How do we know what we know about intuition? Is it a matter of making guesses based on experience? 

Intuition can be related to empirical, or embodied knowledge. It is also identified with an internal or creative aspect of thought. In the latter sense, intuitionism can be described as a "free doing." 

At the very start of the Pensées, Pascal speaks of two fundamentally different types of mind, the intuitive and the mathematical. 

The archetypal (divine) intellect does not take up into itself something lying outside, but produces the object of its own knowledge. Plotinus gave this way of thinking the name "intuitive understanding," which sees the total form of the actual, because it actively produces it each moment and because it is immanent in the formative law which underlies all existence. 

In articulating the aesthetic of the transcendental ideal, Plotinus used Plato's framework to enhance the importance of art rather than to reduce it. Now art could reflect the ideal more accurately than imperfect nature itself. The arts "go back to the Ideas from which Nature itself derives,...and add where nature is lacking." (Enneads) Post-Platonic theorists tended to assign a double residence to the Ideas: both within a transcendental space and in the mind. (see Abrams, Mirror and Lamp, p. 43) The latter possibility, the intuitive ideal, allowed art to conform to a vision both personal and subjective. (see Panofsky, Idea) 

Kant describes intuition as representations of the imagination. "Intuitions are always required to establish the reality of our concepts." (p.196) Apprehension is always bound up with imaginative synthesis, which enables us to combine different representations of a manifold to be apprehended in intuition. In the Critique of Pure Reason, Kant described the a priori schemata, or "pure images" of of understanding: the pure intuitions of space and time. These direct presentations of the categories of the understanding make the mediation of of sense perceptions (images) and concepts possible. 

For Kant, the imagination creates another nature, a representation of nature. The representations of the (productive) imagination are aesthetical EF= "idea.html#1"> ideas . They ressemble the ideas of reason (which cannot be sensibly presented), but they cannot be completely compassed by and made intelligible by language, because no definite thought, or concept is adequate to them. ( see critique of judgement, section 49) 

Many apples fall from trees, but only only rarely is there a Newton to apprehend the laws of the world in such an event. Husserl called this kind of intuition a "Wesensschau." Is the intuition of rationality in nature an essential part of scientific theory? According to Ernst Cassirer, "All concepts, insofar as they claim to give us some form of knowledge of reality, must in the final analysis "fulfill" themselves in intuition." (Logic of the Cultural Sciences, p.56) 

In a Freudian schema, the preconscious is a state of mind which has escaped the primary repression that would consign an idea to the unconscious, but which is still subject to a second censorship and distortion. The intuition is waiting in the "entrance hall" for conceptualization and consciousness

For Bergson, intuitive knowledge is superior to analytic knowledge. In Matter and Memory, Bergson restores perception to an direct "ideo-motor" link to the real. For him, "the reality of things is no more constructed or reconstructed, but touched, penetrated, lived, and the problem at issue between realism and idealism ... is solved, or rather dissolved, by intuition." (p69) (see consciousness) The knowledge we have by intuition is analagous to that we gain by walking around a city and living in it. (cf De Certeau)

"We should have in our minds some conception of a wider field of abstraction, a more concrete analysis, which shall stand nearer to the complete concreteness of our intuitive experience." Alfred North Whitehead. Intuitive thinking has a passionate concreteness of mind. In D+G's words intuition follows the flows of matter. see machinic phylum for "intuition in action."

what did it mean for computer interfaces to become more intuitive? It certainly meant needing to know less how the machine operates, being able to ignore those workings in favor of a sense of direct action in terms of the task at hand. One might say that this apparent concreteness is an illusion, that graphical icons are examples of Whitehead's "misplaced concreteness." 

In Notes on the Synthesis of Form, Christopher Alexander calls the designer's greatest gift "his intuitive ability to organize physical form."In calling for rationnality as "a loss of innocence" , he argues that "Enormous resistance to the idea of systematic processes of design is coming from people who recognize correctly the importance of intuition, but then make a fetish of it which excludes the possibility of asking reasonable questions." (p.9)