This whole post is a good illustration to how math is much more creative and flexible than we are lead to believe in school.
The whole concept of “manifolds” is basically that you can take something like a globe, and make atlases out of it. You could look at each map of your town and say that it’s wrong since it shouldn’t be flat. Maps are really useful, though, so why not use math on maps, even if they are “wrong”? Traveling 3 km east and 4 km north will put you 5 km from where you started, even if those aren’t straight lines in a 3d sense.
One way to think about a line being “straight” is if it never has a “turn”. If you are walking in a field, and you don’t ever turn, you’d say you walked in a straight line. A ship following this path would never turn, and if you traced it’s path on an atlas, you would be drawing a straight line on map after map.
I think those last 2 paragraphs are due to people approximating math that would otherwise be quite complex to calculate, or making models that are approximations due to widespread available technology. Just because I don’t turn if I cross over Mt Everest, does not mean that is the fastest route by foot.
I’m not saying to not use these approximations.
I really recommend the book “Where Mathematics Comes From,” to really think deeply about what math is to us as an animal. Even other animals can do some rudimentary math, and arguably athletes are doing math innately as they perform their sports. Birds and dolphins do physics and calculus. Sort of. In this view, what we teach as math to each other as humans is essentially a language describing these phenomenon and how they work together. Calling this approximation a “straight line” in this language sense isn’t very accurate and it’s what’s causing the debate.
Thank you for the informative answer
This whole post is a good illustration to how math is much more creative and flexible than we are lead to believe in school.
The whole concept of “manifolds” is basically that you can take something like a globe, and make atlases out of it. You could look at each map of your town and say that it’s wrong since it shouldn’t be flat. Maps are really useful, though, so why not use math on maps, even if they are “wrong”? Traveling 3 km east and 4 km north will put you 5 km from where you started, even if those aren’t straight lines in a 3d sense.
One way to think about a line being “straight” is if it never has a “turn”. If you are walking in a field, and you don’t ever turn, you’d say you walked in a straight line. A ship following this path would never turn, and if you traced it’s path on an atlas, you would be drawing a straight line on map after map.
I think those last 2 paragraphs are due to people approximating math that would otherwise be quite complex to calculate, or making models that are approximations due to widespread available technology. Just because I don’t turn if I cross over Mt Everest, does not mean that is the fastest route by foot.
I’m not saying to not use these approximations.
I really recommend the book “Where Mathematics Comes From,” to really think deeply about what math is to us as an animal. Even other animals can do some rudimentary math, and arguably athletes are doing math innately as they perform their sports. Birds and dolphins do physics and calculus. Sort of. In this view, what we teach as math to each other as humans is essentially a language describing these phenomenon and how they work together. Calling this approximation a “straight line” in this language sense isn’t very accurate and it’s what’s causing the debate.