Sign, Identity, Relations #2

A different approach to eliminating or removing Spatial concepts:

1. Perhaps it's too rash to omit or entirely remove all Spatial concepts from a Language.

i. Not just words like 'topology' but the implicit particularities of a Language that are spatial in nature (that it's written, symbols are inherently Spatial, read front to back, etc.).

2. Another approach attempts to minimize (but not eliminate) Spatial concepts in a particular Language.

i. On this view, we might try to compress Spatial concepts into as few such concepts as we can.

ii. Or, compress complex Spatial concepts into more basic ones.

3. Example: a Topological Space can be compressed (abstracted away) into a single Point (akin to Newtonian Mechanical treatments of Mass as Point through the Second Law).

i. On such a reading, a Connector can be thought of as a most minimal Spatial concept (most maximal Eliminative Conception of Space) without completing omitting Spatial concepts (which may not be possible).

ii. Interestingly, such an approach might be akin that taken by Gordon Spencer-Brown in the Law's of Form - e.g. Law of Calling or Sign.

An Approach

1. Let * stand for the most minimal and the most maximal Connector.
2. Let * represent both the most minimal spatial MN and the most maximal spatial concept MX.
3. Such that * ≡ MN, * ≡ MX, and * ≡ MN MX.
4. On this view, what justifies * * | * is just that (* * | *) ≡ (* * | MX) ≡ (MN MX | MX) ≡ (MX | MX).

Existing and "real-world" analogies are ubiquitous: Points in Point-based Geometries, Lattices, 0- and 1- Dimensional Spaces, G. Spencer-Brown's Form of Condensation, etc. I'll set aside a more extensive philosophical justification to get what Crispin Wright would call a "Cognitive Project" off the ground - we'll come back to that later.

Here, maximal and minimal are taken to be dualities (and in the precise manner as defined in a Single Valued Logic - where one Value is a Proper Subset of the other and both stand in a Natural Ordering). In Physics, this approach parallels Wave-Particular Duality as opposed to the traditional Atomistic thinking of the Ancient Pre-Scientific Philosophers.

Linguistic Reduction

Not to be confused with Reductionism (as a substantial metaphysical thesis in the Philosophy of Science, say); Linguistic Reduction, here, refers to eliminating unnecessary linguistic categories in order to have a single kind of thing do all the work.

Why is Linguistic Reduction desireable?

1. Many programming languages divide into Primitives, Data Structures/Container Types, and leave it at that. Others add to Reference Types, and other derived Kinds.
2. Fewer Kinds simplifies the manner of representation, the cognitive overhead, and increases the semantic expressiveness of the more concise syntax.
3. Practically, this means that the physical markers of language are fewer to do the same work (symbols, tokens, Hard Drive file space, file sizes, number of characters, documentation/conceptual exposition, etc.).

Lingering issues:

1. A lingering thorn - consider the diagrammatic (or linguistic) expression: * * | * - how might this be better represented?
2. Intuitively it seems that we're overlaying one entity layer onto another - that is, language to crudely represent some underlying ontology.
3. Here's a seemingly better starting point - and it has its own limitations and interesting advantages:

i. From the above, We might think of * as a single, unique (a Singleton in programming), Maximal Concept of Space MX.

ii. MX therefore would contain all "Dimensions" and/or Spatial Concepts (MN, and any other such Concept).

iii. Independently of these considerations, we observe that the expression: * * | * is projected across Dimensions - at least 2 appear to be required. If we allow for high levels of idealization/abstraction, 2 appears to be sufficient. (And, I think toward the end of maximal Linguistic Reduction - can these 2 be done away with - can we do the same with a single, unitary, Concept.)

iv. We might then think of | as two MX overlayed onto each other - an MX embedded into an MX.

v. So, on this meditation and line of thinking, * * | * can be reduced to MX and can be simultaneously understood as MX MX (and MX MN, MN MX).

The Empty Expression

Does Semantic Immuration have something to say here?

1. Perhaps one were to see * as having both (OR, XOR) of MN or MX embedded within it.
2. In line with the tacit motivations given far above, consider the base case of an Empty Expression ⊡ or ⊡⊡ in which an Empty Language is embedded into an Empty Expression.
3. From the original line of thinking: might we see * in the same way - with MN or MX embedded within it?
4. Perhaps this starting point makes the clearest sense among the many moves I've detailed above.
5. E.g. - MX(MX) ≡ ⊡⊡ ≡ MXMX ≡ MX MX - I only write MX() for convenience since compounded/concatenated expressions don't quite totally convey the idea. That these are general and extend from the base case to other similar spatially defined conceptual relationships (of which MX and MN are of a kind).
6. We observe that 5. more closely mirrors the non-uniform and always free substitution approach taken here. This move is akin to both the rejection of the Euclidean Parallel Line Postulate and represents a legimiate "line of demarcation" between the Lambda Calculus (and its descendant languages, approaches to formal language systems) and those taken in the vein of Connection Theory, etc.

So, key take aways: add to the motivators for the view above - Empty Expressions. That's another seemingly bona fide example.

Concluding Remarks

Does this better address:

1. A single unifying unitary notational mark that has many simultaneous forms. A "multi-mark" if one will.
2. A mark that is at once precise enough to deteriminantly represent complex relationships yet contains within it a multitude of creative potentialities.
3. Expressions, the notational mark, and the ontology are united into a single linguistic layer.
4. In addition to the aim of Linguistic Reduction, the above I think helps to clarify the axiom * * | * (and various equivalents). How can 2 of something reduce to 1 in a non-arithmetic (or pre-arithmetic) way? Some mentioned approaches: using Laws of Form-style Condensation or MX MN Connector Duality.

Some interesting parallels?

1. How does such an approach formally related to G. Spencer-Brown's Laws of Form? Different starting points but some seemingly more than superficial overlapping axioms, inferential profiles.
2. Do these constitute an unknown Class (or species) of language kinds or rule systems (of which the Laws of Form are a part)?