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

["Clément Pit-Claudel" <cpitclaudel@gmail.com>]

Hi again Alan,

Thanks for your thoughtful response!

On 2018-08-31 15:59, Alan Mackenzie wrote:
> 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.

Thinking more about your message, I'm not sure I agree with this point, either.  I don't know about the US or Germany, but at least in France it is common to define inf and sup as functions from ℝ into ℝ ∪ {–∞, +∞}, the extended real number line.  With these definitions, inf(∅) is customarily set to +∞, and sup(∅) is set to +∞.

https://en.wikipedia.org/wiki/Empty_set#Extended_real_numbers says:

    Since the empty set has no members, when it is considered as a subset of any ordered set, then every member of that set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of the real numbers, with its usual ordering, represented by the real number line, every real number is both an upper and lower bound for the empty set. When considered as a subset of the extended reals formed by adding two "numbers" or "points" to the real numbers, namely negative infinity, denoted −∞, which is defined to be less than every other extended real number, and positive infinity, denoted +∞, which is defined to be greater than every other extended real number, then:

        sup ∅ = min({−∞, +∞} ∪ R) = −∞

    and

        inf ∅ = max({−∞, +∞} ∪ R) = +∞

    That is, the least upper bound (sup or supremum) of the empty set is negative infinity, while the greatest lower bound (inf or infimum) is positive infinity. By analogy with the above, in the domain of the extended reals, negative infinity is the identity element for the maximum and supremum operators, while positive infinity is the identity element for minimum and infimum.

I could be convinced that returning −∞ and +∞ is wrong based on the fact that inf and sup are not min and max, and that the outputs of min and max should be elements of the sets they operate on; conversely, I'm also receptive to the convenience aspect or min and max returning infinities; but all in all I don't find the "numberness" argument particularly compelling, based on the reasoning outlined above.

Clément.