Re: Replacement of Cardinality (ubiquitous ordinals)

On 8/1/2024 8:52 PM, Ross Finlayson wrote:

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.

There very often are things outside
the things which we are discussing.
It is perhaps surprising how important it is that,
when we are discussing certain things,
we are not discussing other things, outside.
It is, it seems to me, a key part of a technique for
exploring the infinite by finite means.
That key part seems to me to be
what you are calling hypocrisyᴿꟳ
It strikes me as _at least_ unwise to discard
such a powerful and reliable tool.

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.

I picked Pythagoras as a concrete example of
my best guess at what you (RF) mean by hypocrisyᴿꟳ
If my best guess is wide of the mark,
perhaps you should tell me what you DO mean.

That sounds like you're delivering a value.judgment:
that we _should not_ not.talk about
what's outside the domain of discussion,

Yeah, my mathematical conscience demands that
hypocrisy is bad.