What do we do with all the contradictory statements and approaches in Jewish thought throughout the ages? We're talking massive amounts of cognitive dissonance. Examples hardly need to be provided -- so I won't.
It seems like there are two general approaches, both of which have pros and cons.
1) Pick and choose. Shoot the arrow (that is, have a world view) and paint the target around it (that is, then find the support in Chazal and the later sages to support your view). Of course in some cases this world view may have been shaped by the sources rather than vice versa, but the effect is the same. I think to some extent most of us do this. I certainly do it more than I would wish.
2) The gorgeous mosaic/ tapestry/ cholent approach. This view holds, basically, that all of these contradictory views are part of a greater whole. R. Aryeh Kaplan exemplified this approach. He compared the sum of all the disaparate thoughts and opinions to the rear view of a tapestry, with all the threads and knots hanging out. The real view is on the other side which isn't seen. The contradictions then only are apparent contradictions. Really there is a sort of Theory of Everything that makes sense of it all. R. Adin Steinsaltz speaks of a "conversation throughout the ages". In his telling, Rava and Abaye are talking to Hillel and Shammai who are talking to Rashi and they are all talking to each other. A beautiful idea, I do find it somewhat suspect. Was the Rambam talking to the Ba'al Shem Tov? Its stretching it, to say the least.
But the second approach may be the superior way to go. There is support for it in the Torah, after all. There are three ways that God's relationship to Israel is described. 1) Parent/ child. 2) Master - king/ slave 3) Husband / wife. These are obviously three different kinds of relationships. Yet apparently we can be married to God as well as subjects of God as well as children of God. Which means that they are begging to be harmonized.
Take an issue like Zionism. There was a time when I really felt cognitive dissonance of being in a place where I valued and supported Israel and felt it to be of some religious signifigance yet felt that I had to respect the vehement anti-Zionist position of the Satmar Rav. I felt that I had to reconcile the two views in some way. I reasoned that Israel represents a potential, an opportunity that the Jewish people could use or squander. The maximalist religious Zionist position and the maximalist anti-Zionist position were therefore there to remind us of what Israel could be whichever way it ultimately goes.
Now, I feel fairly comfortable rejecting the Satmar Rav's view that, for example, Israel's military victories have been manifestations of the satanic. Or that Jews must ever remain under the heel of oppressors. I'm not interested in a view like that. But even if my middle view of the two extreme positions did not do justice to either and even distorted both, I don't know if that view could have been reached without the existence of both views forcing me to try and make them co-exist. It may not be an honest reading of either view, but it did give forth a third view.
When it comes to hot potato issues like feminism or the role of women, relations with gentiles,the place of secular learning, stringencies et cetera, it is obvious that one can sift through the vast corpus of literature and pretty much pick and choose an "elu" that you like, ignoring the "v'elu" that you like less. But what if instead of ignoring the things we don't like we make an earnest attempt to reconcile the positions and maybe reach a new, tempered position?