Re: how (quantities and units, implicits and explicits, intensional and extensional)

On 6/18/2024 10:34 PM, Ross Finlayson wrote:

On 06/18/2024 05:45 PM, Jim Burns wrote:

So, I would say
5 elephants ≠ 5 cats
>
But context matters.
I would also say
5 mammals = 5 mammals
>
I will courageously assert: it depends.

"If 1/oo = 0, what if you add oo/oo = 1?"

If  1/∞ = 0  then
not all  x/y = z  imply  x = z⋅y  or
not all 0⋅x = 0  or
not 1 ≠ 0
I would say context matters.
Here, I see the context can't be the real numbers.
Other than that, what can we say?
Not a shrug.
I'm really asking you, Ross:  _What can we say_ ?
You've extended the reals.  _How_ ?
Consider  ∞
as the point at the top of the Riemann sphere,
plugging the hole left when
the complex plane rolls up and covers
the unit sphere sitting on top of 0+0i
We have 1/∞ = 0
We don't have ∞/∞ or 0/0 at all.
So, not that, either.
So, then, what?
https://en.wikipedia.org/wiki/Riemann_sphere

What you got there is
extensionality and intensionality,
that extensionally X mammals is X mammals,
while, intensionally, it depends:
on the individuals.

I really don't know what you're saying.
For 5 individual elephants and 5 individual cats,
5 mammals = 5 mammals.
Not the same mammals.
In many (not all) contexts,
context matters to the point of making
| 5 mammals = 5 mammals
confusing, pointless, or even dishonest.
⎛ One horse won this year's Kentucky Derby.
⎜ One horse didn't win it.
⎜ One horse = one horse?
⎝ Opinions differ.

Nice thing about language:
it's built into the words.

Words or something else.
https://en.wikipedia.org/wiki/American_Sign_Language

"In-di-vidual."

https://en.wikipedia.org/wiki/Individual
| An individual is
| that which exists as a distinct entity.
Nice thing about the English language:
There are separate grammatical categories for
what exists as distinct entities (count nouns)
and what doesn't (mass nouns).
Is the continuum a count noun or a mass noun?
(Not the best question. English ≠ math)
It seems to me that it crosses back and forth.
Points are definitely a count noun.
But the idea of a continuum seems
inescapably not.individuals.
Perhaps that count/mass dimorphism is
why the occasional poster rejects uncountability.