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

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

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

On 5/11/2024 9:17 PM, Ross Finlayson wrote:

On 05/11/2024 04:47 PM, Jim Burns wrote:

On 5/11/2024 7:11 PM, Ross Finlayson wrote:

>

The case is that induction goes through,
an inviolable law you call it:
does it go all the way through?
Does it complete?

>
It is complete.
There is no completing.activity,
so I wouldn't say it completes.
>
Compare to right triangles:
Are all the squares of two shorter sides
summed to the square of the longest side?
>
That's a tricky question to answer because
there is no summing done.
That relationship between the sides
is simply something true about right triangles.
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And it is complete == it is true for each.
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We don't typically ask the tricky question
about right triangles.
We ask the tricky question about cisfinite induction
because we imagine it as a process,
which we don't for right triangles.
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Cisfinite induction is NOT a process.
Cisfinite induction is an argument,
completely correct or completely incorrect.

>

What I recall of the context of the Pythagorean theorem,

>
Let's refresh our memories.
>
  ͨₐ🭢🭕🭞🭜🭘ᵇ  =  ͨₐ🭢🭕ͩ   +  ͩₐ🭞🭜🭘ᵇ
>
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.
>
∠acb = ∠dca
∟cab = ∟cda
🞃cab ≚ ◥cda
c͡b/c͡a = c͡a/c͡d
>
∠cba = ∠abd
∟cab = ∟adb
🞃cab ≚ ◤adb
c͡b/b͡a = b͡a/d͡b
>
c͡b⋅c͡d = c͡a²
c͡b⋅d͡b = b͡a²
c͡b⋅(c͡d+d͡b) = c͡b² = c͡a² + b͡a²
QED
>
[1] needs its own proof,
but that can be done, too.
>

What I recall of the context of the Pythagorean theorem,
was that after algebra already was trigonometry, and
the definitions of the trigonometric functions, for
sine and cosine and tangent, about the opposite and
adjacent and hypotenuse, then as of a right triangle
with its hypotenuse the radius of a unit circle, that
the right angle is as with regards to the abscissa
and ordinates or where the lines drop or slide to
the x or y axis of the usual X-Y coordinate setting
of a circle centered at the origin, it was of the
secondary school's first three years of geometry,
algebra, and trigonometry, or along those lines.
>
So, we computed a bunch of ready things about
those often with the Pythagorean theorem,
which is as an addition-formula, mostly about
30-60-90 triangles, and, isosceles triangles,
or 45-45-90, then those got used throughout
precalculus and a couple years of calculus
or high school.

>
I agree that the Pythagorean theorem
gets used in a lot of different ways.
>
How we know that the Pythagorean theorem
is a fact about each right triangle
has important similarities to
how we know that we cisfinitely.induced claims
are facts about each natural number.
>

So anyways one time I see a diagram about
Pythagorean triples, those being tuples of
three integers that have a^2 + b^2 = c^2,
and what they'd done was right triangle,
then draw a square as of the square alongside
it, and counting the boxes of the squares of
a b c it's that the boxes of the squares of a
and b equals the boxes of the square of c.

>
Actually,
that works in the opposite direction.
We know that 3:4:5 is a right triangle
because of the Pythagorean theorem.
>

If that's not a proof of the Pythagorean theorem
and least it's graphically intuitive for some values,
where of course there are hundreds of known
proofs of the Pythagorean theorem, since the
time of Pythagoras as some even have as from
greater antiquity, then it reminds of things
like Rodriguez formula, Vieta's formulas,
Nicomachus' theorem and formulas,
Pascal triangle and bonomial theorem,
all what are sorts of addition formulas,
like an addition formula of the product
of exponents as the sum of the powers.
>
So, that Pythagorean triples exist, and it results
that the rightness of a triangle with sides length
the Pythagorean triple can be established without
invoking the Pythagorean theorem, doesn't so
much make it so the other way around, from
induction over Pythagorean triples, without
showing as how all right triangles are somehow
as some congruence to what is some Pythagorean
triple,

>
We shouldn't want to show that
each right triangle has a Pythagorean triple,
because we know that isn't true.
Famously, Pythagoras executed one of his disciples
for proving that the right isosceles triangle 1:1:√̅2
has no Pythagorean triple.
>

of the equivalence class of all the triples
and all the congruences to triangles with a
unit length longest side, establishing infinite
expressions, and closures, of completion,
to make a case for the Pythagorean theorem
as via induction from an explication after
the enumeration of Pythagorean triples,
which via inspection have a^2+b^2 = c^2,
as for that it results congruences that
"go to" any given dimensions of a right
triangle.
>
About the cisfinite and transfinite induction,
and I know it's not the languages fault that
there's the associated psychosexual connotation,
I'm glad you make the point though that
it just is what it is, and, a case for induction
more or less needs some reason its tendency,
to succeed as it were, then that induction
is given its course, then that the course-of--passage,
of what the plain old infinite induction, arrives.

>
Induction does not arrive,
unless you are talking about arriving at
the end of _expressing the argument_
>

I.e., it's always "infinite induction", after cause-and-effect,
with that also being induction or a case, mathematical
induction, and there can't be any reasonable counterclaims
or they'd be just as guaranteed as the contradistinct opposite.
>
>
So, it makes for a very strong perceived requirement
for deductive reasoning _why_ convergence criteria
exist, besides that "given an infinite expression,
it's an infinite expression".
>
Here then that's most Zeno's about geometric series,
and then about things like Stirling numbers and of
course the discussions we've been having over the
past few months about the convergence and
the slooowwwly convergent and all this,
the "scaffolding" of the infinite expressions
we've been discussing and at length.
>
Warm regards
>
>

>

>
Briefly, though this user-agent has some support for Unicode,
and pretty thorough support, you've exhausted the capacity of
the default font, the symbols you've entered.
>
Geometry's "similar" is often related to "congruency, thoroughly",
where "congruent" means "similar".
>
>
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", ....
>
>
The idea of finding these independent abstractions of course
is a great part of "deconstructive account", where we've
developed what are the "combined" and "conflated", even,
which are so great because they're conformal mappings and
congruency-preserving and as for tensorial products and
all such great things about the combined milieu of
arithmetic(s), algebra(s), and geometry together,
the "deconstructive account" is an important part of
the thorough analysis and the reconstructive account,
for mathematical platonists and structuralists, in
the language of mathematical formalism and abstraction.
>
Or, you know, "without loss of generality", ....
>
So, a 3:4:5 triangle, can be determined right, from
trigonometric functions, than any two of the angles
add up to a right angle.
>
>
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Now there's never been a "petition", yet I will advise
that there's an entire agen-da.
>
>

The most usual tools, of classical constructions:
are: compass and edge.
The next most usual tool is some pegs, and a loop of string.
There's a plumb-bob. Of course there's usually a ruler,
or meter-stick.
(The Archimedean spiral then provides for example
getting into angle-trisection, cubing squares,
and circle-squaring.)
At some point was invented the cog-wheel, or as of a gear,
or about the sprocket, like other machines, as of a compound
machine, of simple machines, much or altogether as from
the lever, or, the float. The cog-wheel then, thusly makes
as for a gradiated compound action of levers, and what one
might find today, can make for drawing tools, like the wide
variety of common amusements the spriro-graph, often
introduced in the same era as, the kaleidoscope.
So, the pantograph, is a notion of a means to follow motion,
for tracing, and the course-of-passage, of what's drawing,
where here line-drawing itself is its own deliberate act,
the act of putting pencil to paper and moving it without
lifting it, then lifting it, resulting the act of the tracing of
the drawing of a line.
Here then this tool to trace the angles that open together
at the same rate, or as with regards to a common rate in
some gradiated manner, that it draws the curve of the
sinusoidally periodic the curves the charts the graphs
of the function _sine_ and _cosine_, is kind of a great
thing. I'm not sure what its name is, yet, that: is what it does.
N-gon-ometry and n-equi-lateral-ometry after
tri-equi-lateral-ometry: a technique of primary trigonometry.
So, one need not all these tools at once for most things,
and most things don't require yet only their own tools.