Jon, Jerry, List,
I think, what you wrote is interesting: That a (matematical) relation is a subset of the set of possible tuples (cartesian product) that are made from the elements of two sets, a tuple being a pair of one element from set A and one element from set B. Set B can also again be set A, then it is the relation upon A. The subset can be chosen arbitrarily, then the reason for the relation is the free will of the mathematician at work. Or it may be something like "smaller than", then it is the set of tuples in which the first element (from set A) is smaller than the second (taken from B). In this case the relation has a natural reason, but it still takes a person to see, that one element is smaller than the other. And this person also has to define A and B.
Why am I writing this? I was thinking of how can this mathematics be applied to semiotics. Semiotics is mataphysics, that is universal theory of nature. So I was thinking: In which way can nature, reality, be separated into two sets? My answer is: These two sets are the set of events, and the set of entities. This is the most fundamental natural distinction of reality into two sets, that does not need an observer concept, because this distinction has existed yet in preorganic nature. It is natural.
Relation now is in the first place a relation between an event and an entity. It must be a symmetrical relation, because in preorganic nature there was no mathematician to define which is set A and which is set B.
Now there are two ways to continue: Either compare these considerations with Peirce, and ask: Has the triad "Event, entity, relation" something to do with "1ness, 2ness, 3ness", or with "representamen, object, interpretant"?
Or the second way of approach is, to develop a semiotics and a systems theory based on the triad "event, entity, relation", not caring about Peirce first, but waiting to see later, whether there will be parallelities. I think this is a good idea. I want to follow it, but maybe somebody else smarter than me can do that faster and more efficient.
Best,
Helmut
Gesendet: Mittwoch, 18. Februar 2015 um 22:22 Uhr
Von: "Jerry LR Chandler" <[email protected]>
An: "Peirce List" <[email protected]>
Cc: "Jon Awbrey" <[email protected]>
Betreff: Re: [PEIRCE-L] Re: Relations & Their Relatives
Von: "Jerry LR Chandler" <[email protected]>
An: "Peirce List" <[email protected]>
Cc: "Jon Awbrey" <[email protected]>
Betreff: Re: [PEIRCE-L] Re: Relations & Their Relatives
List, Jon
Your post wrt Number Theory is very revealing concerning the origins of your beliefs with respect to matter / material world / reality in contrast to the world of perceptions, thoughts about the world "out-there." Can it be re-evaluated from an alternative perspective of the notion of (biological) relations?
You appear to base your suppositions on the logical operations of Division as opposed to the natural science concept of atoms as being indivisible objects. Indivisible matter means not separable into equal size parts (to be compared with division as merely logical separation of parts of a whole.)
The concept of irreducible atoms (as non-divisible objects) is accepted today and manifests itself in the chemical table of elements as being unique objects, each defined in terms of its perceived and measured qualia,
AND as separable into part-whole relations (ie, Quantum mechanical descriptions may correspond with some data and organization of part-whole relations.)
By way of reference, in the short little book on "Number Theory," by George E Andrews (Dover, 1971),
Number Theory is separated into two mathematical forms.
One form is the theory of division and focuses on the qualia of integers as Prime Numbers and Factorization. These aspects of number theory are critical to group theory, linear algebra, topology, etc., and, for science, the factorization of polynomials so that collections of connected qualia can be separated.
The second form of number theory is addition. The number theory of addition focuses on the study of part-whole relations among the integers.
These two basic forms of number theory are causes for deep conundrums between the logic of the continuum and the propositional logic of discrete terms, as we have previously discussed here. The example to ponder is the nature of biological relatives.
I would conjecture two questions concerning the general nature of your posts, which often befuddle me (and others as well.) My conjectures are both wide and wild (breadth and comprehension of information) so feel free to alter them by offering more realistic/pragmatic hypotheses.
>From this perspective, is it possible that the focus of your life-long writings seeks to illate the pathways from Peician writings to set theoretic terminology? (Possibly your purpose could be to demonstrate a consistency between Shannon's notions used in computer science logic and CSP?)
And that the division operations are the principle mode of constructing such a path, in parallel to the methodology used by Cantor? (Possibly, your beliefs could be that Peircian logic is different from but the same as set theory logic. Just tweak the definitions a tad here, a tad there, another tad here, another tad there, and so forth, until... Pesto! the "big pictures" coincide and, extremely roughly and metaphorically speaking, CSP = Cantor/Russell.
Cheers
Jerry
On Feb 16, 2015, at 11:00 PM, Jon Awbrey wrote:
> Submitted for your approval, two ways of looking at the
> “divisor of” relation, one of the most fundamental examples
> of a binary (dyadic or two-place) relation in number theory.
>
> Table 1 shows the first few ordered pairs of the relation
> on positive integers that corresponds to the relative term,
> “divisor of”. Thus, the ordered pair i:j appears in the
> relation if and only if i divides j, for which the usual
> notation is i|j.
>
> Table 1. Elementary Relatives for the “Divisor Of” Relation
> ☞http://inquiryintoinquiry.com/elementary-relatives-for-the-divisor-of-relation/
>
> Table 2 shows the same information in the form of a logical matrix.
> This has a coefficient of 1 in row i and column j when i|j, otherwise
> it has a coefficient of 0. (The zero entries have been omitted here
> for ease of reading.)
>
> Table 2. Logical Matrix for the “Divisor Of” Relation
> ☞http://inquiryintoinquiry.com/logical-matrix-for-the-divisor-of-relation/
>
> Just as matrices in linear algebra represent linear transformations,
> these logical arrays and matrices represent logical transformations.
>
> Regards,
>
> Jon
>
> --
>
> academia: http://independent.academia.edu/JonAwbrey
> my word press blog: http://inquiryintoinquiry.com/
> inquiry list: http://stderr.org/pipermail/inquiry/
> isw: http://intersci.ss.uci.edu/wiki/index.php/JLA
> oeiswiki: http://www.oeis.org/wiki/User:Jon_Awbrey
> facebook page: https://www.facebook.com/JonnyCache
> <Elementary Relatives for the “Divisor Of” Relation.png><Logical Matrix for%20%74%68%65%20%E2%80%9C%44%69%76%69%73%6F%72%20%4F%66%E2%80%9D%20%52%65%6C%61%74%69%6F%6E%2E%70%6E%67.png>
> -----------------------------
> PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to [email protected] . To UNSUBSCRIBE, send a message not to PEIRCE-L but to [email protected] with the line "UNSubscribe PEIRCE-L" in the BODY of the message. More at http://www.cspeirce.com/peirce-l/peirce-l.htm .
>
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>
-----------------------------
PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to [email protected] . To UNSUBSCRIBE, send a message not to PEIRCE-L but to [email protected] with the line "UNSubscribe PEIRCE-L" in the BODY of the message. More at http://www.cspeirce.com/peirce-l/peirce-l.htm .
Your post wrt Number Theory is very revealing concerning the origins of your beliefs with respect to matter / material world / reality in contrast to the world of perceptions, thoughts about the world "out-there." Can it be re-evaluated from an alternative perspective of the notion of (biological) relations?
You appear to base your suppositions on the logical operations of Division as opposed to the natural science concept of atoms as being indivisible objects. Indivisible matter means not separable into equal size parts (to be compared with division as merely logical separation of parts of a whole.)
The concept of irreducible atoms (as non-divisible objects) is accepted today and manifests itself in the chemical table of elements as being unique objects, each defined in terms of its perceived and measured qualia,
AND as separable into part-whole relations (ie, Quantum mechanical descriptions may correspond with some data and organization of part-whole relations.)
By way of reference, in the short little book on "Number Theory," by George E Andrews (Dover, 1971),
Number Theory is separated into two mathematical forms.
One form is the theory of division and focuses on the qualia of integers as Prime Numbers and Factorization. These aspects of number theory are critical to group theory, linear algebra, topology, etc., and, for science, the factorization of polynomials so that collections of connected qualia can be separated.
The second form of number theory is addition. The number theory of addition focuses on the study of part-whole relations among the integers.
These two basic forms of number theory are causes for deep conundrums between the logic of the continuum and the propositional logic of discrete terms, as we have previously discussed here. The example to ponder is the nature of biological relatives.
I would conjecture two questions concerning the general nature of your posts, which often befuddle me (and others as well.) My conjectures are both wide and wild (breadth and comprehension of information) so feel free to alter them by offering more realistic/pragmatic hypotheses.
>From this perspective, is it possible that the focus of your life-long writings seeks to illate the pathways from Peician writings to set theoretic terminology? (Possibly your purpose could be to demonstrate a consistency between Shannon's notions used in computer science logic and CSP?)
And that the division operations are the principle mode of constructing such a path, in parallel to the methodology used by Cantor? (Possibly, your beliefs could be that Peircian logic is different from but the same as set theory logic. Just tweak the definitions a tad here, a tad there, another tad here, another tad there, and so forth, until... Pesto! the "big pictures" coincide and, extremely roughly and metaphorically speaking, CSP = Cantor/Russell.
Cheers
Jerry
On Feb 16, 2015, at 11:00 PM, Jon Awbrey wrote:
> Submitted for your approval, two ways of looking at the
> “divisor of” relation, one of the most fundamental examples
> of a binary (dyadic or two-place) relation in number theory.
>
> Table 1 shows the first few ordered pairs of the relation
> on positive integers that corresponds to the relative term,
> “divisor of”. Thus, the ordered pair i:j appears in the
> relation if and only if i divides j, for which the usual
> notation is i|j.
>
> Table 1. Elementary Relatives for the “Divisor Of” Relation
> ☞http://inquiryintoinquiry.com/elementary-relatives-for-the-divisor-of-relation/
>
> Table 2 shows the same information in the form of a logical matrix.
> This has a coefficient of 1 in row i and column j when i|j, otherwise
> it has a coefficient of 0. (The zero entries have been omitted here
> for ease of reading.)
>
> Table 2. Logical Matrix for the “Divisor Of” Relation
> ☞http://inquiryintoinquiry.com/logical-matrix-for-the-divisor-of-relation/
>
> Just as matrices in linear algebra represent linear transformations,
> these logical arrays and matrices represent logical transformations.
>
> Regards,
>
> Jon
>
> --
>
> academia: http://independent.academia.edu/JonAwbrey
> my word press blog: http://inquiryintoinquiry.com/
> inquiry list: http://stderr.org/pipermail/inquiry/
> isw: http://intersci.ss.uci.edu/wiki/index.php/JLA
> oeiswiki: http://www.oeis.org/wiki/User:Jon_Awbrey
> facebook page: https://www.facebook.com/JonnyCache
> <Elementary Relatives for the “Divisor Of” Relation.png><Logical Matrix for%20%74%68%65%20%E2%80%9C%44%69%76%69%73%6F%72%20%4F%66%E2%80%9D%20%52%65%6C%61%74%69%6F%6E%2E%70%6E%67.png>
> -----------------------------
> PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to [email protected] . To UNSUBSCRIBE, send a message not to PEIRCE-L but to [email protected] with the line "UNSubscribe PEIRCE-L" in the BODY of the message. More at http://www.cspeirce.com/peirce-l/peirce-l.htm .
>
>
>
>
-----------------------------
PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to [email protected] . To UNSUBSCRIBE, send a message not to PEIRCE-L but to [email protected] with the line "UNSubscribe PEIRCE-L" in the BODY of the message. More at http://www.cspeirce.com/peirce-l/peirce-l.htm .
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