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

["Herring, Davis" <herring@lanl.gov>]

> +/-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)

What is the list of such axioms?  If it includes the existence of an additive identity, you lose the natural numbers; if it includes the trichotomy property, you lose the complex numbers; if it includes commutativity of addition, you lose the ordinal numbers.  Of course we can decide that any or all of those "aren't really numbers", but why?

While addition is one of the operations with the broadest applicability (but see the supernatural numbers) and associativity is one of the properties most widely required of a binary operation (but see the octonions), even associativity of addition is sometimes violated (consider the example of infinite series <https://en.wikipedia.org/wiki/Associative_property#Non-associative_operation>).

>> 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.

Which of those sets aren't sets of numbers?  No one said anything about vector spaces, Lie groups, Sobolev spaces, tangent bundles, etc.

>> With all due respect to your mathematician friend, ....
> 
> That is offensive in the extreme.

I'm sorry, but I don't understand how.  I _meant_ the "due respect" that I wrote.  Perhaps I should not presume that any two people are friends, but the broad use of that term was meant to be generous, not condescending.

>> .... 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?

I don't follow the analogy.  Outside of trivial misconceptions (like "weather" vs. "climate"), I don't know of a situation where the _definition_ of "climate change" is unclear.  On the other hand, I don't know of any rigorous definition for "number" (thus my request for axioms above).

> I would have expected,
> in this mailing list (as contrasted with lesser forums) to see respect
> for expertise, not disparagement.

No one is disparaging expertise, certainly not in general.  Experts routinely disagree, without any disrespect, about the more philosophical topics in their field.  Given (what I consider to be) such a weakness of definition, I remain uncertain why a mathematical expert would so flatly reject a widely-used category of "number".

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

Again, I'm afraid you'll have to explain.  I stated that I considered it unlikely that a mathematician would be unfamiliar with the extended reals (considering that they are used as a notational device even in Calculus I).  I said so because I don't know what other explanation to offer for such surprise at the IEEE rules, based so closely as they are on the extended reals.

Davis