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

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

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

[...]

>
Aristotle has both _prior_ and _posterior_ analytics.

⎛ 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

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,

Wikipedia seems to say that
syllogisms are prior, and
use of syllogisms is posterior.
They don't seem to be 'either.or', but 'both.and'.
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.
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.
However, when I read Wikipedia,
I think that, perhaps,
analysis is not a waste of time and effort, after all.

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.

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.
I have a strong suspicion that things don't work that way.