> Now that I'm taking calculus, I feel like I always have a corresponding application for each topic we cover. It's very exciting!

Mathematical concepts need not necessarily have any practical or real-life applications (pure mathematics), but it's a worthy pursuit on its own. However, every maths concept we learn outside of specialized mathematics courses have some application (applied mathematics). That's how these courses are designed.

It's funny when some people lament that much of the mathematics they learned at school or uni/college have no real-life applications! I wonder what they think about why they learned it. It also shows a big problem with math education and pedagogy. Many people are sailing without a destination in mind, hoping that they'll just hit some land in the future. The academic practice of mathematics has turned into the drudgery of endless symbolic manipulation.

But at the same time, people like you who lean into creativity, hit the limits of their mathematical knowledge early. For them, the problem isn't that the mathematics they learned have no applications. It's that they don't know where to look for the mathematics they need. Here is the failure of math education. If everyone was taught about the different branches of Mathematics, how they relate to each other and where they're applied, I can guarantee that many students will learn those on their own before they're formally introduced to them. That would enable the talented and interested students to shape their own destiny.

Agreed.

> Mathematical concepts need not necessarily have any practical or real-life applications (pure mathematics), but it's a worthy pursuit on its own.

I guess I always had a sense that this was true, but taking a formal proofs class has really opened my eyes to how true this is. I'm actually planning on doing abstract algebra because I've enjoyed proof writing a ton!

> It's that they don't know where to look for the mathematics they need. Here is the failure of math education.

This is a really interesting space imo, because I did try teaching myself calculus through 3blue1brown, who has super cool visualizations, but also isn't rigorous enough to apply it to complicated problems. On the other end, I also tried Khan Academy, but found it too abstract and hard to follow.

Perhaps that was just since I was 14 at the time, and now I'm better at symbolic reasoning. Yet, there's something so enticing about visualization, that I wish there was a way to have the rigour of set theory with the intuitiveness of visualization (that call is a big reason I love the work that dynamicland.org and folk.computer are doing right now).