Numbers and Functions Steps into Analysis

Sometimes authors and lecturers on analysis insist that students must not use geometrical intuition in developing the fundamental concepts of analysis or in constructing proofs in analysis. Such a prohibition appears to be consonant with Felix Klein's description in 1895 of the developments due to Weierstrass, Cantor and Dedekind, namely the Arithmetisation of Analysis.

However, such advice is impossible to implement and is in any case untrue to the origins of the subject. We can hardly conceive of a Dedekind cut, for example, without imagining a ‘real line’, and such imagining was certainly part of Dedekind's own thought. We have eyes, and we have imaginations with which to visualise, and such visualisation is central to much of the development of analysis. Every development of the real number system is a way of formalising our intuitions of the points on an endless straight line. We cannot conceive how the theory of real functions could have developed had there been no graphs drawn.

However, geometric intuition is not always reliable, and knowing when it should be trusted and when it should not is part of the mathematical maturity which should develop during an analysis course.

There are contexts in which geometrical intuition is misleading.

1 When comparing infinities: because there is a one-to-one correspondence between the points of the segment [0, 1] and the points on the segment [0, 2], there appear to be the ‘same’ number of points on both segments. The conflict with intuition here is simply to do with infinity, not to do with rationals and irrationals, because the same paradox arises if we restrict our attention to rational points.

2 When comparing denseness with completeness: because there is an infinity of rationals between any two points on the line there are rationals as close as we like to any point. That most of the cluster points of ℚ are not in ℚ again seems paradoxical. Even the terminating decimals are dense on the line and will give us measurements as accurate as we maywish, yet they do not even include all the rational numbers.