Intuition and comprehension in mathematical education

When the pupil is presented with a logical proof, is it necessary for him to have to add to this his own means of direct comprehension, that is to say, what is known as his intuition?

This is not a new problem. However, it has recently attracted much attention due to the increase in rigour of the mathematical proofs given in teaching situations. In my opinion, in order that these more axiomatised demonstrations do not stifle the mathematical reasoning process, it is absolutely essential to encourage the use of such intuitive support. This is the first problem that I should like to examine.

First, I consider that there are a number of distinctions that should be made: it is necessary to distinguish between what I call intuitions of adhesion and intuitions of anticipation. The intuition of adhesion expresses itself in the feeling of evidence about a certain fact. It is intuitively evident that the relation of equality is transitive, that from any point not on a straight line there can be only one perpendicular to that line, that there is always a natural number greater than a given number, etc. When I say that something is ‘intuitively evident’ I refer to the fact that the need for a mathematical proof is not felt in these cases (though, from the mathematical point of view, such a proof may be necessary).