Appreciating the Why
If I wanted you to calculate a bunch of derivatives for me, I could simply show you the algorithmic approach to the process and leave you to do it. With some practice, you’d get better and better at taking derivatives and could even become better than a mathematician.
However, the drawback is that you wouldn’t necessarily understand why this algorithm worked. You would get the correct answer all the time, but you wouldn’t know why it was workings. In essence, you would simply be defaulting to the process and trusting that this works. In a way, you’d be taking what I say is the method to perform derivatives on faith. Since you trust what I say, you would believe that what you’re doing is indeed taking derivatives of functions.
Unfortunately, this decision to “take the equations on faith” is a reason why so many people in our society look down at scientists and mathematicians who say, “look at the equations” when posed a question. When giving this response, we know that people will mostly not take the time to understand what the equations mean. Instead, they will either take it on faith, or accuse us of taking equations too seriously. This can be seen from those who say scientists are slaves to their equations and that nothing else can change their minds.
Whenever this kind of sentiment is expressed, I try to counter it. The reason is simple: the person making that claim doesn’t fully understand what equations are (at least, in science).
Often (and particularly with phenomena that are discussed in one’s school year), equations don’t just come out of the blue. They aren’t simply handed down from the sky above and become law for the rest of time. Instead, equations are consequences that stem from the relationships of different variables. They aren’t put together because they look nice, or even that they necessarily produce the correct result. Each component of an equation means something very specific. That is, if you want to prove the equation, you need to explain why everything is as it is. There aren’t any free passes. As such, an equation doesn’t force nature to be that way. It is the result of the pattern that variables tend to follow.
This is why I always want to understand how the equations I learn work. I’m currently in a statistics class where all the formulas are given. And while computation is the point of the class, I’d like to understand why they are as they are. This isn’t within the scope of the class, and it has bugged me all the time.
Realistically, I don’t need to fully understand a proof if I just want to get a result. It’s not necessary, but it is nice, because understanding the mechanics of an equation means it is easier to work with and modify. That’s why I try and take the time to really understand a proof when presented with something new.
I don’t trust my equations blindly. I use them because they work and because they mean something. If I had it my way, I’d make every equation super simple. Unfortunately, the reality is that many equations are not like that, and instead are giant messes. However, as my teacher likes to say, “I’d rather have it be ‘right’ than ‘nice’ every day of the week.”