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:

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[...]

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

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You have confused me.
I agree that "similar" is related to "similar"
but I don't see why you tell us this.
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  ͨₐ🭢🭕🭞🭜🭘ᵇ  =  ͨₐ🭢🭕ͩ   +  ͩₐ🭞🭜🭘ᵇ
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The right triangle 🞃cab is split into
two right triangles ◥cda ◤adb
by segment a͞d perpendicular to b͞c
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🞃cab ◥cda ◤adb are _similar_
[1]
Corresponding sides have equal ratios.

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

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Pythagoras says
∠cab = 90°  ⟹  c͡b² = c͡a² + b͡a²
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That means
∠cab = 90°  ∨  c͡b² ≠ c͡a² + b͡a²
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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.
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The most usual tools, of classical constructions:
are: compass and edge.

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