Re: Replacement of Cardinality (ubiquitous ordinals)

On 08/01/2024 05:36 PM, Jim Burns wrote:

On 8/1/2024 3:28 PM, Ross Finlayson wrote:

On 08/01/2024 04:23 AM, Jim Burns wrote:

On 7/31/2024 8:30 PM, Ross Finlayson wrote:

On 07/31/2024 01:21 PM, Jim Burns wrote:

>

If I remember correctly,  your (RF's) name for
not.talking about
what's outside the domain of discussion
is hypocrisyᴿꟳ.
>
That sounds like you're delivering a value.judgment:
that we _should not_ not.talk about
what's outside the domain of discussion,
that we _should not_ for example, not.talk about
_all_ triangles when we discuss whether
the square of its longest side equals
the sum of the squares of the two remaining sides.

>

Yeah, my mathematical conscience demands that
hypocrisy is bad.

>
Bad why?

>
"Wrong", ....

>
It is wrong to treat claims about right triangles
as though they are claims about more than right triangles.
>

Definition usually expands,

>
The hypocrisyᴿꟳ of NOT expanding
  the definition of right triangle ABC
  to encompass triangles without right angles
leaves it NOT wrong that
a segment CH from right angle C
  perpendicular to and meeting side AB at H
makes two more triangles ACH BCH,
  which are both similar to ABC
  which, as similar triangles,
  have corresponding sides in the same ratio
so that
A͞H/A͞C = A͞C/A͞B
H͞B/B͞C = B͞C/A͞B
(A͞H+H͞B)⋅A͞B = A͞C² +B͞C²
and
  A͞B² = A͞C² + B͞C²  is NOT wrong.
>

hypocrisy is bad.

>
If it is, then it isn't for making things wrong,
  which is something hypocrisyᴿꟳ
  (not.talking about outside the domain)
doesn't do.
>
>

There is no "outside" the universe.
Anything else, there is.
For a while we were having a discussion about Pythagorean Triples,
which are integer tuples that happen to be side lengths of right
triangles. The discussion then got into _completions_, that
just like the least-upper-bound not existing in rationals,
yet it's built out to be and usually with an axiomatization
for field-reals then that line-reals have their own trivial sort,
that courtesy unique prime factorization that for right triangles
with side lengths that aren't integer or rational with respect
to each other, there's a sequence of Pythagorean triples that
goes to it.
That it, ..., "goes to".
Whence you might consider our discussion on Pythagorean triples,
and these sequences of them that attain to right triangles of
not-necessarily integer proportion, then also you might recall,
there was the discussion of equi-lateral triangles, and that
un-hinging them and making them their epi-cycles as it were,
that equi-lateral triangles draw sine and cosine which is the
usual role of right triangles, so, all the properties so
accordingly, can be written altogether in terms of equi-lateral
triangles besides these right triangles.
So, that right-triangles and equi-lateral-triangles somehow concur,
isn't that as a fact that it's an emergent property of a proper
deconstruction of them either, both, together?
Also there was the "very-tall-triangles" bit.
It sort of seems the straw-man of you to say I'm disputing Pythagoras
when all I did was point out that Russell was more-or-less lying to you.
Then, if you recall, it was, "Pick one. Ha, I put them together,
you get both or none". Which was it?
It was "anti-diagonal and only-diagonal".