[lemonwaterlime]

You use it like a conceptual dictionary. Say you’re reading a paper or trying to implement some technology that uses a mathematical concept you aren’t familiar with (e.g. a submanifold). You’d look up “submanifold” and see that it is “ subset of a manifold that is itself a manifold, but has smaller dimension.” Okay, that seems to fit the intuition of a “sub”-something. But I don’t know what a “manifold” is. So I’d look that up.

“A manifold is a topological space that is locally Euclidean (i.e., around every point, there is a neighborhood that is topologically the same as the open unit ball in R^n)”

At this point, either you know what all of those words mean or you don’t. If you do, great! You’re done. If not, you either keep digging deeper into the various terms or you start seriously considering reading one or more of the curated reference books listed at the end of each entry.

Over time you develop the “mathematical maturity” that you don’t need to do a deep dive into the books and can mostly just use the reference.

[Tainnor]

> At this point, either you know what all of those words mean or you don’t. If you do, great! You’re done.

I'm not sure. I only have a rather rudimentary understanding of topology, so I do understand the definition of a manifold on a technical level, but I don't know any interesting examples or theorems about them so it wouldn't be immediately clear to me why something being a submanifold is worth mentioning.

Similarly, I don't think that just reading the definition really gives you a good understanding of groups. You probably want to work through some examples of groups, and arguably, the importance of groups doesn't really become clear until you've encountered group actions.

[fpgamlirfanboy]

You skipped over the second sentence of what you're responding to:

> Say you’re reading a paper or trying to implement some technology that uses a mathematical concept you aren’t familiar with

In such a case you're not interested in either manifold or sub-manifold or group in and of itself. So a lack of familiarity with theorems isn't an impediment.

[Tainnor]

I didn't "skip over" anything. If it's irrelevant to you that something is a submanifold or group, you also don't need to look up the definition.

Reading mathematical definitions on their own just doesn't give you a whole lot of context about the objects they're describing.