I am having trouble with finding the best intuition for homology as sometimes I'm able to find for other subjects which is how someone could have come up with it.
Reading the history of homology didn't satisfy me, I was wondering if someone could help me with this. (The other answers on math exchange aren't really what I'm looking for, they somehow already assume we know what the homology groups are).
I would love an answer that is to homology as those short ones (mostly because I can't think of a long one) illustrate the kind of motivation I like best:
Homotopy: It is clearly interesting to see the maps from $S^1$ to our space look like. After some thinking, we realize we can impose a group structure if we fix the base point, but do we really want to? Umm we kind of get stuck otherwise, so let's fix a basepoint to make things simpler, in fact we also see (details left) after some thinking that it doesn't matter which one if the space is connected!
Ok let's take a simple example like the square, umm well this is already very complicated, but we don't want this, since the square deformation retracts to a point so we want stuff like that to be simple. Okay so we actually want maps from $S^1$ up to homotopy, and our group is still works, and there we have given a motivation to the definition of homotopy group.
2.Discrete finite fourier basis: We hate the regular basis, we want orthonormal stuff!!! We know that if we multiply polynomials the convolution theorem exists (which is how we get fast multiplication), maybe this is true in general? Yes!
Hopefully this made the kind of motivation I'm looking for clear. I'm unable to find one for homology :(.
I understand that we first want higher homotopy groups but they turn out to be hard, so the next thing is it's obviously interesting how maps from $[0,1]^n$ to our space look like, and with a bit of a stretch I can believe we want to linear algebra\group everything, so we consider finite sums of those maps, and from technical reasons we may later find out it's easier to work with maps from the simplex.
However, this still leaves open how we can come up with the boundary maps, or why we would consider the homology groups (I understand that in a weird sense they measure holes- including that the first homology is the abelinzation of the fundemental group), but I would like more convincing how someone could think of this is as the "obvious" next step (Even though historically I'm aware this wasn't the case, it's easier for me to find an explanation of course with retrospect which makes this the obvious next step).
edit: I of course don't require that the answer be a continuation of what I started about homology, a lot of times good examples (if you can show how someone could come from Euler's formula to homology) generalize well to give a good motivation, but I haven't exactly understood how this is the case here.