sitting an exam, you solve the geometry problem but uh oh, you walk out the exam and you realised that you fake solved. What you had actually done was assume the answer was true, did a bunch of working, and gotten back to the answer.
Maybe in your quest to prove that three points were collinear, you chased an angle that relied on the fact the three points were collinear. Maybe you reverse reconstructed a point that could have been defined in one of three ways, and you used a property you hadn’t proven yet.
This is surprisingly common. So for the sake of your sanity, please just write down your definitions and once you have these, then go solve the problem.
Defining things is just good problem solving technique anyway. Once you write down something like
let \(S(n,k)\) be the number of ways to partition \(n\) elements into \(k\) non empty subsets.
You free yourself to do higher order reasoning. You could write something like \(S(n,k) = S(n-1,k-1) + kS(n-1, k)\). I don’t think it would be possible to come with a statement like this with just words, but that’s okay. We no longer live in medieval times where all mathematical equations had to be written in prose.
Definitions guide you to solutions. Sometimes defining something one way over another way allows you to do an induction that you otherwise wouldn’t be able to do. In geometry problems often times conditions can be interpreted in multiple ways, and different constructions can be more or less helpful in actually solving the problem.
Usually the best definitions are the ones that give you a lot of useful information or help transform the problem at hand into a more approachable one.
Definitions don’t solve problems by themselves, but definitions serve as great starting points.
Picking a good definition is very important. Definitions are also very cheap, it doesn’t take very much effort to define something, so keep doing it until you get it right.
Ideas like continuity have precise definitions, but the power of these terms arises from the fact that on top of their technicality, they allow one to reason more abstractly.
To use a theorem, we can just use the fact that the function \(f\) is continuous, instead of having to reason through the epsilon delta definition every time. We have the power to both prove continuity precisely and also use continuity abstractly.
Defining things might seem a little pedantic, but definitions aren’t just formalities. They’re tools for thinking and they are important.