# MES: A Mathematical Model for the Revival of Natural Philosophy

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## Abstract

**:**

## 1. Introduction

#### 1.1. Mathematical Models in Physics

#### 1.2. Composed Objects: The Part–Whole Problem

- “The whole is nothing more than the sum of its parts”;
- “The whole is something else than the sum of its parts”;
- “The whole is more than the sum of its parts”.

#### 1.3. Compositional Hierarchy

_{i}. In this case, we say that C is multifaceted and that the P

_{i}’s represent its lower-level multiple realization (compare with Kim [6]). Over time, new multifaceted components of increasing complexity may ‘emerge’, generally due to non-linear phenomena or chaotic dynamics.

#### 1.4. The MES Methodology for Reviving NP

- they have a tangled hierarchy of components which vary over time, with possible loss of components as well as emergence of more and more complex components and processes;
- through learning, they develop a robust but flexible memory allowing for better adaptation;
- the global dynamic is modulated by the interplay between the local dynamics of a net of specialized agents, called co-regulators, each operating stepwise with the help of the memory.

#### 1.5. Outline of the Article

## 2. Categories for Modeling Multi-Level Systems

#### 2.1. Categories for Modeling Complex Systems

- Small categories: A monoid is a category with a unique object. A group is a category with a unique object and in which each morphism has an inverse. A category K with at most one morphism between two objects ‘is’ (associated to) a p(artially)o(rdered)set (K
_{0}, <), where K_{0}is the set of objects of K and where the order < on it is defined by: k < k’ if and only if there is a morphism from k to k’ in K. - Categories of paths: To a graph G is associated the category of paths of G, denoted by L(G): the objects are the vertices of G, a morphism from x to x’ is a path from x to x’ and the composition of paths is given by concatenation.
- Large categories: Sets denotes the category having for objects the (small) sets and for morphisms from A to B the maps from A to B; the composition is the usual composition of maps. Similarly we define categories of structured sets, for instance the category of groups, with homomorphisms of groups as morphisms; the category Top of topological spaces, with continuous maps as morphisms. Cat denotes the category having for objects the (small) categories H and for morphisms the functors between them, where a functor F from H to H’ is a map which associates to an object A of H an object F(A) of H’ and to a morphism f: A → B of H a morphism F(f): F(A) → F’(B) of H’, and which preserves the identities and the composition.

_{i}and P

_{j}, the paths in P from P

_{i}to P

_{j}have the same composite. In category theory, commutative diagrams lead to a kind of calculus in which they play the same role as equations in algebra.

#### 2.2. Interpreting the Part–Whole Problem via the Categorical Notion of Colimit

#### 2.2.1. The Categorical Notion of Sum

**Definition**

**1.**

_{i}, if it exists, is an object S of H such that there exists a family of morphisms s

_{i}: P

_{i}→ S satisfying the ‘universal’ condition:

(Let us note that these equations, which mean that for each i the composite sif (a_{i}) is a family of morphisms a_{i}: P_{i}→ A toward any object A, then there is a unique morphism a from S to A ‘binding’ this family, meaning that a: S → A satisfies the equations a_{i}= s_{i}a for each i (cf. Figure 1).

_{i}a is equal to a, would have no meaning in a network where composites are not defined).

_{i}modeling its parts”. It corresponds to a ‘structural’ reductionism (in a non-relational context).

#### 2.2.2. The Categorical Notion of colimit

_{i}but also ‘something else’, namely some given relations between them.

_{i}and whose morphisms represent the given relations between them, so that the entire P models the parts and their organization.

_{i}) of morphisms a

_{i}from P

_{i}to A such that, for each morphism f: P

_{i}→ P

_{j}in P we have a

_{j}= a

_{i}f (cf. Figure 1).

**Definition**

**2.**

_{i}) from P to cP (called a colimit-cone) satisfying the ‘universal’ condition:

For each cone (a_{i}) from P to any object A there is a unique morphism a from cP to A such that we have: a_{i}= c_{i}a for each i; this a is called the binding of the cone (a_{i}).

**Proposition**

**1.**

**Proof.**

_{i}) of morphisms forming the colimit-cone from P to cP binds into the comparison morphism c from S to cP. □

#### 2.3. Categorification of A Compositional Hierarchy

**Definition**

**3**

**.**A hierarchical category is the data of a category H and a partition of the set of its objects into a finite number of levels of complexity, numbered from 0 to m, verifying the condition: Each object C of the level n+1 is the colimit of at least one pattern P of interacting objects P

_{i}of levels less or equal to n. Then we call P a lower-level decomposition of C. A morphism of level n is a morphism between objects of level n.

_{i}of each object P

_{i}of P, and so on, down, till we reach a set of patterns of level 0 which form the base of the ramification.

**Definition**

**4.**

_{i})) where P is a pattern of level < 2 having C as its colimit, and Π

_{i}for each object P

_{i}of P is a pattern of level 0 with P

_{i}as its colimit. The patterns Π

_{i}(some of which can possibly be reduced to an object of level 0) form the base of the ramification. This base is not sufficient to re-construct C from level 0 up since we need supplementary data expressing the constraints imposed by the morphisms of P.

#### 2.4. Simple and Complex Morphisms: The Reduction Theorem

#### 2.4.1. Morphisms between Complex Objects Deducible from Lower Levels

_{i}of P to objects P’

_{j}of P’, well correlated by a zig-zag of morphisms of P’ (Cf. Figure 3).

#### 2.4.2. Multifaceted Objects

**Definition**

**5.**

#### 2.4.3. Complex Morphisms

**Definition**

**6.**

#### 2.4.4. The Reduction Problem

**Theorem 1**

**(Reduction Theorem [10,21]).**

**Proof.**

_{i})). If all the morphisms f

_{ij}: P

_{i}→ P

_{j}of P are simple morphisms, they bind clusters F

_{ij}of morphisms of level 0 from Π

_{i}to Π

_{j}. We can define a pattern V of level 0 as follows: it contains the Π

_{i}as sub-patterns and has also for morphisms the union of the clusters F

_{ij}. It is proved [10] that the pattern V has also C for its colimit. Since C is colimit of such a pattern of level 0, its complexity order is 1. □

**Corollary**

**1 [pure (methodological) reductionism].**

#### 2.5. The Complexification Process. Main Theorems

- ‘adding’ to H a given external graph A,
- ‘suppressing’ a set E of objects and morphisms of H, eventually thus dissociating a complex object by suppressing its colimit;
- ‘binding’ patterns P of a set U of finite patterns in H so that each P acquires a colimit cP or, if P has a colimit in H, preserves this colimit.

#### 2.5.1. The Complexification Process

To construct a (hierarchical) category H’ and a partial functor F from H to H’ satisfying the objectives (O); in particular, it means that, for each P in U, the image of P by F will have a colimit cP in H’.

**Remark**

**1.**

#### 2.5.2. Construction of the Complexification

- The partial functor F from H to H’ is defined on the greatest sub-category of H not meeting E.
- The objects of H’ are: the vertices of A, the (image by F of the) objects of H not in E and, for each pattern P in U, a new object cP which becomes the colimit of F(P) in H’. This cP is selected as follows: (i) if P in U has already a colimit C in H, we take for cP the image of C by F; (ii) If two patterns P and Q in U have the same functional role in H, we take cP = cQ, so that, if P and Q are structurally non-isomorphic, cP will be a multifaceted object in H’.
- The morphisms of H’ are the arrows of A, the (images by F of the) morphisms not in E and new morphisms which are constructed by recurrence to ‘force’ cP to become the colimit of F(P) in H’ for each P in U: At each step of the recurrence, for each P in U we add morphisms from cP to B to bind cones from P to an object B, then we add composites of all the so obtained morphisms; this operation can lead to the emergence of complex morphisms. Then, repetition of such a step on the category so obtained, and so on.

#### 2.5.3. Main Theorems

**Theorem 3**

**(Emergence Theorem [10]).**

**Theorem 4**

**(Iterated complexification Theorem [10]).**

## 3. The MES Methodology

- a Hierarchical Evolutive System (HES) which describes the components of the system and their variation over time through structural changes, leading to
- a developing flexible long-term memory with emergent properties;
- a network of agents, called co-regulators, which self-organize the system through their cooperation/competition; each co-regulator operating at its own rhythm on its own landscape.

#### 3.1. The Hierarchical Evolutive System underlying a MES

_{t}having for objects the states C

_{t}of its components C existing at t and for morphisms the state at t of the links (or communication channels) between these components. The state at t reflects the static and dynamical properties at t, measured by observables depending of the specific system. Among the observables (represented by real functions), we suppose that, at each time, a component has an activity, and a link between components has a propagation delay, a strength and a coefficient of activity. Over time, the components and the links between them vary, with possible addition or suppression, due to structural changes of the kinds indicated in Section 2.5.

#### 3.1.1. Evolutive Systems (ES)

- the timeline of the system, modeled by an interval T of the real line
**R**; - for each t in T, a category H
_{t}called the configuration of**H**at t which represents the state of the system at t; - for t’ > t in T, a partial functor from H
_{t}to H_{t’}called ‘transition’ from t to t’, which models the changes of configuration; it is defined on the sub-category of H_{t}consisting of the (states of the) objects and morphisms which exist at t and will still exist at t’. These transitions satisfy a transitivity condition.

_{t}from its initial apparition in the system to its ‘death’, is modeled by a maximal family of objects C

_{t}of successive configurations connected by transitions. Similarly, a link f from a component C to a component C’ is modeled by a maximal family of morphisms f

_{t}: C

_{t}→ C’

_{t}of successive configurations, connected by transitions, namely the family of its successive states for each t at which both C and C’ exist. At each time t of its existence, a link has a coefficient of activity 1 or 0, to model if it is active (information transfer at t) or not. To sum up more formally:

**Definition**

**7.**

**H**from the category associated to the order on an interval T of

**R**to the category of partial functors between categories; it maps t in T to the configuration category H

_{t}, and the morphism from t to t’ to the transition from t to t’.

#### 3.1.2. Hierarchical Evolutive Systems

**Definition**

**8.**

_{t}of the HES at a time t, we can always ‘categorically’ construct the complexification H’ of H

_{t}for Pr. However, H’ might become a configuration of the system at a later time t’ (dependent on the material change duration) only if the observables defining the states of components and links (e.g., propagation delays for the links) are extendable to H’ through the partial functor from H

_{t}to H’. This is not always possible because these observables might impose some dynamic constraints which cannot be extended to H’. For instance, if one constraint is that the propagation delay of a composite morphism be the sum of the propagation delays of its factors, we have proved [27] that this constraint can only be extended to H’ if the patterns P in U are polychronous in the sense of Izhikevich [28].

#### 3.1.3. Complex Identity of a Component

_{t}in H

_{t}) is the colimit of a lower-level pattern P in H

_{t}. We say that C is activated by P at t if all the morphisms of the colimit-cone from P to C

_{t}are active at t; roughly, it means that C receives collective information from P itself at t. For instance, the information can be a constraint, or a command imposed on C by lower levels; it can also be an energy supply allowing C to perform a specific action.

_{t’}in the configuration at t’

_{’}and the pattern P is transformed in a pattern P

_{t’}via the transition from t to t’. However, this pattern P

_{t’}may not admit C as its colimit in the configuration at t’. Indeed, some objects or morphisms of P may have been suppressed from t to t’ (we recall that the transitions are only partial functors) and it is not supposed that they preserve all colimits. For instance, at a time t a cell is the colimit of the pattern of its molecules existing at that time, but there is a progressive renewal of these molecules and, after some time, the initial molecules will all have disappeared while the cell as such persists

_{t’}of P still admits C

_{t’}as its colimit, while this is no more the case at t+d. However, if C persists at t+d, it admits at least one other lower-level decomposition Q in the configuration at t+d; this Q can have been progressively deduced from P, or not. In this way, C takes its own complex identity independent from its lower-level constituents; this situation corresponds to the Class-Identity of Matsuno [29].

**Definition**

**9.**

#### 3.1.4. Development of a Memory

#### 3.2. A Memory Evolutive System and Its Multi-Agent, Multi-Temporality Organization

- a sub-HES which models a flexible long-term memory Mem, still called ‘memory’, developing through the emergence of multifaceted components connected by complex links;
- a multi-agent organization consisting of a network of evolutive subsystems, called co-regulators, each operating stepwise at its own rhythm, which self-organize the system through their interactions.

#### 3.2.1. The Coregulators and Their Landscapes

_{t}having those links for components; in particular the CR is itself included in the landscape. Using the differential access of CR to the memory Mem, an adapted procedure Pr is selected on this landscape (via pr), and the corresponding commands are sent to effectors E of Pr (Cf. Figure 6).

- p(t)= mean propagation delay of the links in the landscape,
- d(t) = period of CR at t = mean length of its preceding steps,
- z(t) = least remaining life of the effectors of Pr.

#### 3.2.2. The Global Dynamic

## 4. Discussion

#### 4.1. Emergentist Reductionism

#### 4.1.1. Emergence in Terms of Levels

(E) any cone with basis P uniquely factors through the colimit cone from P to cP.

#### 4.1.2. A Hierarchical Category Resorts to An Emergentist-Reductionism

_{i}of each object P

_{i}of P, and so on down to level 0 patterns forming the basis of the ramification (cf. Figure 2). In this way, C is reducible to P, each object P

_{i}of P is reducible to Π

_{i}, and so on top-down through to level 0. This defines a kind of step-by-step reductionism for the objects, with emergent properties related to the bottom to top formation of the different colimits at each step. This situation corresponds to what M. Bunge [8] has called an ‘emergentist-reductionism’.

#### 4.2. Diachronic Emergence in a MES

#### 4.2.1. Emergent Properties of Complex Links between Components

_{’}, but it does not account for its ‘physical’ implementation nor for the duration of the process because of physical temporal constraints similar to those studied above.

#### 4.2.2. Emergence at the Basis of Unpredictability, Creativity, and Anticipation

- the complexification process combines the specified patterns into more complex objects;
- the selection of procedures via the co-regulators leads to an exploration of different possibilities;
- transformational creativity is characterized by iterated complexification processes leading to successive changes in the conditions of change which make the result unpredictable, allowing for surprising results.

## 5. Conclusion

- K-MES, where K is a category of structures (for instance, topological MES if K =Top, multifold MES if K = Cat) in which the configuration categories are internal categories in K and the transitions partial functors in K [36];
- relational MES in which the transitions are replaced by relations, so that a given object at t is related to different objects at t’, the repartition being assigned a specific probability (not yet published).

To conclude, we propose that the MES methodology could help for a revival of Natural Philosophy, by itself and through some of its possible variants.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**A pattern P, the colimit cP of P and the sum S of the family of objects of P. The colimit cone (c

_{i}) from P to cP binds into the comparison morphism c from S to cP.

**Figure 3.**C is an n-multifaceted object which is the colimit of P and of Q. Then the composite of the n-simple morphism from B to C with the n-simple morphism from C to C’ is a complex morphism.

**Figure 6.**The landscape of CR is an evolutive system whose components b, c, c’, pr,…are representede by the curved arrows (their links being the rectangles between them).

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Ehresmann, A.; Vanbremeersch, J.-P. MES: A Mathematical Model for the Revival of Natural Philosophy. *Philosophies* **2019**, *4*, 9.
https://doi.org/10.3390/philosophies4010009

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Ehresmann A, Vanbremeersch J-P. MES: A Mathematical Model for the Revival of Natural Philosophy. *Philosophies*. 2019; 4(1):9.
https://doi.org/10.3390/philosophies4010009

**Chicago/Turabian Style**

Ehresmann, Andrée, and Jean-Paul Vanbremeersch. 2019. "MES: A Mathematical Model for the Revival of Natural Philosophy" *Philosophies* 4, no. 1: 9.
https://doi.org/10.3390/philosophies4010009