Re: universal quantification, because g⤨(g⁻¹(x)) = g(y) [1/2] Re: how

On 05/12/2024 09:11 PM, Jim Burns wrote:

On 5/12/2024 3:34 PM, Ross Finlayson wrote:

On 05/12/2024 11:21 AM, Ross Finlayson wrote:

On 05/12/2024 10:46 AM, Jim Burns wrote:

>

[...]

>
Geometry's "similar" is often related to
"congruency, thoroughly",
where "congruent" means "similar".

>
You have confused me.
I agree that "similar" is related to "similar"
but I don't see why you tell us this.
>

  ͨₐ🭢🭕🭞🭜🭘ᵇ  =  ͨₐ🭢🭕ͩ   +  ͩₐ🭞🭜🭘ᵇ
>
The right triangle 🞃cab is split into
two right triangles ◥cda ◤adb
by segment a͞d perpendicular to b͞c
>
🞃cab ◥cda ◤adb are _similar_
[1]
Corresponding sides have equal ratios.

>
Because triangles, here "similar" means
corresponding sides have equal ratios and
corresponding angles are equal.
>

You can know that a triangle is a right triangle
if you have the trigonometric functions of its angles,
here as where it doesn't necessarily require
the apparatus of Pythagoren theorem proper,
"its own theory", ....

>
Pythagoras says
∠cab = 90°  ⟹  c͡b² = c͡a² + b͡a²
>
That means
∠cab = 90°  ∨  c͡b² ≠ c͡a² + b͡a²
>
One side of that disjunction is true
for any triangle.
Thus, we don't need to know it's a right triangle
in order to know Pythagoras is correct.
>

The most usual tools, of classical constructions:
are: compass and edge.

>
If 🞃cab CAN be classically constructed
then  ∠cab = 90°  ⟹  c͡b² = c͡a² + b͡a²
>
If 🞃cab canNOT be classically constructed
then  ∠cab = 90°  ⟹  c͡b² = c͡a² + b͡a²
>
>

>
I just showed there's another way to arrive at the objects
of the functions of trigonometry, independent of the
usual way. Now, the usual way what was a uniqueness
result, is now a distinctness result, what was a singularity,
is now a multiplicity, and what had no end, now has two.
>
For any planar triangle, that any two of its angles
sum to a right angle, indicates the triangle is right.
>
So, in this context of universal quantification, you had
brought up the point about the Pythagorean theorem
as if it was only available as a definition, and now I just
showed that it's also as after a derivation, and vice-versa.
>
Then as with regards to their conflation, and combination,
their confluence, "right" and "equilateral" trigonometry,
makes for as simple an example as you presented that
a deconstructive and reconstructive account provides a
greater dialectic and reasoning grounds for disambiguation.
>
>
And Zeno's like "you know these universal quantifiers can
help if you can explain how they neatly encapsulate an
expressed intent the actual vis-a-vis the potential infinity".
>
>
No, I don't know where else such ideas about this novel
"equilateral trigonometry" are. Here though there is one.
>
>