What is the evidence for just 4 dimensions?
There's no good evidence for "just 4", but there's very good evidence that there are at least 4. There is, however, no good evidence that there are more than four, either.
What does evidence for a dimension even look like?
Like observing that objects can have length, width and depth (there's 3 orthogonal dimensions, right there), and that objects change over time (that's one more dimension).
I don't see any observable dimension when I look at my environment. I CAN observe objects having "dimensions" of measurement like length and width and depth, but that's not an omnipresent continuum that I can see like I see those objects. It's more of an abstract generalization of possibilities of measurement projected onto our experience of spacetime.
That sounds to me like a fancy way of saying that our scientific generalisations (theories etc.) are based on empirical observations.
All scientific theories of any value are "abstract generalisations", or rather models that allow us to predict how objects behave and how their possible behaviours are constrained.
There is no empirical evidence for dimensions that we can just look at and say "There's a dimension."
Yes there is. For instance, we find that we cannot uniquely specify the location of a given point in the space we observe (say the location of an object, if you need a concrete example) using fewer than three numbers and some unique fixed reference point,
using any possible method. That's empirical evidence for 3 spatial dimensions, right there.
And as for time being the 4th dimension, we don't observe that empirically either. We do experience the passage of time, and we also measure it, but noone I know of observes time as some actual physical dimension that extends thru space. We're talking conceptual constructs here that we impose over our world to make it seem more rational.
Have you ever noticed how stuff doesn't all happen at once? Go figure.
