Jon, List,
interesting! I have a guess, regarding the possible bridge between the "strong typing" relation concept in mathematics, and semiotics and other theories. I know this is anticipation towards much later, and we dont want to do this now, but first talk about mathematics only. So, it is merely to create a holding power for people who read this, by showing that indeed mathematics might be able to contribute much to semiotics and other theories:
The k+1-tuple reminds me of the re-entry concept by Spencer Brown, and also of the term "sign" being used by Peirce for both the triad and a part of it, the representamen, and also the concept of "self-reference" in systems theories: In the k+1-tuple, there is also the whole thing (the relation "L") a part of itself.
The term "quality" in the context you have used it (fourth-last line) reminds me of secondness having two modes, firstness and secondness of secondness: In Peirces "On a new List of Categories" "Relation" is the second category, so it should have two modes. Maybe the quality (which is the first category in (On a new list...") of a relation (eg. "smaller than", or "random" is the firstness of the relation, and the actual subset (plus the relation, or plus the domains, or neither) is secondness (of the secondness, the relation). Just guesses!
Best,
Helmut
Helmut, List ...
The difference between the two definitions is sometimes
described as “decontextualized” versus “contextualized”
or, in computerese, “weak typing” versus “strong typing”.
The second definition is typically expressed by means of
a peculiar mathematical idiom that starts out as follows:
“A k-place relation is a k+1-tuple (X_1, …, X_k, L) …”
That way of defining relations is a natural generalization
of the way functions are defined in the mathematical subject
of category theory, where the “domain” X and the “codomain” Y
share in defining the “type” X → Y of the function f : X → Y.
The threshold between “arbitrary”, “artificial”, “random” kinds of
relations and those selected for due consideration as “reasonable”,
“proper”, “natural” kinds tends to shift from context to context.
We usually have in mind some property or quality that marks the
latter class as “proper” objects of contemplation relative to
the end in view, and so this relates to the intensional view
of subject matters.
Regards,
Jon
On 4/15/2017 2:49 PM, Helmut Raulien wrote:
> Jon, List,
> Thank you, Jon! Your point No. 2 is new to me, that some
> define relation not only as the subset of the domains'
> cartesian product, but as that plus a list of the domains.
>
> In case the subset is not a random one, but a consequence
> of some reasonable classification, eg. in a dyadic relation:
> "x_1 < x_2", what is this term "x_1 < x_2" called then?
> I am asking, because I think, that in common language this
> is what people might understand as relation. I had called
> it "relation reason" before.
> Best,
> Helmut
>
> 15 April 2017 um 16:30 Uhr
>> "Jon Awbrey" <[email protected]> wrote:
>> Helmut, List,
>>
>> Looking over those articles with fresh eyes this morning
>> I see they are rather thick with abstract generalities at
>> the beginning and it would be better to skip down to the
>> concrete examples on a first run-through. I promise to
>> keep that in mind the next time I rewrite them. At any
>> rate, we can always go through the material in a more
>> leisurely fashion on the List.
>>
>> Looking back over many previous discussions, I think one
>> of the main things keeping people from being on the same
>> page, or even being able to understand what others write
>> on their individual pages, is the question of what makes
>> a relation.
>>
>> There's a big difference between a single ordered tuple, say,
>> (x_1, x_2, ..., x_k), and a whole set of ordered tuples that
>> it takes to make up a k-place relation. The language we use
>> to get a handle on the structure of relations goes like this:
>>
>> Say the variable x_1 ranges over the set X_1,
>> and the variable x_2 ranges over the set X_2,
>> ...
>> and the variable x_k ranges over the set X_k.
>>
>> Then the set of all possible k-tuples (x_1, x_2, ..., x_k)
>> ranges over a set that is notated as X_1 × X_2 × ... × X_k,
>> called the “cartesian product” of the “domains” X_1 to X_k.
>>
>> There are two different ways of defining
>> a k-place relation that are in common use:
>>
>> 1. Some define a relation L on the domains X_1 to X_k
>> as a subset of the cartesian product X_1 × ... × X_k,
>> in symbols, L ⊆ X_1 × ... × X_k.
>>
>> 2. Others like to make the domains of the relation
>> an explicit part of the definition, saying that
>> a relation L is a list of domains plus a subset
>> of their cartesian product.
>>
>> Sounds like a mess but it's usually pretty easy to
>> translate between the two conventions, so long as
>> one remains aware of difference.
>>
>> By way of a geometric image, we can picture the
>> cartesian product X_1 × ... × X_k as a space in
>> which many different relations reside, each one
>> cutting a different figure in that space.
>>
>> To be continued ...
>>
>> Jon
>>
>> On 4/15/2017 12:00 AM, Jon Awbrey wrote:
>>> Helmut, List,
>>>
>>> I think it would be a good idea to continue reviewing basic concepts
>>> and get better acquainted with the relational context that is needed
>>> to ground all the higher order functions, properties, and structures
>>> we might wish to think about. Once we understand what relations are
>>> then we can narrow down to triadic relations and then sign relations
>>> will fall more easily within our grasp.
>>>
>>> I've written up intros to these topics many times before, and you
>>> can find my latest editions, if still very much works in progress,
>>> on the InterSciWiki site, though in this case it may be preferable
>>> to take them up in order from special to general:
>>>
>>> Sign Relations
>>> http://intersci.ss.uci.edu/wiki/index.php/Sign_relation
>>>
>>> Triadic Relations
>>> http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation
>>>
>>> Relation Theory
>>> http://intersci.ss.uci.edu/wiki/index.php/Relation_theory
>>>
>>> I think most of the material you mentioned on Relational Reducibility,
>>> Compositional and Projective, is summarized in the following article:
>>>
>>> Relation Reduction
>>> http://intersci.ss.uci.edu/wiki/index.php/Relation_reduction
>>>
>>> Regards,
>>>
>>> Jon
>>
>
--
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The difference between the two definitions is sometimes
described as “decontextualized” versus “contextualized”
or, in computerese, “weak typing” versus “strong typing”.
The second definition is typically expressed by means of
a peculiar mathematical idiom that starts out as follows:
“A k-place relation is a k+1-tuple (X_1, …, X_k, L) …”
That way of defining relations is a natural generalization
of the way functions are defined in the mathematical subject
of category theory, where the “domain” X and the “codomain” Y
share in defining the “type” X → Y of the function f : X → Y.
The threshold between “arbitrary”, “artificial”, “random” kinds of
relations and those selected for due consideration as “reasonable”,
“proper”, “natural” kinds tends to shift from context to context.
We usually have in mind some property or quality that marks the
latter class as “proper” objects of contemplation relative to
the end in view, and so this relates to the intensional view
of subject matters.
Regards,
Jon
On 4/15/2017 2:49 PM, Helmut Raulien wrote:
> Jon, List,
> Thank you, Jon! Your point No. 2 is new to me, that some
> define relation not only as the subset of the domains'
> cartesian product, but as that plus a list of the domains.
>
> In case the subset is not a random one, but a consequence
> of some reasonable classification, eg. in a dyadic relation:
> "x_1 < x_2", what is this term "x_1 < x_2" called then?
> I am asking, because I think, that in common language this
> is what people might understand as relation. I had called
> it "relation reason" before.
> Best,
> Helmut
>
> 15 April 2017 um 16:30 Uhr
>> "Jon Awbrey" <[email protected]> wrote:
>> Helmut, List,
>>
>> Looking over those articles with fresh eyes this morning
>> I see they are rather thick with abstract generalities at
>> the beginning and it would be better to skip down to the
>> concrete examples on a first run-through. I promise to
>> keep that in mind the next time I rewrite them. At any
>> rate, we can always go through the material in a more
>> leisurely fashion on the List.
>>
>> Looking back over many previous discussions, I think one
>> of the main things keeping people from being on the same
>> page, or even being able to understand what others write
>> on their individual pages, is the question of what makes
>> a relation.
>>
>> There's a big difference between a single ordered tuple, say,
>> (x_1, x_2, ..., x_k), and a whole set of ordered tuples that
>> it takes to make up a k-place relation. The language we use
>> to get a handle on the structure of relations goes like this:
>>
>> Say the variable x_1 ranges over the set X_1,
>> and the variable x_2 ranges over the set X_2,
>> ...
>> and the variable x_k ranges over the set X_k.
>>
>> Then the set of all possible k-tuples (x_1, x_2, ..., x_k)
>> ranges over a set that is notated as X_1 × X_2 × ... × X_k,
>> called the “cartesian product” of the “domains” X_1 to X_k.
>>
>> There are two different ways of defining
>> a k-place relation that are in common use:
>>
>> 1. Some define a relation L on the domains X_1 to X_k
>> as a subset of the cartesian product X_1 × ... × X_k,
>> in symbols, L ⊆ X_1 × ... × X_k.
>>
>> 2. Others like to make the domains of the relation
>> an explicit part of the definition, saying that
>> a relation L is a list of domains plus a subset
>> of their cartesian product.
>>
>> Sounds like a mess but it's usually pretty easy to
>> translate between the two conventions, so long as
>> one remains aware of difference.
>>
>> By way of a geometric image, we can picture the
>> cartesian product X_1 × ... × X_k as a space in
>> which many different relations reside, each one
>> cutting a different figure in that space.
>>
>> To be continued ...
>>
>> Jon
>>
>> On 4/15/2017 12:00 AM, Jon Awbrey wrote:
>>> Helmut, List,
>>>
>>> I think it would be a good idea to continue reviewing basic concepts
>>> and get better acquainted with the relational context that is needed
>>> to ground all the higher order functions, properties, and structures
>>> we might wish to think about. Once we understand what relations are
>>> then we can narrow down to triadic relations and then sign relations
>>> will fall more easily within our grasp.
>>>
>>> I've written up intros to these topics many times before, and you
>>> can find my latest editions, if still very much works in progress,
>>> on the InterSciWiki site, though in this case it may be preferable
>>> to take them up in order from special to general:
>>>
>>> Sign Relations
>>> http://intersci.ss.uci.edu/wiki/index.php/Sign_relation
>>>
>>> Triadic Relations
>>> http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation
>>>
>>> Relation Theory
>>> http://intersci.ss.uci.edu/wiki/index.php/Relation_theory
>>>
>>> I think most of the material you mentioned on Relational Reducibility,
>>> Compositional and Projective, is summarized in the following article:
>>>
>>> Relation Reduction
>>> http://intersci.ss.uci.edu/wiki/index.php/Relation_reduction
>>>
>>> Regards,
>>>
>>> Jon
>>
>
--
inquiry into inquiry: https://inquiryintoinquiry.com/
academia: https://independent.academia.edu/JonAwbrey
oeiswiki: https://www.oeis.org/wiki/User:Jon_Awbrey
isw: http://intersci.ss.uci.edu/wiki/index.php/JLA
facebook page: https://www.facebook.com/JonnyCache
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