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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sun, 12 May 2024 11:08:45 -0700
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On 05/11/2024 06:44 PM, Ross Finlayson wrote:
> On 05/11/2024 06:17 PM, Ross Finlayson wrote:
>> On 05/11/2024 04:47 PM, Jim Burns wrote:
>>> On 5/11/2024 7:11 PM, Ross Finlayson wrote:
>>>> 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
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