| From: Thomas Bushnell BSG <xxxxxx@becket.net>
 | Date: Sat, 22 Oct 2005 17:51:06 -0700
 |
 | Aubrey Jaffer <xxxxxx@alum.mit.edu> writes:
 |
 | >  | From: "Marcin 'Qrczak' Kowalczyk" <xxxxxx@knm.org.pl>
 | >  | Date: Sat, 22 Oct 2005 20:52:50 +0200
 | >  |
 | >  | Aubrey Jaffer <xxxxxx@alum.mit.edu> writes:
 | >  |
 | >  | > The total order of the reals is a crucial property for many
 | >  | > applications.  Any subset of the reals has a total order.
 | >  | > The reals with +inf.0 and -inf.0 have a total order.
 | >  | >
 | >  | > But the reals with +inf.0, -inf.0, and +nan.0 do not have a total
 | >  | > order [because (<= 5 +nan.0) and (>= 5 +nan.0) would both be false].
 | >  |
 | >  | It is well known that the default order on the floating point
 | >  | approximation of reals is not total.

The floating-point approximations to the reals do not include NaN.
(which real number would it be approximating?)

 | >   From Wikipedia, the free encyclopedia.
 | >   <http://en.wikipedia.org/wiki/Total_order>
 | >
 | >   In mathematics, a total order, linear order or simple order on a set
 | >   X is any binary relation on X that is antisymmetric, transitive, and
 | >   total.  This means that, if we denote the relation by <=, the
 | >   following statements hold for all a, b and c in X:
 | >
 | >       if a <= b and b <= a then a = b (antisymmetry)
 | >       if a <= b and b <= c then a <= c (transitivity)
 | >       a <= b or b <= a (totalness)
 | >
 | > Which condition does it violate?
 |
 | Totalness (as Marcin said).  NaN comparisons (other than
 | not-equals) always evaluate false.

My point exactly!  NaN violates totalness.  Thus it is not a real
number.

Having (real? +nan.0) ==> #f and +nan.0 be an illegal argument to >,
<, <=, and >= is compatible with IEEE-754.  Just because IEEE-754
defines a behavior for comparisons with NaN doesn't mean Scheme must
redefine <, ... so that it accepts non-real arguments.  Scheme already
has lots of non-real numbers which are illegal arguments to >:

  (> 5 5.+3.i)