If you think about it, pretty much everyone learns math through example first. You learn arithmetic in elementary school, and it might only be after an additional decade or more of schooling that you actually learn the logical underpinnings which make the basic arithmetic operations work. If you actually started with boolean logic and tried to build up all of mathematics from there without at some point tying it back to a tangible example which can be understood in terms of the real world I'm not sure anyone could successfully learn it.
True. Examples are necessary for both motivation, and for showing how the formalisms work, at this level. One good undergraduate textbook that has plenty of examples is Dummit & Foote. However, that doesn't mean that you don't define anything properly, and solely use examples to show how to crunch certain computations. The student is then stuck with a bunch of patterns in their heads, with little mathematical understanding.
So I actually had a basic algebra system before they taught it to me, for what it's worth, probably something I figured out from the structure of the language my parents were using (both accountants). I'm a pretty rare individual, I've never had to work at math.
I took a long time to learn how to learn from example, it used to be impossible but now it's almost just as natural.
