Wilder R L
Science. 1967 May 5;156(3775):605-10. doi: 10.1126/science.156.3775.605.
"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.
现代数学家所使用的“直觉”，是指源于个人和文化经验的一系列态度（包括信念和观点）的积累。它与数学知识紧密相连，而数学知识构成了直觉的基础。这种知识有助于直觉的发展，反过来又因直觉所提出的新的概念性材料而得到增强。直觉的主要作用是提供一个概念基础，指明新研究应采取的方向。个体数学家对数学概念（数字、几何概念等）存在性的看法就是由这种直觉提供的；这些看法常常被坚信到值得被称为“柏拉图式”的程度。直觉在研究中的作用是提供“有根据的猜测”，其可能被证明是对的或错的；但无论哪种情况，没有它就无法取得进展，甚至一个错误的猜测也可能带来进步。因此，直觉在数学概念的演变中也起着重要作用。数学知识的进步会周期性地揭示文化直觉中的缺陷；这些缺陷导致“危机”，而危机的解决会带来更成熟的直觉。现代数学的最终基础因此就是数学直觉。正是从这个意义上说，布劳威尔及其追随者的直觉主义学说才是正确的。现代教学方法通过用“接下来该做什么？”的态度取代“做这个，做那个”的教学模式来认识直觉的这一作用，这种态度诉诸学生已有的直觉背景。正是通过这种方式，才能将对新数学知识的理解和欣赏恰当地灌输给学生。
