Does religion appeal to people's intuitions?
Is intuition a reliable means of gaining knowledge?
Science very often discusses evidence outside the bounds of human intuition. Scientific evidence is often mathematically based.
Do people need to break off from their trusted intuitions in order to know important things about existence?
Sorry I missed this thread until now...
Here is my understanding of intuition based on Jung, cognitive science and some other perspectives...
Intuition is an "irrational" cognitive function which allows us to perceive "truth". It is a complementary opposite to sensation which aims at differentiating the directly observable facts of our existence. Intuition, like sensation, is irrational in that it does not construct knowledge based on rational argument but rather perception of what is. Where as sensation focuses on allowing us to recognize objects of the senses and to understand their physical progressions through sensory spaces, intuition operates to find persistent patterns across sensory modalities. Mathematics, or the quantification of our sensory experience, is very much a matter of intuition grasping the patterns of our sensory experience. Also the way we often map sensory experiences to other unrelated topics shows how intuition is always creatively looking to connect different things back together in some more abstract context.
Intuition and sensation as complimentary pairs of opposite functions of cognitive perception help to contextualize and validate each other. Too much reliance on one at the expense of the other creates a mal-adaptive attitude in the individual psyche.
Now, religion often addresses things at a level above and beyond the day-to-day practical level. As such it typically relies on intuitive perceptions about the nature of things. In this way intuition is key in religion as it must rely on some commonly intuitable pattern that can be seen to "lie behind" the more "mundate facts" of practical reality. As such these intuitive perceptions (beliefs) give us some sense of freedom from or transcendence of our more practical, sensory experience. This freedom is vital to our personal psychological "hygiene". Well-crafted fantasy even is made such that enough of the patterns of our real lives can be found in the fantasy that we can be momentarily caught up in an alternate reality and be shown a perspective that gives us hope for a betterment of that reality we find ourselves in. This is the ultimate value and function of religion in a healthy society.