Dynamical Systems and the Metaphysics of Substance and Accident

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The Metaphysics of Substance and Accident is one of the long established traditions in Philosophy and dates back to its founder Aristotle. We discussed this metaphysics in our Essay on Substance and Accident . In this metaphysics a distinction is made between properties (' accidents ') and that which has (' carries ') those properties, the (' first ') Substance. This, so (traditionally) conceived Substance, when taken generally (it is then ' second Substance ') was called the ESSENCE of the thing (uniform thing, uniform being) in question.
These notions seem clear when we exemplify them with the state of affairs observable in human beings :
SOCRATES is a first substance, it is the Individual, while (Socrates being) HUMAN is second substance, and HUMAN (HUMANITY) is the Essence of Socrates.
(Socrates being) 1.78 meter long is a property (in this case an accidental property, accidental with respect to HUMAN, because it is not part of being human (not every human is 1.78 meter long).
But when we think about these notions further, and especially when we try other examples (for instance a crystal of a certain sort, a star, an ant or a plant of a certain sort, not to mention individual free molecules or atoms), then things turn out not to be so clear anymore :
What is the status of a caterpillar and a butterfly? What is the status of liquid crystals? And with respect to mixed crystals, are such crystals aggregates of substances, or one  substance? Is the status of properties of all these, and other, things really ontologically different [= different according to their (way of) being] from their substance? Where does evolution fit in? How must we interpret chemical reaction-systems? What is exactly the ' Essence ' of a thing? This last question has been already treated, albeit succinctly, in our Essay on Being and Essence.

But to clarify and deepen those observations we need something to know about DYNAMICAL SYSTEMS. Of course that means : natural dynamical systems, i.e. concrete, material, physical or biological systems. We shall discuss them. But we must emphasize that those natural systems are generally very complex, and only partially understood, especially the biological ones. Therefore we shall concentrate our expositions on abstract dynamical systems in the form of computer simulations. Which in fact means that when we discuss real dynamical systems, we are inspired by those abstract systems. Such systems are fully defined, and because of that better understood. Maybe they can supply us with the proper concepts needed for a revised Substance-Accident Metaphysics. We must thereby keep in mind, however, that those simulations are relatively simple and are not able to supply all the concepts needed.
By means of the study of dynamical systems we hope to find out more about the status of Substance, Accident, Essence, Individual, the per se and the per accidens. Maybe we can do this by means of conceps like Dynamical Law, System State, Initial Condition, System elements, Attractors, Phase-portraits, Attractor Basin Fields, Dynamical Stability, and so on, which in fact means the following :   an ontological interpretation of those concepts.
"Ontology " means here : The study of Being as such, the way of being of an individual thing, the status of being of the Essence, the status of being of the Universal, the status and way of being of properties in relation to Substance, the status of a process in terms of being or becoming, etc.

The Dynamical System

What then is a dynamical system?

A DYNAMICAL SYSTEM is a process that generates a sequence of states (stadia) on the basis of a certain dynamical law.

Such a dynamical law, together with a starting-state (initial condition) is already the whole dynamical system.
When the system states together form a continuous sequence, the dynamical law will have the form of (i.e. will be described with) one, or a set of, differential equations, which describe, and dictate as a law, the changes of one or more quantities in time (and in space), and so (describe and dictate) the continuous sequence of states.
When, on the other hand, those states together form a discrete sequence, then the dynamical law will express a constant relation between (every time) the present state and the next state, and this relation is of such a nature that no infinitesimals are involved (in other words, the differences between successive states are nowhere infinitely small).
Every process state (system state) is a (certain) configuration of, ultimately, system elements , which (configuration) is generally different with each successive process state. The dynamical law according to which those configurational changes proceed, is immanent in the (properties of the) system elements (For where else should it be seated?). The changes in the element configuration, and so also the (implied) succession of process states, is the effect of (i.e. is caused by) interactions between system elements (In such a process it is possible that some system elements disintegrate to other elements or form compounds with other elements, and then those products will interact with one another). These interactions are the concrete embodiment of the dynamical law in action.
The changes of configuration could be such, that we can (after the fact) speak of SELF-ORGANIZATION of the system elements towards a coherent stable PATTERN. This pattern can be a final configuration of the system elements, to which the system clings, i.e. never leaving this configuration anymore. But the above mentioned pattern can also be a dynamical pattern, which also is coherent in itself, but which moreover alternates in a regular and coherent way.
Both cases of self-organisation can be interpreted as the formation of an organized whole, and this we will call a Totality (in all cases of real systems, that means a Uniform Being ), especially when the generated (dynamic or otherwise) pattern shows an intrinsic delimitation (a boundary) with an environment. When this all happens we speak of a Totality-generating dynamical system.

An ontological interpretation of a macroscopic Totality (a uniform thing) in terms of dynamical systems will proceed along the following lines :
We will first of all presuppose (the presence of) a Totality-generating dynamical system.
The process stadia of such a system now imply the corresponding process stadia of the Totality. A process stadium of a Totality is the ' Here-and-now Individual ' -- we can call this also : the Semaphoront -- while, all those stadia taken together, form the ' Historical Individual '.
Generation of a Totality means that the system elements, or a part thereof, together form a coherent whole, a totality-resultant of the dynamical system, coherent, either in space, or in time, or in space and time, and having an intrinsic boundary with an environment. So not just any (arbitrarily chosen) process (dynamical system) generates a Totality. Especially for abstract ' dynamical ' systems -- which are moreover just simulations of dynamical systems -- applies the following : They cannot supply all the revised metaphysical concepts, needed for the establishment of a revised Substance-Accident Metaphysics. The dynamical systems that are involved in the generation of full-fledged Totalities thus have some special properties.
Every process stadium is a certain configuration of system elements and can be considered as an initial state , meaning that the system will be observed from that state onwards.
The real , actual (= ' historical ') initial state also is a configuration of system elements, but as configuration it originates from outside the system. It could even be a configuration which in principle cannot be generated by the system from other configurations (i.e. from other states). Such an actual initial state can also be random (i.e. a random arrangement of elements, or / and random states of the elements themselves). But the system can transform this random configuration, once given, into a (following) PATTERN (= a not-random configuration), and in turn into a next pattern, etc., and so leading to a sequencing of patterned process stadia. In other words the system is then able to organize the constituents into a real pattern. The elements (constituents) are going to take part in (the formation of) a Totality. The sequence of process stadia, taken as a whole, also can be considered as a pattern when it gives a reason to do so. But a real Totality is only formed when such a pattern has a, for the system intrinsic, boundary with an environment.

A local, individual action originating from the environment, thus coming from outside a running dynamical system, can be considered as a perturbation of a current process state. The perturbation then creates, as it were, a new initial state with respect to that process.

The relevant properties of the constituents (the system elements) of that process determine the nature of their interactions. Thus that which determines those interactions as such and such (a way) taking place, is immanent with respect to the constituents. The whole system, and thus the whole process, is further constrained by the general (global) state of the environment and thus by the general nature of physical matter, described by Natural Science in the form of general -- i.e. everywhere operating -- Natural Laws. These global (i.e. operating on a global scale) Laws of Nature are immanent, i.e. inherent in the general properties of physical matter.
The mentioned -- (mentioned) with respect to the taking place of the interaction process -- relevant properties of the system constituents can also in this non-global case -- thus in the case of a special process, taking place somewhere, generating a Totality -- be interpreted as a law, namely the law that is valid for specifically that (type of) dynamical system : The Dynamical Law. This law is, as has been said, immanent in the relevant elements of the system. The pattern, i.e. the arrangement -- at a certain point of time -- of these elements is extrinsic with respect to that Dynamical Law. It even could, as have been said, be an arrangement (configuration of system elements) which cannot even in principle be reached by the system itself from whatever initial condition. Such an unreachable state, which as such can only originate from outside the system (= from outside being imposed on the system), is called a ' Garden of Eden State ' of the system (Theoretical models learn that many systems each for themselves have a large proportion of such Garden of Eden States). Such an unreachable configuration (of system elements) either is a real starting state of the dynamical system (i.e. the system happened to start just with such a configuration), and so is coming from outside the system, or such a state is the result of a perturbation, which took place at some point in time during the running of the system, and so is also coming from outside the system, a perturbation of a process state (situated) higher up in the sequence. Thus by actions from outside, a current process state, itself also being a configuration of system elements, can be changed, resulting in a new, i.e. other, configuration of system elements, which then functions as an initial state with respect to the further history of the process.
So a dynamical system implies a number of types (meanings) of : " outside the system " :

Stability of a Dynamical System

The successive process-stadia are -- when no perturbations, coming from without, have taken place -- as succession (thus insofar as being a certain succession), necessary. The succession necessarily follows from the Dynamical Law.
But, as element-configuration every process-stadium is per accidens with respect to the system, because this configuration varies, while the Dynamical Law remains constant. It depends on (the point in) time (when we happen to observe the system).
If the process leads to the formation of a relatively stable PATTERN -- and only such processes are being considered here -- then the above mentioned perturbations (caused by the environment) will be damped (This is demonstrated by theoretical models, and by observations of such processes occuring in reality) : The system attemps to maintain itself, it reverts to its original course.
This can take place in two ways.

Before we explain these two ways, something must first be said about the status of element :
With " elements coming in from outside the system (and therefore quasi elements) " I mean elements which are not imported by the dynamical system itself, but which, by accident end up within the active domain of the system. If a Totality, or more generally, a pattern (of a higher order than that of the system elements themselves) is being generated within the active medium of the dynamical system, then it is possible that the elements belonging to that active medium, as well as possibly elements coming from without (that active medium), are going to participate in the formation of the Totality (the unified pattern, being generated by the dynamical system). The insertion into that Totality, in this last mentioned case (elements coming in from without), does not happen by virtue of the Dynamical Law of the relevant dynamical system, but is a perturbation from without. The present context is concerned with the effect of elements-coming-from-without on the stability of the dynamical system.

Now we are ready to discuss the above mentioned two ways by which the system tries to maintain itself :

  1. If the elements, incoming from without -- we can call them quasi elements (NOTE 1) -- which are going to participate in the formation of the Totality -- ARE OF THE SAME TYPE as those elements which already belong to the evolving Totality, then the Dynamical Law, which is in operation at that particular moment, will not, because of that, be changed, because it is inherent in the relevant properties of those elements.
    Insertion of such new elements, coming in from without, in the evolving Totality, can change the pattern of this Totality, and that means that the Totality not only changes in that sense that it is a next stadium, but moreover that it also has changed in an extrinsic way. It turns out that there are systems whose course of the process, and so the appearance of the sequence of the successive states, is very sensitive to such changes, in such a way that the mentioned change effects a totally different process course, totally different from which it would have been, had the change not taken place, or had another change occurred instead, in spite of the fact that the Dynamical Law has not changed. Such dynamical systems are called chaotic . The system then goes to another attractor that could result in the formation of another Totality. But it could also be the case that the trajectory, leading to such a Totality, becomes very long, or that no Totality is formed at all. But in all these cases the Dynamical Law stays the same.
    Dynamical Systems lacking this sensitivity, or showing it in a small degree only, thus systems which damp such perturbations of stadia, resulting in an unchanged process course (or quickly restored) are stable with respect to change in initial condition (Every system state, thus every stadium, can be considered as an initial state of the subsequent course of the process).

  2. The elements, coming in from without, taking part in the formation of the Totality, could -- concluded after the fact -- have properties that ARE TOTALLY DIFFERENT from those of the original ' real ' system elements, and in such a way that the overall-garnish of relevant properties obtains another face. And because the Dynamical Law is immanent and inherent in the relevant properties of the elements, it is possible that the mentioned ' otherwiseness ' of the incoming elements is such that from that moment on we'll have to do with an altered Dynamical Law (having replaced the original Dynamical Law), or a multitude of different (new) Dynamical Laws. In this very last case the original dynamical system will degenerate. In the first case the process could acquire a totally different course, but it can also occur that all this does not have any noticeable effect on the course of the process :
    The course of the process restores itself quickly and resumes its original course again, because, for instance, the mentioned alteration of the Dynamical Law was small, or because of whatever other factors in this alteration. When this is the case we have to do with a system that is insensitive or little sensitive to alterations of the Dynamical Law. Such a dynamical system is called structurally stable.

The Totality, the Individual, the Identity, and the Dynamical Law

Any single process stadium (and that means any single element configuration, any system state) is per accidens with respect to the Dynamical Law. And because a Totality stadium is a part of the corresponding system state, this also applies to any single Totality stadium.
" per accidens " here means that a particular system state is, among other things, dependent on the point in time of observation (an observation at another point in time will reveal another system state). The system just happens to be in state (say) S. But because the sequencing of states is per se, such a system state is only partially per accidens (i.e. only so in the sense mentioned), while a perturbed system state (insofar as perturbed) is wholly per accidens.
With an ongoing alteration of system states, the Dynamical Law stays (effectively) the same as long as the system does not disintegrate, or change (into some other system), thus as long as the environment of the system is such that under those conditions the system is structurally fully stable.
The Dynamical Law is the actual Identity (with respect to content) of the dynamical system, and together with that it is the Identity of every process stadium, and then also the Identity of the generated Totality, and then by implication the Identity of every Here-and-now Individual, and in this way also the Identity of the complete sequence of Totality stadia, the Historical Individual, which is the very Individual.
"Individual" here can be considered not only as an undivided something, but also as a space-time being, i.e. a Singular (For the concept of Identity see the Essay on Being and Essence ). A Totality is such a Singular.
When this Identity (= intrinsic Whatness) indeed refers to a Totality, we can call it the Essence of that Totality, and as such it is (also) intrinsic cause. The Dynamical Law S, governing dynamical system M, thus is the Essence of the Totality T generated by that dynamical system. This Essence (because it is intrinsic Whatness) can also be directly related to the Species of the Species-Individuum Structure of every genuine Totality.

Substance and Accidents

(For the philosophical meaning of Substance and Accidents see the Essay on Substance and Accident )

If a given real dynamical system generates a full-fledged Totality, for instance (generating) a crystal from a solution, then this Totality is a Substance (in the metaphysical sense of the term), more specifically, it is a First Substance.
The Essence or (ontological) Second Substance, is the Dynamical Law of such a system.
All the observable properties of such a Totality are generated by the system. These properties are called Accidents (although they do not all have a status of 'generated by accident' ), and will be all kinds of quantitative properties like length, volume and the like, but also qualitative properties like configuration (which can end up as colors, densities and the like).
All these observable entities, the First Substance and its properties, are, as has been said,   generated.
Borrowing terms from Genetics, we could say that those generated entities are seated in the 'phenotypical' domain (a domain of being, a way of being), while the corresponding Dynamical Law is seated in the 'genotypical' domain (another domain of being, another way of being). We discriminate between these domains, because the Dynamical Law as such is not observable. It abides in the collection of system elements, i.e. it is dispersed over those elements, without being the same as those elements because it is only dispersed over some (not all) aspects of every system element. Therefore the Dynamical Law is neither a thing, nor a property. It is abstract.
The First Substance and its properties are, on the contrary, concrete and directly observable.
The mentioned accidents belong to that first substance of which they are accidents. Some of them belong to it per se, others only per accidens.
All those accidents together make up the first substance [NOTE 2]. They can only exist as a first substance. They cannot exist on their own, because they are, each for themselves, just a determination of a first substance.

The Essence and the Attraction-basin Field

Depending on the presence of a certain initial condition (i.e. a start configuration of [states of [NOTE 3] ] system elements), the dynamical system will finally reach one or another "attractor", for instance a periodic cycle, along which the system will then cycle indefinitely. Starting from another initial condition the system may reach another settling pattern (cycle of system states), and so reach another attractor.

Remark :   A different initial condition does not correspond to a different settling pattern in a per se manner. So, often a certain set of different initial conditions exists, each member of which bringing the system to the same settling pattern. But such a set need not be the total set of possible initial conditions with respect to the system.

The total set of states, belonging to, and arranged according to, all possible trajectories, all leading to a certain attractor, say attractor A(1), forms, together with the attractor states themselves, the basin of attraction of the attractor A(1) (analogous to all the rivers that drain a certain area and all end up in the same lake).
The total of all possible basins of attraction belonging to (i.e. corresponding to) a certain dynamical system is called the phase portrait (this term is used in the case of continuous systems) of that system, or the attraction-basin field (term used for discrete systems).
The attraction-basin field represents all possible systemstate-transitions of that dynamical system, and is, in a way, equivalent to the Dynamical Law of that system.
The Dynamical Law is the system law (seated) at a low structural level, while the corresponding attraction-basin field is this same law, but now (seen) from a global structural level.
There exist dynamical systems, for instance abstract Boolean Networks (and their real counterparts), where the dynamical behavior depends on a whole set of dynamical laws, but which nevertheless have only ONE attraction-basin field. It seems reasonable to interpret this attraction-basin field as THE (one) Law of the system, and so also as the Essence of the Totality (when a Totality is indeed generated by the system). But of course the mentioned set of dynamical laws can also be interpreted as THE (one) Dynamical Law and so as the Essence of the (generated) Totality.
Just the set of all possible system states corresponding with a certain dynamical system is called the phase-space (term used for continuous systems) or the state space (term used for discrete systems) of the system. The system thus organizes its state-space into (a relatively small number of) basins of attraction, the attraction-basin field, by establishing all its possible state transitions (which means that the possible states are now related to each other in a specific way). And so the dynamical system ' categorizes ' the state-space and because of that the resulting attraction-basin field can be considered as the ' memory ' of the system, especially when such a system is a Boolean Network. A Boolean Network is a discrete dynamical system with only two-valued variables. Such systems constitute a possible basis for the study of genetic and neural networks (See the Essay on Boolean Systems).

The above given interpretation of the notions Totality, Identity, Essence, Here-and-now Individual ( Semaphoront ), Historical Individual, etc. is inspired by the study of simple abstract models of dynamical systems (in the form of computer simulations), which pretend to represent processes, and aspects thereof, in the Real World, especially those processes which show self-organisation of system elements towards stable coherent patterns. Such are for instance crystallisation processes and ontogenetic processes (the last mentioned are processes relating to the formation of an individual organism).
But we must realize that we, in proceeding along these lines, make use of formidable simplifications of natural real processes (natural real dynamical systems), resulting in such models, i.e. reducing them to such models. This is, according to me, inevitable because the processes in the Real World generally are much too complex and much too strongly interweaved and intertwined with other processes, as to allow directly from them (i.e. them serving as a theoretical point of departure) a definitive ontological interpretation of such Totality-generating processes. The models must be part of the point of departure of such an attempt to an ontological interpretation.

Having studied dynamical systems in general (especially with respect to an ontological interpretation of some concepts) the reader is invited to study a specific example of a dynamical system , in order to make his impressions more lively.

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