I think the reason why it's lovely, because it gives you a perspective shift. When we get used to thinking of something as having "this one nature", and we find out there's another "equivalent and consistent nature" think about it. It gives color to the world.

In this context, we often think of percentages as parts of stuff. I think it's because it was often taught alongside fractions. So in our heads, we intuit and think of percentages as breaking stuff apart. Typically, when we break stuff apart, there's often a low chance that for any number of divisions we pick, there will be combination of parts that sum up to the same amount. With that framing, it's surprising that by virtue of switching the percentage to the other number, we get the same part of the whole! It betrays our intuition.

However, if we re-frame it with our abstractions in math, it makes more sense. Multiplication is often taught as repeated addition. This isn't wholly correct. It's a way to scale a value. But scale goes both ways! You can both scale up to make the value bigger, but you can also scale down! By making the scale a fraction, we can scale down, or break stuff apart. And because we can now rely on the properties of the abstraction, namely commutativity of multiplication over natural numbers, it becomes much easier to see that it must be true that by virtue of switching the position of the percent sign.

It's kinda like when you get that ah-ha moment. A moment of insight. Other people call it making connections. But essentially, it's a new way of seeing something that was there all along.

It's like that game, The Witness. On the surface, it seems like it's about solving line tracing puzzles. But in fact, the entire theme of the game is to see with a new perspective what had been in front of you all along. I won't spoil the game for those of you that haven't played it yet, but if you want a guide tour of repeated ah-ha moments and insights, I suggest you try the game.

Which on the other hand is strange, because its creator Jonathan Blow really rails against functional programming and category theory. To me, as I learn more about it, I see the connections that were there all along by giving me apparatus to think about things in a new light.

The Curry-Howard Correspondence was completely surprising to me--that there's a relationship between logical proofs and programs. Propositions correspond to types, proofs correspond to programs, and proof verification corresponds to type checking.

It was also surprising to me that we can think of functions as exponential types, and the algebra works out!

Alan Kay laments that we build software like ancient Egyptians build pyramids--by piling on rocks. We don't build software with arches. I think that's starting to change as mathematical ideas like monads filter into mainstream programming. A lot of the concepts in functional programming and category theory filter down into language features of mainstream languages. Result and Optional types are prevalent now, and they're monads. But it took leveraging the mathematical apparatus to find them, despite decades of stabbing at it by programmers.

In a practical sense, having the insight and ah-ha moment before you need to use the connection is helpful. In the midst of being busy trying to find product-market fit, you often don't have the state of mind or the time to see the connections. But if you already know them, you can take advantage of the properties and shortcuts that it gives you. This is the answer math teachers give you when you ask "when am I ever going to use this?"

But I don't think that's really not why mathematicians do math. I think having that insight and ah-ha moment feels like an expansion of the mind, a perspective shift feels out-of-body. And it can get addicting, especially if you had to work hard to get it. I think if you want to live a life of insight and ah-ha moments as a priority compared to everything else, then there are probably little other vocations better than mathematics. But you don't need to be a pro-mathematician to have those little ah-ha moments to enrich your life. You just have to be willing to work at it a little, and get that occasional perspective shift. And that's why I think it's lovely.