The question was raised in regard to the parallel postulate in Euclidean geometry, which is perhaps a classic example of an impermeable meta-system boundary suggesting immiscible organization on either side. Kent Palmer has done considerable research in this area as exemplary of many similar differences between ’emergent’ meta-system layers. This question reaches deep into the R-theory world view and it has important implications even for how we think of space and time.

The first four axioms are about the realized world only – about material structures, and they exist in a single mechanical context – plane geometry. The relation between the axioms themselves and their formal context, however, is naturally complex. The 5th axiom is about the general context, where there can be other geometries, and thus the range of possibilities is complex. Plane Cartesian geometry, however, is supposed to be exact within a computable, and by analogy mechanical, context. If that context is taken as the general reality (the 5th postulate) you must obtain a contradiction, an impossibility. That is precisely the problem I tried to solve with contextual entailments and relations between realized and contextualized domains. The logical contradiction disappears when you represent the entailments as related to dual or multiple contexts to explain the complexity.

The short answer in R-theory is:

The first four postulates (axioms) define local geometry, which is constructive. They are about the realized world only – about material structures, and they exist in a single mechanical context – plane geometry. The fifth postulate is a statement about non-local, or universal geometry, which is formative (formal cause). The reason they are incommensurate and that the 5th axiom is not derivable from the first four is that the universe is complex such that there are other alternatives than the parallel postulate at that level. The difference between efficient/material causality (the constructive local domain) and final/formal causality (the general nature of space) is such that no single syntax can describe them; they are immiscible. In other words, local space-time does not extrapolate into non-local space-time in a unique way, and there is no self-consistent description that can reduce them to each other. This does not mean, however, that the boundary between them is completely impermeable, as we will see.

We learned in post-modern physics that there is a non-local reality that has different rules than the local realized existence. This became evident in relativity and quantum theories of wave-particle duality and the quantum vacuum. In everyday life as well, there are many non-local contexts. These are information contexts. Formalization of both domains under one system is fundamentally impossible because of the inverse nature of their entailments (recall the entailment definitions in Tutorial #1 and the synthesis paper). One domain describes state construction and the other describes law construction; or equivalently one describes operational behavior and the other describes system origin; and these are encodings and decodings of each other for which there can be no common syntax, and thus no mutual derivation.

These apparently immiscible domains, however, are permeable in a very specific way. They cannot be bridged syntactically, as Robert Rosen said, but they can be bridged relationally, which is what the modeling relation and R-holon does.

Euclidean Geometry as a simple system:

Previously I said that holons can be reduced to simple systems. That is the case of establishing a single formal system. Simple means there is one self-consistent formal system to consider; or put in terms of how we model simple systems, there is one self-consistent formal description. Complexity means there are at least two that must be considered. Simple systems we define involve only the efficient/material domains, operating according to the constitutive parameters of a single formal domain (traditionally described as Platonic law that is immutable). It turns out that these are only approximations of nature, as precise as they may be.

Simple domains are defined in such a way that they can be described using quantitative methods alone, and in modern science (as opposed to post-modern) number theory was thought, like mechanism, to be a fully closed syntax. We hoped that everything could be described this way, that a fully formalized mathematical model might be shown to be consistent with a mechanistic universe. Unfortunately, that formalization program of Hilbert was demolished when Goedel, working toward the same goal, proved it was impossible, and the same result was found in post-modern physics.

The conclusion: Contexts matter. If we were to accept a Rosen modeling relation between measurable existence and contexts, we would discover that Euclid’s axioms of Euclidean geometry (or Hilbert’s more thorough statement of them) do not uniquely define a universal geometry. The 5th axiom is consistent with the others, and defines a simple computable space (its aim), but other universal geometries are also possible, making the overall relation between local and universal geometry complex. If, for example, local geometry is Euclidean and universal geometry is hyperbolic, the universe is fundamentally complex, which is one of the claims of R-theory.

Now, lest we get confused, we are dealing with three meta-levels here, the physical world where natural systems can be analyzed, the formal world where formal systems can be analyzed, and the human world that contemplates these things in ways we don’t know how to analyze. We can even add another level for mathematical and computer models we create to help formalize our thoughts. None of these levels have been considered miscible to date. What we discovered with incompleteness, complexity, uncertainty and observership, was that complex behavior always involves at least two contexts. (Recall the discussion of 2nd-order closure in Tutorial #1)

The axioms of Euclidean geometry (Euclid’s or Hilbert’s) also involve two contexts, one is local where axioms are about construction, and the other is global where the axiom is about the general nature of space. In R-theory there are always these two contexts in a natural system because natural systems are complex; they are context of local behavior and context of global form or ‘shape’, which is the formal cause quadrant of the holon. The 5th axiom thus establishes a non-local constraint at the limit of local constructions.

OK if you buy that, then am I saying that Euclidean Geometry is complex or simple?  Although its local and non-local domains are fundamentally immiscible, it is possible to find a sub-set of them that yields consistent results. Euclidean geometry does just that with the parallel postulate, which along with the first four, defines a computable syntax; but it is not fundamentally derivable from them because the derivation would necessarily implicate many possible general geometries of the formal domain, not just the one parallel postulate that gives us a simple system.

Manifolds

Where, then, do we get contextual ideas or novelty? One view is that we get inspired – they come from Spirit as de-novo emergences of thought. There’s not much we can do with that in science except marvel at the variety of ideas that are possible. Another view, that R-theory proposed, is that they come from examples — exemplars. Each mathematician built ideas on prior work. As one geometry is constructed, and placed into complex context, new global possibilities are implied. So plane geometry is compatible with its extension to spherical geometry, but more significantly to non-Euclidean geometries. We get inspired, as it goes, by prior exemplars that are transformed by context into a new realized thing. Our own conscious mind, which we don’t otherwise understand, is certainly such a context. If I see a chair, I may get an idea for a new kind of table, and so on. These are functional variations on a theme.

As we applied Euclidean geometry to surveying the land, for example, we ‘discovered’ how to modify the previous exemplary geometry to a higher order geometry because we ran into global contradictions — the Earth wasn’t a flat plane. Similarly, Euclidean geometry, which seems to work locally, is now universally understood to not apply to cosmological scales, even though it may apply locally everywhere in the cosmos. This seems paradoxical, but instead it is a statement of relational complexity between a local, efficient/material domain, and a non-local final/formal domain (context). The theory is actually quite easy to comprehend when we accept that relation.

How, for example, do we derive hyperbolic geometry from plane geometry? By placing plane geometry in a different context where we change the 5th postulate so that parallel lines converge (at infinity, but nevertheless producing a singularity). Papers in the last 15 years have marveled at the fact that Milne’s cosmology, which is essentially hyperbolic, seems to approximate the standard geometry derived from General Relativity. It is considered a ‘toy’ model because it is massless, but how can a massless geometry even come close? The bottom line is that General Relativity is a very clever attempt to cross the impermeable boundary from local to non-local reality by one syntactic means. But it must end up being paradoxical because these domains are immiscible. A better description would be based on duality, which Einstein did not want. Ironically, when it is based on duality, a deeper unity is revealed in the holon, which provides what Einstein really wanted, a unifying principle whereby “God doesn’t play dice”. It is the unified duality of the relational holon.

Incomensurability in R-theory:

I showed some holon composition diagrams in Tutorial #1. The 2nd-order (hierarchical) composition (figure reproduced below) is a picture of complexity in its most basic form. There are four versions of the 2nd order closure, which correspond to closure in each cause/quadrant of the diagram (where the arrows cross). The four kinds of causal closure can be described easily in terms of a meal, for example. Efficient closure is where I cook your meal at your house and you cook mine at my house. Material closure is where I give you a meal from my house and you give me one from yours. Final closure is where you design a meal based on one you saw at my house and I design one based on what I saw at your house. And formal closure is where we swap recipes. The first two are closures in the realized system – we are swapping actual work and materials, collectively, ‘stuff’ or products and services. The second two are closures in the contextual system, we are swapping examples and templates, collectively, models. It should be fairly easy to see from this example why 2nd order closure is always complex; you would not be able to exactly predict what kind of meal you will be having next week.

So we can define relational complexity as the existence of two or more immiscible models. The contexts shown in the 2nd-order composition are the domains of these models. If they were miscible, the diagram would reduce to the one above it – a temporal sequence of states that we normally associate with local dynamics (although discretized). If you reduce it further, you get identities as in the first diagram, which are idealizations (they don’t really exist in isolation). Those reductions are what we find empirically at modest dimensions in a well-defined local world. But as you can see in the diagram, even the supposedly singular context of temporal dynamics is really a result of some perturbation or duality on the contextual side. If that duality is noticable, as in the case of isolated quantum particles, or large cosmological distances, then you have uncertainty and complexity as represented in the bottom diagram, where there is a superposition of possible measurements because of the dual context.

It turns out that if you apply R-theory to cosmology, you get exactly the 2nd-order closure. That is, two immiscible contexts that govern the same efficient domain (where closure takes place). One context is that of local space-time, which I describe elsewhere as Minkowski space, or M-space. It is Cartesian, flat, and pseudo-Euclidean. The pythagorean theorem in M-space gives you the Lorentz transformations of Special Relativity.  If, however, we consider that the universe had an origin, or at least appears to have nad when we look through a telescope; and that origin looks to us like an historical point in space-time, then clearly M-space cannot be compatible with the overall geometry. This is easily seen by extrapolating the lines of light travel (45-deg angles in opposite directions if the axes are normalized) in local space backwards in time to where they apparently came from. In M-space these lines will diverge, not converge to an origin, because they are straight lines in different directions, according to the axioms. To make these lines originate from a common historical point, we must therefore have a non-Euclidean geometry that governs the non-local domain (in this case the domain of historical light travel).

So, nature tells us that Euclidean geometry can only be true locally, and that the parallel postulate is probably wrong universally. We may consider it true in a given artificial context, but even its derivation is incompatible with local reality, because it can’t universally exist and in any case, all statements about context are of a different system type – an inverse causality as I’ve described in the R-holon.

Emergence and Meta-system boundaries:

The first four postulates (axioms) do not predict the fifth and are independent of it. Hence it exemplifies a meta-system boundary.  Where does it come from and is there any idea that bridges that boundary?

When two material systems are said to be related, the full causality of that relation is that each system exists in the context of another. For example, I am perceived according to your models of me, and you are perceived according to my models of you. If we try to develop a common model, say a moral system of human behavior, then we interact contextually and form a new model, the shared morality. Each of us then draws part of our self-definition from that shared context, but probably also our original one. In this case it is not necessary to specify whether it is final or formal closure; it could be either or both. And yet there is an analytical difference: Final closure would be where we each consider the other an exemplar for our behavior and construct our models accordingly. Formal closure would be where we each try to control each other by our moral norms. They are quite different behaviorally, but they are both contextual closures and if both are happening we could diagram each with its own arrows. If you try to represent them simultaneously, you get a full contextual relation (say you define your existence by example of your partner, but your partner does everything you say – a rather extreme symbiosis that could define some forms of marriage dependency).

In the synthesis paper I presented another diagram of contextual closure (not specifying final or formal) to show emergence. Contextual entailment implies emergence of a new system because the extreme form of dependency suggested above is not possible without destroying each identity, so dependency requires some autonomy, and that means there is a third system, C in the diagram, through which the dependency occurs. In the simplest case, C is most parsimoniously constructed from A and B, and it brings about new behaviors when realized by either A or B.

OK, that’s a lot to swallow, but the point is holon interactions can produce new contexts that are by definition different from the ones that made it up. Contexts combine like Venn diagrams and their overlap produces a new system. Hence the contextual domain is where origins can be explained.

Contexts by definition represent constitutive parameters that constrain or enable functions (such as the shape of space-time, the gravitational constant, the limits of social acceptability, body language semantics, etc., but not the details of what happens).

Euclidean geometry thus has a system of operation, say holon A, and a system of origin, say holon B. Its origin is represented by the 5th postulate because that defines the properties of its space and the concept of an invariant distance (the separation of the lines). The other postulates define what you can do in that space, the dynamics. We naively thought that the dynamics should somehow add up to the properties of the space, but they never do – the non-local ontology is separate. The independent proposition that these are compatible systems, however, results in computability, so we like it. In the diagram above, applied to mathematics, the emergent property, C, is computability, because of the way we define A and B.  (I’ll not go into the entropy issue here – it applies to living systems most readily. Here it would perhaps be the creativity that is enabled by computability).

In simple language: “By combining the ability to do constructions in local space, with the assumption that those constructions are scale and shape invariant in all space, one allows computational descriptions of supposedly natural objects.”

It is incidental to this example that scientific testing in real nature did not validate Euclidean invariance. We don’t even know yet what to replace it with (General Relativity, quantum reality, or some other model). But the point is meta-system difference results from ANY such duality between local and non-local systems, where each provides part of the meta-system of the other.

Permeability of the meta-system boundary:

Clearly the operational rules of local space are of a different type from the rules of emergence (origins). So it appears as though meta-systems are immiscible. The key is that they are immiscible within the rules of one system, but R-theory defines a fundamental complementarity between all such immiscible systems. That fundamental complementarity can be used to cross larger meta-system boundaries. For example, if we consider the system of all spatial geometries, Euclidean and non-Euclidean, the boundary between local construction and general shape still appears, but is semi-permeable because we can relate multiple local-general systems to each other by means of the holon. We don’t get rid of the basic complementarity, but we can find rules for transforming between systems of geometry, for example from Euclidean to Hyperbolic.

I suggest that the same is true for all information systems; that the meta-layer boundaries are permeable with respect to holon analysis. This applies to Rosen’s “second dualism” (in his book Life Itself) which is between systems and their environments, which we may also equate with meta-systems. Considering the meta-system as a given-system’s environment opens it to holon analysis in the exact same way as the fundamental relation is between observer and observed or non-local/general and local. The local system is an observable system, a measurable system. The non-local system is that which observes and thus imputes formal boundaries and models for interpretation. Above we saw that the 5th postulate interprets the first 4 as defining a general space of consistent scale and shape. These two dualities are then sufficient for describing all system/metasystem complementarities. And the second dualism is permeable by means of the relational cycle of the first, in which the contextual side is reached by inference (encoding) and the realized side is reached by implication (decoding).

Furthermore, if there is a true origin of the universe we must consider creation of ‘something’ from ‘nothing’ in an ultimate sense, i.e., descent of a knowable reality from ultimate ’emptiness’. But even in Eastern philosophy, it is said that non-existence is “full”, and we see existence today related to the quantum void, which seems energetically full. Hence we may retain non-permeability in both directions if we are willing to entertain an infinite and eternal dual existence, or we may consider an ultimate level that is non-dual, and thus prior to all dual concepts such as substance and emptiness.

This last remark pertains to the ontology of the entire relational model – its own ultimate origin and perhaps principle to make it operate; in a sense a fifth cause, which is the cause of the holon. Just as in Eastern philosophy Brahman is the concept used for what lied beyond description and knowing, so there must be a similar background to the holon theory. No theory can be complete according to R-theory itself.

As a further example of emergent meta-system boundaries, and the fact that in R-theory they are permeable boundaries to the theory (meaning we can apply the same causal analysis to bridge from one side to the other), Rosen defined M-R systems in terms of efficient entailment maps related to each other (his diagram 10C.6 in LI). He claimed that because of causal closure, the M-R system defined a new system type, one that is living and impredicative because of internally produced and applied models. This certainly qualifies as an emergent system in most views, but in R-theory the supposed ’emergence’ of life is actually a construction of prior elelments; it is the organization that emerges. The boundary between non M-R systems and M-R systems is semi-permeable via holon relations, in keeping with the whole/part nature of the holon.