Re: How many different unit fractions are lessorequal than all unit fractions? (constructive)

On 09/07/2024 07:29 PM, Ross Finlayson wrote:

On 09/07/2024 05:29 PM, Jim Burns wrote:

On 9/7/2024 3:13 PM, Ross Finlayson wrote:

On 09/07/2024 12:01 PM, Jim Burns wrote:

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[...]

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Aristotle has both _prior_ and _posterior_ analytics.

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⎛ In the _Prior Analytics_ syllogistic logic is considered
⎜ in its formal aspect; in the _Posterior_ it is considered
⎜ in respect of its matter. The "form" of a syllogism lies
⎜ in the necessary connection between the premises and
⎜ the conclusion. Even where there is no fault in the form,
⎜ there may be in the matter, i.e. the propositions of which
⎜ it is composed, which may be true or false, probable or
⎜ improbable.
⎜
⎜ When the premises are certain, true, and primary, and
⎜ the conclusion formally follows from them,
⎜ this is demonstration, and produces scientific knowledge
⎜ of a thing. Such syllogisms are called apodeictical,
⎜ and are dealt with in the two books of
⎜ the _Posterior Analytics_
⎝
-- https://en.wikipedia.org/wiki/Posterior_Analytics
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So, when you give him
a perfectly good syllogism with which he disagrees,
he has either of prior or posterior to deconstruct
either posterior or prior,

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Wikipedia seems to say that
syllogisms are prior, and
use of syllogisms is posterior.
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They don't seem to be 'either.or', but 'both.and'.
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That cheers me up considerably.
The idea I brought away from your (RF's) post was that,
if Aristotle didn't like an result,
he could ignore it and use a different method,
lather, rinse, repeat unit he got an answer he liked.
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That would make those methods worthless.
If a method or cluster of methods only gives you
what you _want_
throw them all away and go do what you want.
It's the same result, with less time and effort.
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However, when I read Wikipedia,
I think that, perhaps,
analysis is not a waste of time and effort, after all.
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thusly not allowing himself to be fooled
by otherwise perfectly and as-far-as-the-eye-can-see
linear induction,
because that would leave a fool of him.

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It would seem to be impossible to be fooled,
if the "correct" answer always turns out to be
the answer one had before investigating,
if one keeps throwing out and trying again.
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I have a strong suspicion that things don't work that way.
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The "constructive" is a usual idea in common usage
about "being constructive" or "constructive criticism"
then that in logic it's included that for structuralism,
constructivism, that "proof by contradiction", is not
considered constructivist.
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So, being constructive, constructive criticism, when
I look at the outcome of otherwise a proof by contradiction
to be rejected, that a "strongly constructivist" view requires
that it's immaterial the order of the introduction of any
stipulations, where in the usual syllogism's proof by contradiction,
whatever non-logical term is introduced last sort of wins,
when if the terms are discovered and evaluated in an
arbitrary order, it's arbitrary which decides and which
is decided.
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The idea is that by avoiding proof-by-contradiction,
then the terms are independent, and it matters as much
more what are introduced as altogether axiomatic stipulations,
as it results a proper "rule 1" or "rule 0" or if you'd recall
we've already had the conversation and it's the same now
as it was then.
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Syllogism as direct implication, where the non-logical terms
are having a logical model, i.e. plainly just transitivity of
association, does not include proof by contradiction, which
must have as an elementary reasoning because syllogism
is usually considered evaluated in only one order, while,
having a logical and constructivist model they can be
evaluated in any old order and are neither arbitrary nor
ambiguous, constructive syllogisms.
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Then, in non-constructive syllogisms, Aristotle includes
terms that help sort them out in a deductive deconstructive
account, like the one a few paragraphs above here.
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Which seems constructive, ....
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There's Bishop and Cheng they were supposedly going about
arriving at some "constructivists' ZF", and including for example
a novel structure of reals or a different model of the reals than
the usual standards, a ring with a "rather restricted transfer
principle".
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There's no real point to be talking about the soft-ball-straw-man's
non-existent smallest positive real in an Archimedean field,
when there are also constructive models of reals numbers
or continuous domains if you will, like Bishop and Cheng's,
with their notion of a ring and a "rather restricted transfer principle",
which you can associate with the many previous writings here
about Sorites and transfer principle and how least-upper-bound
is an axiomatized stipulation.
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That there's introduced indiscernibles in the "rather restricted"
the transfer principle, shouldn't much make for worry, it's the
properties of continuous domains what get fulfilled, vis-a-vis
the quasi-invariant and the measure theory in worlds with
multiples law(s) of large numbers. At least it begins with
"at least one of these three models of continuous domains
is appropriate". What is this you say, "there's no continuous
domains it's not defined", then it's like, "employ Hilbert's
continuity of postulate if you must then draw one straight line
and that's line reals then imagine infinite classical constructions
to make a field then imagine axiomatizing least-upper-bound for
that to make the complete ordered field the field-reals", then
as with regards to "now that you look at it the Fourier-style
analysis sorts of makes a complete metrizing ultrafilter that
happens also to be these signal-reals".
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So, I hope this is constructive.
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Here it's constructivist and I'm a structuralist.
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It's like, "dihydrogen monoxide", and arguments against it.
In its solid or gaseous states, dihydrogen monoxide is
quite inhospitable to life, and in its liquid state
as well usual natural respiration does not result,
and there's a quantity of dihydrogen monoxide that's
of a lethal dose.
Contrast this with the "all-natural", for example cyanide.
Stipulations, independent, consistent, ....
... not "ultimately true", ....
Then, that Russell's retro-thesis is simply not a fact,
logically, helps reflect on the consideration of what
lets out when it's stipulated to reflect a universal
relation on all objects of an elementary theory, that,
that's at best, "incomplete", of course.