As you learn more ideas in mathematics, it’s easy to start feeling like certain ideas are “below” you. This often comes in the form of saying that ideas are “trivial”, as if they shouldn’t take up any of your time. This can be exacerbated further in mathematics by the idea of rigour. Once we learn that not all proofs are equal, it can be tempting to say, “Okay, I get this proof, but that’s not the whole story. You’re not being fully rigorous here.” We can then get caught in the cycle of thinking of our work as more important than “basic” facts.

When I’m sitting in class and listening to the professor, I get frustrated when they go over a concept once and then continue on as if everything is clear. I think to myself, “Do they really expect us to understand a concept from just one example?”

As a physics student, I’m taught over and over again that science is about checking to see where we have made incorrect assumptions. The goal of science is to correct our assumptions about the world, and find better descriptions for what is going on. Of course, this is a distilled version of the goal of science (and I imagine some would disagree), but my point is that science aims to perform a consistency check on our hypotheses and ideas.

If you look at a typical mathematics class, you will find that they follow a similar rhythm. First, students are presented with a definition. A few examples of how that definition applies might be given, and then the rest of the class is spent proving results based on this definition.