I agree with Mike that even the example of $\mathbb{R}^3$ shows that talking about infinite mathematical objects can still make finite physical sense. For example, in $\mathbb{R}^3$ we can show that there are only so many regular polyhedra, and then we invent the microscope and find out that these polyhedra occur all over the natural world - in crystals, in the shape of viruses, and so forth - and no other ones do. That's a pretty good indication that $\mathbb{R}^3$ is a good model of at least some parts of the physical world. The way I think this result should be interpreted is that, even if you believe that there are only a finite number of possible locations in the physical universe (or whatever), the "mesh size" of the universe is small enough that we can take a limit and work in the infinitary setting, which boils down to discarding some negligible error terms.