*Bits, ink, particles, and words.*

When learning a new topic, there’s always a certain tension between two approaches: going straight to abstraction, or starting off easier with examples. I see this more and more as I learn about more complex and detailed physics and mathematics, and it has always made me wonder which way I should go about trying to learn. Just like anyone else, I want to get to a place where I feel fully comfortable with the concept in abstraction, but I don’t want to subject myself to a painful learning process by hitting myself against the brick wall of abstraction.

This is going to be a very quick post, but it’s something I wanted to share since I think it could give some insight into a concept that is used to make proofs in mathematics. When writing proofs, it is often difficult to show that your proof holds for *every* case (say, if you’re trying to show that the square root of two is not capable of being represented by a ratio of integers). Checking every single combination of integer ratios would take an infinite amount of time, so we want to come up with a strategy that is better. To do this, we try to prove our statement by *contradiction*. The idea is that we negate our conclusion, and from that, we need to show that there is a logical contradiction. Therefore, when we come to a contradiction, we know that our assumption of negating the conclusion was false, so that means our conclusion is true.

As a student, I know what it takes to get good grades. Essentially, you want to be able to reproduce the work that is taught in class during a test. You don’t need to be creative or original in your work. Rather, you simply need to understand the procedures and apply them (for the most part).

I imagine that we do this all the time: you’re talking to someone else about solving a certain equation, and then you tell them something to the effect of, “I had to bring sigma to the other side of the equation.”