Graphics Reference
In-Depth Information
Appendix 2
Geometryand intuition
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 wehaveimaginations with which to visualis, 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 closeas weliketo any point. That most of theclustr
points of Q arenot in Q again seems paradoxical. Even the
terminating decimals are dense on the line and will give us
measurements as accurate as we may wish, yet they do not even
includeall therational numbrs.
