Footnote re the irrationals:
Dedekind stressed the distinction
of category between cut and
number in 1888; against the view
of his friend Heinrich Weber that “the
irrational number is nothing other than the cut itself”, he explained that “as
I prefer it, to create something New distinct from the cut, to which the cut
corresponds. We have the right to
grant ourselves such power of creation”,
and cuts corresponding to both rational and irrational numbers were
examples.
-- Grattan-Guinness, The Search
for Mathematical Roots 1870 - 1940 (2000), p. 87
A seemingly slight, even pedantic distinction; but like many another such, it might
have its point. Cf. my
astonished delight in junior high-school, upon meeting the distinction
between x (the thing itself) and ‘x’
(the name of x) -- already adequately foreshadowed in Alice in Wonderland,
but encountered now in a new context.
Likewise the difference between
x and {x} (the singleton-set of x).
In the case of an algebraic number like √2, a simple number
staring you in the face out of a hypotenuse versus the
infinite train of rational pilgrims (never quite arriving at their destination)
of a Dedekind cut, one is reminded
of the variety of definitions of something so familiar as a tangent: the slope of a curve (at a point); the closest linear
approximation to the curve (at that point); versus the distressing definition
in Loomis & Sternberg as an infinite equivalence-class of curves (through
that point).
