I feel like almost all the descriptions of advanced (university level) math concepts are confusing when you first hear them. For example, the definition of a Taylor series according to Wikipedia:
"In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point."
This kind of description really throws me off.
Especially "terms that are expressed in terms of the function's derivatives at a single point." first of all, superficially, the different meanings of the words 'term' and 'terms' throws me off and then even when I get over that, it's not clear what is meant 'in terms of' in what terms? What kind of relationship are we talking about here? I need to keep reading a lot more to fill that gap... In the meantime, I'm in a state of confusion and will need to re-read that sentence later to make sense of it once I have more info.
I'm not sure how to solve that problem to explain it more clearly but it feels like definitions should gradually build up without gaps. Is it even solvable? I would prefer not having read that definition at all TBH as it only serves to confuse me, it has way too many possible interpretations. I prefer to jump straight to the formula.
I think there is a cultural component that has transferred throughout history whereby people need to signal their level of education to others.
You've probably had it happen yourself without realizing.
You hear someone explaining something in a simplistic way, and notice yourself wondering how deeply they understand the topic. Then when you are explaining the topic, you don't want people to question your own knowledge level like you did to the other person, so you use techniques to signal the depth of your knowledge.
This might be fancy words, or skipping over simplistic things.
And I think this just becomes second nature.
You can see it with programming languages. If I told you to rate a Rust dev vs JS dev, you are thinking Rust is harder to learn so they must be smarter.
It can also just be a challenge to imagine how you thought about a concept when you were initially learning it.
English is pretty terrible for explaining a lot of math too. Math is better understood visually, but back in the day you couldn't exactly share an interactive diagram.
That's some really bad wording indeed. But it may vary from one source to another.
The more intriguing point of your comment is about assumptions. Can you provide a relevant example?
It's hard for me to provide a specific example because (for me) it applies to many different fields of advanced math. It seems like people who are good at math have some preconceived idea in their heads about what the purpose of a math concept is going to be and that helps them to make sense of new concepts faster. Maybe that's what people refer to as a 'mathematical intuition'?
To me (who is not naturally gifted at math), it often seems like math has no specific direction; it appears to explore almost every direction arbitrarily. I can't usually tell which part is supposed to be interesting or potentially meaningful so I don't know what to focus on or what to look for when I'm learning it.
It's like if someone gives you a confusing and vague instruction or question, it helps if you know what the reason is. Like if someone asks you "what day is it?", it helps if you know the intention behind the question or else you can't know for sure if you should answer with "27th of September" or "Wednesday". You don't fully understand the question without knowing the intention behind it. To me, learning math presents a much more extreme variant of that effect.
"In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point."
This kind of description really throws me off. Especially "terms that are expressed in terms of the function's derivatives at a single point." first of all, superficially, the different meanings of the words 'term' and 'terms' throws me off and then even when I get over that, it's not clear what is meant 'in terms of' in what terms? What kind of relationship are we talking about here? I need to keep reading a lot more to fill that gap... In the meantime, I'm in a state of confusion and will need to re-read that sentence later to make sense of it once I have more info.
I'm not sure how to solve that problem to explain it more clearly but it feels like definitions should gradually build up without gaps. Is it even solvable? I would prefer not having read that definition at all TBH as it only serves to confuse me, it has way too many possible interpretations. I prefer to jump straight to the formula.