SUO: Re: Membrance Of Things Peirced
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
| I would he had continued to his country
| As he began, and not unknit himself
| The noble knot he made.
|
| Shakespeare, 'Coriolanus', Sicinius, 4.2.32-34
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
Pat Hayes wrote:
>
> > Yang Yun wrote:
> > >
> > > > > It also argues for a divide and conquer approach.
> > > > > This is why I support Nicola Guarino's recent call
> > > > > for a modular approach to reference ontologies.
> > >
> > > > Modular is nice, of course, if and when you can get it.
> > > > There is a catch here though, "triadic irreducibility".
> > >
> > > Google search for "triadic irreducibility"
> > > did not match any documents. whatever ...
> > > yy
> >
> >¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
> >
> > Y^2,
> >
> > FAST Search:
> >
> > http://www.alltheweb.com/cgi-bin/advsearch
> >
> > gave this very interesting one from the CG List:
> >
> > http://www.virtual-earth.de/CG/cg-list/msg02192.html
> >
> > If you use the exact phrase option then you will probably also
> > need to try "triadically irreducible" and "irreducibly triadic".
> > Or you could always try reading about it in Peirce -- his work
> > is in all of the big libraries -- you remember libraries dontcha?
>
> Allow me to interject a note of caution,
> before we waste yet more time on this
> old chestnut.
Darn, I was gonna to let this go --
but now you hadda go and make me --
> Peirce proved a result which he thought very important and fundamental.
> It is indeed correct, but it isnt deep or very important, so he was both
> right and wrong about it. The result was this. Consider nodes in a graph
> and classify them by their 'degree', ie the number of arcs that link to them.
> Then in order to build 'arbitrary' graph structures one must have at least one
> node of degree three. (Roughly, if you only have nodes of degree 2 then all
> you can do is make long rows of nodes; but once you have degree-three nodes,
> you can build arbitrarily branching structures by imitating degree-n nodes
> by sequences of n trinary nodes linked together, as in LISP or RDF.)
I agree with this much, and compliment you on the succinct presentation.
This is a very sincere appreciation, since I have been through many the
long wrangle about this issue that never quite managed to get this far.
> This is Peirce's 'irreducible triadicity',
> and he thought that it meant that there was
> something fundamentally trinary about identity.
Well, sorta. The piece about "teridentity", to which you advert here,
is only a corollary of the more general result about k-adic relations.
What is really at stake, for us, for our purposes,
or for the sake of our mediate time-frame of work,
the way that I see it, anyway, is a bit like this:
It has to do with the relative complexity of the things
that we with start with and the things that we end with,
in a piece of construction work that works on structures.
I made that description as open as I could so that we not
start out being distracted by all of the eventual details.
At this level of abstraction, generality, vagueness, or whatever,
we could express the typical form of the questions that we would
like to ask somewhat as follows:
| What is the biggest bang that we can get for our buck?
More formally:
| What is the greatest complexity of structured thing
| that we can get by a specified type of construction
| from structured things of given maximum complexity?
To express the question in this form is already to have answered
the aspect of it that seemed to be in doubt when we started this.
That is, as far as the bedevilled question as whether the cloven hoof
of triadicity has to turn up somewhere or other, well, the very idea of
taking many things and making another thing is already a matter of three
things at minimum. QED. And that is really all there is to it, except for
the circumstance that we usually have many more questions about the details
of particular cases and the wherewithal of concrete constructions.
That said, let us inquire further into the cases at hand.
Here, the "structured things" that we are working with are k-adic relations.
The measure of "relative complexity" that we attend to here is the "adicity",
the "arity", the "valence", or whatever you want to call it, of the relations.
The "construction process" is to be named -- I think that we need to consider
at least a couple of different species of operations:
1. Relational Composition.
2. Inverse Projection.
I am still looking for good names, ones that make for good compromises
between 19th & 20th Century usages, so these names may change with time.
We had a long discussion of this once before, so I will look up the links.
There are many reasons, beyond the settling of old historical turf battles,
why we ought to be interested in these sorts of questions -- these subjects
of analysis, basis, composition, construction, decomposition, deconstruction,
determination, factorization, irreducibility, modularity, primality, synthesis,
and several other similar types of characteristics that I have probably forgotten --
for they have to do with what sorts of complex relations are constructible and/or
definable, in very concrete ways, from other sorts of typically simpler relations.
I think that the utility of such a study for us ought to be obvious, but maybe not.
Our interest in the arity measure of relatively complexity
commonly splits into what is happening across two borders:
a. The border between 2 and 3.
b. The border between 3 and all higher k.
Crossed with the two forms of derivation operation that I named above,
this produces four different cases to organize the remaining questions.
1a. When it comes to reducing 3-adics to relative compositions of 2-adics,
conversely, of composing 2-adics to get 3-adics, this is a no-brainer.
The question is precisely analogous to the asking if you could ever
multiply two rectangular matrices and get anything other than yet
another rectangulat matrix. No way.
1b. This is the question of whether 3-adics, that is to say,
the right collection of 3-adics, is enough to compose
any k-adic relation. Yes.
When it comes to "projective reducibility", that is, the question
as to whether all of the j-adic projections of a k-adic relation T,
for a specified collection of j's < k, can supply enough information
to "determine", to "reconstruct", or to "recreate" T uniquely, then
we get a more varied series of answers. In fact, we found examples
of 3-adics that were reducible to their 2-adic projections, and we
found other 3-adics that were not uniquely reconstructible from
their 2-adic faces, that is, projectively irreducible triadics.
> However, the result is very fragile.
> If you allow hyperlinks in the graphs,
> it fails immediately. If you take Peirce's
> own algebra of identity and close it under
> inverses, there is an operation that merges
> two binary links into a trinary one, so that
> irreducibiity fails. And in any case the
> logical conclusion only follows if you
> restrict yourself to algebraic notations:
> as soon as you allow quantification into
> the language, everything can be said using
> binary and unary relations, as has been
> well-known now for about 50 years or more.
> So for this particular issue, don't bother
> with the library: just ignore it. It has
> absolutely nothing whatever to do with
> modularity.
All of these arguments are variations on a common theme.
Peirce himself once described them as falling under the
rubric, or aubric, of the "Alchemist's Argument".
The French explain it to their children as the
the "Stone Soup Story". All fairy tales.
I do not think that they need to be
dignified with a further comment.
But I could be wrong about that.
> Peirce was misled by an analogy between logic and chemistry,
At least he did not leave off with alchemy ...
> which wasnt a bad idea in 1885, but seems kind of daft in hindsight.
Some people have more hindsight than others ...
> He seems to have thought that the associations between relations
> and their instances, which he was encoding as arcs in his graphs,
> were like valency in chemistry, so that relations (including that
> of identity) had a kind of intrinsic associated 'atomic number',
> from which it follows that the relation of two things being
> identical has to be different from that of three things being
> identical. This particular analogy was similar in some ways
> to Kepler's idea that the planetary distances from the sun
> arose from packing the regular solids into nesting spheres:
> neat, ingenious, apparently successful; but wrong.
Do you get all your history from Classic Comics, or something?
Yeah, this gullible old dodder Peirce, who worked out a version
of non-standard analysis that compares favorably in its rigor
and its scope with those devised thirty years later, was so
easily suckered by this or that half-baked analogy. Really!
Jon Awbrey
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤