Numbers

[bawden.pa@xerox.com]

    Date: Mon, 21 Aug 89 11:34:37 EDT
    From: Guy Steele <gls@think.com>
    I'm back from an exhausting, and therefore very restful, vacation.
    Back into the fray!

I was wondering why I hadn't heard from you.  I couldn't imagine that you
had given up the argument!

    Here is a new proposal.  Given a set of points within a partial order,
    there are two interesting kinds of "max"-like operation.  One is to find
    some point (not necessarily within the set) that is >= all points in the
    given set; this is SUP....

    Another is to find some point in the set such that no point in the set is >
    that point....

But the version of MAX I am advocating isn't either of these!  I would
allow:

  (max 1.0 549755813889)  ==>  549755813888.0

Where presumably (< 549755813888.0 549755813889) is true.  Thus MAX can
return an object that, according to `<', is smaller than one of the
arguments.

Now I'm sure that about half of you are shouting at your terminal about how
I couldn't -possibly- mean that, but I do.  Understand me once and for all:

		      INEXACT NUMBERS ARE NOT NUMBERS

		      INEXACT NUMBERS ARE NOT NUMBERS

		      INEXACT NUMBERS ARE NOT NUMBERS

They do not obey the rules followed by numbers, because they cannot.
Inexact numbers only represent our -approximate- -knowledge- about numbers.
In

  (max 1.0 549755813889)  ==>  549755813888.0

we strongly suspect that the first number is 1, but we don't know for sure.
Thus we strongly suspect that the answer is 549755813889, but again we
don't know for sure (the first number could have actually been
549755813890).  Since we don't know the answer for sure, we must return an
inexact representation of our best guess.  549755813888.0 is the closest we
can come.  (The next larger might be 549755879424.0, assuming 23 bits of
floating point precision, which is not very close at all.)

Indeed, we could -specify- that MAX always ``round up'' in the inexact case
so that the result always appears greater (accroding to `<') than any of
the arguments.  This would be similar to specifying that `<' must always
behave transitively.  It would be a feeble attempt on our part to paper
over the ugly fact that -in- -general-, inexact numbers do not obey the
axioms of arithmetic.