The role of intuition.
"Intuition," as used by the modern mathematician, means an accumulation of attitudes (including beliefs and opinions) derived from experience, both individual and cultural. It is closely associated with mathematical knowledge, which forms the basis of intuition. This knowledge contributes to the growth of intuition and is in turn increased by new conceptual materials suggested by intuition. The major role of intuition is to provide a conceptual foundation that suggests the directions which new research should take. The opinion of the individual mathematician regarding existence of mathematical concepts (number, geometric notions, and the like) are provided by this intuition; these opinions are frequently so firmly held as to merit the appellation "Platonic." The role of intuition in research is to provide the "educated guess," which may prove to be true or false; but in either case, progress cannot be made without it and even a false guess may lead to progress. Thus intuition also plays a major role in the evolution of mathematical concepts. The advance of mathematical knowledge periodically reveals flaws in cultural intuition; these result in "crises," the solution of which result in a more mature intuition. The ultimate basis of modern mathematics is thus mathematical intuition. and it is in this sense that the Intuitionistic doctrine of Brouwer and his followers is correct. Modern instructional methods recognize this role of intuition by replacing the "do this, do that" mode of teaching by a "what should be done next?" attitude which appeals to the intuitive background already developed. It is in this way that understanding and appreciation of new mathematical knowledge may be properly instilled in the student.
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