Hello everyone, I have been trying to teach myself GR for sometime now, and while I started to learn through books like Schutz and Hartle, I reached a point where the motivation behind the mathematics they used didn't satisfy me nor did it appeal to me. I think the worst I remember is when Schutz defines a manifold along the lines of: manifold is a fancy word for space. I could only wonder just exactly what kind of space that was. While this approach might prove beneficial during an introduction, I never felt I was learning, well basically anything at all regarding the mathematics. I have been trying to teach myself the differential geometry along with the physics of GR, I've used a number of sources but currently I'm working through the mathematical chapters of "Relativity on curved manifolds" by De Felice et. al., and the appendices in General Relativity by Wald. This is my first time dealing with abstract math, I took a linear algebra course before but it never got so much into the abstract side. Which brings me to my question: How do I make the transition from more applied math such as boundary value problems and ODE's to the more abstract thing? I think this is especially difficult since I am self studying. What I gather so far from differential geometry is that the whole thing is about starting with primitive notions such as sets, and giving them different kinds of structures, such as a differentiable structure or a metric structure via a metric tensor. And it seems that abstract mathematics is pretty much like that, identifying and isolating the essential and building upon that. I still struggle reading proofs and sometimes it takes me a while to get what the authors mean when they make mathematical statements, perhaps I should start with something lighter to get used to the general spirit of abstract math?