[tsimionescu]

None of these examples contradict what I've said. The historical fact is that continuous functions and vector addition were discovered as interesting mathematical properties based on certain visual intuitions. Only later were did they come to be extended to more general notions applicable to wildly different conditions.

What I claim in addition is that it's still useful for your intuition to understand this history and the leaps that were made by extremely talented mathematicians of the past who came up with these formalizations of intuitive properties.

I'd also claim that they couldn't have ever arrived at the current formal systems if they hadn't started with certain intuitions for simple systems.

[Tainnor]

You're arguing for including historical motivation in order to make a definition seem more immediately useful and intuitive. I agree with that.

What the person you replied to is suggesting, however, is that this can only get you so far. At some you have to sit down and study the formal definition and understand how it relates to the examples you've seen so far. Otherwise it's very easy to have your intuition lead you astray.

[tsimionescu]

I'd that's what they're arguing, then I fully agree. My fear was that they're arguing that the formal definition is the only thing that you should spend time on, that the original motivation is irrelevant - this is what I was trying to argue against.