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 spacepervadesall 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: anemptydistinction indicates a void content. Since the void can be indicated, it can be used semantically, to (non)represent concepts in the modeled domain.

Boundary mathematicsis 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] https://ralfbarkow.wordpress.com/2006/05/19/multiple-form-logic/

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

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

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

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

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

Well, thanks for your kind words.

Excellent thought-provoking post, Ralf !

Well, it seems fortunate that you are… tolerant of my ignorance about Pile, at this very early stage. As G.S.Brown said „ignorance is a prerequisite of wisdom“, so I am blessed with the… bliss (of ignorance) which is (sadly -hehe) irreversibly lost by knowledge (later on). 🙂

As regards Wiliam Bricken’s system, you know the proof in my site about it being a „special instance of Multiple Form Logic.

http://multiforms.netfirms.com/more_theorems.html

The „XOR operators“ are intimately related to a kind of „unrestricted self-reference“. (A „XOR“ is like „an attempt of the Void to become distinct from itself“, poetically speaking). Anyway…

I was just looking at some old material about this today (from 2003) and about the fact that (despite my ignorance) there must be deep similarities between Pile and Multiple Forms. I can sense it; it must be there; in a few days we will _all_ know more. I don’t think we can stop it from being known; it’s inevitable.

Some thoughts (for the moment)

1) I need a Prolog-based or Formal Logic formulation of „Pile“, if possible; this might speed up everything (in my learning process) quite a lot.

2) Has anybody else implemented any other theorem-proof-related stuff using Pile?

3) As regards implementations of Pile, I can do _yet_ another, in Assembly Language 100% (quite suited for the task) but this does not seem our _current_ biggest priority.

Pile seems a process rather than a state; dynamic rather than static.

Pile seems more algorithmic; O.T.O.H.

Multiple Form Logic is more structural or axiomatic: Algorithms have little to do with it, except concerning proofs & efficiency of proofs. Well…

ANY algorithm that can locate a „sub-tree inside a tree“ can be used in optimizing Multiple Form Logic proofs, because this is the essence of the 3rd Axiom, used repeatedly as a rewriting-rule in proofs.

The whole thing together, Pile and MF logic, is beginning to look beautiful. Because I also tried to get rid of data, even the simple data of the „constant form“ used by G.S.Brown: Only operators exist, as relations. No „1“ is even necessary, except as a „construction“.

In the case of a finite number of truth-values or states, instead of „1“ we can use the _UNION of these values_; this „reeks of associative children“ in PILE, perhaps? or not?

Best wishes and thanks

George S.

after all, there is a new way to make things right, it is the common companies way!

as you all now so well what Mr KRIEG say that erez said, let me say here clearly to what erez invite others for to develop together the pile technology, it is specified in the site http://iswith.info and any other way would go against the intellectual properties right of erez elul, so please try this one, thanks.

Hi Ralf and Erez.

Actually Erez’s alternative business model, the „common company“ is very interesting and much more practical than it initially seemed, when I first looked into Erez’s ideas a VERY long time ago.

So I hope that… later on today or tomorrow, I will find the time to write an extensive review and presentation of Erez’s alternative business model, which (as I understand it) tries to safeguard profit-sharing and a fairer distribution of revenue, as well as the rights and incentives of investors. And… in view of the big changes expected to come soon due to PeakOil, Erez’s ideas may become suddenly widely applicable in society.

However, all this has little to do with Multiple Form Logic. More posts on Multiple Form Logic will appear in my blogs, particularly the dedicated technical blog

http://prologsource.wordpress.com

(where a brief description of „Dream Prover“ is given, a nice theorem prover with a graphic editor and a Logic slide-show utility). Nobody supports this work financially, so I have to work during my free time, earning a living through part-time work in the Semantic Web, in a company who need fast Assembly-language-based Natural Language Analysis and Data-mining code.