Don’t you dare back down on the tyranny of consistency, the twin of the tyranny of hypocrisy!
This is something that’s bugged me for a long time, one of the main causes behind e.g. “Something so ridiculous only an academic could believe it.”
One thing that doesn’t seem to get mentioned often is the difference between a contradiction and a paradox. A paradox is only a seeming contradiction, based on incomplete knowledge. Take Zeno’s paradox, for example, (where the hare can never catch up to the tortoise, because the tortoise will always have moved forward by the time the hare catches up to where he was; related to not being able to cross a room because you are always halving the distance, etc.). The paradox was thought up to prove Xeno’s idea that motion and change are mere illusions. We have a contradiction, therefore the premise must be false.
The point, of course, is that apparent contradictions or inconsistencies may only prove a lack of knowledge or understanding (in Zeno’s case, of differential calculus).
In philosophy, the idea is to sit in a dark room and understand the universe using reason alone. In science, the idea is to ask “OK, what actually happens” i.e. do an experiment and see what reality says about your ideas. I’ll take reality over intellectual consistency any day (although, of course, most scientists are perhaps a bit distant from reality).