We can correct a small mistake in that one year old post (see George’s recent comment  ). 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 ,
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  ), 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  ). 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  and the complement in  ) — 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
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!