Re: inexactness vs. exactness

[William D Clinger <will@xxxxxxxxxxx>]

Aubrey Jaffer claims to have proved that the language of
the R5RS not only regards inexact numbers as neighborhoods,
but that no other interpretations of the R5RS are tenable.

Jaffer's alleged proof contains many errors of logic, which
I will happily enumerate if anyone claims to remain convinced
by the alleged proof.  Below I merely offer a simple argument
that the "inexact numbers denote neighborhoods" interpretation
is itself untenable.

Suppose (for a contradiction) that inexact numbers do denote
neighborhoods.  Then let [x, y] be the neighborhood denoted
by the inexact number 1.0.  If 0 < x <= y, then the inexact
number (* 1.0 1.0) denotes [x*x, y*y].  If (* 1.0 1.0)
evaluates to 1.0, then 1.0 denotes both [x, y] and [x*x, y*y],
hence x = x*x and y = y*y.  Therefore x = 1.0 = y, so under
our assumptions, the inexact number 1.0 really denotes only
itself.  (Had we considered an open neighborhood (x, y) with
0 < x <= y, we'd have concluded that the neighborhood denoted
by 1.0 is empty, which is even less satisfactory.)

The argument above generalizes to arbitrary positive inexact
numbers z by considering the expression (* (/ z z) (/ z z)),
which probably evaluates to the inexact number 1.0.  Therefore,
under the assumption that every positive inexact real number
denotes a neighborhood [x, y] with 0 < x, we can conclude that
every positive inexact real number really denotes only itself.

That argument generalizes to arbitrary inexact real numbers
by considering the expression (- z).  The generalization to
inexact complex numbers is left as an exercise.  Hence, under
reasonable assumptions, every inexact number really denotes
only itself.

There are only two ways to avoid this conclusion.  One way
is to conclude that every inexact real number denotes the
neighborhood consisting of all real numbers, and that every
inexact complex number denotes the neighborhood consisting
of all complex numbers.  The other is to abandon the idea
that inexact real numbers denote neighborhoods.

Will