By Hilda and Stu - August 23, 2005

Consider a smooth manifold of thoughts, T, embedded within some wider universe U.
Now consider a thought t, within T.

Also consider a language L which is both an operator and a set of rules not assumed to be consistent or finite.

Definition

V(L,t) - a verbal space generated by L at t - is a linear tangent space of T in U, generated using L.

• The origin 0 of this space is the unquestionable context at t, that is, the null space of L at t.
• The vectors of V are then the expressible utterances in L which are conceptually compatible with t. These do not necessarily reside within T, and they may or may not be "near" T in some sense.
  
• Without supplying a-priori norms or basis for U or T, we assert the notion that given appropriate structure of U,T and the rules of L (see Refinement below), utterance vectors in V may be found to be metrically "near" or "far" to elements of U in general, and of T in particular.
• Vector addition in V: v1 + v2 corresponds to disjunction of utterances in a linear superposition. L. E.g. "The pendulum was constructed by Larry OR the Flying Spaghetti Monster". Believers in Larry AND believers in FSM can both believe in this statement without needing to resolve it to remove disonance. That is, we do not need to answer whether Larry is-same-as FSM in order to accept v1 + v2 as a legitimately comprehensible vector.
  
  • Note dim(v1 + v2) <= dim(v1) + dim(v2)
• Wedge product in V: v1 ^ v2 corresponds to conjunction of utterances, producing (in the thinker or listener) a new meta direction inquiring into implications of compatibility of v1 & v2. For example, consider the conjunction of "Hector is brave" and "Hector is meek". We are immediately led into consideration of the compatibility of bravery and meekness, are we not? This inquiry may well lead us to need more basis from L (more words besides "Hector", "brave", "meek", "is ").
• Multiplication in V by scalars can be interpreted as weighting with some (possibly complex) numeric quantity, often "importance" and/or "truth". (Many " false" statements at t are still "understandable" at t). It seems useful to assume in some research models that these scalars may in fact be complex, capturing some dual-faceted nature understandable in terms of magnitude+phase and/or (x+iy). Simpler systems may fall back on real weights as prudence dictates. The exact choice here is a key design parameter in verbal space design.

Conjecture

Goedel's incompleteness theorem is relatable to the notion that a particular V(L,t) may not, in general,
capture all the dynamics of T, and certainly not if T contains a complete and consistent L. Thus the generation of V should be considered a product of both L and T, without subsumption of one into the other.

This idea may be considered similar to the notion that language L interacts with the mental state of organisms T to produce culture V. Note that L, T, V are all potentially within U, but we do not attach particular significance to this notion.

In special cases, we may formulate tautological, flat thought spaces with total verbal coverage in some L, but these are not complex enough to describe their own dynamics properly (i.e. they are
"incomplete" systems). For example, almost any verbal thought like "all X is-always Y" is doomed to miss some behavior x which is conceivable and probably documented, somewhere, as not-y or unprovably-y. (waves hand at blackboard, drinks from bottle of grog with FSM pirate logo)

In general, a "proper" thought space will contain many curvy dynamics that are not flatly expressible
in any linearizable verbal system with a comfy truth center to squat in. We will return to the idea of "proper" thought spaces later, in the context of morality.

Refinement of T, L, V

But what do we mean above when we say that V is the expressible utterances in L which are conceptually compatible with t? And of what use is this concept? We now return to some of the issues postponed above.

L is a language operator in some universe U that is expressed from some point of view t in T. The idea at t may be arbitrarily complex, but is assumed to be part of a large manifold of thought T, which represents the mental (and any qualized paramental) activity of some individual or group of organisms.

We do not consider here in detail the shape of T, although we conjecture that it may be not unlike a generalized space filling curve or fractal, to the extent permitted by the smoothness criterion postulated above. The seeming contradictions in this paragraph are intentional, and are to be left behind here.

Now, since T is smooth at t, we know it has a well defined tangent space in any suitable embedding E(t)
Here E is the set of expressibles, relevant expressions of thought we are to consider available from a mind-state t, whether internal to an organism or external. (Not all expressibles need fit in our
language L). Then L generates some subspace of E(t) which might be
expressed or understood by an organism at t using L. We make no restriction here on the availability of descriptors for L within E(t), nor do we assume L is well-defined, only that it exists as an entity we may attempt to model (and hence "give birth" to the language model L for our purposes here).

The definition of a linear V model at t within E serves the purpose of creating locally indexable sets of disputable thought for evaluation and use by organisms. The mechanics of linearity allow for numerous techniques of combination, leading all the way to observation, identification, and control of stochastically
modeled systems by organisms, but are valid only within very limited domains.

Next consider the vector interpretation of V as the using the disjunction operator OR. What is meant here is that any set of statements v1, v2, v3,... may be combined with the OR operator to produce a new statement that may be further disjoined. The resultant statement will reflect "all" the reality in it's component statements, without necessarily making sense (since it may contain contradictions) or being useful (since it's components may be irrelevant to each other).

We also notice that the degrees of freedom of v=v1,v2,v3...vn will generally increase as the number of components n increases. However, when statements va,vb overlap, then the increase is less.

Finally, we can see that a reduction in degrees of freedom of v corresponds to a kind of disambiguation, and a corresponding increase in syntactic simplicity, although L may generate non-trivial spaces with small amounts of syntax in some cases.

Next we consider C(T), a set of cohomology groups giving the geometric structure of T. How do these relate to the utterance stru ctures V, locally and globally? The answer depends on how topologies are defined in both T and E (and thereby V).

Typically a useful topology, metric or norm in E(t) cannot be well defined verbally without some reference to T, and vice versa.
In a physical manifestation we might anticipate unexpected relationships between the structures of E and T that would defy the pitiful mechanisms introduced here so far. Nevertheless, we now consider some simple examples of mappings from E(t) to T that reflect a relationship of norms.

• Common words from L can make sentences in E "related", and can reflect connected clusters of concepts in a verbally influenced mind T.
• Patterns of expression (e.g. grammatic, rhythmic) in E can be recognizable in T, and thus are close under some conjectured pattern norms.
• We know that experiental spatial norms (e.g. distance between objects in R-n) may be both precisely described verbally and precisely conceived intellectually, under certain conditions.

These last notions have all been formalized in many ways before and we will not attempt same here. Rather our point is to introduce a usable framework for description of experiments and results in
knowledge capture and evaluation. By abstracting U, T, L, E, and V in this loose manner we attempt to give room for particular scenario definition without repetition of the foregoing.

In particular scenario descriptions then, we shall move directly to the discussion of the following "verbal theory components"

• Assumptions regarding structure of T, L, E, V (mainly operators and norms)
• Relationship of norms on T to norms on E under certain transformations.
  
• Relationship of cohomology groups of T to the cotangent bundle of utterance sample spaces V(L,t).

This last mouthful means simply that geometric reasoning regarded thought is hypothesized to be mappable to/from sets of verbal statements taken from a range of points of view.

Punchline

As a special case, we may take the basis for L to be a set of RDF/OWL triples representing conceptual categories.