Well, the set of all numbers is infinite, so you would wouldn't expect to find just any number appear in a lot of different contexts by chance. Some numbers do

Well, the set of all numbers is infinite, so you would wouldn't expect to find just any number appear in a lot of different contexts by chance. Some numbers do tend to show up over and over again. That makes those numbers special. φ is such a special number. But there are a lot of special things about 3 though too: It's an integer (which is a pretty special property for a number to have, considering there are an infinite number of real numbers between each integer). What's more, it's a very small integer (the fourth smallest) and it's the second smallest prime number. In fact, we could say that 3 is a very special number, in the scheme of things, and has lots of reasons to show up in particular contexts. Now φ being the golden ratio is pretty much all that's unique about it. Otherwise, it's just one of the uncountably infinite set of real numbers between 1 and 2. So if you find φ show up some place, almost surely you're dealing with the golden ratio. The same is true of π. It's just some real number that happens to be the ratio of a circle's circumference to its diameter. I've heard of mathematicians working on problems and having π appear in their calculations, and then suddenly realizing that they're working on a harmonics problem. The fallacy of numerology is getting attracted to numbers that have lots of reasons to show up in various problems (usually small integers, like 3) and then taking whatever one of their many properties is most convenient and treating it as an iron link to whatever they're trying to prove.