AN article by a mathematician, "The Role of Intuition," in Science for May 5, brings home the fact that whenever the higher reaches of learning and discovery are discussed by men who have had some experience of them, what they say is pretty much the same. The writer of this article is R. L. Wilder, research professor of mathematics at the University of Michigan. At the outset, Prof. Wilder does the best he can to establish what he means by intuition, drawing on Descartes, Kant, and a few others. I believe [he writes] that the intuition about which some philosophers speak is—if not wholly, at least partially—a "native intuition."

Thus Descartes stated:

"By intuition I understand, not the fluctuating testimony of the senses, nor the misleading judgment that proceeds from the blundering constructions of the imagination, but the conception which an unclouded and attentive mind gives us so readily and distinctly that we are wholly freed from doubt about that which we understand."

And Kant, as I interpret him, conceived of the concepts of both time and space as deriving from an a priori intuition which is independent of experience. Among the more modern philosophers, especially those of a mystical bent, knowledge imparted by this native intuition may be considered more valid than that gained from observation and experience. The "intuitionism" of Brouwer and Poincaré, insofar as it conceived of the natural numbers as "intuitively given," seems to proceed from this native intuition.

Probably a very large book could be made simply by collecting other definitions of intuition which parallel what has been said here, but for practical purposes it seems likely that the definition made by Descartes is as good as any. Prof. Wilder's article is essentially a discussion of the respect the teacher of mathematics ought to show for the intuition of his pupils, but before we see what he says on this subject it may be valuable to recall the similar views of Albert Einstein. Wilder observes: "It is almost a truism that without intuition, there is no creativity in mathematics." And Einstein, writing in the Journal of the Franklin Institute in 1936, in the paper, "Physics and Reality," said:

Physics constitutes a logical system of thought which is in a state of evolution, whose basis cannot be obtained through distillation by any inductive method from the experiences lived through, but can only be attained by free invention. The justification (truth content) of the system rests in the proof of usefulness of the resulting theorems on the basis of sense experiences, where the relations of the latter to the former can only be comprehended intuitively. Einstein also wrote: That the totality of our sense experiences is such that by means of thinking it can be put in order is a fact that leaves us in awe but which we shall never understand. . . . nothing can be said concerning the manner in which the concepts are made and connected and how we are to coordinate them with the experiences. Well, to call these concepts "intuitions" is to say something. By perceptive accounts of the meaning of intuition, such as Prof. Wilder's article, aided by the framing effect of each man's own subjective experience, we assemble a kind of unified constellation of subtle feelings about "knowing," and these feelings gain some authenticity from the "awe" of which Einstein speaks. All that we are saying is that it may be useful to think about the "intuition." Or that "something" may be said on the subject, provided it is said carefully, and without matter-offact presumption. This seems exactly the case in respect to Prof. Wilder's remarks about how to teach mathematics to the young. He begins by saying that "individual mathematical intuition is not a static but growing thing." He continues: It starts developing when we are children, during the time when we learn to distinguish shapes and sizes (geometric intuition) and to count (arithmetic intuition). We are not born with it, for without a cultural basis for its development, there can apparently be no mathematical intuition. By the time the child starts in school in our culture, however, he usually has some basis to build on—his parents have probably taught him to count, for example—and the continuing development of this basis undoubtedly forms one of the central responsibilities of primary teachers. What is this "basis" for intuitive development? (We do not really know we are not born with it!) It lies, Prof. Wilder shows, in the relationship between

two components—the "knowledge" component and the intuitive component. The knowledge component is, so to speak, both the raw material and the fruit of the intuitive component. The relation between these components, as maintained by the teacher, or as suppressed, determines the quality of the education that takes place. The two cannot really be separated, of course, but the relation needs to be understood. Prof. Wilder explains: Perhaps I can make this clearer by stating my conception of what the new curricula being developed today should accomplish in contrast to the old, standard, mathematical curriculum. The old curriculum was designed chiefly for the knowledge component; the student was taught how to perform arithmetic and algebraic operations and how to prove theorems. But little conscious development of mathematical intuition took place; what there was of this seemed to find expression chiefly in the problems that were given to be solved. . . . For example, while under the old system the student was told the formula for carrying out a process, under the new he should be invited to do a little guessing as to what form the process should take. This guessing and the accompanying experimentation, resulting in a decision as to the final result develops and strengthens the mathematical intuition. Prof. Wilder then points out that this procedure is exactly that followed by the research mathematician, and he holds that "all concepts should be introduced in this way." When the intuitive capacities of the student are bypassed in education, no real or "creative" teaching occurs: To explain a concept to a student adds to his knowledge component, perhaps, but does not strengthen his intuition. Probably the worst example of this kind of thing is the writing of a definition on the board, then explaining what it means and how it is used. There is a striking parallel, here, to points made by Frank Lindenfeld and Peter Marin in their article on "open field" teaching (MANAS, Sept. 7, 1966), in which the writers say: We view the process of education as involving a flow from experience to perception to abstraction. . . . Perhaps a good example of this is the concept of "alienation." Students, as we all know, hear a good deal about this "condition" of modern man. But their understanding of the concept and the condition to

which it refers is much richer if the discussion of it has emerged naturally, as a result of the confrontation of their own experiences. They may not have the word at first to describe what they are talking about, and the teacher may then want to supply it, but we feel that he should supply the word and the abstract concept only after the students have provided the opportunity. If the concept comes first, the student will apply it like a "title" to their experience without ever letting the experience itself emerge—and their knowledge will tend to remain "abstract," without roots in their personal experience. The correspondence, here, with what Prof. Wilder says, is obvious. (It is incidentally interesting to wonder how intuitive operations may differ in relation to a "social" subject such as Lindenfeld and Marin are concerned with.) A parallel more specific in its relation to teaching mathematics is found in an article by J. J. Gordon in Education of Vision (Braziller, 1965), in which the writer tells how he encouraged his students to use analogy and metaphor to solve scientific problems by unconventional and new means. Mr. Gordon makes this general observation: Perhaps the greatest danger in the teaching of science is to present students with a fait accompli universe. It is a didactic tradition that undergraduate students must accept the phenomenological universe as described by someone with special knowledge, i.e., the teacher. The teacher is saying to students that they must surrender to his rules or they can't play in his backyard. By the time a student has clerked his way through his undergraduate work in a science, it may not be possible for him to tolerate the ambiguity of constructing his own ways of understanding. The point is made—heaped up, pressed down, running over. But where did we get all these terrible habits that genuine teachers have to spend their lives trying to correct? Is it conceivable that the religious background of Western man, with its precise "creedal" formulations of religious truth, inscribed on the tabula rasa of small children's minds in the form of "correct belief," is a basic cause of these difficulties in education? If so, then what might be the application of "open field" teaching in this crucial area of learning?