Formal comment #230 (defect)

NaNs are not real numbers
Reported by: 	Aubrey Jaffer 	
Component: 	arithmetic 	
Version: 	5.92

   1. 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?.


RESPONSE:

This is related to formal comments #143 and #167.  The
comment's argument is based upon a close reading of the
draft R6RS.

The editors believe it is important to distinguish between
the mathematical numbers and the Scheme numbers that attempt
to model them.  The editors note the presence and importance
of the word "attempt".

If NaNs were not regarded as a real number, then the domains
and ranges of the flonum procedures would not be a subset of
the Scheme numbers that are regarded as real, which would
make the flonum operations less compatible with the generic
operations.  The proposal would also have an adverse effect
upon performance, as noted in the response to formal comment
#143.

The word "indeterminate" does not imply a probability
distribution.  In fact, it correctly connotes that there
is not enough information to construct a probability
distribution.

The editors believe it is better to continue to regard quiet
NaNs as real numbers, and to leave the semantics of signalling
NaNs to some future version of the report.

The next draft should note that the flmin and flmax procedures
always return an inexact NaN when one or more of the arguments
is a NaN.