A more familiar, if not necessarily more precise, term for the faculty involved in the act of artistic creation John Dewey identifies as "intuition" is "imagination," the latter of which Dewey discusses immediately after introducing the former in Chapter XI of Art as Experience:
In what precedes, I have said nothing about imagination. "Imagination" shares with "beauty" the doubtful honor of being the chief theme in esthetic writings of enthusiastic ignorance. More perhaps than any other phase of the human contribution, it has been treated as a special and self-contained faculty, differing form others in possession of mysterious potencies.
While Dewey himself is not above invoking "mysterious" processes such as "flash of revelation" in describing intuition, he does hesitate to attribute magical properties to imagination.
It is the large and generous blending of interests at the point where the mind comes in contact with the world. When old and familiar things are made new in experience, there is imagination. When the new is created, the far and strange become the most natural inevitable things in the world. There is always some measure of adventure in the meeting of mind and universe, and this adventure is, in its measure, imagination.
The use of passive voice here--"old and familiar things are made new in experience," "the new is created"--is not simply clumsy writing (although Dewey's prose does sometimes have a clumsily hurried quality, as if he is choosing the words that most immediately come to mind), but expresses Dewey's restraint in considering the nature of imagination. He resists the idea that it is a "power" that acts on experience but instead sees it as a function of experience: "[A]n imaginative experience is what happens when varied materials of sense quality, emotion, and meaning come together in a union that marks a new birth in the world."
Unless regarded as this kind of "union," imagination becomes merely the "imaginary," which "gives familiar experience a strange guise by clothing it in unusual garb, as of a supernatural apparition." With the imaginary, "mind and material do not squarely meet and interpenetrate." The artist "toys with material rather than boldly grasping it." A truly imaginative artist does not distort or supersede experience for the sake of fancy. (Dewey cites Coleridge's distinction between imagination and fancy.) However much the "real" may be transformed by imagination (Dewey is not making a case for realism), it is not reduced to mere fantasy. Imagination makes the intangible tangible because "possibilities are embodied in works of art that are not elsewhere actualized." Art makes the real visible.
Dewey more specifically identifies the difference between the imaginative and the imaginary by making a further distinction between what he calls "inner" and "outer" vision.
There is a stage in which the inner vision seems much richer and finer than any outer manifestation. It has a vast and enticing aura of implications that are lacking in the object of external vision. It seems to grasp much more than the latter conveys. Then there comes a reaction; the matter of the inner vision seems wraith-like compared with the solidity and energy of the presented scene. The object is felt to say something succinctly and forcibly that the inner vision reports vaguely, in diffuse feeling rather than organically. The artist is driven to submit himself in humility to the discipline of the objective vision. But the inner vision is not cast out. It remains as the organ by which outer vision is controlled, and it takes on structure as the latter is absorbed within it.
The artist who insists on his "inner vision," who remains satisfied with that inner vision, is likely to only indulge in the imaginary. The artist who is willing to "submit himself in humility to the discipline of the objective vision" (who must accept the demands of the outer vision or there is no art) will, potentially at least, discover the fuller possibilities of the imagination. The imagination isn't confined to the reveries of the fantasist. It requires the "solidity and energy" of the "objective vision," of the art object itself, the making of which is the ultimate exercise of imagination.
The artist who devotes his/her attention to the "objective vision" finds "the object is felt to say something." Dewey is probably using the construction "say something" very loosely, to indicate that the work as shaped turns out to express the sharpest and most far-reaching vision, but it might mislead us into thinking that the vagueness of the inner vision becomes the more clearly enunciated "theme" through outer vision. Something closer to the opposite is true. The disciplined artist allows the work itself to find what it will say; its meaning will develop "organically," not as the figural rendering of the artist's "intention." The artist feels that the object has spoken. If inner vision "takes on structure as the [outer vision] is absorbed within it," the artist ends up "saying" what the work has said.