One of the major divides among statisticians is in the philosophy of probability. On one hand there are frequentists who like to place probabilities only on data; there is some underlying process that the data describes, but one doesn't have knowledge of all data, so there only exists uncertainty about what unknown data the process could generate. On the other hand there are bayesians who, like the frequentists, place probabilities on data for the same reason, but who also place probabilities on the models they use to describe the underlying process and on the parameters of these models. They allow themselves to do this because they concede that, although there does exist one true model that could perfectly describe the underlying process, as the modeler they have their own uncertainty about what that model may be.
I think this is an interesting divide because it highlights two different sources of uncertainty. Some uncertainty exists because there exists true randomness in the world (or so physicists believe), while other uncertainty exists solely due to our own ignorance -- the quantity in question may be completely determined! We are just forced to work with incomplete information. I think it is fascinating that there exists a unified framework that allows us to manage both sources of uncertainty, but it does make me wonder: do these two sources of uncertainty warrant their own frameworks?