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

On 06/19/2024 01:29 PM, Ross Finlayson wrote:

On 06/19/2024 09:43 AM, Jim Burns wrote:

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
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I will courageously assert: it depends.

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

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If  1/∞ = 0  then
not all  x/y = z  imply  x = z⋅y  or
not all 0⋅x = 0  or
not 1 ≠ 0
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I would say context matters.
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Here, I see the context can't be the real numbers.
Other than that, what can we say?
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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.
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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.
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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)
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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.
>
>

>
Well good sir, mostly it's that firstly there's
that the "infinite limit" must concede that it's
actually infinite and that the limit is not only
"close enough" yet actually that it achieves the
limit, the sum, because deduction arrives at
that otherwise it's no more than half, and,
not close enough.
>
Then there's for division and divisibility,
the "infinite-divisibility" and for this
sort of "actually complete infinite limits"
the "infinitely-divided".
>
Then it's pretty much exactly most people's
usual notion of that an infinitude of integers,
regular both in increment and in dispersion,
so equi-distributed and equi-partitioning
the space of integers, is the same kind of
thing when shrunk to [0,1], the space of [0,1]
as by the same members, that it fulfills
extent, density, completeness, measure,
thusly that the Intermediate Value Theorem
holds, then thusly any relevant standard
analysis about calculus holds, or has forms
that hold.
>
>
The indididuals of a continuum are called "individua".
It's called individuation these sorts of notions,
for example "quantization" and "renormalization", sometimes.
>
>
>

Ah, one thing that I wanted to mention was about
"the complete ordered field being unique up to
isomorphism", about the usual assignment as showing
the reals and complex numbers as "unique up to
isomorphism" about the usual analyticity of the
complex numbers, about the most usual sort of
ubiquitous Eulerian-Gaussian complex analysis,
after Euler's formula and Gaussian integral.
What it is is that at one point I wrote non-standard
field axioms for [-1, 1], so, now the usual "the
complete ordered field being unique up to isomorphism"
is a distinctness result instead of a uniqueness result.
Then, another thing is about a deconstructive account
of complex analysis about the very definition of complex
numbers a + bi and the definition of the operations upon
them. The thing is that division, for complex numbers,
the definition of division, can be de-constructed, left
and right, so that now there are non-principal branches
of division, in complex numbers.
Then, besides that the hypercomplex numbers or the
Cartanian style in reflections and rotations making
for algebras without so much caring about Eulerian-Gaussian
complex analysis after Euler's formula, the Gaussian
integral, then the screw period about the Wick rotation,
then also there's a deconstructive account of division
about extended complex numbers, and, field axioms for [-1, 1].