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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sat, 11 May 2024 18:17:50 -0700
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On 05/11/2024 04:47 PM, Jim Burns wrote:
> On 5/11/2024 7:11 PM, Ross Finlayson wrote:
>> On 05/11/2024 02:44 PM, Ross Finlayson wrote:
>>> On 05/11/2024 02:05 PM, Ross Finlayson wrote:
>>>> On 05/11/2024 12:24 PM, Jim Burns 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.
>
> And it is complete == it is true for each.
>
> 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.
>
> Cisfinite induction is NOT a process.
> Cisfinite induction is an argument,
> completely correct or completely incorrect.
>
>> See, the contrary inductive analyst just says
>> "in case you don't have a deductive argument why
>> something is so, induction is so much shifting-sands
>> and slippery-slope." He just has "the base case is
>> you haven't completed induction, and so is the
>> subsequent case, case closed: case not closed".
>
> When the argument is completed,
> induction is completed.
>
> ----
> There is something completely different
> which is also called induction.
> The completely.different induction is physics.
>
> Physics.induction is not unbreakable.
> Physics.induction isn't cisfinite or transfinite induction.
>
>




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.

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.

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, 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.

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,
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