What is intuition?

One can have memory of the future as well as of the past. Memory of the future is usually called instinct in animals, intuition in human beings.

—Robert Graves

Pure analysis puts at our disposal a multitude of procedures whose infallibility it guarantees; it opens to us a thousand different ways on which we can embark in all confidence; we are assured of meeting there no obstacles; but of all these ways, which will lead us most promptly to our goal? Who shall tell us which to choose? We need a faculty which makes us see the end from afar, and intuition is this faculty. It is necessary to the explorer for choosing his route; it is not less so to the one following his trail who wants to know why he chose it…Logic, which alone can give certainty, is the instrument of demonstration; intuition is the instrument of invention.

—Henri Poincaré

Intuition is a mode of gathering.

—John Sallis

Mathematical intuition need not be conceived of as a facility giving an immediate knowledge of the objects concerned. Rather it seems that, as in the case of physical experience, we form our ideas also of those objects on the basis of something else which is immediately given.

—Kurt Gödel

[Intuition] grasps a succession which is not juxtaposition, a growth from within, the uninterrupted prolongation of the past into a present which is already blending into the future. It is the direct vision of the mind by the mind.

—Henri Bergson

Intuition is the ability not to construct solutions to problems in a rational manner, but rather to produce them spontaneously (holistically) according to situational demands…Intuition is, we could perhaps say, a fire that lights itself.

—Wolfram Wilss

My intuition made me work…Intuition makes us look at unrelated facts and then think about them until they can all be brought under one law.

—Albert Einstein

By intuition I do not mean the fluctuating testimony of the senses or the deceptive judgment of the imagination as it botches things together, but the conception of a clear and attentive mind, which is so easy and distinct that there can be no room for doubt about what we are understanding…Intuition is the indubitable conception of a clear and attentive mind which proceeds solely from the light of reason.

—René Descartes

Intuition [in Nobel laureates] is closely associated with a sense of direction; it is more often about finding a path than arriving at an answer or reaching a goal. The ascent of intuition is rooted in extended, varied experience of the object of research: although it may feel as if it comes out of the blue, it does not come out of the blue.

—Ference Marton