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Of course, there’s a ton more, but that’s the gist of what PDEs can look like. Honestly, it’s pretty fun stuff. You get to wrestle with tough problems, draw from a very diverse toolkit, and Awesome That’s What I Do I Ride Horses I Play Guitars I Know Things Shirt has practical relevance. It’s a cool field. You need both intuition and rigor in order to reach your full potential in math. Do not dismiss the need for, or appreciation of, rigor just because you’ve never seen it before. Likewise, do not think that mathematics is all about rigor, or that you should abandon your intuition. You need both. By the way, I’m about as close to the opposite of Hardy as you can get. I’ve spent my whole career breaking Hardy’s famous maxim by finding new applications for parts of “pure” mathematics that previously had no applications.

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I suspect the reason it wasn’t found sooner is that you can’t even see the statement of the theorem unless you use constructive logic. In classical logic, the law of the excluded middle is tautologically true. Most mathematicians use exclusively classical logic in all of Awesome That’s What I Do I Ride Horses I Play Guitars I Know Things Shirt. Someone else wrote you a great reply about PDE. But I’ll just say this: PDE looks like they’re the 2 or 3-dimensional version of ODE. They’re only that on the surface. In reality, they’re the infinite-dimensional version of ODE. This makes them very, very, very, very hard. There is essentially no hope at ever recovering a general theory for solving. PDE, unless someone divines one of the most brilliant ideas ever conceived. Right now there are a lot of useful tools that apply to large classes of important PDE. And it let you solve them / prove solutions exist/prove solutions are stable.

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