I know it seems silly, but recently I’ve been really impressed by the mechanism of division in math. Maybe it’s because in my head I’ve always conceptualized it as breaking up a larger group into parts, and it’s only lately that I’ve started to think of it as standardization. When dividing a sum for a mean, you’re standardizing the numerator by the denominator. The denominator becomes 1, and for all the groups. Another example is when calculating out a conditional probability:
In this case, dividing by the probability of event E is standardizing — or conditioning on — the numerator by that probability. The probability of E and F is what it may be, but once divided by P(E) it is standardized to what P(E∩F) would be if P(E) were equal to 1; that is, it’s the P(E∩F) if the probability of E were the only universe that exists. So simple and so damn brilliant. If I were to go back in time and decide I wanted to live in the world of philosophy, I would have studied math.