> Like with programming, if you learn one language, you can usually learn another in a jiffy.
I actually agree with you in general but, this isn't a guarantee.
I've known plenty of people who "only" know how to "program" in Visual Studio. They don't understand how to use git outside of a helper GUI, they didn't understand the distinction between a compiler and a linker, or build options, or pathing issues--at least not down to the bare wires of what's actually going on.
I'm not saying "everyone" is like that, but when you give exhaustive encapsulating tools, oftentimes people will only learn what they have to, to get running and never delve deeper. Of course, never delving deeper is the key issue but some frameworks/systems make delving deeper much easier than others, and better yet, encourage it.
For an example, it was way easier for me to learn deep aspects of "what is really going on" when I started learning Linux--even immediately when I had to decide how to partition my hard drive correctly just to install it. I "could" just hit auto format but... what are all these buttons? Why is there different types of partitions? Is there a benefit? Not hiding me the inherent complexity allowed me information to better my understanding.
Perhaps this applies more to programming than math, as, the core is to do math... right? The rest is just implementation details of "how do I tell this [math thing] to do [thing I want]". So as long as you know what you're asking "I need a line integral" that's a general enough term. You're learning the _techniques_ not only the framework itself.
I actually agree with you in general but, this isn't a guarantee.
I've known plenty of people who "only" know how to "program" in Visual Studio. They don't understand how to use git outside of a helper GUI, they didn't understand the distinction between a compiler and a linker, or build options, or pathing issues--at least not down to the bare wires of what's actually going on.
I'm not saying "everyone" is like that, but when you give exhaustive encapsulating tools, oftentimes people will only learn what they have to, to get running and never delve deeper. Of course, never delving deeper is the key issue but some frameworks/systems make delving deeper much easier than others, and better yet, encourage it.
For an example, it was way easier for me to learn deep aspects of "what is really going on" when I started learning Linux--even immediately when I had to decide how to partition my hard drive correctly just to install it. I "could" just hit auto format but... what are all these buttons? Why is there different types of partitions? Is there a benefit? Not hiding me the inherent complexity allowed me information to better my understanding.
Perhaps this applies more to programming than math, as, the core is to do math... right? The rest is just implementation details of "how do I tell this [math thing] to do [thing I want]". So as long as you know what you're asking "I need a line integral" that's a general enough term. You're learning the _techniques_ not only the framework itself.