I am attaching a good text on research methodology that was used as a guide in the Computer Science PhD I am attending. If I can remember you will see there positivism as a philosophical viewpoint positioned against other eventually more open views like the one of Karl Popper.  

In research about faith and religion (Theology, etc) you will probably find non-positivistic research inclination. Faith by definition precludes proof (there is no faith once you have seen the proof) therefore this might be a fertile land for you. In this context I am very interested in whether there is some structure in the possible interactions of God with the world (after John Polkinghorne's work).

If I have the time I might add something later to this answer, from my own (though modest) experience as a researcher, since it might be a bit different from most Phd Students. I have a part-time job and I am not attached to any research center or any scolarship. I have two advisors (1 formal and 1 informal), but in part I work as an amateur, with lots of independence.

My major worry is never whether I can or not prove something or ever will. I am just trying to understand what I find and solve problems as I progress. My most important steps have never been aware of whether they were provable. Even if not, I often find them important, for example as a connection (a way) to something else (eventually provable in turn)

I am not following a very structured aproach, with strict and frequent deliverables obligations, subordinate to performance indicators. I might have come across some interesting (to be verified) results, but if I did, it was only because I was given freedom. Subordinating research to timetables, funding or verifiability requirements is surely necessary and important. But it is just one way. It is not the only one.

Another sign of non-positivist inclinations in research methodology (which could be nice to consider including in research evaluation) is the tendency to include "beauty" as a criteria (or auxiliary criteria) to whether a scientific theory is true or not. It is known that in the mathematical stage of a theory (before experiments) scientists tend to resort to these criteria (beauty, parcimony, etc..) to prefer some models to others.

About this check the following wonderful conference of Michael Atiyah on the beauty of mathematics:

http://www.heidelberg-laureate-forum.org/blog/video/lecture-tuesday-september-24-sir-michael-atiyah/