This is an explanation of the modification I made in category theory symbols in R-theory from .
Rosen used category theory to loosen up the formalism of cause and effect. Entailments are more general. Then one can decide what kind of rules it has, depending on the kind of sets and morphisms. The idea of a homomorphism is important, and the best I can figure is that it invokes the idea of formal cause (although the mathematicians are not clear about these causalities). In other words, if a morphism between two sets (an entailment) preserves the ‘structure’ of the sets, it is a homomorhism. This is mathematical structure, which is mathematical formalism. So, for example, if you have a machine the laws are purely mechanical. There is a homomorphism between machine A and mechanical result B. The results will be mechanical results. That keeps it within one formal cause system, one formalism. To describe something like General Relativity you no longer have a homomorphism because the formal definition of space-time, the mechanical coordinate system, has to change. So, now you can describe a morphism on the entailment – now it changes formal structure. That’s my limited understanding, and it could be wrong or naive.
In any case, I saw a problem in the logic. We have one kind of entailment, a morphism on states. I did not see how one could get from states to a morphism; how a morphism is generated. Rosen gave a long discussion about that in Essays in which he cited Schroedinger as talking about inertial and gravitational objects. These words come from physics but were used metaphorically for anything in nature that is forced (inertial object) by anything that does the forcing (gravitational object). The question was how to get from an inertial object to a gravitational object. The standard entailments don’t seem to do that, but perhaps there is something about it that I missed. If so, then what I did in the 2011 paper isn’t wrong, but already handled and my proposal is then simply to adopt different notation to make the meaning clear and to establish the holon theory. What I did was this:
A mapping is normally drawn with solid lines using solid heads to indicate the morphism and open heads to indicate the set transition. The open arrow thus describes an inertial object being pushed. The solid head might be taken to describe a gravitational object, generation of a morphism, because it has a beginning and an ending too. So, you can draw a gravitational arrow between sets, indicating that one set produces a force or a function. Hence you have a much more general way of describing systems.
However, it is not clear how the magic occurs. You can draw:
more commonly drawn as
to indicate an efficient mapping where a function (vertical solid head) is responsible for the inertial transition, say causing a ball to accelerate.
And you can draw the gravitational map:
where an inertial transition is responsible for producing a function.
Then you can put the two together to get a closed loop of efficient causation
and this is the convention in Rosen’s diagram of life, where he describes a closed loop of three efficient maps (with an implicit fourth and fifth with the environment).
However, this causal structure cannot exist as a mechanism and it has a bias built into its logic. Namely, we understand the forcing of an object in the first diagram. State a pre-exists and is acted upon by a function to reconfigure it to state b. But what is the interpretation of the second mapping? A function pre-exists and is changed into another function by the existence of a state? It says that new functions arise simply from states. Since we imagine the world to consist of a set of states at any moment in time, it means simply that “functions happen”, and it doesn’t say how or why. It doesn’t answer Schroedinger’s question. In fact it is a trick. The closed loop diagram is really only showing the combination of two efficient maps, functions on states, and saying those functions generate each other via the states. If we define the background of the diagram a space where efficient maps can occur, the diagram becomes like an Escher drawing – it is impossible in any space defined by efficient maps.
So, the key to answering Schroedinger’s question, how a state produces a function, lies in changing the background, the context. I didn’t see any way to indicate that change in standard category theory (but there may be a way that I’m not aware of, because like I said, I didn’t go very far into it).
There was another problem in the standard conventions: the idea of a “pre-existing state”. What the H#@$ is that? It is a highly reductionistic concept. States don’t pre-exist, they are abstracted from natural systems by interaction or measurement, according to Rosen’s own theory. So an open headed arrow from state ‘a’ to state ‘b’ is technically either meaningless or a gross summary of something else that is happening.
So, I adopted a different labeling convention, first to drop the convention of mapping between states and instead map between sets of states to a given state. So the set transition becomes drawn in this way:
Which is heresy, but is says that we don’t get ‘a’ from another state, but from a whole system, A, that has the potential state ‘a’ and what the morphism does is to abstract (remove) the state from the system, or, in fact, produce it from where it did not actually exist as such before, only as potential state. This is consistent with quantum mechanics, where we cannot say that a particle state pre-exists before measurement.
The second change was to indicate the realized system background vs. the formal system background (context). The former would be shown with solid lines and the latter with dashed lines.
No damage to category theory done there if it turns out these contextual maps are reducible to realized ones, but they are not.
But you can’t simply identify the implicit portion of the realized closed efficient loop above, like this, because then the original realized efficient maps (solid to open head arrows) are lost. You have to replace all the arrows with dashed arrows and the entire diagram becomes an inverse (formal-final) diagram instead of an efficient diagram. You therefore have to draw one or the other but they are immiscible because they exist in different domains of reality (different formal domains):
We know there is good solid meaning to the original efficient map. The double labeling that is thus required clearly demonstrates why the closed loop is impossible in any homomorphic domain, why it is an Escher diagram. Part of the loop follows one topology and the other part follows a different topology, just as Escher did.
So, the only solution is to expand the loop to include complete mappings in each domain, labeling them to identify which domain they belong to, the realized material world or the contextual formal world. like this:
so that you have contextual and realized maps alternating. Then state transitions do not come from states but from whole systems exactly like this one, which exist between the function (solid head) arrow and the set transition (open head arrow). That tells you that this diagram itself has become holographic. It composes and decomposes into self-similar diagrams infinitely. And that, I think, is what nature does. It says clearly there are two complementary domains of reality – the realized and the potential or contextual. You get a new function when you put a state into a new context. What could be more obvious? Put a screwdriver in the kitchen and someone might use it to open cans instead of to tighten screws. The arm-waiving is gone, we can say exactly how states produce new functions, and its not only mathematically tight, it is intuitively obvious.
One more thing, once we have done the above, it becomes clear that the diagram exactly matches Rosen’s modeling relation. Hence the two theory tracks he followed are now unified.
Here you can also see the holographic quality in the red loops that indicate internal relations of each system, which must also be whole like main diagram. And, by extension, larger diagrams are also self-similar and holographic.
You can also see that I split the encoding and decoding arrows from Rosen’s original modeling relation diagram. He noted with some emphasis that encoding and decoding are necessarily outside the formal and natural system boxes, so what are they? They are clearly information relations that exist within the holographic nature when you combine diagrams. Here they are clearly identified as relations BETWEEN the two domains of reality, realized and formational, and as such they are technically functors in Category theory. Each side of the modeling relation is a category and functors exist between categories. Categories contain homomorphic entailments, i.e., entailments of the same logical type.
Thus we have way of analyzing whole systems.
But the next thing I want to do is to work with Rosen’s detailed mappings in his books to see if there is any inconsistency with this extension of the theory. I don’t yet understand the normal logic in those more detailed diagrams well enough to do that. I suspect I will find something quite similar, where I can inject the contextual maps.
Please cite:
1.
Kineman, J. Relational Science: A synthesis. Axiomathes 21, 393–437 (2011).
