A visual proof of the existence on an anti-symmetric complementary space S* of gaps (an other space) within the traditional mathematical continuum X=[0,1].

Found in (Anishchenko, Zenkin et al. 2003 – Semantic symmetry of a spiritual)

Jessicatz Fairymeadow tries to do another Pile Implementation with a view to implement the used data types as C++ Templates. The code is meant for Linux32- and 64-Bit (not yet tested for Win).

Please find it in the SVN Repository with the URL: http://data.fieldsofnoise.org/repository/jessicatz/pile.

The main program is pile_test.cpp, the other six files having some resemblance to a “framework”.

Cf. also the thread “Logical Domains in Pile” in the pile development mailing list.

Just revisited George’s open public letter and read the comments from May 17th ff., which I had not noticed but now. I feel, George understood me correctly. I maintain a mostly academic inspiration about the subject.

In Dominique Paß’s book “Bewußtsein und Ästhetik” ¹ you find a short excerpt of L. H. Kauffman’s paper “Self-reference and recursive forms” ².

Figure: “primary (primordial) self-reference”

The same visualization could be found in (Westphal 2006 – Freeing Data From the Silos)³, but without the mathematical background provided by Kauffman’s paper.

Kauffman’s visualization of the “primary (primordial) self-reference”, where the aspect of the original self-reference is stressed, has the same qualities as Spencer-Brown’s “mark of distinction”. Besides “mark”, Kauffman denotes the marking, that is given in the form of the arrow here also pointing the way ahead, as “barb” or “pointer” as well. (cf. combinative pointer)

Paß writes on p. 162 quoting Kauffman:

“Therefore, self-reference and the idea of distinction are inseparable (hence conceptually identical).” (…) Die Markierung (“pointer”) wird konstruktiv, insofern mit der Markierung der Raum entsteht, auf den die Markierung verweist. Mehr noch: “the mark points [...] to its own location, and in this process becomes a locus of reference. The mark referes to itself. The whole refers to itself through the mark.” (…)

1Paß, Dominik (2006): Bewusstsein und Ästhetik. Die Faszination der Kunst. Bielefeld: Aisthesis. (http://www.aisthesis.de/titel/3895285889.htm)

2Kauffman, Louis H. (1987): Self-reference and recursive forms. In: Journal of social and biological structures, 10, pp. 53–72.

3Westphal, Ralf (2006): Freeing Data From the Silos. A Relationistic Approach to Information Processing. Available online at http://www.pilesys.com/new/Documents/Freeingdatafromthesilos-Pileexplained.pdf, esp. pp. 13–14.

Perl 6 introduces the concept of junctions: values that are composites of other values. [24] In the earliest days of Perl 6’s design, these were called “superpositions”, by analogy to the concept in quantum physics of quantum superpositions — waveforms that can simultaneously occupy several states until observation “collapses” them. A Perl 5 module released in 2000 by Damian Conway called Quantum::Superpositions [25] provided an initial proof of concept. While at first, such superpositional values seemed like merely a programmatic curiosity, over time their utility and intuitiveness became widely recognized, and junctions now occupy a central place in Perl 6’s design.

(Source: http://en.wikipedia.org/wiki/Perl_6)

Keywords: junctions, composite value, superposition, Perl 6, design

Since the action of joining or being joined is evidently one of the central issues of this blog, a definition of junction might be useful (definition understood as communicative act to describe a phenomenon).

WordNet 2.1 Synonyms Only says:

/noun./ spot, topographic point, place, unification, union, connexion, link, connection, connective, connector, connecter, joining,

. …

The Quantum::Superpositions module provides a new scalar data structure: the superposition. In a metaphor drawn from quantum mechanics, superpositions store a collection of values by overlaying them in parallel superimposed states within a single scalar variable.
. …

A junction is a superposition of data values pretending to be a single data value.
. …

junction – the shape or manner [sic!] in which things come together and a connection is made
. …

–> http://multiforms.netfirms.com/multiforms_1.html#download

[...] A Prolog Theorem Prover simplifying Logic, by using Multiple Forms The strategy of this Prolog program (“mflogic.exe”, which you can download) is essentially the same as the strategy of a (human) theorem-prover, who knows the Axioms of Multiple Forms: As much as possible, all logic formulae are progressively reduced, by cancelling out “Outer Parts” if these are also found in “Inner Parts” of expressions (using Axiom 3). They are also reduced by “ the All” (=One) “absorbing anything” that “exists outside itself” (using Axom 1), or (finally) reduced by pairs of identical forms “ collapsing” when they apply to each other, “distinguishing each other” (XOR-wise, by Axiom 2).

The Prolog program “includes some extensive automatic comments, sprinkled over the derivation steps, so it can become an educational tool for learning Multiple Form Logic™.”

Find a screenshot here:

George, a.k.a. Omadeon,

“wrote it recently, coming back to Multiple Form Logic™ after a long period of absence from this field, and it is still Version 1. Future versions planned may include graphic representations of Multiple Form simplifications, which are quite spectacular, even on paper, like watching an avalanche of ‘bubbles’ breaking and re-organising themselves.”

By the way, what are traditional Propositional Logic formulae? The Prolog program translates them into Multiple Form Logic™, and then uses repeatedly the Three Fundamental Axioms of Multiple Form Logic as re-write rules, until the resulting expression is irreducible. (You can either pick a formula from a library, or write your own). Then, the result is converted back into Propositional Calculus. Multiple Form Logic™ is supposed to do better than just ‘prove theorems’ to be ‘true’ or ‘false’; but actually “optimizes logic expressions, regardless of whether or not they are reducible to true or false.”

Had a closer look at the video Entailments_Context.mov of Dr Paul Pangaro teaching Cybernetics and Pask’s Entailment Meshes at Stanford:

• Coherence
• Unit of memory

Going through omadeon’s LoF bookmarks on del.icio.us, I found Nick Green’s paper: (Green 2004 – Axioms from Interactions of Actors). It deals with Gordon Pask’s Interactions of Actors Theory and his earlier Conversation Theory.

The introduction of Nick’s paper talks about a special form as a computing element comparable to the “Pile object”:

The central object is the stable concept triple which takes the form of the Borromean link. The potential of this form as a concurrent computing element and a model of continuity is discussed.

Cf. the earlier item Symbols And Borromean Links inserted in this blog, and the videos of Dr Paul Pangaro teaching Cybernetics and Pask’s Entailment Meshes at Stanford:

[Regarding Semantic Nets]

In a sense, there is too much freedom. And if you have too much freedom, there is no way to go.

A decision restricts the degree of freedom of another decision thus constructing a beginning-midpoint-end configuration, i.e. one way. (German: Eine Entscheidung begrenzt den Freiheitsgrad der anderen Entscheidung, konstruiert aber eine Anfang Mitte Ende – Konfigurtation, d.h. einen Weg.)

And then one day, these entailment meshes came along.

One year after the “Multiple Form Logic” post (see [1] ), George A. Stathis alias omadeon is part of the “Pile” community for some inspiring days now. You are welcome!

We can correct a small mistake in that one year old post (see George’s recent comment [2] ). The mistake is all about boundary (and a self-reference operator).

Excursus Boundary Mathematics

• (Kauffman 2002 – Box Arithmetic)
• (Bricken 1995 – Distinction networks)
• (Laws of Form – Wikipedia, the free encyclopedia)

Boundary mathematics provides new conceptual tools, such as <void>, representational space, distinction, observer perspective, pervasion, ensemble, and environment, which permit modeling strongly parallel problem solving with locally coordinated systems of simple agents. (Bricken 1995 – Distinction networks, p. 14)

The origin concept of ‘boundary mathematics’ is the void:

“Distinctions are constructed in an empty context, a representational space. Drawing a closed loop on the page, for example, indicates a distinction, apparently cleaving the page into two parts, an inside and an outside. The page itself, however, is not torn asunder, it remains whole. Distinctions do not interact with their substrate. The representative space has no metric, it is devoid of characteristics and thus transparent to operations on distinctions. The representational space pervades all distinctions it contains; it is both the outside and the inside. Pervasiveness means that a distinction does not create a Cartesian duality; context and content are not separated by EXCLUSIVE-OR, they are associated by INCLUSIVE-OR.

The representational space provides a unique tool which is not present in traditional notations: it can be void (unmarked or empty). The void cannot be accessed directly, since it is not perceptible. Distinctions provide an indirect access: an empty distinction indicates a void content. Since the void can be indicated, it can be used semantically, to (non)represent concepts in the modeled domain.

Boundary mathematics is based on representations of distinctions as containers. Its origin concept is the void. The introduction of a useful void is analogous to the introduction of a useful zero in number systems. Just as zero overcomes a weakness in Roman numerals by permitting efficient algorithms for multiplication, the void overcomes a weakness in logical notation by permitting efficient algorithms for deduction.” (Bricken, William (1995): Distinction networks. Available online at [4],
p. 4)

Drawing a closed loop on the page…
And it’s one, two, three, what (notation) are we fighting for?

There are mainly two kinds of notations used by the “Pile” community:

• Ralf Westphal’s shoe->string notation and
• the one inspired by Miriam (see [5] ), the PILE v0.1 Notation.

I slightly modified the latter so that we end up with the following image, where we are drawing three closed loops, which hang together in an ensemble called “the Pile object” (with the dotted horizontal line symbolizing an extra distinction, ‘child’/'parents’):

Drawing a closed loop on the page, for example the one marked by z, indicates a distinction, whereas the shoe->string notation doesn’t deal with distinctions at all.

Finally, a “Pile” pervades all distinctions it contains. It is sort of a page Bricken is talking about, i.e. one incarnation of the representative space mentioned by him, but a ‘page’ by means of computer sand.

Anyway, all conflicts of the past aside (no more fighting!), George A. Stathis alias omadeon began to relate Multiple Form Logic (MFL for short) with “PILE objects” in a post to his blog (see [6] ). There, he confronted his MFL notation with the “Pile” shoe->string notation.

George’s/omadeon’s first result is this:

The emerging fundamental conceptual difference between Pile Objects and Multiple Forms is that Pile objects are built on spaces which are a priori unique and distinct (the “Terminal Values”, TV’s) and they are described as “directed relations”, whereas Multiple Forms are constructed always on the same (undistinguished) Void Space, and they are described as (multiple) Distinctions, which are -of course- also directed by virtue of their implicit distinction between inside and outside.

Not half bad! But I hesitate to buy this story. “Pile” objects aren’t build on spaces (plural). It gave me some trouble to boot-strap one unique “Pile” object, (see [7] and the complement in [8] ) — with the help of three others (four ensembles in sum). But, note that all this is done in the same representational space (singular), in the same frame of reference. (The same page!)

It is a matter of taste, if you begin with an a priori ‘outside’ (the “Terminal Values”, TV’s — e.g. a spoon), or let a thing called outside emerge as a simulation of nature constructed out of a “Pile” (“There is no spoon.”). Terminal values (TVs) are distinctions and connections again — but on the other hand. TVs are turned inside out, without lo(o)sing the connection to the other side. They are the constructed components with which the “Pile” assembles its ‘outside’. (Pretty much the same as the brain generates its environment.)

So, I hesitate to see “fundamental conceptual difference[s] between Pile Objects and Multiple Forms”. I agree with George/omadeon that

both systems deal with relations (or distinctions) which are (1) Multiple, and (2) based on undefinded _distinct_ irreducible [...] entities, which are never assumed to be the same

but deleted the “external” in the comment quoted above (see [9] ), and demonstrated in [7] and [8] how we could define a “Pile” ensemble unique to a “Pile” consisting of at least four such ensembles.

Much more could be said, e.g. about “directed relations”, the self-reference operator of a “Pile”, and… But, I feel it’s enough for the 365th day after my first MFL post, and I’m looking forward to continue the exchange with omadeon and our new team. Make our minds work in new directions!

Thanks
Ralf B.

References

[1] http://ralfbarkow.wordpress.com/2006/05/19/multiple-form-logic/

[2] http://ralfbarkow.wordpress.com/2006/05/19/multiple-form-logic/#comment-3898

[3] http://ralfbarkow.wordpress.com/2006/05/20/first-axiom-of-george-alexander-stahis-multiple-form-logic-as-pile_objects/

[4] http://tech.groups.yahoo.com/group/lawsofform/files/distinction-networks.pdf

[5] http://ralfbarkow.wordpress.com/2006/06/28/pile-v01-notation/

[6] http://omadeon.wordpress.com/2007/05/13/beginning-to-relate-multiple-form-logic-with-pile-objects/

[7] http://ralfbarkow.wordpress.com/2006/05/21/bootstrap-of-a-pile-system/

[8] http://ralfbarkow.wordpress.com/2006/06/04/pile-space-or-numbers-as-points-on-a-line/

[9] http://ralfbarkow.wordpress.com/2006/05/19/multiple-form-logic/#comment-3893

Starting from what he calls the “Container-Reference Dichotomy” since an earlier paper (Westphal 2006 – Freeing Data From the Silos, pp. 1-3), Ralf now tries to get along with ‘containers’ only. [1] [2]

He questions the necessity of making a decision between container and reference. Further on, he conceives of ‘data’ and ‘containers’ as a (only one?) hierarchy of values in a “universe of unique values“ (a distant salute to the Pile space).

In Ralf’s new model there are two different sorts of values: atomic values (AV) and composite values (CV). I find it hard to understand his concept of atomic values. Ralf writes:

„Während der Wert eines AV jedoch noch quasi gleichgesetzt werden kann mit den rohen Daten, die er enthält (z.B. ein Name oder eine Telefonnummer oder ein Preis), [...]“

The value of an atomic value could be quasi equaled with the raw data that it contains (e.g. a name or a telephone number or a price).

If an atomic value contains data, i.e. a hierarchy of values, how could it be atomic then? I think, the problem lies in the putative need for another word for terminal value (TV).

In Ralf’s model there is no outside regarding atomic values (data is inside again!?), but in the illustration of a composite value there are call outs („Vera“, „42“, etc.), lines crossing the boundary of a composite value.

In contrast with Ralf’s model, the qualities ‘inside’ and ‘outside’ are key to understand the meaning of ‘terminal value’. A TV is both: an inside value and a connector, port, or connection to the outside (where the data live).

References
[1] http://ralfw.blogspot.com/2007/01/das-intuitive-datenmodell-i-motivation.html

[2] http://ralfw.blogspot.com/2007/02/das-intuitive-datenmodell-ii-es-kann.html