I am not a mathematician, but my son is, and I parrot back what he told
me. N.B., I may have it garbled:
The real numbers are not naturally compact, in a topological sense,
because they have no end. The real numbers may be compactified either at
one point or at two. One point compactification introduces infinity with
both positive and negative numbers ending at infinity. (The number line
is closed in a circle.) Two point compactification introduces positive
and negative infinities; their inverses produce positive and negative
zeros. Both approaches are acceptable. It is incorrect to consider them
as right and wrong, though some theorems are more conveniently solved
with one or the other.
Don Guinn wrote:
One of the reasons that APL and now J have been so appealing to me is
that
they adhere to mathematics more than to computers. The developers
have hidden the peculiarities of particular hardware platforms while
maintaining reasonable efficiency. Taking advantage of peculiarities
of IEEE
floating point is simply locking one into a particular hardware platform.
Ok, IEEE floating point is common to many platforms and probably on
all that
J support, but there are platforms that represent floating point
differently. And IEEE floating point may be superseded in some future
hardware. But if one really wants to take advantage of the
peculiarities of
IEEE then there are bunches of goodies in it beyond having both a
positive
and negative zero. Personally, I am glad that J does not.
I have not followed this discussion too closely because it is trying
to take
advantage of a hardware peculiarity, but really - zero is not a positive
number and neither is it a negative number. Just because it is internally
formatted as a positive or negative number internally is just a
convenience
of a hardware design.
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