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

From: Herring, Davis
Subject: Re: Making 'eq' == 'eql' in bignum branch
Date: Sat, 1 Sep 2018 01:05:33 +0000

> +/-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 

>> 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.


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