I find that almost all books I read do discuss "applications". No mathematics is an island. But as you say, the problem is that a lot of those applications don't really count, because the domain of the application is as foreign as the original concept.
For example, knowing that the snake lemma, a purely categorical statement, is most useful in homological algebra (such as, for example, simplicial or singular homology), is utterly useless if you don't already have an interest in algebraic topology. There really is almost no other motivation for the snake lemma, so now we're faced with the problem of trying to convince you that it's interesting by trying to convince you that algebraic topology is interesting. It can be done, and maybe we'll eventually bottom out in something like financial statements or bridge-building, or another topic that is widely recognised as "useful" and very far-removed from the snake lemma. Either way, it will be a long and arduous path, and I hardly think mathematicians can be blamed for this or be dismissed as elitists for the inherent difficulties of the subject.
But even for "first year stuff", the applications are kind of pointless. Do you really want to learn calculus because of physics? That's the most obvious and most historical application, but calculus is so foundational that you might as well motivate addition of real numbers by saying that numbers are added in physics too. More likely for the HN crowd, you want to learn calculus because you want to know how a neural network's backprop algorithm works, but how is the first year teacher going to anticipate that this is your particular interest in calculus?
At some point, I think there has to be a little "trust me, this is useful" and you just struggle through the subject until you can see on your own, after the fact, what the struggle was about. First wax on, wax off, Daniel-san. Then you will learn how you really were learning how to block karate blows.
For example, knowing that the snake lemma, a purely categorical statement, is most useful in homological algebra (such as, for example, simplicial or singular homology), is utterly useless if you don't already have an interest in algebraic topology. There really is almost no other motivation for the snake lemma, so now we're faced with the problem of trying to convince you that it's interesting by trying to convince you that algebraic topology is interesting. It can be done, and maybe we'll eventually bottom out in something like financial statements or bridge-building, or another topic that is widely recognised as "useful" and very far-removed from the snake lemma. Either way, it will be a long and arduous path, and I hardly think mathematicians can be blamed for this or be dismissed as elitists for the inherent difficulties of the subject.
But even for "first year stuff", the applications are kind of pointless. Do you really want to learn calculus because of physics? That's the most obvious and most historical application, but calculus is so foundational that you might as well motivate addition of real numbers by saying that numbers are added in physics too. More likely for the HN crowd, you want to learn calculus because you want to know how a neural network's backprop algorithm works, but how is the first year teacher going to anticipate that this is your particular interest in calculus?
At some point, I think there has to be a little "trust me, this is useful" and you just struggle through the subject until you can see on your own, after the fact, what the struggle was about. First wax on, wax off, Daniel-san. Then you will learn how you really were learning how to block karate blows.