>So by the intuitive explanation we can make this single line any dimension that we want.And if this makes sense in the given context - sure, why not?
You seem to be missing my point. What I'm saying is that it's useless to have totally generalized abstractions (outside of pure math) since more often than not, they are so far removed from reality that most people can no longer make any connection to use cases.
My entire argument is that there's a heavy communication failure between mathematicians and scientists of every other field where math is used as a tool. Sure, it's convenient for a mathematicians to be able to use shorthand gibberish to talk to other mathematicians. It doesn't justify pushing this jargon on other fields.
Besides which, convenience is no excuse for making something hard to understand. Sure it's convenient to name all your variables in a program a, b, c etc but you're going to get lynched by any programmer that tries to read your code later, including yourself.
When it comes to a point where gibberish becomes the only way to explain mathematical abstractions, then you should step back and ask yourself "where the hell did this go wrong?".
>This isn't just a problem within mathematics. For a simple programming example:[...]
I believe in giving simple explanations and expanding them whenever there's a loophole that needs to be fixed. You don't make programs that cover every single niche use case, either (if you don't have to, that is). That's the problem with math - the generalizations, while useful in math itself, are sheer overkill in many situations outside of math.
Lastly, sorry taking this out of order, but:
>An isolated concept is worthless.
So is a generalized abstraction without any context. I'm saying that the sweet spot is somewhere in between for most people to understand and apply concepts, and that it's better to generalize upwards from reality and actual use cases instead of starting utterly removed from reality and trying to apply the generalizations downwards.
And if this makes sense in the given context - sure, why not?
You seem to be missing my point. What I'm saying is that it's useless to have totally generalized abstractions (outside of pure math) since more often than not, they are so far removed from reality that most people can no longer make any connection to use cases.
My entire argument is that there's a heavy communication failure between mathematicians and scientists of every other field where math is used as a tool. Sure, it's convenient for a mathematicians to be able to use shorthand gibberish to talk to other mathematicians. It doesn't justify pushing this jargon on other fields.
Besides which, convenience is no excuse for making something hard to understand. Sure it's convenient to name all your variables in a program a, b, c etc but you're going to get lynched by any programmer that tries to read your code later, including yourself.
When it comes to a point where gibberish becomes the only way to explain mathematical abstractions, then you should step back and ask yourself "where the hell did this go wrong?".
>This isn't just a problem within mathematics. For a simple programming example:[...]
I believe in giving simple explanations and expanding them whenever there's a loophole that needs to be fixed. You don't make programs that cover every single niche use case, either (if you don't have to, that is). That's the problem with math - the generalizations, while useful in math itself, are sheer overkill in many situations outside of math.
Lastly, sorry taking this out of order, but:
>An isolated concept is worthless.
So is a generalized abstraction without any context. I'm saying that the sweet spot is somewhere in between for most people to understand and apply concepts, and that it's better to generalize upwards from reality and actual use cases instead of starting utterly removed from reality and trying to apply the generalizations downwards.