Jon, List,
is it true, that, other than in dyadic relations, in triadic relations there are much more than one kind of symmetry? if the sets are X, Y, Z, and variables of elements of either sets are x, y, z with x E X, y E Y, z E Z:
three kinds of dyadic symmetry or linear symmetry:
-(x,y) implies (y,x) in L
-(y,z) implies (z,y)
-(z,x) implies (x,z).
And many kinds of triadic or rotatory symmetry:
-(x,y,z) implies (z,y,x)
-(y,z,x) implies (x,z,y)
-(z,x,y) implies (y,x,z)
-all three of the mentioned as one relation
-(x,y,z) implies (y,z,x)
-(y,z,x) implies (z,x,y)
-(x,y,z) implies (y,z,x) and (z,x,y)
-and all of these examples vice versa
and perhaps many more I have not seen?
So, is this really a complicated subject? Is it relevant for signs I dont know, but I always suspect, that mathematics are relevant for nature: I just have read in a book about plants, that flower organs, eg. the seeds in a sunflower, are arranged due to something known as Fibonacchi-row, triangle row, hare-problem, or golden cut. The evolution of categories 1-3-6-10-15-... also follows the Fibonacchi or triangle row. I just have the impression, that, once something gets triadic, it becomes quite complex.
Best,
Helmut
Gesendet: Montag, 28. Dezember 2015 um 05:10 Uhr
Von: "Jon Awbrey" <[email protected]>
An: "Helmut Raulien" <[email protected]>
Cc: "Peirce List" <[email protected]>
Betreff: Re: Relations & Their Relatives
Von: "Jon Awbrey" <[email protected]>
An: "Helmut Raulien" <[email protected]>
Cc: "Peirce List" <[email protected]>
Betreff: Re: Relations & Their Relatives
Inquiry Blog
http://inquiryintoinquiry.com/2015/12/08/relations-their-relatives-16/
http://inquiryintoinquiry.com/2015/12/10/relations-their-relatives-17/
http://inquiryintoinquiry.com/2015/12/12/relations-their-relatives-18/
http://inquiryintoinquiry.com/2015/12/22/relations-their-relatives-19/
Helmut, List,
I used braces {...} to indicate a set {x, y} of two elements.
Written order makes no difference to sets, so {x, y} = {y, x}.
I used parentheses (...) to indicate an ordered pair (x, y).
The ordered pairs (x, y) and (y, x) are distinct if x ≠ y.
We say that a dyadic relation L is "symmetric"
if (x, y) being in L implies that (y, x) is in L.
Regards,
Jon
On 12/23/2015 11:54 AM, Helmut Raulien wrote:
> Supplement: I think, in my below text there are a lot of mistakes: I mixed up
> tuples with products, I am not sure when to use round and when winged brackets,
> and the set of unordered triples does not consist of three, but of six sets of
> ordered ones, and I dont know what else. I must read your wikipedia papers about
> relations first, I think. It is very interesting, eg. it seems to me, that other
> than with dyadic products, with triadic ones there are a lot of different kinds
> of symmetry, rotational and linear. Happy Christmas and happy new Year!
> Jon, list,
> about ordered and unordered pairs: In the mathematical books I had read in,
> there was only the way of writing ordered pairs. And symmetry was only explained
> by the example of a subset of a product of two same sets (A x A). I had thought
> then, if you have two different sets, A and B, symmetry makes no sense, lest you
> look at a subset of {(A x B) U (B x A)}. You wrote, the short way of writing
> this, is {A x B}. That would be an aggregate of unordered pairs. Now, if you
> have three sets, A, B, C, then how do you write the unordered cartesian product?
> Should be {A, B, C} written in a triangle, with one "x" in the centre, or "{A x
> B x C}"? (short form) ? Or {(A x B x C) U (B x C x A) U (C x A x B)} (long form)?
> I still find relation reduction interesting. I must read your paper again, about
> the projective reduction, and try to find out, whether it is possible to
> projectively reduce a triadic relation of R, O, I to three dyadic ones, but not
> R-O, O-I, I-R, but R-R, R-O, R-I. Because I think, this is the Peircean way.
> Have you tried that? Im not sure, when I will have time resp. overcome my
> laziness to, well, just take a piece of paper and start. To just start is always
> the most difficult part, like with the tax declaration, but other than with
> that, there is no penalty if I dont.
> Best regards,
> Helmut
--
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http://inquiryintoinquiry.com/2015/12/08/relations-their-relatives-16/
http://inquiryintoinquiry.com/2015/12/10/relations-their-relatives-17/
http://inquiryintoinquiry.com/2015/12/12/relations-their-relatives-18/
http://inquiryintoinquiry.com/2015/12/22/relations-their-relatives-19/
Helmut, List,
I used braces {...} to indicate a set {x, y} of two elements.
Written order makes no difference to sets, so {x, y} = {y, x}.
I used parentheses (...) to indicate an ordered pair (x, y).
The ordered pairs (x, y) and (y, x) are distinct if x ≠ y.
We say that a dyadic relation L is "symmetric"
if (x, y) being in L implies that (y, x) is in L.
Regards,
Jon
On 12/23/2015 11:54 AM, Helmut Raulien wrote:
> Supplement: I think, in my below text there are a lot of mistakes: I mixed up
> tuples with products, I am not sure when to use round and when winged brackets,
> and the set of unordered triples does not consist of three, but of six sets of
> ordered ones, and I dont know what else. I must read your wikipedia papers about
> relations first, I think. It is very interesting, eg. it seems to me, that other
> than with dyadic products, with triadic ones there are a lot of different kinds
> of symmetry, rotational and linear. Happy Christmas and happy new Year!
> Jon, list,
> about ordered and unordered pairs: In the mathematical books I had read in,
> there was only the way of writing ordered pairs. And symmetry was only explained
> by the example of a subset of a product of two same sets (A x A). I had thought
> then, if you have two different sets, A and B, symmetry makes no sense, lest you
> look at a subset of {(A x B) U (B x A)}. You wrote, the short way of writing
> this, is {A x B}. That would be an aggregate of unordered pairs. Now, if you
> have three sets, A, B, C, then how do you write the unordered cartesian product?
> Should be {A, B, C} written in a triangle, with one "x" in the centre, or "{A x
> B x C}"? (short form) ? Or {(A x B x C) U (B x C x A) U (C x A x B)} (long form)?
> I still find relation reduction interesting. I must read your paper again, about
> the projective reduction, and try to find out, whether it is possible to
> projectively reduce a triadic relation of R, O, I to three dyadic ones, but not
> R-O, O-I, I-R, but R-R, R-O, R-I. Because I think, this is the Peircean way.
> Have you tried that? Im not sure, when I will have time resp. overcome my
> laziness to, well, just take a piece of paper and start. To just start is always
> the most difficult part, like with the tax declaration, but other than with
> that, there is no penalty if I dont.
> Best regards,
> Helmut
--
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
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