Is it possible to have different ontologies that account for all phenomena?
Yes, if by "all" we mean practically (what humans can now do), instead of theoretically (see, for example, John Barrow's New Theories of Everything). See the following, from Massimo Pigliucci's Essays on emergence, part I:
Batterman takes on an alleged (as it turns out) case of reduction of a phenomenological to a more “fundamental” theory: the relationship between classical thermodynamics (phenomenological) and statistical mechanics (fundamental). The fact is, “the quantities and properties of state in orthodox thermodynamic equations appear largely to be independent of any specific claims about the ultimate constitution of the systems described,” which would seem to cast some doubts on the simple version of the reduction story. As Batterman puts it, “Reduction in this context typically is taken to mean that the laws of thermodynamics (the reduced theory) are derivable from and hence explained by the laws of statistical mechanics (the reducing theory) ... [but] there are very good reasons to deny that all thermodynamic (and hydrodynamic) phenomena are reducible to “fundamental” theory,” and these reasons have to do with phase transitions (solid and liquid, liquid and gas, etc.).
Batterman continues: “The renormalization group explanation provides principled physical reasons (reasons grounded in the physics and mathematics of systems in the thermodynamic limit) for ignoring details about the microstructure of the constituents of the fluids. It is, in effect, an argument for why those details are irrelevant for the behavior of interest.” [Italics in the original]
Pigliucci is clearly concerned with talking about emergence, but two bits come out of the discussion above which bear on the cardinality of the phenomenology–ontology relationship:
- "the quantities and properties of state in orthodox thermodynamic equations appear largely to be independent of any specific claims about the ultimate constitution of the systems described"
- "ignoring details about the microstructure of the constituents of the fluids"
Now, there is some wiggle-room, here. A scientific theory can be fantastically successful even if it is not precise. So, perhaps the microstructure—the ontology—is only relevant when one goes down to a high enough precision. To use another example, in physics, there are open questions at energy levels well past what can currently be tested in the laboratory and perhaps even what is generally observed from cosmic rays (see the Greisen–Zatsepin–Kuzmin limit). So, it seems that multiple different ontologies are viable options.
Switching to philosophy of mind, you may find Putnam and Fodor's multiple realizability of interest:
Multiple realizability, in the philosophy of mind, is the thesis that the same mental property, state, or event can be implemented by different physical properties, states or events.
P.S. My answer to What is the difference between Fact and Truth? may also be relevant.
What is the value of interpretational parts in theories if they have no effect on predictions?
From reading a decent amount of the history of the philosophy of science, I suspect these matter when forming new hypotheses. Karl Popper, in The Logic of Scientific Discovery, famously stated:
I said above that the work of the scientist consist is in putting forward and testing theories.
The initial stage, the act of conceiving or inventing a theory, seems to me neither to call for logical analysis nor to be susceptible of it. The question how it happens that a new idea occurs to a man—whether it is a musical theme, a dramatic conflict, or a scientific theory—may be of great interest to empirical psychology; but it is irrelevant to the logical analysis of scientific knowledge. The latter is concerned not with questions of fact (Kant's quid facti?), but only with questions of justification or validity (Kant's quid juris?). (7)
I do not know of any good, systematic treatment of how hypotheses are formed. I do know that this is a deep desire of machine learning folks; much ML is currently pattern matching; making the transition to hypothesis formation would likely mark a major leap forward. The following, from What emotions really are, may be of interest:
Children do not create concepts simply by grouping particulars on the basis of overall similarity. Instead, they create causal explanatory theories of particular domains and cluster instances according to their possession of theoretically significant properties in the these schemes of explanation (Keil 1989). (6)
I do not recall whether there was any particular connection to emotions here; what I do know is that Griffiths is concerned to group emotions into natural kinds. That being said, studying early development of humans may shed light on the use of non-mathematical parts of hypotheses.
How do different philosophical traditions address this issue?
I don't have the time to get into detail, but Michael Friedman's Dynamics of Reason may help point the way, especially his use of "constitutive principle" and "correlative principle". A bit on those terms:
Michael Friedman in his Dynamics of Reason is intrigued by the physical theory of a time being composed of a mathematical language, coordinating principles, and empirical laws and regularities, and how their status changes through a revolution.