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Aventurine

puisque le salut est en nous, n'est il pas assuré que l'armée des esprits débouche dans l'éternité, pourvu que nous ayons soin de maintenir à la notion d'éternité sa stricte signification d'immanence radicale ?

L'homme OCCIDENTAL

«L'homme occidental, l'homme suivant Socrate et suivant Descartes, dont l'Occident n'a jamais produit, d'ailleurs, que de bien rares exemplaires, est celui qui enveloppe l'humanité dans son idéal de réflexion intellectuelle et d'unité morale. Rien de plus souhaitable pour lui que la connaissance de l'Orient, avec la diversité presqu'infinie de ses époques et de ses civilisations. Le premier résultat de cette connaissance consistera sans doute à méditer les jugements de l'Orient sur l'anarchie et l'hypocrisie de notre civilisation, à prendre une conscience humiliante mais salutaire, de la distance qui dans notre vie publique comme dans notre conduite privée, sépare nos principes et nos actes. Et, en même temps, l'Occident comprendra mieux sa propre histoire: la Grèce a conçu la spéculation désintéressée et la raison politique en contraste avec la tradition orientale des mythes et des cérémonies. Mais le miracle grec a duré le temps d'un éclair. Lorsqu'Alexandre fut proclamé fils de Dieu par les orientaux, on peut dire que le Moyen Age était fait. Le scepticisme de Pyrrhon comme le mysticisme de Plotin ne s'explique pas sans un souffle venu de l'Inde. Les "valeurs méditérranéennes", celles qui ont dominé tour à tour à Jérusalem, à Byzance, à Rome et à Cordoue, sont d'origine et de caractère asiatique...... quant à l'avenir de l'Occident, il n'est pas ici en cause : une influence préméditée n'a jamais eu de résultats durables, et prédire est probablement le contraire de comprendre. Toute réflexion inquiète de l'Européen sur l'Europe trahit un mauvais état de santé intellectuelle, l'empêche de faire sa tâche, de travailler à bien penser, suivant la raison occidentale, qui est la raison tout court, de faire surgir, ainsi que l'ont voulu Platon et Spinoza, de la science vraie la pureté du sentiment religieux en chassant les imaginations matérialistes qui sont ce que l'Occident a toujours reçu de l'Orient» Léon BRUNSCHVICG

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    Category-functor modelling of systems | 26 octobre 2005

    (cliquer aussi sur le titre pour visionner un  document de Burgin sur l'entropie)

    CATEGORY-FUNCTOR MODELLING OF NATURAL SYSTEMS

    A. P. LEVICH, A. V. SOLOV'YOV

     

    ABSTRACT

    An approach to the derivation of dynamic equations for natural systems modelled by mathematical structures is suggested. The approach rests on an extremum principle which postulates that among all possible states of a system those ones are actually realized which correspond to an extremal (in a rigorous mathematical sense) structure. The suggested method of ordering the structured sets with the aid of category and functor theory generalizes the cardinality ordering of structureless sets. The method makes it possible to determine the functionals for the variational problem which describes a system under study. The approach is illustrated by a model of an ecological community.

    INTRODUCTION

    A theoretical description of any natural system includes two important aspects at least. First, one should construct a concrete mathematical model of both the admissible states of the system and its transitions between these states. Second, one should establish the choice rules (often in the form of an extremum principle) selecting, among the many theoretically admissible states of the system, only those states which are realized in nature under the given external conditions. In the present paper, we suggest that both these problems be solved by the methods of category theory.

    Apparently, the first to widely use set theory and topology for modelling the biological processes was Rashevsky (1954, 1955) laying the foundations of abstract biology. Later, in his works on relational biology, Rosen (1958, 1959) made the next step and applied category theory to the description of biological systems, in particular, a living cell. In subsequent years, an abstract category-theoretic approach to problems of mathematical biology was developed in various ways among which we would like to note primarily: the molecular set theory (Bartholomay, 1960, 1965), the organismic supercategories (Baianu, 1968, 1970), and the energy theory of abstract ecological systems (Leguizamon, 1975, 1993).

    In this work, we have attempted to construct a category-functor model of abstract natural systems (not necessarily biological) and to apply it to the description of ecological communities. The paper contains a development of ideas presented in Levich (1982).

    A CATEGORY-THEORETIC MODEL OF NATURAL SYSTEMS

    An analysis shows that, in all practically interesting cases, an arbitrary state or even the whole state space of a natural system can be identified with a set equipped with a certain mathematical structure: algebraic, topological, etc.

    For example, it is convenient to describe a state of an ecological community by means of a set with a partition into mutually disjoint classes where elements of the set are organisms forming the community, while partition classes are populations of biological species (Levich, 1982). Below, we will illustrate the basic statements of the general theory with the ecological community example.

    Returning to an arbitrary natural system, we note that a mathematical structure, specified on sets-states, models only those properties of the real system which are invariant under its transitions from one admissible state to any other. Precisely such a property is the existence of a species structure in an ecological community. Thus, in the general case, the state space of a natural system is simply a class of uniformly structured sets (such as groups, topological spaces, sets with partitions, etc.).

    Let A and B be two uniformly structured sets. Consider their direct product and fix a certain subset in it. Any subset is called a correspondence from A to B. In this connection, one uses the following notation: for the complete image of the element with respect to the correspondence and for the complete preimage of the element with respect to the same correspondence. Situations are possible when or/and for certain a and b. Let us list the most important types of correspondences from A to B:

    1. Everywhere defined correspondences , such that for any ;

       

    2. Surjective correspondences , such that for any ;

       

    3. Functional correspondences , such that either is the empty set or consists of one element for any ;

       

    4. Injective correspondences , such that either is the empty set or consists of one element for any
    .

    Moreover, various combinations of types 1-4 are also admissible; in particular, everywhere defined functional correspondences which are identified with ordinary mappings. All of them will be needed below.

    Following Levich (1982), we call correspondences preserving a mathematical structure specified on the sets A and B structure morphisms and denote them by Greek letters: For example, structure morphisms of sets with partitions are correspondences transforming each partition class of one set as a whole into some partition class of another set, so that different classes are transformed into different ones. Other examples of structure morphisms are homomorphisms of groups and continuous mappings of topological spaces.

    According to the aforesaid, we will identify transitions between admissible states of an arbitrary natural system with corresponding structure morphisms. It is clear that each system is characterized by a specific class of structure morphisms reflecting its concrete “transitional” properties; even if two systems have the same state space, but distinct classes of structure morphisms, then one must distinguish such systems.

    Thus, we suggest to associate with each natural system a mathematical construction S consisting of two classes: and . Here is a fixed class of uniformly structured sets which will be also called objects; is a fixed class of structure morphisms between these sets. It is not hard to see the following:

    1. With each ordered pair of objects one can associate the set of structure morphisms from A to B.

       

    2. Each structure morphism of the class belongs to one and only one set , where .

       

    3. For any three objects , one can introduce the multiplication of structure morphisms: , . Here is the product of structure morphisms and defined by the following condition: (, ) if and only if there exists an element such that and simultaneously.

       

    4. The multiplication of structure morphisms is associative, i.e., for any , , , and .

       

    5. For each , there exists a structure morphism called the unit morphism such that , for any , , and . This is the identity correspondence:

    Therefore, S is a category (Bucur & Deleanu, 1968; Goldblatt, 1979). Thus, the proposed model of natural systems has a simple category-theoretic interpretation according to which admissible states of a system are objects of the category S and permissible transitions between these states are morphisms of the category S.

    In what follows, in general considerations, S will designate a category such that the class consists of all uniformly structured sets, while the class consists of all structure morphisms between them.

    A FUNCTOR METHOD OF COMPARISON OF STRUCTURED SETS AND AN EXTREMUM PRINCIPLE FOR NATURAL SYSTEMS

    The next important step in describing a natural system is to establish an extremum principle selecting from the class of admissible states of the system only those states which are realized under the given external conditions. To this end, it is necessary to construct a function on the system state space taking values in a linearly ordered set (class) because the concept of “extreme state” is meaningful only in this case. The present section is devoted to a search for such a function defined on the class and having the most natural form from the viewpoint of category theory.

    First of all, let us recall some information about cardinal numbers of nonstructured sets. Consider the category Set whose objects are arbitrary sets and morphisms are arbitrary correspondences between these sets. Below, we will call subclasses of the direct product binary relations on the class .

    Let be the binary relation on defined by the rule:

    Û there is an injective mapping of into .

    Here . It is obvious that has the following important properties:

    1. for any (reflexivity);

       

    2. if and , then
    (transitivity);

    i.e., is a preorder relation on . We factorize this preorder by introducing the binary relation on the same class according to the rule:

    Û and .

    It is not difficult to verify that the so defined relation is reflexive, transitive, and symmetric (the latter signifies that if , then as well). Therefore, is an equivalence relation on . One can show (Levich, 1982) that if and only if there exists a bijection between the sets and .

    For each , we denote by the class of all the sets which are -equivalent to the set , i.e., . One usually calls this class the cardinal number of the set and writes (if is a finite set, then is identified with the number of elements in ). In other words, cardinal numbers of nonstructured sets are elements of the factor class . We simultaneously have the canonical surjection , .

    It is natural to define an order relation on the factor class , namely:

    Û .

    The reflexivity and transitivity of follow from the corresponding properties of the preorder . The antisymmetry of is established by the simple reasoning: and Û and Û Þ . Thus, is indeed an order relation on and, moreover, this order is linear (i.e., any two cardinal numbers are comparable) (Bourbaki, 1960).

    Now, let us try to extend the above constructions to the case of the category S of uniformly structured sets. Let . By analogy with universal algebra, injective mappings of A into B (and vice versa) preserving a mathematical structure specified on these sets will be called structure monomorphisms.

    Consider the binary relation defined on the class by the following rule:

    Û there exists a structure monomorphism of A into B.

    Evidently, the relation is reflexive and transitive; hence is a preorder on . Factorization of this preorder creates the equivalence relation on the same class in the usual way:

    Û and .

    For each , let us introduce the class of uniformly structured sets which are -equivalent to A. Following Levich (1982), we call this class the structural number of the set A and use the notation for it. Thus, by definition, structural numbers are elements of the factor class . We simultaneously have the canonical surjection , .

    With the aid of one can define the order relation on the factor class by setting

    .

    The reflexivity, transitivity, and antisymmetry of are verified in an elementary way. However, unlike cardinal numbers, structural ones are ordered only partially. For example, there are no structure monomorphisms between two sets with the partitions: and (i.e., the corresponding structural numbers and are incomparable). Thus, functions defined on and taking values in cannot be used for formulating an extremum principle.

    Let us seek a way out of the above situation by constructing a suitable generalization of structural numbers. To this end, one should recall that S is a category and the factor class is ordered linearly.

    Given an arbitrary object A in the class , consider the mapping associating with each the set of structure morphisms from A to B and with each the everywhere defined functional correspondence from to for any . It is obvious that the conditions

    1. if , then ;

       

    2. if , then ;

       

    3. for any ;

       

    4. if and , then

    are valid. Therefore, the mapping is a one-place covariant functor (Bucur & Deleanu, 1968) of the category S into the category Set. It is known as a representing functor.

    Following Levich (1982), we call the cardinal number the invariant of the object with respect to the object . It is not difficult to prove the proposition:

    if then .

    Indeed, the inequality means that there exists a structure monomorphism of B into C. Let be such a monomorphism and be arbitrary structure morphisms from A to B. Since b is an everywhere defined functional injective correspondence, the equality implies . Consequently, is an injection of into and . Q.E.D.

    In particular, if , then . This justifies the term “invariant” introduced above for . Since is a linearly ordered factor class, we have: or or for any . Thus, one can compare objects of the category S by comparing their invariants. The last circumstance forms the basis of the functor method of comparison of structured sets (Levich, 1982).

    Having fixed , we obtain the function , which is the composition of the injection , and the canonical surjection . The function is defined on the whole class and takes values in the linearly ordered factor class , i.e., it can be employed perfectly well in an extremum principle.

    Now, keeping in mind the subsequent applications to ecological communities, we concentrate attention on finite sets. Instead of S, consider the category whose objects are finite nonstructured sets and morphisms are correspondences between them having the type “a” (the multi-index a takes one of sixteen values which are subsets of the set of the main types of correspondences listed in the previous section). Assume , , and , where are nonnegative integers. Let us denote by the invariant of the object with respect to the object in the category . It is obvious that coincides with the number of morphisms from to and is a function of m and n. The following is a list of the results of calculating the invariants for all the sixteen values of the multi-index a:

    ,

    ;

    ,

    ;

    ,

    ;

    ,

    ;

    ,

    ;

    ,

    ;

    ,

    ;

    ,

    .

    Here is the system of all coverings of the set , , and is an arbitrary k-element set.

    As a special case of S, consider the category such that the class consists of finite uniformly structured sets and the class consists of structure morphisms which have the type “a” as correspondences between sets (see the previous paragraph). The category already gives sufficient possibilities for modeling a broad spectrum of natural systems.

    Let and be the invariant of the object B with respect to the object A in the category , i.e., . (For example, if is the category of finite sets with partitions, then , where is the i-th partition class of the set A and is the partition class of the set B into which the class passes as a result of morphisms from (Levich, 1982)). Moreover, let be the same sets A and B, but only deprived of the mathematical structure on them (also known as supports of the structure). Consider the quantity called the specific invariant of with respect to ; it has the meaning of the number of correspondences from per one structure morphism from . It is convenient to take the specific invariant as a quantitative measure of deviation of the structured sets A and B from the corresponding supports and of the mathematical structure. Applied to natural systems, this means that the states of a system with greater values of the specific invariant are “stronger” structured than the states with smaller ones.

    According to the aforesaid, we introduce the following definition:

    For a natural system admitting a description within the framework of the category

    If a class of states between which transitions are permitted from the viewpoint of some substantial reasons, for example, due to conservation of macroscopic parameters of the system, is interpreted as a macrostate of the system and a result of an arbitrary transformation of the system is interpreted as its microstate, then is a generalization of Boltzmann's entropy which is conventionally defined as the logarithm of the number of distinct microstates corresponding to a given macrostate. The possibility of the above interpretation and the coincidence of special cases of the quantity with the traditional formulae for Boltzmann's entropy justify the term “entropy” applied to . Note, however, that, in the present approach, the entropy appears without any probabilistic considerations.

    For fixed , the entropy is a function of taking values in the positive real semiaxis . Indeed, this follows from the obvious inequalities and , where the supports of the structure correspond to the objects A, B and the variable .

    It is now clear that the desired extremum principle can be formulated in the form of the postulate:

    In reality, a natural system passes from a given state

    to the state for which the entropy is maximal within the bounds defined by the external conditions (for example, available power and other resources).

    The suggested principle admits the following interpretations:

    1. Since, for fixed , the specific invariant is regarded to be the measure of deviation of the structured set from the support of the structure, the extremum principle selects those states of the natural system which are deviated most strongly from their nonstructured analogs , i.e., are “maximally” structured.

       

    2. For fixed , the invariant depends only on the number of elements in the set . Therefore, if is also fixed, then the entropy will be the greatest when the invariant is the smallest. However, we can treat a small number of structure morphisms from A to B as the high “stability” of the state . Thus, the extremum principle realizes the states of the natural system with both the maximum number of elements in the set-support of the mathematical structure and the maximum “stability” with respect to morphisms of the structure.

       

    3. It is also possible to interpret the entropy
    as the amount of information associated with the states A and B of the natural system (Levich, 1978). Hence, the extremum principle admits the informational formulation: a real system passes from a given state A to the state B with the maximal amount of information.

    In the next section we will discuss an ecological application of the above extremum principle.

    THE EXTREMUM PRINCIPLE IN ECOLOGY OF COMMUNITIES

    Consider an ecological community consisting of organisms of the same trophic level without an age structure (for example, cells of phytoplankton) which belong to w species and consume m mutually irreplaceable resources. Assume that in this community fission of cells and their death are admissible but absorption of one organism by another and introduction of organisms from outside are inadmissible. Since, in the adopted category-theoretic model of natural systems, states of the ecological community are described by means of sets with partitions, transitions between states of the community satisfying the above requirements are injective surjective structure morphisms of sets with partitions.

    We will seek the stationary final state of the community. The following is a modification of the extremum principle for this special case:

    In the course of time, the ecological community passes to the state with the maximum value of the entropy possible under the given resource restrictions.

    Let be the number of organisms of the species i in the community, be the vector of the species sizes, be the total number of organisms, be the amount of the k-th resource in the environment, be the requirement of an organism of the species i for the k-th resource. Evidently, any admissible state of the community is characterized completely by the vector. One can show (Levich, 1982) that where A corresponds to . On the other hand, we know from the previous section that . Thus, a conditional extremum problem appears (Levich, Alekseev, and Nikulin, 1994):

    A solution of this problem is yielded by the species structure formula (Levich, 1980)

    which connects the species sizes in the ecological community with the resources on which n and depend. Levich (1980) and Levich,. Zamolodchikov, and Rybakova (1993) have demonstrated the adequacy of the species structure formula to the empirical data (see also Lurie, Valls, and Vagensberg (1983)).

    It has been shown as well that:

    A solution of the extremum problem exists and is unique under any () and realizes a maximum of the functional (the existence and uniqueness theorems) (Levich, Alekseev, and Nikulin, 1994).

    The space of environmental resources consumed by the community splits (stratifies) into non-intersecting subsets, any of which corresponds to a unique collection of resources consumed completely; and in these subsets the state of the community depends on the same resources as arguments and only on them (the stratification theorem) (Levich, Alekseev, and Nikulin, 1994).

    The fractional sizes of the species depend only on the resource amount ratios in the environment and take the greatest values under the resource ratios which are equal to the given species' ratio of demands for them (the optimization theorem) (Levich, Alekseev, and Rybakova, 1993; Alexeyev & Levich, 1997). This optimization theorem creates an efficient method of management for the species structure of ecological communities with the aid of environmental resource factor flows (Levich & Bulgakov, 1993; Levich, Khudoyan, Bulgakov, and Artiukhova, 1992; Levich & Bulgakov, 1992; Levich, Maximov, and Bulgakov, 1997).

    The variational problem of finding the maximum entropy under restricted (from above) environmental resource consumption turns out to be equivalent to the variational problem of finding the minimum environmental resource consumption by the community under restricted (from below) “structure entropy” of the community (“the Gibbs theorem”). This allows the suggested extremum principle to be interpreted as the principle of minimum environmental resource consumption (providing a sufficient degree of system organization) (Levich & Alexeyev, 1997).

    , the logarithm is called the entropy of the state with respect to the state .

     

    Publié par topos à 17:26:29 dans Catégories et foncteurs | Commentaires (0) |

    Théorie des "named sets" de Burgin | 26 octobre 2005

    (cliquer aussi sur le titre pour visionner un autre document .pdf de Burgin sur "The essence of information")

    In: Structures in Mathematical Theories, San Sebastian, 1990, pp. 417-420

    Burgin M.S.
    THEORY OF NAMED SETS AS A FOUNDATIONAL BASIS FOR MATHEMATICS





    The most recognized basis of modern mathematics is set theory. But at the same time other mathematical constructions appeared (some of them recently and others many years ago) that in some sense are more general than sets. These are fuzzy sets, multisets, L-fuzzy sets etc. Categories and algorithms were taken as (alternative to sets) bases of mathematics. All this was a symptom that there exists a more fundamental structure than set and it may be taken as a basis for mathematics. It appeared that such structure is a named set.

    In order to give the exact definition of a named set, let us introduce three collections   Ens, Set, Col . Each of them consists of sets or classes (that are called objects) and their morphisms. Totalities of sets (classes) from   Ens, Set, Col are denoted by   ObEns, ObSet, ObCol,  respectively. Totalities of morphisms (i.e., mappings or binary correspondences between sets or classes) from  Ens, Set, Col  are denoted by   MorEns, MorSet, MorCol, respectively. If   X, Y Î Ob K, then all morphisms from X to Y are denoted by Mor (X,Y) where K is one of the following Ens, Set, Col.

    Suppose that the following conditions are valid:

    1) ObEns, ObSet Í ObCol;

    2) MorEns, MorSet ÍMorCol;

    3) the totality   MorCol  is closed in respect to the product of morphisms (it means that if   a,b Î MorCol   and their product ab is defined, then ab Î MorCol).

    Let us select some subclass M from the class MorCol. This selection may be to some extent arbitrary that makes possible, using different conditions on M, to define constructions necessary in each concrete case.

    Definition 1. A named set (with respect to M ) (or N-set) is a triad  X = ( X, r, I )  where XÎ ObEns,   I Î ObSetrÎ Mor (X,I) and  rÎM.

    Let X = (X, r, I ) be a named set.

    Definition 2. 1) The set  is called the support of  X  and is denoted by  S(X);   2) the set  is called the name set or the set of names of  X  and is denoted by  N(X);  3) the set   Nf (X) = { a ÎI; (xÎS(X) & (x.a) Îr }  is called the set of nonvoid (factual) names of the named set X;  4) the mapping (correspondence) r is called the naming mapping (correspondence) of  and is denoted by  n(X); 5) the element r(x) Î I  (for a single valued mapping r) and the set r(x) = { a ÎI; (x.a)Îr } (for a binary relation or multivalued maping r) is called the complete name of x Î in  X; if r is not single valued, then any element a Î r(x)  is called a partial name of x in  X . Otherwise, it is called simply a name of x.

    Many important mathematical notions may be modeled as special cases of named sets as it is demonstrated by the following examples.

    1. A multiset is a collection that is like a set but can include identical or indistinguishable elements (Knuth, 1973). For instance, {a,a,b,b,b} is a multiset that contains two elements a and three elements b . Thus a multiset is obtained if we take a named set X and add an axiom demanding that elements from the support S(X) of X are distinguishable if and only if they have different names in X .

    2. According to M.Aigner (1979), a multiset on a set S is a function r: S ® N that defines multiplicity of the elements from S (here N = {0,1,2, ..., }). Such multisets are the named sets in the case when Ens consists of arbitrary sets and their maps, Set contains the single object N and all binary relations on it, while Col is equal to Ens .

    3. In (Hickman, 1980), multisets are defined like in (Aigner, 1979) but instead of N the class Card of all cardinals is taken. So multisets, in this extended sense, are also special cases of named sets.

    4. A fuzzy subset A of a universe U is a pair (A, m ) where m is the membership function of A (Zadeh, 1965). If the universe U may be an arbitrary set, then the above definition gives us the general notion of a fuzzy set. Thus fuzzy sets are the named sets when Ens consists of arbitrary sets and their maps while Set contains the single object [0,1] and all relations on it.

    5. Taking instead of the interval [0,1] a complete lattice L we receive the notion of L-fuzzy set (Goguen, 1967) or when U = X ´ Y we have L-fuzzy relation (Salii,1965). So, L-fuzzy sets and relations are also special cases of named sets.

    Definition 3. A named set X is called: 1) normalized if Nf (X) = N(X); 2) a singlenamed set if Nf (X) consists of a single element; 3) an individually named set if n(X) is a bijection; 4) a one-to-one named set if both sets S(X) and N(X) consist of one element.

    Examples of single named sets give us usual sets. Really in the framework of named set theory we can have a new understanding of different trends in the mathematical set theory. The first fact that lies on the surface is that any usual set is a single named set because we cannot speak about, use or construct any set without giving a name to it. This name may be a single sign (M, for example), a logical formula {x Î X; P(x) & (y Î Y ((x,y) Î A ÍX ´ Y)} or an algorithm (some partial recursive function etc.). However, this name always exists. So all elements from the set with this name (M, for example) have the same common name ("an element from the set M").

    Definition 4.   A named set    Y = (Y, b, J)   is called a named subset (a weak named subset) of the named set    X = (X, a, I)   if  Y ÍXJÍI and  b = a|(Y,J)   ( and b Ía Ç (Y ´ J) ). Such relation between named sets is called the inclusion (the weak inclusion) and is denoted by Y Í X   (Y Íw X).

    Definition 5. If   X = ( X, r, I )  and  Y = ( Y, q, J ) are named sets then a morphism from X to Y is a pair   F = (f, g ) where  f: X ® Y, g: I ® J   for which the equality   fq = rg   is valid, i.e., the following diagram is commutative

    For morphisms   F = (f,g): X ® Y   and  J = (t,s): Y ®Z, their product (composition) is defined in a natural way as  FJ = (ft, gs): X ® Z.

    Theorem 1. If the classes Ens, Set and Col are categories, then the collection NSet of all named sets and their morphisms is also a category (Burgin, 1984).

    If we take modern analysis the main structure on which functions are defined and studied are different kinds of manifolds: topological, differentiable, smooth etc. The most general kind is a topological manifold. Each of them is a named set X = (X, r, I) where X is a topological space, I is some n-dimensional vector space R and a is a continious relation that is a local homeomorphism, i.e. for each point x from X there exists an open neighborhood that is homeomorphic to some open subset of R . Conditions that define special cases of topological manifolds (differentiable, analytical etc.) may be also formulated in the language of the named set theory, i.e., as conditions on named sets that are obtained by application to these named sets definite set-theoretical operations.

    Scalar, vector and tensor fields on manifolds are also named sets having the form X = (X, n, D) where X is the same as above, D is some set of scalars (real or complex numbers), vectors or tensors and n is a function defined on X.

    Categories play an important role in modern mathematics. Any category K consists of two classes  Ob K of objects from and  Mor K  of morphisms from  K. For any two objects A and B from  K the set  H(A,B) from the class  Mor K  is singled out. A partial binary composition of morphisms is defined in  MorK. Then to each morphism f from H (A,B) a one-to-one named set ({A}, f, {B}) is corresponded. In such a way, a system T(K) of one-to-one named sets is related to the category K. The system T(K) satisfies three conditions (i) - (iii) (Burgin, 1988). Any system T of one-to-one named sets satisfying conditions (i) - (iii) is called categorical. It is shown that there exists a one-to-one correspondence between abstract categories and categorical systems of named sets. By this correspondence, any functor between abstract categories is mapped into a homomorphism of categorical systems of named sets.

    If we take mathematical logic we also see a lot of different named sets. As an example we can take such important construction as model. It should be noted that there exists no generally acceptable and exact definition of a model in mathematical logic. Some authors treat a model simply as a mathematical structure, exactly as a set with a system of relations (may be functions) on it (Malzev, 1970; Shoenfield, 1967). Other authors interpret a model as a pair consisting of a mathematical structure (the same as above) and a partial map of a logical language into this structure (Chang and Keisler, 1973; Mendelson, 1963). It is necessary to note that in the second case we again have some named set. Really, when one speaks about a model of some logical language then he factually bears in mind the named set (L, i, M) of an interpretation of the language L into some mathematical structure M. As mathematical structure M (that is called a model of the language L) the set on which the predicates and functions are defined usually is taken. The map i is built in such a manner that there are correspondences:  (i) between predicate symbols from the alphabet of the language L and predicates having the same number of variables defined on M;  (ii) between functional symbols and functions on M; (iii) between constants and elements of the mathematical structure M .

    In a similar way an interpretation of a formal theory (or a deductive calculus) T into a model M is built. The corresponding modeling named set has a form  (T, p, M)  and the truth is such a property that its conservation is demanded from the naming relation p. In other words, if all formulae from T are considered as true, then M is a model of T if and only if the images of these formulae are true in M. It should be noted that when in logic (considered as a model), it is taken a pair consisting of the interpretation map and a mathematical structure, then such model is the part of the modeling named set.

    Likewise, any formal calculus  C is also a special kind of named sets. Really, it may be considered as a triad   C = (A, R, T) where  A is the set of axioms,  R are rules of deduction by which from axioms the theorems of the calculus are deduced. These theorems form the set T. The named set  (A, R, T)  is called a named set of calculus rules. The same calculus may be represented by another (deduction) named set  (A, d, T)  where the relation d connects any axiom a from A with such theorems t from T that a is used in a process of deduction of t.

    The main progress of contemporary mathematics may be explained as a process of transition from usual (singlenamed) sets to more general cases. It is characteristic for the most mathematical fields. For example, in topology a lot of achievements is connected with the introduction of fibers and their special cases fiber spaces, bundles, smooth fibers etc. But any fiber F is a topological functional named set  F = (E, p, B) ,  i.e., a named set in which the base B and the fiber space E are topological spaces and  p is a continuous projection of  E onto B. It is possible to demonstrate that analogous situations are characteristic for all other branches of mathematics.

    Publié par topos à 17:16:57 dans Mathesis universalis | Commentaires (0) |

    Ensembles, classes et catégories | 26 octobre 2005

    (cliquer sur le titre pour visionner un document pdf : "Sets, classes and categories")

    On sait que la philosophie de Badiou (qui de l'aveu même de ses détracteurs, et Dieu sait s'il y en a, à commencer par Sokal, possède une culture mathématique impressionnante) a des vues très arrêtées, et "révolutionnaires", sur le lien entre philosophie et mathématiques...passons les en revue brièvement :

    1 les mathématiques sont l'ontologie, qui ne fait donc pas partie de la philosophie

    2 les topoi (ces catégories découvertes il y a 40 ans par Grothendieck et Lawvere) forment le cadre de la logique de l'apparaitre, les ensembles celui de l'ontologie (discours sur l'Etre)

    3 le système axiomatique retenu par Badiou est celui de Zermelo-Fraenkel, à l'exclusion semble t'il des autres "solutions" au paradoxe de Russell : theorie des types, axiomatique NBG de Von Neumann, Bernays et Gödel, "New foundations" de Quine, "Non well founded sets" d'Aczel, "univers" de Mac Lane ou Grothendieck, etc...

    Bien entendu, on comprend que le philosophe fasse un choix (trace une diagonale dirait Badiou) dans la "forêt" proliférante des découvertes mathématiques.

    Néanmoins il est absolument nécessaire, en ces matières, de laisser aussi la parole aux mathématiciens (ce que n'est pas Badiou, malgrès sa virtuosité technique), c'est bien la moindre des choses s'agissant...des mathématiques !

    On doit rappeler par exemple que la plupart des mathématiciens "catégoristes" n'ont pas du tout la même évaluation épistémologique que Badiou : ils ne veulent tout simplement plus entendre parler des ensembles, ni en mode ZF ni en autre !

    Publié par topos à 14:41:01 dans Catégories et foncteurs | Commentaires (0) |

    LE BUT DE NOTRE CROISADE

    Notre but, notre tâche, est le réarmement intellectuel, moral et spirituel de l'OCCIDENT chrétien, et donc de toute l'humanité: passée, présente et future. Car les fruits de l'esprit sont éternels et divins, puisque Dieu est Esprit. Le monde est tout ce qui arrive, tout ce qui est le cas, tout ce qui est un fait, et glisse immédiatement au passé et semble t'il au Néant : mais l'esprit est SENS, et les oeuvres, évènements, penseurs du passé peuvent toujours être réinterprétés par l'Esprit qui construit et déconstruit. Aussi le passé vit-il toujours et éternellement sous forme de SENS, et infléchit-il ainsi le présent et le futur, qui à leur tour le modifieront en créant les conditions de sa réinterprétation. UN est le TOUT. "le chemin qui mène à ce but, au savoir absolu, ou encore à l'esprit qui se sait comme esprit, est le souvenir des esprits tels qu'ils sont chez eux mêmes et accomplissent l'organisation de leur royaume. Leur conservation selon le côté de leur libre existence dans son apparition phénoménale sous la forme de la contingence, est l'Histoire, tandis que du côté de leur organisation comprise de manière conceptuelle, c'est la science du Savoir dans son apparition phénoménale; l'une et l'autre réunies ensemble, l'Histoire comprise conceptuellement, constituent le souvenir et le GOLGOTHA de l'Esprit Absolu, l'effectivité, la vérité et la certitude de son trône sans lequel il serait solitude sans vie: Et c'est seulement du calice de ce Royaume d'esprits que monte vers Lui l'écume de Son Infinité"

    Moi

    Comment j'ai appris à ne plus m'en faire et à aimer D-ieu plus que l'être-pour-la-mort...


     


    une fourmi noire, 


     


    dans la nuit noire,


     


    sur la terre noire,


     


    sous une pierre noire,


     


    D-ieu seul la voit


     


    et ici le diable souffle : Dieu....et la police, peut être ?


     


     


     


     

    Notre CREDO

    "le propre de l'esprit est de s'apparaitre à lui même dans la certitude d'une lumière croissante, tandis que la vie est essentiellement menace et ambiguïté. Ce qui la définit c'est la succession fatale de la génération et de la corruption. Voilà pourquoi les religions, établies sur le plan vital, ont beau condamner le manichéisme, il demeure à la base de leur représentation dogmatique... ce qui est constitutif de l'esprit est l'unité d'un progrès par l'accumulation unilinéaire de vérités toujours positives. L'alternative insoluble de l'optimisme et du pessimisme ne concernera jamais que le centre vital d'intérêt; nous pouvons être et à bon droit inquiets en ce qui nous concerne de notre rapport à l'esprit, mais non inquiets de l'esprit lui même que ne sauraient affecter les défaillances et les échecs, les repentirs et les régressions d'un individu, ou d'une race, ou d'une planète. Le problème est dans le passage , non d'aujourd'hui à demain, mais du présent temporel au présent éternel. Une philosophie de la conscience pure, telle que le traité de Spinoza "De intellectus emendatione" , en a dégagé la méthode, n'a rien à espérer de la vie, à craindre de la mort. L'angoisse de disparaitre un jour, qui domine une métaphysique de la vie, est sur un plan; la certitude d'évidence qu'apporte avec elle l'intelligence de l'idée, est sur un autre plan" Léon BRUNSCHVICG

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