*Bits, ink, particles, and words.*

When you learn a new concept, chances are that there’s some sort of procedure to follow in order to come up with an answer to a problem. This helps students when they are first learning, because it lets them follow clearly laid out steps that will culminate in the correct answer. For example, if we were trying to add two fractions together, we know that a common denominator is needed. As a result, students might be told that they should multiply each fraction by the other’s denominator, which will guarantee that the denominators are the same. It might even be written in a nice three-step method like this:

When studying rings in abstract algebra, one also learns about subrings. They are pretty much exactly what you would expect: subsets of a ring with the same operations defined on this subset. However, a more interesting type of ring is an *ideal*.

If you have ever come across someone talking about special relativity, there’s a good chance you will be able to tell me one of the two fundamental axioms in the subject: the fact that the speed of light is constant in all frames of reference.

I’ve always been a bit wary about intuition. For a long time, I would avoid using the term, because I found it was creating a dangerous habit in terms of thinking clearly while solving problems. Your intuition isn’t always right. As such, instead of trying to guess when your intuition is correct, my advice was simply to throw it out, and work with what you have. This seemed like a much easier and straightforward way to go about reasoning, particularly within mathematics.