Helmut, List,
That's not really supposed to be a recursive definition.
It's just my sloppy notation or lack thereof that makes
it seem so. I wrote it more precisely in my article on
Relation Theory:
http://intersci.ss.uci.edu/wiki/index.php/Relation_theory
<QUOTE>
Following the pattern of the functional case, let the notation
“L ⊆ X × Y” bring to mind a mathematical object specified by
three pieces of data, the set X, the set Y, and a particular
subset of their cartesian product X × Y}. As before we have
two choices, either let L = (X, Y, graph(L)) or let “L” denote
graph(L) and choose another name for the triple.
</QUOTE>
For a k-place relation, then, we let “L” denote the (k+1)-tuple
and use a new name “graph(L)” for the subset of X_1 × ... × X_k,
yielding L = (X_1, ..., X_k, graph(L)).
Regards,
Jon
On 4/19/2017 4:24 PM, Helmut Raulien wrote:
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
17. April 2017 um 16:05 Uhr
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
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