Why should it buy you something is the real question.
You don't need to understand it the way the "initial" author thought about it, should that person had given it more thoughts...
History of maths is really interesting but it's not to be confused with math.
Concepts are not useful as you think about them in economic opportunity case. Think about them as "did you notice that property" and then you start doing math, by playing with these concepts.
Otherwise you'll be tied to someones way of thinking instead of hacking into it.
I know more math than the average bear, but I think the parent has a point even if I don’t totally agree with them.
Take for instance the dual space example. The definition of it to someone who hasn’t been exposed to a lot of math seems fine but not interesting without motivation — it looks just another vector space that’s the same as the original vector space if we’re working in finite dimensions.
However, the distinction starts to get interesting when you provide useful examples of dual spaces. For example, if your vector space is interpreted as functions (for the novice, even they can see that a vector can be interpreted as a function that maps an index to a value), then the dual space is a measure — a weighting of the inputs of those functions. Even if they are just finite lists of numbers in this simple setting, it’s clear that they represent different objects and you can use that when modeling. How those differences really manifest can be explored in a later course, but a few bits of motivation as to “why” can go a long way.
Mathematicians don’t really care about that stuff — at least the pure mathematicians who write these books and teach these classes — because they are pure mathematicians. However, the folks taking these classes aren’t going to all grow up and be pure mathematicians, and even if they are, an interesting / useful property or abstraction is a lot more compelling than one that just happens to be there.
Your post represents a common viewpoint, but I don't agree with it. I'm a retired programmer trying to learn algebra for the purposes of education only. I am not supposed to take an exam or use the material in any material way, so to speak. I'd like to understand. Without understanding motivations and (on the opposite end) applications I simply lose interest. I happen to have a degree in math, and I know for the fact that when you know (or can reconstruct) the untuition behind the theory - it makes a world of a difference. If this kind of understanding is not a goal, then what is?
BTW, by "buying" I din't mean that it should buy me a dinner, but at least it's supposed to tell me something conceptually important within the theory itself. Example: in the LADR book, the chapter on dual spaces has no consequences, and the author even encourages the reader to skip it :).
You don't need to understand it the way the "initial" author thought about it, should that person had given it more thoughts...
History of maths is really interesting but it's not to be confused with math.
Concepts are not useful as you think about them in economic opportunity case. Think about them as "did you notice that property" and then you start doing math, by playing with these concepts.
Otherwise you'll be tied to someones way of thinking instead of hacking into it.