[r6rs-discuss] [Formal] NaNs are not real numbers

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	         submitter's name: Aubrey Jaffer
	submitter's email address: agj_at_alum.mit.edu
		    type of issue: Defect
		         priority: Minor
		   R6RS component: Arithmetic
	    version of the report: 5.92
one-sentence summary of the issue: NaNs are not real numbers
    full description of the issue:
  2. Numbers
  This chapter describes Scheme's representations for numbers.  It is
  important to distinguish between the mathematical numbers, the
  Scheme numbers that attempt to model them, the machine
  representations used to implement the Scheme numbers, and notations
  used to write numbers.  This report uses the types number, complex,
  real, rational, and integer to refer to both mathematical numbers
  and Scheme numbers.
So the term "real" must conform to mathematical usage.
  2.4. Infinities and NaNs
  Positive infinity is regarded as a real (but not rational) number,
  whose value is indeterminate but greater than all rational numbers.
  Negative infinity is regarded as a real (but not rational) number,
  whose value is indeterminate but less than all rational numbers.
These "regards" can be made less vague:
    Positive infinity denotes an irrational real number whose value is
    greater than all rational numbers representable by the
    implementation.
The number (expt 2 (expt 10 10000000000.1)) would be a candidate value
for all existing Scheme implementations.
  A NaN is regarded as a real (but not rational) number whose value is
  so indeterminate that it might represent any real number, including
  positive or negative infinity, and might even be greater than
  positive infinity or less than negative infinity.
Here R6RS runs afoul of mathemtatics.  A probability distribution is
not a real number.  While there are an infinitude of numbers which can
serve to denote positive infinity, there is no real number whose
ordering behavior matches R6RS_92 NaN's behavior in comparisons.
Mathematically, a real number must be negative, zero, or positive;
thus NaN is not a real number; thus the text of section 2 is
incompatible with the report referring to NaN as a real number.
Then what about IEEE-754 flonum NaN?
  2.1. Numerical types
  ... Likewise, every implementation is required to designate a subset
  of its inexact reals as flonums, and to convert certain external
  representations into flonums.
By rewording this sentence, some flonums may be used to designate
NaNs, even though NaNs are not real:
    Likewise, every implementation is required to designate a subset
    of the union of inexact NaNs with inexact reals as flonums, and to
    convert certain external representations into flonums.
The "fl" comparison operations, which operate on flonums, are then
free to follow IEEE-754 NaN comparison behavior.
But the "Arithmetic Operations" <, <=, >, and >= on page 42 must
signal an exception when applied to non-real arguments like 5+4i or
+nan.0.
By the way, the behavior of "flmin" and "flmax" are not specified when
applied to NaNs.