Relational Science

Triggering Questions (Kent Palmer)

1. Is there meta-causality, i.e. do causes have causal affects on each other.
2. If causes cause each other, rather than effects on something else but themselves, how is this causality trasmitted from cause to cause?
3. What does it mean for a cause to cause another cause to do something or to be changed in some way?
4. Can this causation be transmitted across multiple causes of different kinds in a series?
5. For Holonic Circularity that you posit, is it possible for a set of causes to have causal flow between them such that the causation completes a circuit?
6. Does this causal circularity when completed make the four causes a whole, in the sense of a holon, and what does that mean?
7. What are the implications of this causal circularity for things outside that holonic system?
8. Is this causal circularity autopoietic, or self-bootstrapping, or does it come from outside the holon?
9. Is there some source of casuality of this kind, some fifth cause that causes this causation?
10. Does this causality of causation need to be mediated, or can it be direct. If it needs to be mediated what is the nature of that mediation?
11. What is the relation between this kind of causality and the concepts of possibility, necessity, potentiality, actuality, and in general the coming into existence of the holon and other things produced by the holon?
12. What is the nature of Emergence, of the holon and things coming from the holon via this meta-causation?
13. What is the ultimate nature of the holon itself. Is it pure meta-causality, is it something that is materialized in a system, in general what is the ultimate nature of the holon that allows it to have this meta-causality
14. What are some examples of meta-causality working in the world, based on Aristotle’s four causes.

I decided to deal with these questions as a set, and draw some fundamental answers from them, so I’ll take them in groups.

Question #1, 2, &3:  Meta-causality

The basic approach to causality in R-theory is that it refers to ways of understanding nature. I believe it was Aristotle’s idea as well, that the term ’cause’ refers to a means of ‘explanation’. No one can say what is ‘really’ in nature – we only have percepts of nature to deal with. Hence, anything we say is in the descriptive system. It is at most a pointer to something we make a leap of faith to believe is “out there”. For all we really know, our essence is alive in a jar in some laboratory being fed percepts of a body and a natural world. As Descartes said, all we can really know is that we exist; all else must be subject to inquiry.

A slight aside: Even Descartes’ justification of existence is questioned by philosophers and theologians, but Descartes’ point was that we have to make that much of a leap of faith in order to justify knowing anything, and the fact that we are conscious justifies the leap. History suggests that to him it was a form of faith. Reportedly, Descartes was quite spiritual and retained his deep connection with Catholicism (despite some books questioning that on slight evidence). His science got him in trouble with the Church, but he saw his work as non-contradictory. He apparently thought his famous quote “Cogito ergo sum” was a proof of the existence of God, because our own existence as suggested by mind must imply a conscious source. I believe he was philosophically on solid ground in that thinking; that awareness of existence implies origin that is causally rich enough to allow consciousness, whether it pre-exists in some ultimate form or is a principle that develops (or both, as in ancient Eastern philosophy that equates the ultimate with the essence of the personal, as Christianity also claimed to do more literally).  In Descartes’ time “le cogito” was a term that referred to conscious awareness of existence generally, not just analytical reasoning or information processing. It very much meant existence of the ‘self’.

Continuing ….

We tend to explain nature using the word “because”. In Rosen’s writings he says that causes are answers to the question “why” in addition to answering “how”. The supposed ‘higher’ causes of Aristotle, final and formal, are answers to “why”, so indeed our traditional sense of causality suggests that we should ask how the why (Kent’s questions about flow). That can be done, but it does not imply any explanation other than another layer of holon related causalities. They implicate each other going up the hierarchy, and they ‘mean’ each other going down the hierarchy (or my R-theory counter-clockwise around the cycle). Efficient cause ‘means’ necessary state change because that is the supposed organization of the material world.  That’s its definition and the efficient cause itself is just a label for its regularity. Material cause means a measurable pattern.  Final cause means a pattern that is an example in a context, because that is the supposed nature of context, to interpret, just as it is the supposed nature of the realized world to act and manifest. The “Old West” is an interpretation of the events and things that characterize it, and yet it was a natural phenomenon in its own right that also determined those events. It was a context. It is obvious in mental examples but less so in physical cases. A middle ground is in ecology, where it is near inescapable to consider the ecological niche, which becomes defined through adaptation by current distributions and acts through selection to constrain future ones, in some sense real (as real as genes, by the way). Formal cause means function because it gives contextual parameters to natural process. So water can flow due to gravity, but the force of gravity is constrained by the gravitational constant.

I’ll go over the four causes again, because it is so fundamental to relational theory and the R-theory synthesis. An easy example is a spaghetti dinner. The ingredients of the meal are material in the usual sense. They are measurable states of matter and energy, even as temporal sequences of states. The pasta, the tomatoes, etc. Material can be defined at any level. We can talk about the molecules the atoms or the whole vegetables or the substance after they are made into sauce, or the whole meal. At any system scale we are talking about material explanation, or material cause of what the meal is or consists of. The processes that resulted in those material states can also be an explanation of how the meal came to be as it is. This is efficient cause and it refers to reconfiguration of states. As such, it is incapable of explaining the origin of a material object except as a transformation of another pre-existing material object. Hence theories that employ only efficient and material cause (the standard category theory entailment map) cannot address the context of origin. We thus have a theory of dynamics in local space, motion of planets, formation of stars, chemical interactions in a test tube, etc.; but these mechanistic explanations run into trouble when the system it is used to describe involves some novelty or its own origin. That, most obviously, is the case with living systems via reproduction and evolution and “origin of species”, and it turned out to be the case in physics as well in the origin of quantum particles and the origin of the universe. None of these fields of inquiry can be adequately addressed by a mechanistic theory for that precise reason. But there is no question that we use this causal system of entailment throughout science. The equation F=ma is an entailment of efficient (the force) and material (the state variables) cause. Differential equations such as d/dt(position) = velocity define an efficient cause as a temporal change of state. Geometric construction is a procedure in space, and is also an efficiency defining a material object.

But continuing with the meal analogy, if we ask further of the Chef who performed all the cooking processes that transformed the material into a meal, how he knew what to do, we may get several kinds of answers. If he is a professional Chef he may say it was Divine inspiration. His creation has that certain ‘Je ne cest qua’! This was Aristotle’s final cause (as distinct from R-theory, note below) extending from God in a hierarchy down to the ‘lower’ material level. However, a down-home country mom might admit that she used grandma’s recipe. That is formal cause. Aristotle, or his later interpreters, gave different descriptions of formal cause, trying to explain it. One analogy was like this one, it provides the constitutive parameters of the efficient processes. The range of temperature during cooking that results in just this meal, the amount of tomatoes, the kind of mashing or slicing, the limits of force and shape that produce pasta, etc. It does not specify what those processes are, but what their constraints are. One may use a fork or a spoon to beat the eggs for the pasta or mix the sauce, or perhaps shaking it all together would work too, but somehow the processes must be selected and constrained in a certain way to get the right result.

A curious pasta lover might then ask grandma where she got the recipe. She got it from her mother, who got it from a neighbor, and on eventually to Italy and China. Perhaps many of these people were inspired, but they also had examples to work with. That was the key feature of R-theory holon construction; to take Aristotle’s hierarchy and turn it into a natural holarchy. Final cause is not the inspiration, it is the example. Now it could also be an exemplary thought if we are applying it in that domain, and indeed exemplary thoughts trace back to prior exemplars too, and perhaps ultimately to Divine inspiration, or at least one can say so. But even that follows from an exemplar which religions are quick to describe. The big question then, is where does the exemplar come from?

R-theory’s solution was that, certainly in the case of any natural analysis, it comes from prior states – that’s what an exemplar is, a set of previous measurable conditions. Those conditions are placed into a new context to produce a new recipe. The context went for China to Italy to neighbor to grandma to mom in many cycles of the holon and many versions of the recipe and many final results that were good meals and became examples. So, then the remaining question is what about Divine inspiration? Where does the theory say that comes from, if it is to be a complete philosophy not leaving out even conceptual systems? I’ll comment more on this in the future, but it is the four-cause holon itself; we are the inspiration that performs the loop, and so is nature. This could be called a fifth cause, but we can’t work with it in science because it is science happening, along with everything else. R-theory says up front that no theory can have a complete explanation without being the thing it is explaining, so this is where the buck stops. In the most ancient wisdom; “Tat twam asi”; “Your essence is that essence”, or later in more tangible terms, Brahman (ultimate existence) and Atman (personal existence) are the same — Descartes’ point.

But I don’t mean to say that the holon works because of its divinity. It is a system of explanation. It is not intended as a divine pattern any more than all of science attempts to discern patterns that are true descriptions of nature. If we are close to the mark, and if one chooses to ask the question of ultimate meta-causation, then perhaps we can say something like our theory (R-theory?) exists at the heart of nature….until someone finds a better description (better in terms of epistemological criteria, otherwise it may be ‘better’ for something other than science). The alternative is some other way of explaining nature that will have the exact same question of it; why is it a proper explanation? What makes it so? These are empirical matters, not ontological ones. We seek the best explanation meeting a set of knowledge criteria. There are six epistemological criteria (in the synthesis paper but also a future post in more detail). The meta-causality can be addressed only by testing these criteria; otherwise it is a direction into unknowable territory that can only be answered by faith. But we should be clear that there is no proposal that the R-holon is a natural object either; it is a formal object attempting to bootstrap knowledge as science must do, from initial assumptions. It is a means of causal analysis and synthesis together.

Another way to say this is that there is no more question if the causes are real than if nature is real – neither can be proven. Also the ideas did not begin with Aristotle, he acquired them from the Egyptian and Indus philosophies that were less dualistic. I am growing convinced that his four causes are reinventions of the “four faces of Brahman” from the Vedic civilization or prior. Interestingly this idea was carried into Buddhism as well, with Buddha depicted with four faces. It is far too trivial an interpretation of these extremely introspective traditions that talked about the most fundamental non-dual relation between existence and non-existence, and how that starts to manifest, to attribute this four-ness to cardinal directions, as is often done. However, with the Abrahamic religions the concept of a personal-impersonal non-dual systemic reality was lost and God became an external agent. Hence with Aristotle these causalities became hierarchical with God at the top and the world at the bottom. That was both appropriate and necessary theologically at the time. However, our current scientific and ecological separation from nature is a result of that basic philosophical split. Various attempts to return to Eastern mysticism are popular now, but fail to see it outside of the power of thinking and myths of oneness without separateness – just a spiritual ego as opposed to a material one, but not a true unification. The Vedic belief saw us as intimately part of the whole – and that was also changed in later Hinduism to develop a sense of fatalistic destiny. So, you get a split between fatalists and constructivists (not that these cover the territory). Some traditions describe this age of humanity as chaotically driven by dualism as we struggle to resolve that into a true whole.

So, in most of science causality is losing its popularity because of the failure of Hilbert and other’s attempts to formalize it as a complete 2 cause system. Efficient/material causes primarily, within an immutable, pre-given Platonic domain of formal laws waiting to be discovered. By not varying, the formal domain is not a natural cause, but a set of Divine or Anthropic parameters (raising the question of the Anthropic Principle). When our thinking is opened up to a four cause system, nature becomes self-generating and self-explaining, aside from its own necessary meta-level of origin, as discussed above.

Given the above understanding, the question of “transmission” of the causes, or how one cause causes the other, is clearly inconsequential. It is asking why and explanation explains. The answer is “because that’s what explanations do” and to such a question that answer too does not explain. It is simply the nature of explanation to create a descriptive system that seems to act, at least in our minds, like the system it is describing. The answer is empirical testing.

Questions #4,5:  Transitivity of causes

This was addressed in tutorial #1. The causes are transitive around the cycle as many times as an analysis requires, linking to elements in other holons as needed. There is, in a sense, “transmission” of information from material to final cause (encoding), and transmission from formal to efficient cause (decoding). But I would caution about the concept of ‘transmission’, as in Shannon-Weaver information theory, because it is not really a thing being transmitted. It is an induction between inverse systems. Unlike an efficient cause, for example, it may not be possible to associate an object with the act of encoding or decoding. For example, the shape of space-time is a formal cause with respect to dynamics. It is not necessary to consider space-time as a substance to discuss its formal properties as a coordinate system. We simply are looking for a mathematical description that fits the general constraint on behavior, then implicitly accepting there is some similar principle in nature. To search for every such principle in terms of material particles or patterns would lead to mechanical reductionism. In the case of efficient causes we already have accepted the idea that they can be discussed as causes without asking what causes them or how laws are transmitted to objects to make them move accordingly. While we may be able to discuss particle mediators for atomic forces, it is more difficult to discuss the particle mediator for laws of scale or geometry, for example. What ‘transmission’ makes nature aware of the Pythagorean theorem? It is simply a metric that describes whatever is there under certain conditions. We have to assume the ‘whatever’ in most cases.

Question #6: Do the four causes make a whole, a holon? What does this mean?

Yes, as in the diagrams. The four-cause holarchy defines holism, what we must mean by the word “whole”. It is a bit different from the usage as in “whole milk”, meaning nothing has been removed from the original, but analogously the same. The holon is a causal explanation of nature from which nothing natural has been removed, for example as we purposely removed final and formal cause to create more precise descriptions of material events as mechanisms. In the same sense, 2% milk is skimmed so that we won’t get fat or build up too much cholesterol; but as our thinking has been trained, we tend to want such material metaphors for non-material analogies.

Questions #7,8, and 9:  What is outside the holon?

Nothing we can describe. All modes of description are included in the holon. The inside is more important – what is at the center of it. There you have some consciousness-like reality that cannot be described because it is pure happening. Also, at the center is the concept ‘now’. Nothing in normal science defined ‘now’. There are representations that suggest it, like t = 0, but every observation is of the past and probabilities are of the future; very little is said about ‘now’ – almost as if it is a myth. Some in fact believe it is. Our way of looking at reality has made past and future real, and ‘now’ unreal, along with our own experiences considered mainly myths or epiphenomenon, incidental and reducible to the happenings in the physical substrate. The center of the holon is “now-here” (as opposed to “no-where”). it is the essence of being and happening – best I can do with words.

Question #10: Is the holon mediated by anything else?

Of course it is. But we’ve exhausted our intellectual tools for describing natural causes with the holon itself, so what mediates it can’t be described, but is implicitly ‘real’. It can be experienced, however, and we do all the time. If one admits to experiential evidence, the holon is empirical. Philosophies vary, but in the ancient view it was described as Atman in the sense of a “Self” or Brahman in the sense of ultimate unknowable source of existence. Today it is described as the relation between measurable existence and the quantum vacuum. We cannot explain why particles pop out of nothingness of the vacuum, and then disappear again. But we have to believe, in that model, that something allows that to happen, because we observe it. In R-theory, that is represented as encoding and decoding between local and non-local reality; “existence with attributes” and “existence without attributes”.

Again, most scientists and others do not consider the idea of an efficient or material cause problematic because we are used to talking about them. But the effect of a recipe on dinner should not be a difficult concept either. In that transitive example, the recipe constrained efficient processes which transformed material states, resulting in dinner. The effect of being hungry should not be a mystery either, in evoking a pattern of behavior to apply a recipe for food. We have the capacity to interpret exemplars in our body that correlate with behaviors for obtaining food. At this gross level of explanation, it is a simple matching exercise between patterns felt as conditions of the body and patterns learned for addressing them, and in that case we are the mediators. But in proposing the exact same kind of entailment at all levels, we cannot clearly identify the agent. The truth is we don’t clearly know what we mean by us, either. The knowable facts are that it happens for these reasons (i.e., “causes” constitutes the way of reasoning).

Question #11: Coming into being: possibility, necessity, potentiality, actuality.

The non-dual identity is the Being. The four quadrants are its coming into being. Since we can’t include the Being itself in the four quadrants, we can’t describe it. Also, since the holon is both part and whole, the Being is both what is created and what does the creating. Best to just think of it as nature unless you prefer a religious view.

Question #12: What is emergence? Is it via meta-causation?

This was partly addressed in Tutorial #2. Emergence is normally thought of as the spontaneous creation of a new kind of system that cannot logically come from its ground of Being. For example, “emergence of life”. Rosen showed that life is indeed logically connected to predecessors in complex relations. R-theory proposes that the “Ground of Being” is relational complexity rather than mechanical simplicity. In that case, there is no mysterious emergence; it is now understandable. Also, there is no more necessary meta-causation than I discussed above. The holon itse

lf can explain how life developed from complex relations (see Synthesis paper for life-holon diagrams). Since it abandons the idea of a material ground (fundamental particles as building blocks), one can naturally ask where those ideas come from, since the existence of physical objects is pretty obvious in our world. R-theory does not say we should disbelieve in the material world. It says it is a reduction of the complex ground beneath it, just as life is a construction on that ground (note the top red arrow in the diagram does not point down from “mechanical”).

What we previously called emergence is now a phenomenon explained by the relational holon. I’ll be a little suspenseful and lead into it. Can there be true emergence in a mechanically defined domain, i.e., the left side of the holon when there is only one formal cause on the right? Is that possible? The answer is no. It is the same question as existence of Rosen’s closed loop of efficient entailment, or the 2nd-order holon discussed in Tutorial #2. It is a logical structure that violates the logic of the domain as defined by any single formal system. Is emergence logically possible on the right, in the contextual domain? The answer is yes. It is characteristic of that domain, as shown in this diagram below (repeated from Tutorial #2):

Non-local potentials for existence necessarily overlap because they are not distinct entities with separate location as we see in our sensory concept of the realized world. Also, they are not destroyed by that overlap but co-exist. We can think of them as combining in the same sense as Venn diagrams. In computer graphics it is easy to visualize. Take a partially transparent blue circle and overlap it with a yellow circle, and there will be a greenish region produced. If we try to do the quantitative addition of two systems we have the ambiguity that 1 + 1 might equal 1, 2, or 3!!  That is related to Godel’s incompleteness proof for number theory; the meanings (semantics) are necessary. For quantitative number theory to work we have to count discrete entities and think of them as objects or things. If they are not discrete, there is the possibility of different results, and emergence is one of them.

I re-visited the contextual emergence diagram because in the other tutorial I said I would address the claim in the diagram about “entropy reversal”. While entropy increases in the realized world according to the 2nd law of thermodynamics, it decreases in the contextual world. This is part of the “inverse” entailment of going from structure to function in the contextual world, as opposed to going from function to structure in the realized world. Because contexts are indiscreet (non-discreet?) and thus overlapping there are many functions that could result from placing some structural pattern into them (final cause). In the usual concept of entropy, it is negatively equated with order. Full equilibrium that exhausts all usable energy is equated with high entropy. But if there is a non-random distribution, that represents order and decreased entropy. So organisms reduce entropy by establishing internal order that is balanced with a corresponding increase in entropy in the environment. That internal order is accomplished by closure contextual combinations. Hence, just as the 2nd law applies to mechanically closed systems only, its inverse applies to contextually closed systems. R-theory proposes that these two domains actually exist as complementarities at every level, so our previously idea that the universe must eventually run out of energy, reasoning only from the mechanical side, is not supported when we consider the combination.

Question #13: What is the nature of the holon?

It is a model, and therefore an explanation for natural complementarity, also referred to as duality. Because of its closures, however, it provides for both dual and non-dual descriptions of nature. It is both epistemology (the vertical components) and ontology (the horizontal components). The most ‘real’ aspects about it are the four causal quadrants, because they implicate the four elements sitting between the causes as their beginnings and ends, and they establish the cycle of causes that is self-generating. The holon is a meta-model for science and nature. It does not, however, objectify the causes themselves, it objectifies the relations, and the causes are the descriptions of those relations. For example, the formal/efficient relation refers to the relation between what is allowed in nature and what happens – our usual concept of natural law. The material/final relation refers to the relation between what did happen and what that enables as future possibilities. As far as its meta-causality, which I addressed above, the best we can say of it is probably that it is consciousness.

Again, we have been moving away, in science, from concepts of causality in favor of heuristics and statistics, because the implications of adopting a fully causal view are seen as unpalatable. My aim in explaining it is to help demystify it so people don’t think it is an more religious than current science, perhaps even less so. But it does force us to consider some very deep questions about our own reality as well as nature

Giving up on causality, many physicists today argue that we can’t know, for example, Quantum reality, and ultimately any reality; we can only have calculations that seem to work. It has been proposed as a “model based realism”. R-theory goes a step farther and adopts that as a realistic model. Nature models in the same way we do, and we are seeing the results. The alternative is to simply say we can’t know, but there are mathematical procedures that seem to get right approximately answers. I’m not aware of anyone who is comfortable with that, and I am certain we would not be getting answers anywhere near as good as we are if we had abandoned realism earlier. Somehow, chasing the carrot is good for the rabbit.

Do the causes ‘flow’?  I know what you mean but that is obviously not a proper description of it. Flow is an efficient/material concept. Is there a fifth cause that makes them work?  I’m not sure that matters to science, because the four causes are already more than science wants. But for fireside chats, yes, I think there is a fifth cause or fifth element as you wish, but it can’t be described. Its the genie in the lamp, the prime mover, the eternal Being in the Upanishads, Brahman – behind everything without any knowable attributes at all. Such a background must exist for any view of nature. Once we have a system of knowing, we automatically imply a source, or at least the possibility of the question. But why saddle this theory with that problem? We already accept that causes do things without asking why. That’s what the word means, the buck stops with the cause. Cause is explanation of how/why <something in nature>.

Question #14: Examples

The claim is that everything is an example, so the better question is what is not an example, and preferably something that is so elegant in its construction that it can point out a weakness in R-theory itself. I employ various examples all through the Tutorials and the papers. See a recent example in the R-theory interpretation of Learning Organization, on this website. The Tutorial examples are to support the argument, but the examples addressed more thoroughly in papers, such as space-time theory, are meant to test the extreme application of the view to see if it can hold up.

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.

Recall the definitions from the 2011 synthesis paper:

My understanding of transitivity is rather simple. If A entails B and B entails C then A entails C transitively. That is the case for all the entailment transitions in either half (left or right) of the diagrams, including the identity diagram (b). A closed loop of causation cannot exist in either domain it has to go through the opposite domain in what I suppose we can call an inverse transitivity. For the relations in the holon, we have to write “If A is related to B and B is related to C than A is related to C”, and that looser form of transitivity is also true as long as we understand that relation is information which means that it is an abstraction for a previous set of conditions to establish new boundary conditions for what can happen on the dynamical side. It therefore does not specify the dynamics, it changes their constraints so the system behaves differently, perhaps with a different set of attractors.

While the realized entailment is a necessity according to the laws of a given domain, relations involve selection and interpretation by possibly different systems because contexts are non-local and non-discrete, and thus always shared to some degree, whereas within the realized domain entailments can be exact because states are discrete and the rules are defined for one domain of realization. More specifically, when we write that f entails f contextually, we have to say that f makes f possible, not that it necessitates f. Sustainability of the identity then becomes a genuine question that can be address, involving other relations.

Hence, f:(A,s) defines an efficient/material entailment in category theory [using the new conventions above]. We can also write f  |– s, meaning “f entails s“.  I’ll use words from now on because it is easier. Between holons we can use the traditional category theory to construct open-ended entailments. But, if we try to implement the Schroedinger/Rosen hypothesis (see Rosen’s book, Essays on Life Itself) that a material result (inertial system) can become an efficient cause (gravitational system) we should be able to write that s entails f. If we then construct these hierarchically, remaining in the realized world of efficient entailments, we have a description of f transitively entailing itself, which is indeed a logical contradiction in the mechanistic notion of the realized world, because that kind of organization cannot exist in the realized domain alone. This is a flaw in the mechanistic world view, or as Rosen calls it, “the Newtonian paradigm”.

But for that reason we must propose a way for a material result to become an efficient cause. In other words, how a ‘thing’ becomes an agent, not just saying that it does paradoxically. This is done in R-theory by considering inverse contextual relations. In plain language, a ‘thing’ placed in a different context, acquires new functions according to the nature of the context. This is directly analogous to saying a function causes a material result according to the nature of the material. Clearly the same force applied to wood produces a different result when applied to steel. In the same way a structure produces a different function in different contexts. For example: a screwdriver taken from the shop to the kitchen can acquire a new function for opening cans.

Note that in Rosen’s work, and Louie’s that followed, this contextual entailment was not proposed. It is an addition I made in R-theory, in fact what distinguishes R-theory from its predecessors. The Rosen complexity school simply worked with the contradiction of a closed loop of efficient entailment IN the realized domain, supposing that it is simply indicative of what can really happen in the natural world except that it was overlooked by mechanistic science. We know that a massive object has both inertial and gravitational properties, and so it is simply a fact that a material result in the efficient map (at the head of an open arrow) can also be the start of a closed headed arrow (efficient cause). But that merely throws a monkey wrench into quantitative science because it can’t be handled in any equation to create a loop from result to cause. Therefore, a context is required to save computation of observed temporal properties while still allowing for complex loops. That was the resolution R-theory came up with and it profoundly implies that no physical event takes place except within a context that determine its universal constraints; and that allows the possibility that those constraints can vary, or be altered by the system. That allows us to model obvious ways they are altered, such as by thought, but also un-obvious ways we have discovered empirically, as in quantum theory, relativity, ecology, sociology …. all fields that are characterized by complexity.

I label the inverse entailment by using dashed instead of solid lines, and because it is inverse the arrow conventions also switch positions. Thus you have s entailing f. Because the transitive relation between f and itself happens through context there is no logical contradiction — the problem of transitivity between a structure and a function has thus been removed because context inverts structure via context to entail function.

There is no difference between identity and interactive holons in this regard. The only difference is that identities define a special system and context that are associated with the origin of a specific holon, whereas the links shown externally (for convention) define a systems interactive properties as a whole. F in diagram (b) above has four facets that entail or relate each other, and thus define the identity in precisely the same way that they act between holons.

Relations are not a necessity but a possibility, as discussed above. For example, fish occurring out of water in the realized domain may place them in an unsuitable context (of existence factors) for their survival.  That contextual relation will then not result in any realizations. However, if there is some pre-adaptation that allows some fish to survive well enough to reproduce and evolve, then there may indeed be realizations of a new kind of amphibian. The relational arrows in this case do not say that a new kind of fish must necessarily emerge from that contextual matching, but that one can emerge. How we analyze that possibility can vary. We might give it probabilities or suitabilities, or we might try to analyze the internal physiology to see if survival is possible. But we do not know of a set of entailments that will necessarily predict the result – that is the whole point of relation vs. entailment. If we did know of such efficient maps that would explain the phenomenon adequately, that could be represented entirely in the realized domain because its very existence means that it shares a common context. The loop through context still theoretically exists, but it makes no difference in that special case. Again, this is not a flaw in the causal architecture, but a description of natural complexity and way of analyzing it.

Holon relations can then have multiple forms depending on how they are combined. There are many ways to combine them — I have not counted the ways. I showed these three in the paper [left], and additionally dealt with the case of shared contexts. I call identity a 1st-order closure and the hierarchical entailment you see at the bottom is a 2nd-order closure (meaning it closes two holons with each other), which is necessarily complex because it then has a dualistic context. The sequential composition in the middle is a 1.5 order holon. It is not fully complex because its context is not a dualism but a temporal sequence. Instead of being dual, the context is bifurcated according to our traditional concept of time. This is a mechanistic sequence where s1 is transitively related to s2 by a time domain. But notice there is no smooth transition between s1 and s2, the sequence of states is described in discrete steps as a result of the causal loop. Similarly, if we were describing the state-bifurcated context of a quantum particle — its various energy states — the sequence would depend on energy transitions between various quantized state potentials.

 I further argue that all natural systems have at least two fundamental contexts; that of their origin and that of their operation (which is the reason I also show the way holons interact with internal vs. external links). These are never commensurable contexts, so reality is complex. Naturally any syntactical interpretation of the holon loop that does not take into account contextual relations as an expression of semantics, must reveal a contradiction; but the contradiction is the result of the method of analysis. Specifically, any reductive method of analysis will reveal a contradiction in the logic, which is really a contradiction of, and in a natural sense disproof of, reductionism. The combination of like-headed arrows (see legend) is therefore quite unlike the forward and inverse entailments where unlike headed arrows are combined. The former is an information encoding and decoding between categories – the real meaning of ‘relation’ – whereas the later is an entailment within a category.