In philosophy intuitions are usually regarded as the most basic unit of our thinking, like the atomic units of the mind. When arguments reach the point of basic intuitions, it is thought that the two parties will have to agree to disagree. This is because the premises of arguments are thought to be constructed from intuitions, so if you are attempting to argue about intuitions there are no more fundamental intuitions to appeal to you have reached a dead end. This view of intuitions is incoherent and inconsistent with the way we actually think and argue.
Our thoughts are in constant competition for survival in the ecosystem of our minds. This framing is more accurate than the atomic model where intuitions are the basic units from which other thoughts and arguments are constructed. It should be noted first that size in an ecosystem is irrelevant to survival, so unlike the atomic model intuitions are not analogous to the smallest members of their physical world counterpart. Intuitions are not fundamentally different from any other knowledge, they are simply the aspects of our mental ecosystem which have developed the most robust methods of survival. In this respect intuitions are most analogous to humans in the evolutionary environment.
Why are intuitions usually misunderstood? In part it is because the atomic model of intuitions contains some truth. The truth it contains is that while under the sway of some intuition it is difficult to imagine that the intuition could be displaced, just as it is difficult to imagine how humans could be displaced as the dominant species in the Earth’s ecosystem. Because the intuition contains some of our most useful and true knowledge, we are right to refuse to give it up without a good alternative.
The atomic model also attempts to offer an explanation for why some subjects, such as taste, are subjective. If one person likes chocolate ice cream and another likes vanilla, we usually do not think that appealing to argument will bring them into agreement. This is often true in our current situation, but there is no fundamental roadblock that will prevent the advance of argument (perhaps with the development of brain-machine interfaces) to the point where these disagreements will be resolvable. We might even say that today a chef’s dish or an artist’s piece are forms of argument well suited to resolving these disagreements. For instance, I have long disliked the color purple but various paintings have helped soften this position.
Intuitions are also useful in argument for at least two reasons: they are, to varying degrees, widely held across the population and a conclusion argued to from intuitions will be less likely to be rejected. A common and effective approach to argument involves figuring out what the other person’s intuitions are (either by talking with them or assuming they have the same or similar intuitions to others in their culture) and using them to derive conclusions which we would like them to see.
Another possibility is that we introduce some new idea that disagrees with some (or all) of another’s intuitions and although they do not agree with it, they understand and remember it. Over time, they apply the theory as a sort of game, not taking it seriously because they see no way in which it could be true. As time goes on, however, the idea may find new applications and turn out to work as well or better as one or more intuitions. The original intuitions may end up being discarded, either because it is noticed that they are useless or because they wither and die from disuse. Later the new idea might itself be considered an intuition.