I am quite used to "counting"/computing finite dimensions. For example, one would expect a hypersurface in $\mathbb{C}^3$ to have dimension $3 - 1 = 2$. But it is often the case that the space one is interested in computing the dimension of is the quotient of an infinite-dimensional space, often a space of functions, by the action of an infinite-dimensional Lie group. For example, many moduli spaces are of this type.
I am interested in the heuristic way of computing sizes of spaces of functions, which goes roughly as follows. Let's take, say, the space of connections on Minkowski spacetime, modulo gauge equivalence. The space of connections on (4-dimensional) Minkowski spacetime can be thought of as $4$ functions of $4$ variables (it has the same "size" as the space of $1$-forms $A_0 dx_0 + A_1 dx_1 + A_2 dx_2 + A_3 dx_3$, which is parametrized by the $4$ functions $A_0$, $A_1$, $A_2$ and $A_3$). I am assuming the gauge group is $U(1)$. But then the group of gauge transformations has the size of a single function from $\mathbb{R}^4$ to $U(1)$, thus, since $U(1)$ is $1$-dimensional, it has the same size as a single (real-valued) function of $4$ variables.
So one would heuristically expect the space of connections on Minkowski modulo the group of gauge transformations to have the size of $3$ (real-valued) functions of $4$ variables (unless I am making some silly mistake). And so on.
Well, I am interested in getting some heuristic way of counting dimensions of moduli spaces of Yang-Mills instantons, or Yang-Mills-Higgs instantons and so on. It would also be interesting if one could sort of "understand" this way what happens when one does a symmetry reduction.
Another example I am interested in, from this point of view, is first order and linear second order PDEs. I have heard some practitioners of Cartan's methods use some similar heuristic way of counting (especially when talking about jet spaces).
I think there is nothing wrong in developing heuristics, because it leads to some clever guessing of a true statement, which could then be proved by more rigorous methods.
I am interested in examples of such a heuristic way of counting dimensions. References, or some detailed examples as answers are more than welcome. Thank you!
