Re: Making 'eq' == 'eql' in bignum branch

[Alan Mackenzie]

Hello, Davis.

On Wed, Aug 29, 2018 at 03:42:26 +0000, Herring, Davis wrote:
> > Such a name should make clear that we're talking about a signed
> > infinity, not one that makes 1/x continuous by adding a point at
> > infinity to the real line.

> Sure: the extended reals
> <https://en.wikipedia.org/wiki/Extended_real_number_line>, not the
> projective reals
> <https://en.wikipedia.org/wiki/Projectively_extended_real_line>.

> > Signed infinities are convenient symbols, but they're not numbers; I
> > recall the horrified reaction of a mathematician when I told her
> > about IEEE infinities (she thought I was making it up about NaNs).

> Of course +/-inf are numbers.

I'm speaking not as a mathematician, but as somebody with a degree in
maths.

+/-infinity are not numbers.  They do not satisfy the axioms which
define numbers.  For example they do not satisfy

    (x + y) + z = x + (y + z)

.  They cannot be made to satisfy these axioms.

> They're not real numbers, but neither are complex numbers,
> split-complex numbers, dual numbers, p-adic numbers, quaternions,
> octonions, sedenions, hyperreal numbers, or (please no) surreal
> numbers.

That's a strawman.  The issue being discussed here is numbers, not
arbitrary algebraic structures.

> With all due respect to your mathematician friend, ....

That is offensive in the extreme.

> .... she has no exclusive claim over the definition (such as it is) of
> "number"

That's analagous to saying that climate scientists don't have the
exclusive say-so about climate change.  Ha ha, who needs experts?

Mathematicians are _experts_ on numbers, with a deep understanding of
their definitions, properties, and limitations.  I would have expected,
in this mailing list (as contrasted with lesser forums) to see respect
for expertise, not disparagement.

> (and I'm a bit surprised if she hadn't heard of at least one of the
> extended real lines I linked).

Again, offensive.  "Infinity", with or without a sign, might be classed
as an "extended number", but as noted above, is not a number.

> NaN, on the other hand, is of course a floating-point value that
> explicitly claims not to be a number.  I'm not sure whether to believe
> it; it seems very useful that the operations on floating-point
> "numbers" are closed, but that depends on NaN being one of them...

NaN is not a number, either, for much the same reasons.  If you actually
attempt to do anything numerical with NaN you will run into trouble.

There have been discussions about the Lisp functions max and min being
called without arguments.  Some people have advocated that non-numbers
be returned in such cases.  I request most earnestly that this is not
done.  Callers of these functions have a right to assume that returned
results are mathematically correct and that they can do arithmetic with
them.

Currently, max and min throw errors if called with no arguments.  This
is surely the right thing to do, and we should carry on doing it.

> Davis

-- 
Alan Mackenzie (Nuremberg, Germany).