On 04/23/2024 07:05 PM, Ross Finlayson wrote:
> On 02/23/2024 11:08 AM, Ross Finlayson wrote:
>> On 02/13/2024 04:39 PM, Ross Finlayson wrote:
>>> On 02/13/2024 03:47 PM, Jim Burns wrote:
>>>> On 2/13/2024 4:38 PM, Ross Finlayson wrote:
>>>>> On 02/13/2024 01:15 PM, Jim Burns wrote:
>>>>>> On 2/13/2024 2:05 PM, Ross Finlayson wrote:
>>>>
>>>>>>> What I have in mind here is that
>>>>>>> the continuous domain has sorts of topologies, that
>>>>>>> make it so, that every open set is also
>>>>>>> a continuous domain, and, to make it so, that
>>>>>>> only two open sets that are "contiguous", are in
>>>>>>> this sort of topology.
>>>>>>
>>>>>> I vaguely recall from a topology course that
>>>>>> my instructor blended "contiguous" and "continuous"
>>>>>> into "contiuous" for some theorems.
>>>>>>
>>>>>> There might be a notion of "contiguous" that's
>>>>>> useful here. If there is, I'd like to know it.
>>>>>>
>>>>>> Please define "contiguous" for the context
>>>>>> in which you are using it here.
>>>>>>
>>>>>> Googling "contiguous" or "contiuous"
>>>>>> could in principle resolve my quandary,
>>>>>> but there are too many other uses.
>>>>>> I tried it and got no joy.
>>>>>> (For "contiuous", too many typos of continuous")
>>>>>
>>>>> Contiguous basically means "in all neighborhoods
>>>>> together".
>>>>
>>>> So, for 𝒪 = {∅,X}
>>>> all points in X are contiguous to each other?
>>>> Do I have that right?
>>>>
>>>> This seems related to Hausdorff space.
>>>> | for any two distinct points, there exist
>>>> | neighbourhoods of each that are disjoint from each other.
>>>> |
>>>> https://en.wikipedia.org/wiki/Hausdorff_space
>>>>
>>>> With your interest in defining ℝ,
>>>> that seems right up your (RF's) alley.
>>>> | Of the many separation axioms that can be imposed on
>>>> | a topological space, the "Hausdorff condition" (T2) is
>>>> | the most frequently used and discussed.
>>>> | It implies the uniqueness of limits of
>>>> | sequences, nets, and filters
>>>> |
>>>> ibid.
>>>>
>>>> ----
>>>> Side note:
>>>> You know what's a connected topology?
>>>> 𝒪 = {∅,X} is a connected topology.
>>>>
>>>> Condition #4 is vacuously satisfied.
>>>> 4. B ∈ 𝒪\{∅,X} ⇒ X\B ∉ 𝒪
>>>>
>>>> But 𝒪 = {∅,X} isn't Hausdorff.
>>>>
>>>> It makes me go "Hmmm."
>>>>
>>>>> Not sure about "contiuous", "conti?uous".
>>>>
>>>> I would not be surprised if "contiuous"
>>>> was an invention of my instructor,
>>>> and not particularly widely used.
>>>>
>>>>> Mathematics: both sides to anything.
>>>>
>>>> ...unless only one side exists.
>>>>
>>>> Mathematicians do agree on some things.
>>>> But those seem to only be things upon which
>>>> there is no rational alternative to agreeing.
>>>>
>>>>
>>>
>>>
>>>
>>>
>>> Consider for example the matrix product.
>>> It's well known that aligning m x p and p x n
>>> results an m x n matrix, its product.
>>>
>>> Yet, there are as well generalized products,
>>> for matrices of any dimensions, and as well
>>> inner and outer products, for example the
>>> determinant, or as with regards to bivectors,
>>> for examples, these fuller and scalar and
>>> inner and outer: "complementary duals",
>>> the duals of the complements.
>>>
>>>
>>> There's as well sort of generalized inverses,
>>> with regards to principal values p.v. for example,
>>> as with regards for example to complex numbers,
>>> or for example the postive and negative roots of even
>>> and odd powers of non-negative numbers.
>>>
>>> Again this is as with regards to "complementary duals",
>>> and about the usual notion of the inverse as axiomatic.
>>>
>>> Another key notion is for the deconstructivist account,
>>> for as what results what was elementary, for example
>>> atoms, has "sub-atomic particles", where for example
>>> the very theory of atoms is a deconstructivist account,
>>> and the very theory of axiomatic set theory making
>>> a descriptive set theory a model of models of mathematics,
>>> is as of deconstructive and constructivist accounts,
>>> for the constructivist side that it is structure, and
>>> for the intuitionist side that there's more than one
>>> way to look at things, then as with regards to the
>>> results of existence, distinctness, and uniqueness.
>>>
>>> Here then this "continuous topology" notion is rather
>>> for a sort of inner product already, if a wider and more
>>> general class of structures the topologies.
>>>
>>>
>>> Here, pair-wise contiguity is sharing all neighborhoods,
>>> while transitive contiguity has that contiguity is associative.
>>> It seems an example of a usual property like transitivity
>>> that happens to also model and be modeled by an other
>>> usual property like associativity, sort of in simile to how
>>> here the notion of a continuous domain's continuous topology,
>>> has that composing and decomposing them is just like as
>>> of the regions or analytical regions of the surfaces or bodies
>>> they represent, being together, and meeting.
>>>
>>>
>>> So, I think this is a good idea and part of mathematics.
>>> It's a bit more involved than topology which is connected
>>> enough to establish usual results, while it provides for
>>> various certain stronger and more direct results.
>>>
>>> When I think of algebraic geometry, I sort of have the
>>> geometry first, because it's Euclidean and there's the
>>> Cartesian and R^N. The algebras are a much richer
>>> milieu for the deconstructivist and analytical accounts,
>>> and models of all the products, whose images, are geometric.
>>>
>>> So, let's consider further this notion of a definition of
>>> "continuous topology", and why it's a central and primary
>>> feature in the structure of the objects of mathematics.
>>>
>>>
>>
>>
>> Here then let us consider furthermore this notion
>> of a "continuous topology" it being embodying contiguity,
>> and of the sort of least-tenuous "tenuous topology",
>> as with regards to the myriad kinds of "torsional topology",
>> that vis-a-vis the initial and final and vacuous and discrete
>> and indiscrete sorts of topologies, arises these particular
>> varieties that do and don't relay and maintain the notion
>> of the neighborhoods, and neighborliness, of points,
>> in contiguity, among most usual abstractions that represent
>> the surfaces of the manifolds of the geometry.
>>
>>
>
>
>
>
> Jakubowicz points out that Aristotle introduces a notion of contiguity.
> It's quite a great survey of Aristotle on continuity and the bibliography.
>
> A Problem in Aristotle's Continuity Theory, Sammy T. Jakubowicz
> The Trouble With Touching: A Problem in Aristotle's Continuity Theory
>
> https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=de4554185846fe34baf94cbb1ce77530ebe32ec7
>
>
> This is where, these days, people will often find "Aristotle must be our
> master so what he says must be the same as what is our standard today",
> yet that is mostly Eudoxus about the complete ordered field, that
> Aristotle's continuum also includes notions of the contiguity of points,
> so, it is so that both modern echoes of the coat-tailing wall-papering
> sort dont' say much at all and indeed obscure the fuller dialectic, and,
> such notions as the "divided, ad infinitum" follows the "infinitely
> divisible" concept, explored at least since Aristotle.
>
> So, when I say that "line-reals are as an Aristotle's continuum", it is
> like so.
>
>
> Then, this notion of "continuous topology" is a great complement
> to the notion of "continuous domain" that we have here today.
>

Points a lot at Ian Mueller: "Aristotle on Geometrical Objects".

https://philpapers.org/rec/MUEAOG

Published Online: 2009-07-17
Published in Print: 1970

"[Jakubowicz has] attempted to go beyond those writers who make
the claim that Aristotle's ideas of continuity, geometry, etc.,
were essentially those that _we_ hold."