Why do they have to be geometries of their own kind? For example, instead of talking about lines in spherical geometry, can’t we just imagine circles on a literal sphere in Euclidean space?
In: 4
Sure, you can, in the same way that you can think of 9x as x+x+x+x+x+x+x+x+x. But you wouldn’t want to, because it either denies, or makes very inconvenient, a lot of tools you’d want to use to attack problems.
Ultimately, everything in math is a huge pyramid scheme built on top of only a few basic logical symbols. The point of math is to abstract away the lower levels of symbol with simpler, higher levels of symbol that reduce the number of steps and the amount of effort you need to see and talk about the behavior of mathematical objects.
This approach turns out to be especially inconvenient when you’re talking about curved geometry that isn’t somehow “sitting in a higher dimensional space”. For example, space-time in our Universe is curved, in the sense that its geometry is non-Euclidean, but it’s not “inside some bigger space” as far as we know. We can mathematically describe it as one if we really want to (it’s called an [embedding](https://en.wikipedia.org/wiki/Embedding#Differential_topology)), but then we’re inventing some higher-dimensional fiction for no reason (and it turns out that, in general, you [need twice as many dimensions for this to work](https://en.wikipedia.org/wiki/Whitney_embedding_theorem) – a sphere can do with 3, but other curved 2d surfaces like the Klein bottle need 4; 4d spacetime requires, in general, eight!).
Latest Answers