Path: ...!Xl.tags.giganews.com!local-1.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail
NNTP-Posting-Date: Mon, 30 Dec 2024 02:13:03 +0000
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (constructive)
Newsgroups: sci.math
References: <vb4rde$22fb4$2@solani.org>
 <3d1a8334-deee-45c6-ae03-340cd8551908@att.net> <vbafj7$3vd6q$1@dont-email.me>
 <69325e33-6b9a-4c2f-a0e3-25508d41b114@att.net>
 <rMATvapsf5bpmLmDOt3mDtI5bcA@jntp>
 <d7e0b83e-66ca-4d1f-a165-69c0dd47718e@att.net> <vberjd$qdqn$1@dont-email.me>
 <4eab0882-4a5f-4848-a09b-485714912695@att.net> <vbfq2l$utdu$3@dont-email.me>
 <75c9d649-d983-47fb-95e6-d2743b889481@att.net>
 <205758fd-bfcb-4bbf-9dd8-845cb3df8c4f@tha.de>
 <44111bb0-2d71-4714-8c0c-0fdd0a6d1355@att.net>
 <9pGdnUAZb9ReOEH7nZ2dnZfqn_GdnZ2d@giganews.com>
 <115eed3f-4e3d-4a2b-a4e7-6a1656341011@att.net>
 <EJGdnTn5QaKAkUD7nZ2dnZfqn_SdnZ2d@giganews.com>
From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sun, 29 Dec 2024 18:13:16 -0800
User-Agent: Mozilla/5.0 (X11; Linux x86_64; rv:38.0) Gecko/20100101
 Thunderbird/38.6.0
MIME-Version: 1.0
In-Reply-To: <EJGdnTn5QaKAkUD7nZ2dnZfqn_SdnZ2d@giganews.com>
Content-Type: text/plain; charset=utf-8; format=flowed
Content-Transfer-Encoding: 8bit
Message-ID: <DIWdnblEbcMynO_6nZ2dnZfqnPqdnZ2d@giganews.com>
Lines: 147
X-Usenet-Provider: http://www.giganews.com
X-Trace: sv3-EKokrvI4smlmEoqtodJ/cSb2hgQOdSB0lC23SNAqZnMStm92czl3bfMXGW+zeXV2u9abk4cK1ukv542!Hm5vepa4r/vknjNqPy1rMsy6clnHinOfNj47Om3gVb5NF3igK5l4F5FLCTBWyzw394EhOIZlkh0=
X-Complaints-To: abuse@giganews.com
X-DMCA-Notifications: http://www.giganews.com/info/dmca.html
X-Abuse-and-DMCA-Info: Please be sure to forward a copy of ALL headers
X-Abuse-and-DMCA-Info: Otherwise we will be unable to process your complaint properly
X-Postfilter: 1.3.40
Bytes: 8004

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:
>>
>>>> [...]
>>>
>>> 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.
>>
>>
>
>
> 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.
>
> 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.
>
> 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.
>
> 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.
>
> 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.
>
> Which seems constructive, ....
>
>
> 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".
>
> 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.
>
> 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".
>
>
> So, I hope this is constructive.
>
> Here it's constructivist and I'm a structuralist.
>
>
>